Biomechanical Systems Technology: Cardiovascular Systems

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Biomechanical Systems Technology: Cardiovascular Systems

Cardiovascular Systems 6506-Cardiovascular tp.indd 1 9/28/07 10:07:17 AM BIOMECHANICAL SYSTEMS TECHNOLOGY A 4-Volume

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Cardiovascular Systems

6506-Cardiovascular tp.indd 1

9/28/07 10:07:17 AM

BIOMECHANICAL SYSTEMS TECHNOLOGY A 4-Volume Set Editor: Cornelius T Leondes (University of California, Los Angeles, USA)

Computational Methods ISBN-13 978-981-270-981-3 ISBN-10 981-270-981-9 Cardiovascular Systems ISBN-13 978-981-270-982-0 ISBN-10 981-270-982-7 Muscular Skeletal Systems ISBN-13 978-981-270-983-7 ISBN-10 981-270-983-5 General Anatomy ISBN-13 978-981-270-984-4 ISBN-10 981-270-984-3

A 4-Volume Set

Cardiovascular Systems

Editor

Cornelius T Leondes University of California, Los Angeles, USA

World Scientific NEW JERSEY

6506-Cardiovascular tp.indd 2



LONDON



SINGAPORE



BEIJING



SHANGHAI



HONG KONG



TA I P E I



CHENNAI

9/28/07 10:07:20 AM

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

BIOMECHANICAL SYSTEMS TECHNOLOGY A 4-Volume Set Cardiovascular Systems Copyright © 2007 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN-13 978-981-270-798-7 (Set) ISBN-10 981-270-798-0 (Set) ISBN-13 978-981-270-982-0 ISBN-10 981-270-982-7

Typeset by Stallion Press Email: [email protected]

Printed in Singapore.

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PREFACE Because of rapid developments in computer technology and computational techniques, advances in a wide spectrum of technologies, and other advances coupled with cross-disciplinary pursuits between technology and its applications to human body processes, the field of biomechanics continues to evolve. Many areas of significant progress can be noted. These include dynamics of musculosketal systems, mechanics of hard and soft tissues, mechanics of bone remodeling, mechanics of implant-tissue interfaces, cardiovascular and respiratory biomechanics, mechanics of blood and air flow, flow-prosthesis interfaces, mechanics of impact, dynamics of man-machine interaction, and many more. This is the second of a set of four volumes and it treats the area of Cardiovascular Systems in biomechanics. The four volumes constitute an integrated set. The titles for each of the volumes are: • • • •

Biomechanical Biomechanical Biomechanical Biomechanical

Systems Systems Systems Systems

Technology: Technology: Technology: Technology:

Computational Methods Cardiovascular Systems Muscular Skeletal Systems General Anatomy

Collectively they constitute an MRW (Major Reference Work). An MRW is a comprehensive treatment of a subject area requiring multiple authors and a number of distinctly titled and well integrated volumes. Each volume treats a specific but broad subject area of fundamental importance to biomechanical systems technology. Each volume is self-contained and stands alone for those interested in a specific volume. However, collectively, this 4-volume set evidently constitutes the first comprehensive major reference work dedicated to the multi-discipline area of biomechanical systems technology. There are over 120 coauthors from 18 countries of this notable MRW. The chapters are clearly written, self contained, readable and comprehensive with helpful guides including introduction, summary, extensive figures and examples with comprehensive reference lists. Perhaps the most valuable feature of this work is the breadth and depth of the topics covered by leading contributors on the international scene. The contributors of this volume clearly reveal the effectiveness of the techniques available and the essential role that they will play in the future. I hope that practitioners, research workers, computer scientists, and students will find this set of volumes to be a unique and significant reference source for years to come.

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CONTENTS

Preface

v

Chapter 1 A Simulation Study of Hemodynamic Benefits and Optimal Control of Axial Flow Pump-Based Left Ventricular Assist Device Huiting Qiao, Jing Bai and Ping He Chapter 2 Techniques in Visualization and Evaluation of the In Vivo Microcirculation Shigeru Ichioka Chapter 3 Impedance Cardiography: Development of the Stroke Volume Equations and their Electrodynamic and Biophysical Foundations Donald Philip Bernstein and Hendrikus J. M. Lemmens Chapter 4 Indicator Dilution Techniques in Cardiovascular Quantification Massimo Mischi, Zaccaria Del Prete and Hendrikus H. M. Korsten Chapter 5 Analyzing Cardiac Biomechanics by Heart Sound Andreas Voss, R. Schroeder, A. Seeck and T. Huebner Chapter 6 Methods in the Analysis of the Effects of Gravity and Wall Properties in Blood Flow Through Vascular Systems S. J. Payne and S. Uzel Chapter 7 Numerical and Experimental Techniques for the Study of Biomechanics in the Arterial System Thomas P. O’Brien, Michael T. Walsh, Liam Morris, Pierce A. Grace, Eamon G. Kavanagh and Tim M. McGloughlin

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CHAPTER 1 A SIMULATION STUDY OF HEMODYNAMIC BENEFITS AND OPTIMAL CONTROL OF AXIAL FLOW PUMP-BASED LEFT VENTRICULAR ASSIST DEVICE HUITING QIAO∗ and JING BAI Institute of Biomedical Engineering, School of Medicine Beijing, 100084, P. R. China ∗[email protected] PING HE Department of Biomedical, Industrial and Human Factors Engineering Wright State University Dayton, Ohio, 45435, USA

This chapter gives an overview of various ventricular assist devices with a particular focus on axial flow pumps. Using the Hemopump as an example, we developed a model of a canine circulatory system assisted by an axial flow pump, and used computer simulation to predict the effects of the assist device under various hemodynamic conditions. In general, the results from the simulation are in good agreement with that observed in clinical and animal experiments. The same model is further used to explore the techniques and strategy for optimum control of the assist device by introducing an objective function, and choosing suitable membership functions with associated weighting factors, the model offers great flexibility in choosing the targeted hemodynamic variables, in specifying the particular way that each of these variables is to be optimized, and in assigning the relative importance of each targeted variable. The methods used and the results obtained in this study can be incorporated into the design of an advanced physiological controller for a long-term operation of the axial flow pump-based assist device as well as other types of continuous-flow LVAD. Keywords: Ventricular assist device; axial flow pump; model; simulation; optimum control.

1. Introduction Heart failure is a major public health issue in the developed world. It is estimated that in the United States alone, heart failure affects nearly 5 million patients.1 For patients with the most advanced heart failure, heart transplantation has been the only means that prolongs and improves the quality of life. On the other hand, because of the restrictive criteria for heart transplantation and the chronic shortage of available donors, significant amount of research has been devoted to develop mechanical cardiac assist devices that provide circulatory support to a failing heart either as a destination therapy, or to serve as a bridge to heart transplantation or recovery.2,3 Many different types of ventricular assist devices have been developed and significant amount of research has been devoted to improve the material, structure and 1

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H. Qiao, J. Bai & P. He

control strategy of these devices.4–8 Since many of these devices have been extensively reviewed recently,3,9–12 in this chapter, we will just give a brief discussion of several ventricular assist devices, and then focus on one particular type of left ventricular assist devices that uses axial flow pump. We will use a computer model to study the hemodynamic effects of the device in supporting a failing heart, and to explore the techniques and strategy for optimum control of its operation.

2. A Brief Review of Ventricular Assist Devices 2.1. General objectives of ventricular assist devices The heart serves as a mechanical pump of the circulatory system. In heart failure, its ability to pump the blood can be greatly diminished which can lead to death. The best solution to a severely damaged heart is surgical replacement, or heart transplantation. Despite the great success in cardiac transplantation, healthy donor hearts are always in short supply. For example, in the United States, each year approximately 40,000 healthy donor hearts are needed while only 2,300 are available.13 The shortage of healthy heart donors and the restrictive criteria for heart transplantation have prompted continuous research and development of various mechanical cardiac assist devices. Ventricular assist device (VAD) is a kind of cardiac assist device that can partially or totally replace the functions of the natural heart in pumping blood. The general objectives of ventricular assist devices include improving the circulatory state of the patient, maintaining adequate blood circulation throughout the body, reducing the preload and afterload of the failing heart, decreasing myocardial oxygen consumption, increasing oxygen supply to the heart by increasing coronary flow, improving the contractibility of myocardium, strengthening the pump function of the heart, and even replacing the failing heart temporarily before surgical heart transplantation. According to the state of the cardiovascular system, ventricular assist device may be used to provide left-ventricular assist, right-ventricular assist and double-ventricular assist. Due to the high incidence of left-ventricle failure, the great majority of ventricular assist devices are used for left-ventricular assist. For simplicity, when ventricular assist is mentioned in this chapter, it generally refers to left-ventricular assist, and the device is called the Left-Ventricular Assist Devices (LVAD).

2.2. Classification of ventricular assist devices Ventricular assist devices are commonly composed of a pump, a control console, a power supply and cables. The pump is the central part of the ventricular assist device. Ventricular assist devices are classified according to different criteria.

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2.2.1. Pulsatile and nonpulsatile assist devices Based on the blood flow produced by the assist devices, ventricular assist devices can be classified as pulsatile assist devices and nonpulsatile assist devices. Since the flow pumped by a natural heart is pulsatile, it is generally considered that pulsatile assist is preferred, but clinical evidences are not conclusive. For example, Nose14 reported that long-term nonpulsatile biventricular bypass did not produce harmful physiological effects in animals, and nonpulsatile assist devices can function as well as pulsatile assist devices if 20% higher blood flows are pumped. On the other hand, Nishimura15 and Nishinaka16 suggested that prolonged nonpulsatile left ventricular bypass may produce certain effects on the arterial structure and function, such as on the wall thickness, smooth muscles and vasoconstrictive function of the vessels. Structurally, while the devices that produce a pulsatile flow are relatively large, the nonpulsatile ventricular assist devices can be much smaller and simpler, and have a smaller thrombogenic foreign surface. 2.2.2. Parallel and sequential assist devices Based on the relation between the blood flow in the assist device and the blood flow in the ventricle, ventricular assist devices can also be classified as parallel assist devices and sequential assist devices. When the pump of an assist device is connected directly between the left atrium and the aorta, a portion of the total aortic flow is delivered by the assist device and the remaining portion is pumped by the left ventricle. These devices are called Parallel Left Ventricular Assist Device (PLVAD). An example is the atrio-aortic LVAD (AA-LVAD), which withdraws blood from the left atrium and pumps it into the ascending aorta.17 In the parallel assistant mode the assist device can take over varying percentages of the pumping workload and blood flow. This kind of devices aids the heart by pumping a portion of, or in some cases, all of aortic blood flow. Another kind of ventricular assist devices withdraws blood from the left ventricle and delivers it to the aorta. Since the blood pumped by this kind of device first passes through the left ventricle, such a device is called Sequential Left Ventricular Assist Device (SLVAD). SLVAD can also be called as Serial Left Ventricular Assist Device. Comparing with PLVAD, SLVAD is easier to implement. 2.2.3. Axial blood pump and other types of ventricular assist devices Based on the structure of the pump, ventricular assist devices can be labeled as the impeller type,18 the diaphragm type,19 the sac type,20 and the pusher-plate type.21 The advantages of the impeller type pump include simple structure, small volume, the ease of implant, low thrombogenic possibility and so on. The impeller type pump can be further divided into centrifugal pump and axial flow pump. A centrifugal pump moves the blood by the centrifugal force generated by the rotating impellers. In general, a higher aortic blood pressure can be produced when

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the rotating speed of the impellers increases. In an axial flow pump, the rotating impellers initially move the blood spirally. A diffuser is used to guide the blood to finally move along the axis of the pump. Compared with the centrifugal pumps, the axial flow pumps have smaller sizes, require less time to implant, consume less power, and operate at much higher rotational speeds to deliver the desired blood flow.13 Over the past 20 years, many intracorporeal axial flow pumps have been developed and tested clinically that include the Hemopump, MicroMed DeBakey VAD, Jarvik 2000, HeartMate II, and Berlin INCOR I.12,13,22,23 Combining with other advantages, axial flow pumps have become the mainstream of ventricular assist device used in clinic today. 2.3. Intra-aortic balloon pump, Hemopump and dynamic aortic valve In the earlier years of the development of ventricular assist devices, great efforts were concentrated on developing pulsatile assist devices. Prolonged clinical trials however, also exposed some problems with these devices, such as the complicated structure, the large volume, heavy weight and so on.24 The research efforts were then turned to develop smaller and lighter ventricular assist devices with a simple structure and low power-consumption. Since the ventricular assist devices are mostly used as an implanted device, axial flow pumps have received much attention. The Hemopump is an early type of LVAD that uses axial flow pump. Dynamic aortic valve is a new type of axial flow pump-based LVAD that is still at the stage of research and development. Though different ventricular assist devices are based on different principles and are used for different cardiovascular situations, there are several common goals of using these devices that include decrease of the workload of the left ventricle, increase of the cardiac output, and increase of the oxygen supply to myocardial tissues by increasing the blood flow to the coronary arteries. In the following paragraphs, we will first give a brief description of three ventricular assist devices, IntraAortic Balloon Pump, Hemopump and Dynamic Aortic Valve. We will then develop a dynamic model of an axial flow pump and incorporate it into a canine circulatory model to investigate the hemodynamic benefits and optimal control of the assist device. 2.3.1. Intra-aortic balloon pump Intra-aortic balloon pump (IABP) has been widely used for cardiac assist since Kantrowitz reported the first successful clinical trial in 1967.25 IABP played an important role in the early development of cardiac assist devices, and it is the most widely used cardiac assist device in clinic. The IABP consists of an inflatable balloon, a pressure source, and a control unit that causes inflation and deflation of the balloon. The balloon is surgically inserted into the femoral artery and advanced to the descending aorta, just distal to the

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Fig. 1.

5

Illustration of the placement of intra-aortic balloon pump.

aortic arch, as shown in Fig. 1. The intra-aortic balloon pump is operated in a counterpulsation mode, and its inflation and deflation are timed according to the cardiac cycle: the balloon is deflated just before the opening of the aortic valve and is inflated after aortic valve closure. Inflation of the balloon in early diastole displaces a volume of blood back towards the aortic root, increasing the blood pressure there and thus enhancing the coronary flow. Deflation of the balloon in end diastole or early systole reduces aortic pressure and consequently reduces LV after-load for the succeeding ejection. The most critical control of the operation of the IABP is the timings of balloon inflation and deflation. For example, while active balloon deflation in early systole greatly helps the left ventral to pump the blood, a delay of balloon deflation into later systole would have much less beneficial effects. Similarly, while the balloon inflation in diastole can greatly increase the coronary flow, the balloon inflation during systole will cause a severe harm to the failing heart. To search for the optimal control of IABP, various approaches have been used that include mathematical modeling using lumped parameters,26 computer simulation with a multi-segment arterial model,27 and direct experimental measurements using an performance index.28 Though IABP is widely used to assist a failing heart during and after Cardiotomy, the support provided by the IABP is sometime insufficient as it can only increase cardiac output by 10% to 15%. More powerful assist devices are needed to increase the survival rate of patients suffering from severe heart failure.

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2.3.2. Hemopump The Hemopump was designed around 1982 while Dr. Wampler worked with the Nimbus Corporation in California and a patent was filed in 1983.29 Clinical trials of the Hemopump were conducted under an investigational device exemption (IDE) approved by the FDA in 1988 and the preliminary results of the clinical trials were encouraging.8,13,30 The Hemopump has been applied to the patients undergoing typical cardiac surgery, minimally invasive coronary artery bypass graft,31 patients suffering cardiogenic shock and in studies of percutaneous transluminal coronary angioplasty.29 Though the clinical study with this device has been discontinued in the United States, the experimental results with the Hemopump provided a valuable basis for the design and development of other axial flow ventricular assist devices that are undergoing clinical tests today.13 The Hemopump consists of a disposable pump assembly, a high-speed motor and a console, as shown in Fig. 2. The miniature axial flow pump is inserted into the femoral artery and threaded up towards the aorta root. The inlet flow cannula of the pump passes through the aortic valve with its tip situated inside the left

Fig. 2.

The composition of Hemopump.

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ventricle. Power to drive the pump is transmitted from an external electric motor with a flexible cable threaded through a sheath. When the device is activated, the pump impellers rotate and blood is drawn through the inlet cannula into the pump and then discharged into the aorta. In this process the left ventricle is relieved of its major workload. The capability of being inserted into the circulatory system without major surgery makes the Hemopump as safe and convenient as IABP. On the other hand, due to its ability to actively pump the blood and automatically adjust the flow rate, the Hemopump is more effective than IABP in assisting left ventricle. The rotating speed and pumping capacity of the Hemopump vary according to the type of the device. For example, the Hemopump type 14F has a rotation speed ranging from 27,600 rpm to 45,000 rpm, and the pump generates 1.3 L/min ∼ 2.3 L/min of blood flow. For type 24F (HP31), the rotation speed ranges from 17,000 rpm to 26,000 rpm, and the flow ranges from 3 L/min ∼ 5.1 L/min. The rotation speed of the impeller is controlled by a control console. In general, a higher rotation speed produces more blood flow. Yet a high speed is often associated with some problems such as increased power consumption and the risk of blood cell damage. As a result, optimum control of the Hemopump is an important area of research. 2.3.3. Dynamic aortic valve As a new type of a LVAD, Dynamic aortic valve (DAV) was proposed by Li et al. in 2001.32 Though the research of DAV is still in its early stage in laboratory, it has caught much attention from researchers and surgeons due to its simple structure. The DAV consists of an external drive unit and an internal pump, which are working together without any power cable connecting them. The pump is made up of a support cage and a magnetic rotor-impeller, as shown in Fig. 3. The magnetic rotor rotates in the presence of a revolving magnetic field generated by the external drive unit. A typical DAV is 22 mm in diameter and 30 mm in length, and is inserted to replace the natural aortic valve. The principle of the operation of DAV is to pump the blood out of the left ventricle by rotating its impeller at a high speed as well as to replace the function of the aortic valve. This latter function is achieved by maintaining a pressure difference

Fig. 3.

Schematic show of dynamic aortic valve.

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between the left ventricle and aortic chamber all the time, and therefore preventing blood regurgitation during diastole. Owning to its small volume, compact structure and enough flux, DAV has a great potential to become an effective cardiac assist device.

3. Simulation Study of the Hemopump Assessment of the performances and assistant effects of an axial flow pump-based assist device under various hemodynamic conditions in a clinical setting is difficult, while the method of modeling and simulation is more flexible and convenient. Using an adequately developed model of the axial flow pump (in the following text, we may simply call it the flow pump, or the pump) and the circulatory system, the operation mode of the pump and the parameters of the circulatory system can be adjusted easily, and the hemodynamic variables of the system at various locations can be observed conveniently. Consequently, the performance and assistant effects of the pump can be assessed under different conditions. Such a model can also be used for the study of the optimal control of the pump.

3.1. Modeling the Hemopump To evaluate the assist effects of the pump, a static model and a dynamic model of the pump are established based on the experimental data obtained from a particular kind of axial flow pump, the Hemopump type HP31, and the dynamic model is incorporated into a multi-element cardiovascular model.33

3.1.1. Static model of the pump Assuming that the inflow of the pump is not hampered, the flow produced by the pump is a function of the rotation speed of the pump and the pressure difference between the outflow and inflow of the pump.34 In a typical clinical application, the pressure difference between the outflow and inflow of the pump is the aortic pressure minus the left ventricular pressure. The idealized relation between the static pump flow, Qstat , and the pump pressure head, ∆h, can be described by the following Euler equation35 : ∆h =

u2 u cot β ∆Pstat = − · Qstat , ρg g g πd b

(1)

where ∆Pstat is the static pressure difference, ρ is the density of the fluid, g is the gravity acceleration, u is the linear velocity of the liquid (e.g. blood), β is the angle of the impeller outlet, d is the diameter at the impeller output, and b is the width of the pump impeller.

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If we express u as a function of the rotation speed n in rpm u=

πdn 60

(2)

Equation (1) can be transformed into Qstat = An − Bn · ∆Pstat ,

(3)

nbπ 2 d2 60 cot β

(4)

where An =

and Bn =

60b . nρ cot β

Equation (3) shows a linear relationship between the pump flow and the pressure difference for a fixed rotation speed n. Given the values of An and Bn at a certain rotation speed, the static pump flow under various pressure differences can be determined. The Hemopump type HP31 can be operated at 7 different rotation speeds ranging from 17,000 rpm (Speed 1) to 26,000 rpm (Speed 7) with an interval of 1,500 rpm. Meyns et al.34 performed in vitro measurements of the static flow–pressure difference relations of HP31 Hemopump at the above 7 speeds up to a pressure difference of 230 mmHg. The data (Fig. 2, in Ref. 34) showed a nearly linear relation between the flow and pressure difference in the region of low-pressure difference. In the region of high-pressure difference, this linear relation broke down due to flow separation within the rotor. To establish a static model of the Hemopump, an up-limit of the pressure difference of 130 mmHg is used so that no flow separation occurs. Consequently, the values of the two coefficients, An and Bn in Eq. (3), are obtained by fitting a least squares line to the corresponding data in the range of 0–130 mmHg. Figure 4 shows the seven fitted straight lines and Table 1 lists the values of An and Bn for each of the seven pump rotation speeds. A static model of the pump based on Eq. (3) is shown in Fig. 5, which consists of a constant flow source Qn and a resistance Rn . The model gives the following relation between the pump flow Qstat and the pressure difference ∆Pstat : Qstat = Qn −

1 ∆Pstat . Rn

(5)

By comparing Eq. (5) with Eq. (3), we have: Qn = An

and Rn = 1/Bn ,

(6)

both depend upon the rotation speed of the Hemopump, as shown in Table 1. In the static model shown in Fig. 5, Qn represents the blood flow of the pump with rotation speed n when pressure difference between inlet and outlet is zero, while Rn represents the total resistance to the blood flow in the flow pump, including both the effect of the impeller and blood viscosity.

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Fig. 4. Results of least squares fit to the reported data obtained from in vitro measurements of Hemopump HP31 operated at seven rotation speeds.

Table 1. Values of the two parameters of the linear model shown in Eq. (3) for the Hemopump HP31 operated at seven levels of rotation speed. Speed level

1

2

3

4

5

6

7

An Bn

56.744 0.5381

64.056 0.5072

69.655 0.4851

75.118 0.4260

80.928 0.3726

85.105 0.3014

87.598 0.2274

Fig. 5.

The static model for the axial flow pump.

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3.1.2. Dynamic model of the axial flow pump Equation (3) is for static flow under constant pressure difference. When the pump works in a living circulatory system, the pressure at inlet (Pin = blood pressure in left ventricle), the pressure at the outlet (Pout = blood pressure in aorta), and the pump flow all change with time during a cardiac cycle, and the inertial property of the liquid needs to be considered. To include the effects of the inertial property of the blood in the pump, the static model of the pump is modified by including an inductor H, as shown in Fig. 6. According to the mechanical property of the blood, the inertial property of the blood manifests itself as a force that opposes the change of the flow velocity36 : F =

∂m ∂v dv d (m · v) = v+ m= m, dt ∂t ∂t dt

(7)

where F is the force, v and m are, respectively, the velocity and mass of the fluid within the cannula of the pump. Equation (7) is applicable only for incompressible fluid. Equivalently, the inertial property of the blood can be modeled by a pressure difference ∆P  , which is induced by the change of the flow rate. Numerically, ∆P  is equal to the force F divided by the cross section area A of the cannula of the pump: ∆P  =

1 dv ρ dQH F = · ·m= L· · , A A dt A dt

(8)

where L is the length of the cannula; ρ is the fluid density; and QH is the volumetric flow rate. Based on Eq. (8), the value of H in Fig. 6 can be determined as: H =L·

ρ . A

Using the data from Meyns et al.37 and Reul38 : L = 8.5 cm

Fig. 6.

The dynamic model for the axial flow pump.

(9)

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A ≈ πd2 /4 = π · 8.12 /4 mm2 ρ = 1.06 · 103 kg/m3 we obtain: H = 1.748 Pa · S2 /ml 3.1.3. Cardiovascular model including the axial flow pump A canine cardiovascular model has been developed previously that can simulate the dynamic relationship between the cardiac function and the vascular function, as well as be used as a tool to investigate the mechanism of heart failure and the effects of cardiac assist device.26,39,40 As shown in Fig. 7, the model consists of four heart chambers, a pulmonary circulation, 11 aortic segments, and lumped venous and peripheral vascular systems. The detailed structure and operation of each block of the model can be found in Bai and Zhao40 and Zhou et al.39 Here we just give a brief description of the model for the left ventricle with regional ischemia. The left ventricle is functionally divided into two non-physiological compartments, one consisting of all the normal myocardium and the other consisting of all the ischemic myocardium. The ratio of the ischemic myocardial mass to the total myocardial mass, denoted by Rm , is set to be 30% in this study. During each cardiac cycle, each compartment generates its own pressure-volume loop, which in turn determines the pressure and volume of the entire left ventricle. The maximum elasticity (which is related to myocardial contractility) of the normal compartment (Ees1 ) is 5 mmHg/ml and the maximum elasticity of the ischemic compartment (Ees2 ) is 1.5 mmHg/ml. The canine circulatory model shown in Fig. 7 also includes the dynamic model of the pump shown in Fig. 6. To simulate the typical clinic setting,30 the inlet of the pump in our model is inserted into the left ventricle where the pressure is denoted as PLV , and the outflow of the pump is discharged into the descending aorta where the pressure is denoted as Pa [4] (the number 4 refers to the 4th aortic segment). As a result, Pin and Pout in Fig. 6 become PLV and Pa [4], respectively. 3.2. Computer simulations and results Simulations were performed using the above model which represents a canine cardiovascular system assisted by the axial flow pump. The heart rate was set to be 120 beats/min. The computer program for the simulation study was written in Delphi language and run on a PC. The time interval for computation was 0.001 second. During each time interval, the pressure, flow and volume in each block of the model were computed and updated sequentially, starting from the left and right ventricles. Using the above model and computer simulation, the system parameters can be changed easily and the hemodynamic variables in various parts of the circulatory system can be observed conveniently. Consequently, the effects of the assist device

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Fig. 7. Block diagram of the model for the canine circulatory system including a axial flow pump (AFP).

under different cardiovascular conditions can be assessed conveniently. In general, the results obtained by computer simulation are consistent with the experimental results reported previously.41,42 3.2.1. The pump flow and pressure difference during one cardiac cycle The simulation results show that the change of flow (Q) lags behind that of pressure difference (∆P ) by about 0.5 s at a heart rate of 120/s. The time course of Q versus ∆P during a cardiac cycle forms a loop which proceeds clockwise. This phenomenon, which is due to the inertial property of the blood within the cannula and pump, was also observed previously by Meyns et al.34 in their in vivo study of the Hemopump.

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The hysteresis of the loop reduces as pump speed increases. This can be explained by the fact that when the pump speed is increased, more blood is expelled by the pump and the ventricular cavity is reduced. As a result, the contribution of the heart to the pump flow is reduced, which is illustrated by the decrease in the upstroke in flow during the ejection phase at high speeds. At low pump rotation speed, the flow shows significant pulsation. As the pump speed rises, the pulsation of pump flow decreases.

3.2.2. The effects of the pump on various hemodynamic variables Figure 8 shows the effects of the pump on various hemodynamic variables when the pump speed is increased from 1 to 5. Figure 8(a) plots the aortic pressure during a cardiac cycle. As the rotation speed increases, the mean aortic pressure increases while the pulsation of the aortic pressure diminishes. The same trends have been reported in clinical experiments.37,41,43 Figure 8(b) shows the blood flow pumped out by the left ventricle. Only at Speed 1, there is noticeable blood flow pumped out by the left ventricle during early systole. As the pump rotation speed is further increased, the amount of blood pumped out by the left ventricle becomes negligible. This observation is consistent with the results reported by Peterzen et al.41 They noticed that the aortic valve was almost always closed when the Hemopump was operating. The change in the volume of the left ventricle during each cardiac cycle is shown in Fig. 8(c). The volume of the left ventricle decreases with the increase of the rotation speed of the pump. At Speed 5, the volume of the left ventricle decreases to a rather small value of 20 ml. The pressure-volume loop of the left ventricle is depicted in Fig. 8(d). As the pump speed increases, the loop moves to the left (the mean volume decreases) and the area of the loop decreases, indicating a decrease in the workload of the left ventricle.44 Figure 8(e) shows the change of the left atrial volume during the cardiac cycle. As the pump speed increases, the left atrial volume decreases, which may explain the potential collapse of the left atrium at high speeds reported in literature.42

3.2.3. The beneficial effects of the axial flow pump on a failing heart Figure 9(a) shows the simulation results of the effects of the pump on the stroke volume which is the summation of the blood pumped out by the left ventricle and the blood pumped out by the pump during a cardiac cycle. Since the contribution of the left ventricle in pumping blood quickly becomes insignificant as the pump rotation speed increases (see Fig. 8(b)), the increase in the stroke volume is almost entirely due to the increase in the pump flow.

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Fig. 8. The effects of the pump on various hemodynamic variables when operated at five different speeds. (a) aortic pressure (b) blood flow pumped out through the aortic valve by the left ventricle (c) volume of left ventricle (d) pressure-volume relationship of left ventricle during the cardiac cycle (e) volume of left atrium.

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Fig. 9. The assistant effects of the pump at different rotation speeds: (a) stroke volume (b) myocardial oxygen consumption (white bars) and oxygen supply (black bars).

The total myocardial oxygen consumption, VO2 , of the left ventricle is considered as the summation of two parts45 : VO2 = VO2 (1) + VO2 (2)

(10)

VO2 (1) = B · PVA(1) + C · Ees1 · (1 − Rm) + D · (1 − Rm)

(11)

VO2 (2) = B · PVA(2) + C · Ees2 · Rm + D · Rm

(12)

where VO2 (1) and VO2 (2) are the volume of oxygen consumed by the normal compartment and the ischemic compartment of the left ventricle, respectively; PVA(1) and PVA(2) are the area of the individual pressure-volume loop of each compartment, and Ees1 and Ees2 are the maximal elasticity of the normal region and the ischemic region of the left ventricle, respectively. The values of the three coefficients used in the simulation are obtained based on the work of Suga et al.46 : B = 1.65 × 10−5 /mmHg C = 0.0024 ml2 /mmHg D = 0.0177 m The volume of myocardial oxygen supply (VOS ) is calculated from the total coronary flow during one cardiac cycle (FT C ): (A · T )O2 · FT C (13) 100 where A is the volume percent of the oxygen content in the coronary arterial blood and T is a variable that depicts the ability of the myocardial oxygen absorption. The typical values of A and T are determined from the work of Walley et al.47 and a final value of VOS = 0.0015 FT C is used. Figure 9(b) compares the myocardial oxygen consumption (VOC ) and the volume of myocardial oxygen supply (VOS ) at each of the 5 pump speeds. The myocardial VOS =

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oxygen consumption (VOC ) of the left ventricle is directly related to the area of pressure-volume loop. Figure 8(d) indicates when the pump speed increases, the area of the pressure-volume loop of the left ventricle decreases. As a result, the oxygen consumption decreases as the pump speed increases. Alternatively, the decrease in oxygen consumption of the left ventricle can be explained by the reduction of its preload that is related to the left atrium volume and the left ventricular enddiastolic volume. With the increase of pump speed, both the left atrium volume and the left ventricular end-diastolic volume decrease, as indicated in Figs. 8(e) and 8(c). Consequently, the oxygen consumption is decreased. As the pump rotation speed increases, the aortic pressure increases (Fig. 8(a)). As a result, the coronary flow increases, so does the oxygen supply. Figure 9 depicts the three most important beneficial effects of the pump to a failing heart: an increase in the stroke volume, an increase in the oxygen supply, and a reduction in the oxygen consumption. These benefits were also reported by Peterzen et al.41 based on their clinic experiments. Figure 9 also shows that each of the effects becomes more significant when the pump rotation speed increases. Finally, the results of the computer simulation regarding the beneficial effects of pump in terms of the values of several relevant hemodynamic variables at five pump speeds are summarized in Table 2.

Table 2. Values of various hemodynamic variables when the pump is operated at five different rotation speeds. Pump Speed

1

2

3

4

Qmpm (ml/s) Pmao (mmHg) Paosy (mmHg) Paodi (mmHg) Ppao (mmHg) Plvsy (mmHg) Vlvsy (ml) Vladi (mmHg)

19.917 99.419 108.31 90.402 17.913 108.33 51.035 9.3042

27.693 105.23 110.72 100.90 9.8177 99.500 46.200 8.1257

30.783 113.54 116.33 111.15 5.1769 68.586 35.601 6.3562

33.819 122.24 123.63 121.22 2.4047 31.273 23.350 4.7155

37.924 133.70 133.92 133.58 0.3369 4.4318 17.032 3.1842

SV (ml/beat) VOC (ml/beat) VOS (ml/beat)

12.962 0.0683 0.0579

13.951 0.0568 0.0673

15.380 0.0421 0.0830

16.865 0.0311 0.0998

18.853 0.0274 0.1163

Qmpm = mean pump flow Pmao = mean aortic pressure Paosy = aortic peak systolic pressure Paodi = aortic minimum diastolic pressure Ppao = pulsation of aortic pressure (Paosy – Paodi ) Plvsy = left ventricle peak systolic pressure Vlvsy = left ventricle end systolic volume Vladi = left atrium end diastolic pressure SV = stroke volume VOC = volume of myocardial oxygen consumption VOS = volume of myocardial oxygen supply

5

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4. Optimum Control of the Axial Flow Pump The results of the simulation study indicate that the direct effects of the axial flow pump include an increase in stroke volume, an increase in mean aortic pressure and coronary blood flow, a decrease in the area of the pressure-volume loop of the left ventricle and a decrease in left-atrial volume. The results also show that all these changes are further enhanced when the rotation speed of the pump impellers is increased. In practical applications however, it is not always beneficial to operate the pump at the highest possible speed. For example, in animal experiments with the Hemopump, Siess et al.36 found that at high pump speeds, a total collapse of the left ventricle can occur that leads to a sudden decrease in stroke volume. In addition, it was suggested that the shear forces inside the high-speed pump can cause blood trauma and thrombosis.48 Finally, a high pump speed requires greater power consumption. An important question is therefore how to achieve the clinical objectives of the assist device with the lowest pump speed. In the following sections, we describe a general framework for designing an optimum control strategy for the axial flow pump and test the strategy using the simulation model described in the previous sections.49 An objective function is first defined that includes four hamodynamic variables with suitable weighting factors: stroke volume, mean left-atrial pressure, aortic diastolic pressure and mean pump speed. The flow pump is then allowed to operate at either a constant speed or two different speeds during a cardiac cycle. The goal is to maximize the objective function by varying the magnitude and timing of the pump speed. The results of simulation indicate that in general, different clinical objectives or different cardiac conditions require different operation parameters. The results also suggest that it is more beneficial to operate the pump at two different speeds than to maintain a constant speed throughout the cardiac cycle.

4.1. The objective function for optimal control of the axial flow pump A general approach in optimal control is to maximize the value of an objective function (or a performance index) by adjusting a certain set of operation variables.50 The objective function (OF) has the following general form: OF = w1 · µ(v1 ) + w2 · µ(v2 ) + · · · + wn · µ(vn ),

(14)

where v1 , v2 , . . . , vn are the members of the objective function, µ(vi ) represents the membership function of vi , and w1 , w2 , . . . , wn are the weighting factors. To quantitatively evaluate the objective function, three questions need to be answered: how to choose the members, how to construct the membership functions, and how to determine the weighting factors.

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4.1.1. Determine the members of the objective function The general clinical objectives of the flow pump-based assist device are to increase the cardiac output, to increase the oxygen supply to the failing heart, and to decrease the workload of the heart. The actual effects produced by the assist device can be assessed by measuring the changes in the major hemodynamic variables from their preoperative values. In choosing the members of the objective function for optimal control of the pump, a logical approach is to include a minimum number of hemodynamic variables that best represent the clinical objectives and are relatively easy to measure clinically. The selected members of the objective function are: stroke volume (defined as the combined volume of the blood pumped out by the left ventricle and the blood pumped out by the pump during a cardiac cycle), mean left atrial pressure, minimum aortic (diastolic) pressure, and mean pump speed. The first two variables are directly related to the clinical objectives of the assist device. In fact, a low cardiac index (< 2.0 L·min−1 · m−2 ) and a high pulmonary wedge pressure (> 20 mmHg), which is directly related to mean left atrial pressure, are two important criteria for selecting patients for receiving LVAD.51,52 After the assist device is implanted, these two variables can readily be measured in vivo for hemodynamic assessment.22 The other two members, minimum aortic pressure and pump speed, are also easily measurable. The choice of minimum aortic pressure can be justified as follows. As mentioned previously, increase of coronary blood flow is one of the important objectives of the assist device. Coronary flow is mainly determined by mean aortic diastolic pressure. An increase in minimum aortic pressure directly increases coronary flow. On the other hand, an abnormally high aortic diastolic pressure increases the afterload of the left ventricle, and can produce damaging effects to certain organs such as eyes, brain and kidney. Consequently, there is a desired range of aortic diastolic pressure for a healthy cardiovascular system, and this information can be used to construct the membership function. Finally, the mean pump speed is included in the objective function as a penalty term. 4.1.2. Establish the membership function The four members of the objective function, stroke volume (SV ), mean left atrial pressure (PMLA ), minimum aortic pressure (PMAO ) and mean pump speed (M P S) have different desired ranges and different units. In order to have a uniform treatment of their individual contributions to the objective function, the membership function for each member is established. The value of each membership function ranges from 0 to 1, with 1 represents the condition that the member reaches its desired value. In general, the desired value is the one found in a healthy cardiovascular system. The membership function for stroke volume (SV ) is chosen to be a sigmoid function: 1 , (15) µ(SV ) = −a(SV −b) 1+e

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where a and b are two parameters which can be determined by two boundary values of SV . Let SV2 represent the desired stroke volume (upper bound) that µ(SV2 ) = 1 − ε where ε is a small positive value (e.g. 0.01), and let SV1 represent the lower bound that µ(SV1 ) = ε. Then a=

2 ln[(1 − ε)/ε] SV2 − SV1

and b =

SV1 + SV2 . 2

(16)

Based on out simulation results from a health canine heart, SV2 = 15 ml/beat, SV1 = SV2 /3, and ε = 0.01. That makes a = 0.919, and b = 10. The actual curve is shown in Fig. 10(a). The membership function for mean left atrial pressure (PMLA ) is defined as follows:  1 PMLA ≤ 8 mmHg , (17) µ(PMLA ) = −(PMLA −8)2 e 2σ2 PMLA > 8 mmHg

Fig. 10. Membership function of (a) stroke volume (SV ); (b) mean left atrial pressure (PMLA ); (c) minimum aortic pressure (PMAO ); (d) mean pump rotation speed (MPS ).

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where PMLA = 8 mmHg is considered as the normal value for a healthy canine heart. The parameter σ determines the rate of decrease of the membership function as PMLA increases. Based on the clinical criterion of patient selection52 as well as simulation results, a σ = 7 mmHg is chosen. The actual curve is shown in Fig. 10(b). The membership function for minimum aortic pressure (PMAO ) is determined based on the following considerations. The International Society of Hypertension, World Health Organization defined the desirable range of PMAO in human as 70 mmHg – 80 mmHg.53 Considering that in this study the optimum control strategy is tested with a canine model and the health canine model produces a PMAO of 93 mmHg, the desirable range of PMAO is shifted to 90 mmHg–100 mmHg. More specifically, the membership function for PMAO have a value of 1 when PMAO falls into this range, and have a descending branch when PMAO is either below 90 mmHg or above 100 mmHg. Mathematically, the membership function for PMAO is defined by the following expressions:  −(PM AO −90)2   2σ2 (σ = 20) 0 ≤ PMAO < 90 mmHg e (18) µ(PMAO ) = 1 90 ≤ PMAO ≤ 100 mmHg  −100)2   −(PM AO 2σ2 (σ = 30) PMAO > 100 mmHg e The actual curve is shown in Fig. 10(c). Finally, a straight line is used to define the membership function of mean pump speed (MPS). It is assigned that µ(MPS ) = 1 when MPS = 0, and µ(MPS ) = 0.5 when the pump is operated at the highest speed (26,000 rpm, Speed 7).34 The expression for the membership function is therefore: µ(MPS ) = 1 − MPS /52, 000,

(19)

and the actual curve is shown in Fig. 10(d).

4.1.3. Determine the weighting factor for each membership The weighting factors in Eq. (14) represent the relative importance of each member in achieving the clinical objectives of the assist device. Depending upon the cardiovascular conditions of an individual patient and the specific clinical objectives, a different set of the weighting factors may be assigned. In this study, two cases are demonstrated. In the first case, the main clinical objective is to increase cardiac output, and the membership function µ(SV ) receives the largest weight. Without quantitative justification, the following set of weighting factors is defined for the purpose of demonstration, w(µ(SV )) = 1, w(µ(PMLA )) = 1/3, w(µ(PMAO )) = 1/3, and w(µ(MPS )) = 1/5. This set of weighting factors means that to increase SV is 3 times more important than to increase PMAO or to lower PMLA , and 5 times more important than to lower MPS. In the second case, the main clinical objective is to increase coronary flow and the membership function µ(PMAO ) receives the largest

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weight. In this case, the relative weights become [1/3 1/3 1 1/5]. After normalization (the sum of the four weighting factors equals 1), the objective function in each case can be expressed as: Case 1 (emphasizes SV ): OF = 0.5357µ(SV ) + 0.1786µ(PMLA) + 0.1786µ(PMAO ) + 0.1071µ(MPS ),

(20)

Case 2 (emphasizes PMAO ): OF = 0.1786µ(SV ) + 0.1786µ(PMLA) + 0.5357µ(PMAO ) + 0.1071µ(MPS ).

(21)

4.2. The model used in the simulation The canine cardiovascular model shown in Fig. 7 is used again in this part of the study. The model includes a failing heart assisted by the axial flow pump. Three conditions of the left ventricle are simulated in this study: normal (the ratio of the ischemic myocardial mass to the total mass, denoted by Rm , is 0), minor ischemia (Rm = 25%), and severe ischemia (Rm = 50%). When the model simulates a normal left ventricle, it produces the following hemodynamic values: SV = 13.4 ml, PMLA = 10.8 mmHg, and PMAO = 93.1 mmHg. These values are used as the baseline, preoperative hemodynamic values.

4.3. Optimum control of the axial flow pump using a single pump speed Table 3 lists the values of SV, PMLA , and PMAO for a heart with minor ischemia (Rm = 25%) when the Hemopump is off as well as when it is operated at Speed 1 to Speed 4. The last two rows of the table list the values of the objective function calculated using the two sets of weighting factors defined by Eq. (20) and Eq. (21). As shown in Table 3, when the ischemic left ventricle is not assisted by the pump, the SV is decreased by 10%, PMLA is increased by 19%, and PMAO is decreased by 11%, as compared with the baseline values of a normal heart. When the pump is turned on, these variables move towards the normal values. The objective function is maximized at Speed 2 under both sets of weighting factors. Table 4 lists the values of SV , PMLA , and PMAO for a heart with severe ischemia (Rm = 50%) under various operating speeds of the pump. As compared with the baseline values of a normal heart, the SV is decreased by 19%, PMLA is increased by 37%, and PMAO is decreased by 22% when the pump is turned off. When the pump is operating, the objective function is maximized at Speed 3 when SV is emphasized (Case 1), and maximized at Speed 2 when PMAO is emphasized (Case 2).

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Table 3. Single pump speed and minor ischemia. SV, PMLA , and PMAO at different pump speed and corresponding values of the objective function (OF ) using two sets of weighting factors (Case 1 and Case 2). Rm = 25%

Pump is off

Pump Speed 1

Pump Speed 2

Pump Speed 3

Pump Speed 4

SV (ml/beat) PMLA (mmHg) PMAO (mmHg)

12.0 12.8 82.4

13.5 10.7 91.5

14.3 9.68 101

15.7 8.07 112

17.2 6.61 123

OF (Case 1) OF (Case 2)

0.877 0.901

0.932 0.946

0.946 0.953

0.942 0.915

0.910 0.819

Table 4. Single pump speed and severe ischemia. SV, PMLA , and PMAO at different pump speed and corresponding values of the objective function (OF ) using two sets of weighting factors (Case 1 and Case 2). Rm = 50%

Pump is off

Pump Speed 1

Pump Speed 2

Pump Speed 3

Pump Speed 4

SV (ml/beat) PMLA (mmHg) PMAO (mmHg)

10.8 14.8 72.9

12.4 12.3 87.5

13.7 10.6 98.7

15.3 8.68 110

17.0 6.89 122

OF (Case 1) OF (Case 2)

0.707 0.712

0.880 0.913

0.932 0.944

0.943 0.925

0.912 0.827

4.4. Optimum control of the axial flow pump using two pump speeds When the pump is operated at a constant speed throughout the cardiac cycle, it is referred to the non-synchronous (with the heart) operation. Considering the prospective long-term usage of the assist device, we cannot ignore the concern regarding the long-term physiological effects of a nonpulsatile circulation. In the synchronous pumping mode, the pump is operated at a lower speed during diastole (to avoid backflow) and is operated at a higher speed during systole. Figure 11 depicts the general timing of the pump speed in one cardiac cycle which is 500 ms in our simulation. Speed A represents the lower speed and Speed B represents the higher speed; t1 is the time when the pump speed is switched from Speed A to Speed B, and t2 is the time when the pump speed returns from Speed B to Speed A. For each heart condition (Rm = 25% and Rm = 50%) and each weighting set (Case 1 and Case 2), all possible combination of two speeds and all possible timing of each speed — from 0 to 500 ms with an increment of 20 ms, are systematically tested and the corresponding value of the objective function is calculated. Table 5 gives the optimal pump speed and timing that maximize the value of the objective function for the case of minor ischemia (Rm = 25%), and Table 6 gives the results for the case of severe ischemia (Rm = 50%). In both tables, Speed A represents the lower speed and Speed B represents the higher speed; t1 is the

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Fig. 11. Schematic two speed control strategy during one cardiac cycle with the left ventricular pressure (Plv ). Table 5. Optimal pump speeds and timings that maximize the objective function (OF ) in Case 1 and Case 2 for minor ischemic heart. Rm = 25%

t1 (ms)

t2 (ms)

Speed A

Speed B

OF

Case 1 Case 2

40 40

400 440

1 1

4 3

0.954 0.956

Table 6. Optimal pump speeds and timings that maximize the objective function (OF ) in Case 1 and Case 2 for severe ischemic heart. Rm = 50%

t1 (ms)

t2 (ms)

Speed A

Speed B

OF

Case 1 Case 2

40 20

380 400

1 1

5 4

0.953 0.955

time when the pump speed is switched from Speed A to Speed B, and t2 is the time when the pump speed returns from Speed B to Speed A. When the time course of the pump speed is plotted together with the time course of the left ventricular pressure PLV during a cardiac cycle, it is found that in general, optimal pump operation calls for a higher pump speed during systole and early diastole and a lower pump speed during late diastole. In addition, a severely ischemic left ventricle requires a higher pump speed during systole than a left ventricle with a minor ischemia (Speed B = 5 & 4 versus 4 & 3). The results also indicate that if the

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increase of SV is more important (Case 1), the pump should be operated at a higher Speed B but with a shorter duration (a more ‘pulsatile’ operation). On the other hand, if the increase of PMAO is more important (Case 2), a smaller Speed B can be used with a prolonged duration (a less ‘pulsatile’ operation). This observation is consistent with the results reported by Klots et al.54 which showed that left ventricular volume unloading is more pronounced in pulsatile LVAD as compared with nonpulsatile LVAD. Finally, by comparing the values of the objective function in Tables 5 and 6 with that in Tables 3 and 4, one can see that by allowing the Hemopump to operate at two different speeds, a slightly larger value of the objective function can be obtained.

5. Conclusion Despite extensive research and great advancement in recent years, the search for ideal mechanical cardiac assist device continuous. While the early left ventricular assist device (LVAD) was mainly devised to serve as a bridge to heart transplantation, the recently completed REMATCH (the Randomized Evaluation of Mechanical Assistance for the Treatment of Congestive Heart Failure) trail has clearly demonstrated the feasibility of using LVAD as destination therapy. The REMATCH trail however, also revealed many problems such as infection, bleeding and a high probability (35%) of device failure.2 Another concern is the cost of these devices. For example, cost data from REMATCH showed that the median overall cost was about $250,000 as compared with the current US cost of $205,000 for each cardiac transplantation.9 These problems have prompted increasing development of smaller, simpler, and more reliable devices that are more affordable, easy to use and have less risk for infection or hematologic aberrancies. The axial flow pump possesses many promising advantages to meet these demands. The dynamic model for the axial flow pump described here was developed based on a theoretical analysis as well as in vitro and in vivo experimental data. Incorporated in a canine circulatory system, the model has been used to predict the effects of the pump on various hemodynamic variables that are in good agreement with the reported results in clinical and animal experiments. After the above validation, the model was further used to study the optimal control of the assist device through the objective functions. By using the membership functions and the associated weighting factors, the method offers great flexibility in choosing the targeted hemodynamic variables, in specifying the particular way that each of these variables is to be optimized, and in assigning the relative importance of each targeted variable. Although the present study is carried out in the canine model, the general conclusions should be applicable to a human model. The methods used and the results obtained in this study can be incorporated into the design of an advanced physiological controller for a long-term operation of the axial flow pump-based assist device as well as other types of continuous-flow LVAD.13,55

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Acknowledgments This work is partially supported by the National Nature Science Foundation of China, the Tsinghua-Yue-Yuen Medical Science Foundation, the National Basic Research Program of China, and the Special Research Fund for the Doctoral Program of Higher Education of China.

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20. E. Okamoto, T. Hashimoto, T. Inoue et al., Blood compatible design of a pulsatile blood pump using computational fluid dynamics and computer-aided design and manufacturing technology, Artif. Organs. 27 (2003) 61–67. 21. P. M. Portner, P. E. Oyer, D. G. Pennington et al., Implantable electrical left ventricular assist system: bridge to transplantation and the future, Ann. Thorac. Surg. 47 (1989) 142–150. 22. S. Westaby, O. H. Frazier, F. Beyersdorf et al., The Jarvik 2000 Heart. Clinical validation of the intraventricular position, Eur. J. Cardio-thor. Surg. 22 (2002) 228–232. 23. G. M. Wieselthaler, H. Schima, M. Hiesmayr et al., First clinical experience with the DeBakey VAD continuous-axial-flow pump for bridge to transplantation, Circulation 101 (2000) 356–359. 24. S. Takatani, Can rotary blood pumps replace pulsatile devices, Artif. Organs. 25, 9 (2001) 671–674. 25. P. J. Overwalder, Intra Aortic Balloon Pump (IABP) Counterpulsation, The Internet Journal of Thoracic and Cardiovascular Surgery 2 (1999) 2. 26. D. Jaron, T. W. Moore and P. He, Theoretical considerations regarding the optimization of cardiac balloon pumping, IEEE Trans. Biomed. Eng. 30 (1983) 177–186. 27. W. Ohley, C. Kao and D. Jaron, Validity of an arterial system model: a quantitative evaluation, IEEE Trans. Biomed. Eng. BME-27 (1980) 203. 28. M. J. Williams, Experimental determination of optimum performance of counterpulsation assist pumping under computer control, Comput. Biomed. Res. 10 (1977) 545– 559. 29. M. S. Sweeney, The Hemopump in 1997: A clinical, political, and marketing evolution, Ann. Thorac. Surg. 68 (1999) 761–763. 30. O. H. Frazier, R. K. Wampler, J. M. Duncan et al., First human use of the Hemopump, a catheter-mounted ventricular assist device, Ann. Thorac. Surg. 49 (1990) 299–304. 31. S. J. Phillips, L. Barker, B. Balentine et al., Hemopump support for the failing heart, Asaio Trans. 36, 3 (1990) 629–632. 32. G. R. Li, W. Ma and X. Zhu, Development of a new left ventricular assist device: the dynamic aortic valve, Asaio J. 47 (2001) 257–260. 33. X. Li, J. Bai and P. He, Simulation study of the Hemopump as a cardiac assist device, Med. Bio. Eng. Comput. 40 (2002) 344–353. 34. B. Meyns, T. Siess, S. Laycock et al., The heart-Hemopump interaction: a study of the Hemopump flow as a function of cardiac activity, Artif. Organs. 20 (1996) 641–649. 35. I. Ionel, Pumps and pumping (Elsevier Science Press, 1986), p. 104. 36. T. Siess, B. Meyns, K. Spielvogel et al., Hemodynamic system analysis of intraarterial microaxial pumps in vitro and in vivo, Artif. Organs. 20 (1996) 650–661. 37. B. P. Meyns, P. T. Sergeant, W. J. Daenen et al., Left ventricular assistance with the transthoracic 24F Hemopump for recovery of the failing heart, Ann. Thorac. Surg. 60 (1995) 392–397. 38. H. Reul, Technical requirements and limitations of miniaturized axial flow pumps for circulatory support, Cardiology 84 (1994) 187–193. 39. X. Q. Zhou, J. Bai and B. Zhao, Simulation study of the effects of regional ischemia of left ventricle of cardiac performance, Automedica 17 (1998) 109–125. 40. J. Bai and B. Zhao, Simulation evaluation of cardiac assist devices, Methods of Information in Medicine, 39 (2000) 191–195. 41. B. Peterzen, U. Lonn, A. Babic, H. Granfeldt et al., Postoperative management of patients with Hemopump support after coronary artery bypass grafting, Ann. Thorac. Surg. 62 (1996) 495–500.

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42. E. Chen, and X. Q. Yu, The Hemopump’s design and its experimental study, J. Biomed. Eng. 17 (2000) 469–472. 43. G. D. Dreyfus, The Hemopump 31, the sternotomy Hemopump clinical experience, Ann. Thorac. Surg. 61 (1996) 323–328. 44. F. R. Waldenberger, P. Wouters and E. Deruyter, Mechanical unloading with a miniaturized axial flow pump (Hemopump): an experimental study, Artif. Organs. 19 (1995) 742–746. 45. H. Suga, Total mechanical energy of a ventricular model and cardiac oxygen consumption, Am. J. Physiol. 236 (1979) H498–505. 46. H. Suga, Y. Yasamura, T. Nozawa, S. Futaki, Y. Igarashi and Y. Goto, Prospective prediction of O2 consumption from pressure-volume area in dog hearts. Am. J. Physiol. 252 (1987) H1258–1264. 47. K. R. Walley, C. J. Beeker, R. A. Hogan et al., Progressive hypoxemia limits left ventricular oxygen consumption and contractility, Circ. Res. 63 (1988) 849. 48. M. Loebe, A. Koster, S. S¨ anger, E. V. Potapov, H. Kuppe, G. P. Noon and R. Hetzer, Inflammatory response after implantation of a left ventricular assist device: comparison between the axial flow MicroMed DeBakey VAD and the pulsatile Novacor device, Asaio J. 47 (2001) 272–274. 49. P. He, J. Bai and D. D. Xia, Optimum control of the Hemopump as a left-ventricular assist device, Med. Bio. Eng. Comput. 43 (2005) 136–141. 50. T. L. Saatty and J. M. Alexander, Thinking with models: mathematical models in the physical, Biological and Social Sciences (Pergamon Press, 1981), pp. 148–151. 51. P. M. McCarthy, HeartMate implantable left ventricular assist device: bridge to transplantation and future applications. Ann. Thorac. Surg. 59 (1995) S46–S51. 52. J. Parameshwar, and J. Wallwork Left ventricular assist devices: current status and future applications, Int. J. Cardiol. 62 (1997) S23–S27. 53. WHO-ISH The Guidelines Subcommittee of the WHO-ISH Mild Hypertension Liaison Committee. 1993 Guidelines for the Management of Mild Hypertension. Memorandum from a World Health Organization — International Society of Hypertension Meeting, J. Hypertensions 11 (1993) 905–918. 54. S. Klotz, M. C. Deng and J. Stypmann et al., Left ventricular pressure and volume unloading during pulsatile versus nonpulsatile left ventricular assist device support, Ann. Thorac. Surg. 77 (2004) 143–150. 55. Y. Wu, P. Allaire, G. Tao, H. Wood, D. Olsen and C. Tribble, An advanced physiological controller design for a left ventricular assist device to prevent left ventricular collapse, Artif. Organs. 27 (2003) 926–930. 56. D. D. Xia and J. Bai, Current status and typical technique of axial flow blood pumps, Foreign Medical Biomedical Engineering Fascicle 28, 6 (2005) 366–369 (in Chinese). 57. D. D. Xia and J. Bai, Simulation study and function analysis of the dynamic aortic valve. Progress in Natural Science (accepted). 58. D. D. Xia and J. Bai, Simulation study and function analysis of micro-axial blood pumps, 27th Annual International Conference of the IEEE Engineering in Medicine and Biology Society, pp. 2971–2974. 59. J. Li, Dynamics of the vascular system (World Scientific Publishing Company, 2004), pp. 234–244. 60. D. B. Olsen, The history of continuous-flow blood pumps, Artif. Organs. 24 (2000) 401–404.

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CHAPTER 2 TECHNIQUES IN VISUALIZATION AND EVALUATION OF THE IN VIVO MICROCIRCULATION SHIGERU ICHIOKA Department of Plastic and Reconstructive Surgery, Saitama Medical University 38 Morohongo, Moroyama, Iruma-gun, Saitama, 350-0495, Japan [email protected]

Microcirculation plays a direct role in the accomplishment of the principal purpose of the circulation system. The functions of the microcirculation incorporates oxygen supply, transport, diffusion, and exchange of nutrients and metabolites between blood and tissue, maintenance of body temperature, regulation of blood pressure, tissue defense and repair. These complex functions are carried out in the microcirculation by a number of dynamic changes in the blood within the vessels, the vessels themselves or the tissues surrounding the vessels. Intravital microscopic approaches have greatly contributed to the advancement of research in the fields of microcirculation. The techniques represent the only method that allows direct visualization and quantitative analysis of the microcirculation. In this chapter in vivo microscopic techniques with useful experimental models, which provide significant information regarding the microcirculation, are introduced. Keywords: Microcirculation; intravital microscope; visualization; experimental model.

1. Introduction Although there is no universally accepted definition of the microcirculation, it is widely taken to encompass the flow of blood in the circulatory system of small vessels less than 150 µm in diameter. It includes arterioles, capillaries, and venules. In spite of their small size in diameter, the capillaries are enormous in total length, which is about 90–110 thousand kilometers i.e. two and half times the perimeters of the earth. The microcirculation constitutes a huge biological system essential for maintenance of life. The microcirculation plays a direct role in the accomplishment of the principal purpose of the circulation system. The functions of the microcirculation incorporates oxygen supply, transport, diffusion, and exchange of nutrients and metabolites between blood and tissue, maintenance of body temperature, regulation of blood pressure, tissue defense and repair. These complex functions are carried out in the microcirculation by a number of dynamic changes in the blood within the vessels, the vessels themselves or the tissues surrounding the vessels. Such changes include vasospasm, vasomotion, blood flow, velocities, leukocyte-endothelium interaction, macromolecular leakage, angiogenesis, microembolic and thrombotic events, and functional capillary densities, and so forth. A number of expedient techniques have enriched the armamentarium for the evaluation of the microcirculation. Most of them have used methods such as microspheres,1 xenon washout,2 tissue oxygen levels,3 laser Doppler4,5 and dye 29

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diffusion methods,6 but these methods characteristically measure indirect indicators of blood perfusion, and do not allow distinct analysis of microcirculatory or microhemodynamic mechanisms within individual segments of the microvasculature. The use of these indirect techniques has been criticized for providing only speculation to the microcirculatory status. In contrast, intravital microscopic approaches have overcome the limitation of the indirect methods and have greatly contributed to the advancement of research in the fields of microcirculation. The techniques represent the only method that allows direct visualization and quantitative analysis of the microcirculation. In this chapter in vivo microscopic techniques with useful experimental models, which provide significant information regarding the microcirculation, are introduced.

2. Intravital Microscopic System 2.1. Microscope In vivo microcirculation can be studied directly with an ordinary light, preferably stereo, microscope with variable magnifications. The microscope used in our laboratory is the stereo microscope (SZH 10, Olympus, Japan), with a useful zoom function (Fig. 1). 2.2. Suspension of the microscope Since small movements and vibrations of the microscope would interfere with the observations, the microscope should be mounted on a firm stand. The stand should include a movable arm so that the whole apparatus can be moved in any direction.

Fig. 1. Equipment for intravital microscopy. (a) Stereo microscope with variable magnifications and a zoom function. (b) Halogen projection lamp for trans-illumination. (c) Mercury lamp for epi-illumination. (d) Charged-coupled device (CCD) camera. (e) Hard disk video recorder. (f) TV monitor.

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2.3. Illumination A 150 W halogen projection lamp (Nikon) or a 100 W mercury lamp achieves sufficient illumination as a light source. In vivo microscopy includes trans- and/or epi-illumination techniques for the direct visualization of the microvasculature. The trans-illumination technique requires thin, translucent preparations for the investigation to guarantee exact imaging of microvessels and blood cell flow. Narrow band filters (of a wave length between 400 and 440 nm) allow for contrast enhancement by dark staining of erthrocytes, facilitating quantitative analysis of the microvascular flow. Epi-illumination may be performed without contrast enhancement; however, optimal images and analysis of the microcirculation are best achieved using fluorescent markers. By staining different components of the blood, such markers permit visualization of plasma [fluorescein-isothiocyanate (FITC)-Dextran, FITCAlbumin], red cells (FITC-labeled), and white cells (rhodamin 6G). 2.4. Imaging and recording Microvascular appearances are imaged by charged-coupled device (CCD), intensified charged-coupled device (ICCD) or silicon-intensified-target (SIT) cameras connected to the stereo microscope. The images are transferred to video systems and stored on videotapes or hard disk video recorders, permitting quantitative offline analysis of microcirculatory parameters by the use of computer-assisted image analysis systems.

3. Tissue Preparation Intravital microscopic techniques allow study in a variety of experimental models. Microcirculation is visualized mostly by means of placing the surgically prepared tissue of the experimental animals under the microscope. 3.1. Anesthetics Several anesthetics commonly used to immobilize small rodents, and other small to medium-sized mammals for surgical procedure as well as microvascular observations include sodium pentobarbital, urethane, and inhalation anesthetics. Sodium pentobarbital is the most commonly used anesthetic. The animals are anesthetized with an intraperitoneal injection of 50–75 mg/kg body weight or an intramuscular injection of approximately 30 mg/kg body weight. It is the most frequently used anesthetic during surgical intervention, but researchers should take into consideration the fact that sodium pentobarbital generally depresses arterial blood pressure and respiration below normal when it is used for hemodynamic measurements. Urethane is a favorite choice for acute experiments. It is injected intraperitoneally in a dose of 1 g/kg body weight. Urethane anesthesia has been verified to be

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suitable for physiopharmacological investigation in cardiovascular system, because it preserves a number of cardiovascular reflexes.7 But, we recommend it only for acute studies and not for chronic experiments. Our experience is that repetitive intraperitoneal administrations of urethane solution tend to induce inflammation of the abdominal cavity or peritonitis. The commonly used inhalant anesthetics include isoflurane and sevoflurane. Inhalation anesthesia offers the advantage of accurately controlling anesthetic depth with the safety of being able to discontinue the administration of the inhalant anesthetic immediately if any problem should arise. The depth of anesthesia during maintenance is easily controlled by adjusting the vaporizer output and the total flow rate. Inhalant anesthetics can be eliminated quickly and rapid recovery is expected when compared to the injectable agents. Inhalant anesthetics are mostly eliminated through ventilation, whereas injectable anesthetics rely on the liver and kidney for metabolism/elimination. For hemodynamic measurements care should be taken because all inhalant anesthetics depress cardiopulmonary function a dose-dependent manner as shown by the decreases in cardiac output, blood pressure, and respiratory rate. 3.2. Experimental models There are two types of experimental models; acute models and chronic models. The former type is available for single use in an acute experiment principally based on exteriorization and in situ visualization of inner organs and tissues. The latter mainly utilizes chronically instrumented animals the aim of which is microcirculatory observation for prolonged periods of time. 3.2.1. Acute experimental models (i) Mesentery Rats are used for studies of the mesenteric microcirculation.8 The abdomen is opened along the midline by using a radiocautery, and a loop of small intestine is exposed. A section of the small intestine is carefully drawn through the incision and positioned on a transparent observation stage (Fig. 2a). The mesentery is covered with a piece of film dressing (e.g. Saran wrap) to prevent drying of the tissue. The model with transillumination provides best-contrasted microvascular images among the various experimental models (Fig. 2b). (ii) Cremaster muscle Both rats and mice are suitable for studies of the cremaster muscle microcirculation.9,10 The cremaster muscle surrounds the testes. It receives its vascular supply from the pudic-epigastric truncus. Distal to the external iliac-femoral intersection, the pudic-epigastric truncus lies in the abdominal muscles parallel to the inguinal ligament. The trunk subtends many branches at the perineal region. One of these terminal branches forms the cremaster muscle first order vessels.

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Fig. 2. Observation of the rat mesentery. (a) The small intestine positioned on a transparent observation stage. (b) Microvascular appearance of the rat mesentery providing excellent contrast. Scale bar = 50 µm.

In preparing this model, the scrotum is pulled away from the underlying cremaster and testes while a longitudinal incision is made in the ventral scrotal sac. The exposed cremaster muscle is continuously irrigated with suffusate, and is then freed from the scrotal sac and surrounding fascia by blunt dissection. The thin fascia covering the cremaster muscle is carefully removed with forceps. A ligature is attached to the extreme distal tip of the muscle. The testicle and cremaster are pushed into the abdomen through the inguinal canal, which everts the cremaster muscle. An abdominal incision is then made and the vessels connecting the cremaster to the epididymis are cauterized. Using the distal ligature, the cremaster muscle is then exteriorized through the annulis inguinalis, leaving the freed testicle in the abdominal cavity. The cremaster muscle is positioned over an optical port for transillumination (Fig. 3). (iii) Hamster cheekpouch Golden hamsters or Syrian hamsters are used.11,12 The cheek pouch is gently everted and pinned with 4–5 needles into a circular well filled with silicone rubber to provide a flat bottom layer, thus avoiding stretching of the tissue, but preventing shrinkage. In this position, the pouch is submerged in a superfusion solution that continuously flushes the pool of the microscope stage. Before the pouch was pinned, large arterioles and venules are located with the aid of an operating microscope. In order to produce a single-layer preparation, an incision is made in the upper layer so that atriangular flap can be displaced to one side. The exposed area is dissected under the microscope, and the fibrous, almost avascular, connective tissue converging the vessels is removed with ophthalmic or microsurgical instruments. The dissected part of the pouch is 125 to 150 µm thick. Dissected pouches with petechial hemorrhages and those without blood flow in all vessels are discarded. The preparation is placed under an intravital microscope (Fig. 4).

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Fig. 3.

Skeletal muscle microcirculation of the mouse cremaster. Scale bar = 100 µm.

Fig. 4.

Fluorescent micro-angiogram of the hamster cheek pouch. Scale bar = 100 µm.

3.2.2. Chronic experimental models (i) Rabbit ear chamber A transparent chamber is implanted in the surgically prepared rabbit ear.13–15 The chamber is made of transparent acryl-resin, and is composed of a disk with a central round table and three peripheral pillars, a cover plate, and a holder ring (Fig. 5). White rabbits weighing 3.0–3.5 kg are the appropriate size for most studies. The ear is shaved, treated with a depilatory cream and sterilized with povidone-iodine. Four holes are punched through the cartilage and skin of the distal portion of the ear with a specially designed puncher (Fig. 6a). The size and position of the holes are adjusted to match those of the round table and pillars of the disk. Care should be taken to avoid large blood vessels. The epidermis on both sides of the ear around the central hole is carefully retracted so as to leave the subcutaneous blood vessels intact (Fig. 6b). Three peripheral pillars of the disk are inserted through the outer

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Fig. 5. Complete rabbit ear chamber set. The chamber is composed of a disk with a central round table and three peripheral pillars, a cover plate, and a holder ring. A 10×10 grid (grid constant 500 µm) is superimposed on the cover plate for microcirculatory analysis.

Fig. 6. Procedure for rabbit ear chamber installation. (a) Four holes are punched through the cartilage and skin of the distal portion of the ear. (b) The epidermis on both sides of the ear around the central hole is carefully retracted so as to leave the subcutaneous blood vessels intact. (c) Installed rabbit ear chamber. (d) The disk is designed to leave a 50-µm-thick space into which the new microvessels sprout.

small holes of the ear to position the chamber and the central round table plugges the large puncture in the center. A cover plate is fixed on the pillars by the holder ring (Fig. 6c). The disk is designed to leave a 50-µm-thick and 6-mm-diameter space between the central round table and the cover plate. The new microvessels begin to sprout into the space in several days after implantation (Fig. 6d). An intravital microscopic system under transillumination allows observations of angiogenesis during wound healing as well as microcirculation after completion of repair process (Fig. 7).

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Fig. 7.

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Gross appearance of microcirculation in the rabbit ear chamber 3 weeks after implantation.

Fig. 8. Aluminum chamber inserted into the dorsal skinfold. (A) Side view of the chamber frames. The skinfold chamber consists of two frames held together with polyethylene tubes. (a) Opening (7 mm on each side) for intravital microscopic investigation of underlying tissue; (b) holes drilled for polyethylene tubes to allow fixation and adjustment of both chamber frames. (B) Transectional view of the aluminum chamber. (c) Opening for intravital microscopic observation; (d) coverglass; (e) polyethylene tube; (f) fluke for fixing the chamber to the observation stage.

(ii) Mouse skinfold chamber The basic structure of a mouse chamber (Yasuhisa-koki, Ltd. Japan) is shown in Fig. 8. The skinfold chamber is inserted using the following procedure (Fig. 9).16,17 The dorsal skin is pulled and fixed to a board with 26-gauge injection needles. Four small holes are punched to penetrate the double-layered skinfold. In the center of the four holes, a square area of one layer of skin (about 7 mm in one side) is removed, and the remaining layer, consisting of epidermis, subcutaneous and thin striated skin muscle, is covered with a coverglass incorporated in one of the frames

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Fig. 9. Skinfold chamber installation. (a) The dorsal skin was drawn upward and fixed to the board with 26-gauge injection needles. (b) A square area of one layer of skin was removed (about 7 mm on each side), maintaining the other layer intact. (c) The frames were implanted so as to sandwich the stretched double layer of skin. The intact layer, consisting of epidermis, subcutaneous tissue, and a thin layer of striated skin muscle, was covered with a coverglass incorporated in one of the frames. (d) General appearance of a mouse with the skinfold chamber in place.

of the chamber. A polyethylene tube is passed through each small hole and melted at both ends with heat to hold the two chamber halves together. The frames are implanted so that they sandwich the extended double layer of skin perpendicular to the animal’s back. A recovery period of 72 hours between chamber implantation and the microscopic investigation is generally allotted to eliminate the effects of immediate surgical trauma on the chamber tissue. The microcirculation of the skin as well as the striated skin muscle can be inspected with an intravital microscope and transillumination (Fig. 10). Similar skinfold chambers are applicable for rats and hamsters.18–21

Fig. 10. The microcirculation of the skin (a) as well as the striated skin muscle (b) is inspected with the mouse skinfold chamber. Scale bar = 50 µm.

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Fig. 11. (a) The hairless mouse ear spread on an observation stage. (b) Microvascular appearance of the hairless mouse ear. Scale bar = 50 µm.

(iii) Hairless mouse ear The homozygous (hr/hr) hairless mouse, although similar in appearance to the nude mouse, is immunologically intact, with a normally functioning thymus and T-cells. The use of the hairless mouse ear allows studies of intact skin microcirculation without surgical intervention.22,23 The ear of the hairless mouse measures 10–13 mm in width and length and comprises approximately 6% of the animal’s total body surface area. It presents a central cartilaginous sheet (approximately 50 µm thick) sandwiched between two full thickness dermal layers together, giving the ear a total thickness of approximately 300 µm. The relatively large size and thin structure of the hairless mouse ear make it easy to study the ear microcirculation using vital microscopy. Its nourishment is supplied by three to four neurovascular bundles entering the ear at its base and branching out periphery into descending orders of arterioles that feed the capillaries. The capillaries form characteristic loops around empty but otherwise normal hair follicles and drain into post-capillary venules and veins. This entire vascular network can be visualized by transilluminating the spread ear through the vital microscope (Fig. 11). Although the optical resolution is not so good due to the overlying skin, the hairless mouse ear offers the advantage of a simple and reproducible inspection of the microcirculation of intact mammalian skin.

4. Microcirculatory Measurement Visualized microcirculatory images are analyzed using the following measurable parameters. 4.1. Vessel diameter The measurement of the diameter of microvessels is a key feature in the classification of the vessels and in the design of physiological theories that address homeostatic

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phenomena such as blood flow distribution, the regulation of blood pressure, and fluid exchange. Video-recorded microvascular images captured and stored as static digitalized images onto a personal computer are analyzed using an image analysis program (e.g. Scion Image for the Windows, NIH image for the Macintosh platform). After setting the scale per pixel using a calibration function, measurement of the transverse distance of the vascular regions of interest provides the vessel diameter. If the boundary of the vessels is unclear in the original captured images, contrast modification and an accurate trace of vascular regions with the pencil function of an image processing software (e.g. Adobe Photoshop) helps detection of the vessel walls by the analysis software.

4.2. Tissue vascularity Capillary perfusion is a prerequisite for the transport of oxygen, nutrients and metabolites to and from the tissue cells. Therefore, reliable quantification of capillary perfusion represents an important goal of experimental and clinical research. The nutritive perfusion of a tissue depends on the number and anatomical distribution of capillaries and on the blood flow within these conduits. As a quantitative parameter, functional capillary density (FCD) has been proposed as a measure of capillary perfusion. FCD is defined as the total length of red cell-perfused capillaries per unit area, and it has been used as an indicator of the quality of tissue vascularity in various animal models.24 In FCD measurement capillaries are conventionally distinguished from other microvessels i.e. arterioles and venules. However, capillaries are not only sites that supply oxygen to the tissue. Microvascular PO2 measurements have revealed the longitudinal perivascular PO2 gradient along the arteriole and have suggested a significant amount of O2 diffusion from the arteriolar network.25 In this sense, the total length of all the microvessels including arterioles, capillaries and venules per unit area or the functional microvascular density properly represents tissue vascularity. The term FCD was recently used to indicate functional microvascular density.17 This interpretation is essential in the quantification of angiogenesis where the capillaries can hardly be distinguished from other microvessels due to immaturation of the microvasculature.26 A microvessel is defined to be functional if passing RBCs are noted in the dynamic microvascular image. FCD is assessed by an image processing software (e.g. Adobe Photoshop CS) (Fig. 12).27 The Pencil tool is selected from the toolbox and the capillaries (microvessels) depicted on the screen are manually redrawn in a layer superimposed over the image, using a distinctive color. Because the redrawn pencil line is automatically shown on the computer screen, it can be controlled at every step and necessary corrective measures can be taken. The color line is then selected with the Magic Wand tool and the Select Similar command in the Select menu, and the area covered by the pencil line is quantified

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Fig. 12. (a) Representative image of fluorescently stained microvessels. (b) FCD is assessed by Photoshop image analysis. Microvessels depicted are manually redrawn in a layer superimposed over the image, using a distinctive color and the total length is calculated.

by using the Histogram command in the Window menu. On the basis of the resolution and magnification of the image and the predefined width of the pencil line, the length of the pencil line is calculated and the length of the capillaries (microvessels) is expressed relative to the area covered by the image (given in cm/cm2 ).

4.3. Blood flow velocity Recently developed image acquisition and analysis software (CapiScope: KK Technology, UK) provides a feasible measurement of red cell velocity using a spatial correlation technique. Measurements can be made in real time, directly from the subject. However, it is usually more convenient to record onto good quality video tape or, for best quality, directly into the computer memory, or for longer sequences, directly to a hard disk. The CapiScope also has functions of morphometric analysis including vessel diameter and capillary density. Velocity is measured by using the mouse to draw a line along the vessel. The gray level profile along the line is taken for each field. The gray level pattern along each line is compared to the pattern from the next field (or several fields later for very low velocities). The comparison is performed by calculating the correlation coefficient for every possible shift of the previous gray level profile relative to the new profile. The shift which produces the highest correlation, and which can be seen as the peak in the correlation in the following figure, indicates the distance that the pattern travelled between the two gray level profile measurements. Since the time lapse between the two gray level profiles is known (i.e. 1/60th second for NTSC based systems) the velocity can be calculated (Fig. 13).

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Fig. 13. Blood flow velocity measurement using the Capiscope. (Above) The gray level pattern along each line is measured. (Below) The pattern is compared to that from the next field (or several fields later for very low velocities). Velocity can be calculated based on the distance that the correlated pattern has travelled between the two gray level profile measurements.

4.4. Leukocyte behavior Circulating leukocytes play a key role in the pathogenesis of microvascular injury through a process first involving leukocyte rolling on the venular endothelium, which is then followed by firm adherence of leukocytes to endothelial cells. Adherent leukocytes may promote venular injury by releasing reactive oxidants and proteolytic enzymes, which then further damage underlying endothelial cells. Rhodamine 6G is a cationic fluorescent dye that selectively accumulates in the nuclei and mitochondria of living cells.28 Intravenous administration of rhodamine 6G (ICN Biomedicals Inc.) (0.5 mg/kg body weight) allows visualization of fluorescently labeled-leukocytes under intravital fluorescence microscopy (Fig. 14).29 The straight portions of venules are usually selected as sites of interest. Leukocyte behavior is classified as nonadherent, rolling, or sticking. Nonadherent leukocytes are defined as cells passing the observed vessel segments without interacting with the endothelial lining and are expressed as cells per unit time (i.e., 30 seconds). Rolling leukocytes are defined as cells moving along the endothelial lining at a velocity markedly less than that of the surrounding red cell column and are given

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Fig. 14. Fluorescently visualized leukocytes labeled by rhodamine 6G under an intravital fluorescence microscopy. Scale bar = 50 µm.

as the number of cells traversing an observed vessel segment within 30 seconds. The rolling fraction (%) is calculated as the ratio of rolling cells/rolling cells plus nonadherent cells times 100. Sticking leukocytes are defined as cells that do not detach from the endothelial lining within 30 seconds of observation time and are expressed as cells/mm2 . 4.5. Oxygen measurement One of the most important molecular species which are subject to convective transport within the macro- and microcirculatory system is oxygen. Several methods have been utilized in efforts to measure tissue oxygen using microelectrodes, spectrophotometry, etc. Among these the O2 -dependent quenching of phosphorescence technique,25 which allows regional determination of intravascular and interstitial PO2 values, is herein explained. Pd-meso-tetra (4-carboxyphenyl) porphyrin (Pd-porphyrin, Porphyrin Products, Logan, UT) bound to bovine serum albumin is used as the phosphorescent probe for the O2 -dependent quenching. The basis of this technique is a well-known reaction in which Pd-porphyrin is excited to its triplet state by exposure to pulsed light, after which phosphorescence intensity is reduced by emission and energy transfer to O2 . The quenching of the phosphorescence by the O2 is diffusion limited. Thus, if the PO2 distribution is homogeneous, it can be described as: I(t)/I0 = exp[−(1/τ0 + Kq × PO2 )t] = exp(−t/τ ),

(1)

where I(t) is the light intensity at time t, and I0 is the initial value of light intensity at t = 0. The Stern-Volmer equation can be written as 1/τ = 1/τ0 + Kq × PO2 ,

(2)

where τ0 and τ are the phosphorescence lifetimes in the absence of O2 and in the area being measured, respectively, and Kq is the quenching constant. The PO2 values can

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be obtained from the measurement of τ , because Kq and τ0 are substance specific. However, Golub et al.30 pointed out that the rectangular distribution model should be used to analyze the data when the PO2 distribution is heterogeneous in the measuring area. The fitting equation they proposed is I(t)/I0 = exp[−(1/τ0 + Kq × PO2 )t] × [1 + (Kqσt)2 /2].

(3)

By using this equation to calculate the PO2 values, the accuracy of the curve fitting is improved compared with a conventional curve fitting, especially at the initial segment of the decay curve. Figure 15 shows a schematic diagram of phosphorescence quenching laser microscope system. The Pd-porphyrin phosphorescent probe is excited by epi-illumination using a N2-dye pulse laser (LN120C, Laser Photonics). The area of the epiilluminated tissue is 10 µm in diameter on the surface. The phosphorescent emissions from the tissue are captured by a photomultiplier (C6700, Hamamatsu Photonics, Hamamatsu, Japan) through a longpass filter at 610 nm. Signals from the photomultiplier are converted to 10-bit digital signals at intervals of 3 µs. The averaged 20–40 data are calculated by mathematically fitting the decay of the phosphorescence to the rectangular model equation30 using online computer analysis. For regional analysis, intravascular PO2 measurements are carried out immediately, whereas interstitial PO2 measurements are started 30 min after injection of the Pd-porphyrin solution (25 mg/kg body wt) usually into the cannulated jugular vein of the experimental animal. 4.6. Permeability measurement One of the functions of the microcirculatory vessel wall is to act as a molecular sieve, so that many plasma constituents have limited access to the surrounding tissue. This barrier function of the vessel wall is compromised in certain conditions (e.g. acute inflammation), and the resulting leakage of fluid and protein into the interstitium can result in morphological and functional disturbance such as edema and exudates. Therefore, a frequent assay of microvascular integrity is the rate at which protein leaks from plasma to tissue.31 Intravascular injected fluorescein isothiocyanate (FITC)-labeled macromolecules (e.g. FITC-Dextrans or FITC-Albumins) have been used to study vascular permeability under an intravital fluorescence microscope. The injected fluorescently labeled molecules trace the movement of macromolecules across the vascular wall. Animals receive FITC-labeled macromolecules intravenously and fluorescein angiograms are recorded and digitized. In general, the fields selected for investigation are the areas around the postcapillary venules that are relatively free of capillaries and other leaking postcapillary venules. Fluorescence intensities in the intravenular and corresponding perivascular area are measured using a computerassisted digital image processing and analyzing software (Scion Image or Adobe Photoshop).

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Fig. 15. Arrangement of the intravital laser microscope using O2 -dependent phosphorescence quenching. A general observation of the microcirculation is performed using an intravital microscope coupled with CCD camera. Pd-porphyrin is excited by epi-illumination with a N2 -dye pulse laser with a 535-nm line at 20 Hz via the objective. The epi-illuminated tissue area is 10 µm in diameter. Phosphorescence emission is captured by a photomultiplier through a long-pass filter at 610 nm, and signals from the photomultiplier are converted to 10-bit digital signals at intervals of 3 µs. ADC, analog-to-digital signal converter; PMT, photomultiplier tube.

Leakage is quantified by measuring the gray levels in selected areas directly over and immediately adjacent to the blood vessels. The areas are specified by the Select Regions tool (Scion Image) or by the lasso tool from the tool box (Adobe Photoshop). Mean gray levels within the predefined area are assessed using the Measure command in the Analyze menu (Scion Image) or the Histogram command in the Window menu (Adobe Photoshop). The difference in the gray levels of areas overlying and adjacent to the blood vessels is used as a measure of vascular permeability (Fig. 16).

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Fig. 16. Permeability measurement. (a) Microvascular image immediately after injection of FITClabeled macromolecular dextran. The fluorescent marker is retained within the vessel. (b) Microvascular image 15 minutes after the marker injection. Leakage is quantified based on the difference in the gray levels of areas overlying and adjacent to the blood vessels.

5. Application of the Microcirculatory Experimental Models Visualization models of the microcirculation can be applied to the studies of diverse pathophysiological phenomena. The current section introduces two examples of this application; experimental models of angiogenesis and ischemia-reperfusion that have been matters of concern in various fields of medicine. 5.1. Angiogenesis model Angiogenesis is the process of new vessel formation. It is essential in all connective tissue healing, as well as a wide variety of other physiologic and pathologic processes. Application of the mouse skin fold chamber model and the hairless mouse ear model allow direct visualization of angiogenesis during wound healing. 5.1.1. Angiogenesis model using mouse skinfold chamber A wound is created in the mouse skin fold chamber model (Fig. 17).26 The area selected for the wound on the intact side of the folded skin (opposite the coverglass) is marked with a round-shaped ink stamp having a diameter of 1.6 mm. A circular area of skin is then excised down to, but not including, the underlying striated skin muscle layer using microforceps and scissors under an operating microscope. The preparation provides a full-thickness dermal wound having a surface area of approximately 2 mm2 and a depth of approximately 0.3 mm. The agent to be studied and/or vehicle are applied topically in the wound defect if necessary. The wound is covered with a dressing and the plastic cap is used to fix the dressing in place throughout the experimental period. The model allows direct and repeated observation of microcirculatory changes during the wound healing angiogenesis process in the same animal over an extended period of time (Fig. 18).

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Fig. 17. Wound creation procedure. Transectional schema (above) and appearance under an operating microscope (below). A circular area of skin was excised with microforceps and scissors down to, but not including, the underlying striated skin muscle layer. A dressing (a) covered the wound, and the plastic cap (b) fixed the dressing.

Fig. 18. Changes in microvasculature during wound healing angiogenesis in the same animal on day 3 (a), day 5 (b), day 7 (c), and day 10 (d) after wounding.

5.1.2. Angiogenesis model using hairless mouse ear The hairless mouse ear also offers an angiogenesis model.32,33 Circular wounds are created on the dorsal aspect of the ears, down to, but not including, the pineal

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cartilage layer. The wounds are positioned between the anterior and middle principal neurovascular bundles approximately 1.5 mm from the ear border. After wounding and hemostasis, the agent to be studied and/or vehicle are applied topically in the wound defect and the wound is covered with a thin sheet thus helping to maintain a stable environment in the wound. Over the agent and the plastic covering, all ears/wounds are covered in their entirely with a bioadhesive dressing. Wound epithelization and neovascularization are directly visualized and measured using an intravital microscopic system. The digitized images of the epithelial and neovascular edges are traced and the area within the tracing is measured and used to calculate the progression of the epithelial or neovascular edge at the time the measurement is performed. 5.2. Ischemia-reperfusion Ischemia-reperfusion injury has become of the major concerns in many fields of medicine. A number of investigations have proved that impairment mechanisms of various organs are quite attributable to the process of ischemia-reperfusion injury. Ischemia-reperfusion injury has been defined as cellular injury resulting from the reperfusion of blood to a previously ischemic tissue. When a tissue has been depleted of its blood supply for a significant amount of time, the tissue may reduce its metabolism to preserve function. The reperfusion of blood to the nutrient- and oxygen-deprived tissue can result in a cascade of harmful events. Most of the microcirculatory models provide modes for studies of ischemiareperfusion. Clamping of the pedicled nutritional vessels or direct compression of the observed microvessels induces ischemia, and release of the obstruction results in reperfusion. Various microcirculatory measurements have been performed after the ischemia-reperfusion procedure.

References C. Y. Pang, P. Neligan and T. Nakatsuka, Plast. Reconstr. Surg. 513 (1984). P. M. Hendel, D. L. Lilien and H. J. Buncke, Plast. Reconstr. Surg. 387 (1983). B. M. Achauer, K. S. Black and D. K. Litke, Plast. Reconstr. Surg. 738 (1980). M. D. Menger, J. H. Barker and K. Messmer, Plast. Reconstr. Surg. 1104 (1992). L. Heller, L. S. Levin and B. Klitzman, Plast. Reconstr. Surg. 1739 (2001). C. Y. Pang, P. Neligan, T. Nakatsuka and G. H. Sasaki, J. Surg. Res. 173 (1986). C. A. Maggi and A. Meli, Experientia 292 (1986). B. W. Zweifach, Anat. Rec. 277 (1954). M. Pemberton, G. Anderson and J. Barker, Microsurgery 374 (1994). M. Shibata, S. Ichioka and A. Kamiya, Am. J. Physiol. Heart Circ. Physiol. H295 (2005). 11. B. R. Duling, Microvasc. Res. 423 (1973). 12. T. J. Verbeuren, M. O. Vallez, G. Lavielle and E. Bouskela, Br. J. Pharmacol. 859 (1997). 13. M. Asano, Jpn. J. Pharmacol. 225 (1964).

1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

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14. S. Ichioka, M. Shibata, K. Kosaki, Y. Sato, K. Harii and A. Kamiya, J. Surg. Res. 29 (1997). 15. S. Ichioka, M. Shibata, K. Kosaki, Y. Sato, K. Harii and A. Kamiya, Microvasc. Res. 165 (1998). 16. S. Ichioka, T. C. Minh, M. Shibata, T. Nakatsuka, N. Sekiya, J. Ando and K. Harii, Microsurgery 304 (2002). 17. S. Tsuji, S. Ichioka, N. Sekiya and T. Nakatsuka, Wound Repair Regen. 209 (2005). 18. D. Nolte, M. D. Menger and K. Messmer, Int. J. Microcirc. Clin. Exp. 9 (1995). 19. S. Ichioka, M. Iwasaka, M. Shibata, K. Harii, A. Kamiya and S. Ueno, Med. Biol. Eng. Comput. 91 (1998). 20. S. Ichioka, M. Minegishi, M. Iwasaka, M. Shibata, T. Nakatsuka, K. Harii, A. Kamiya and S. Ueno, Bioelectromagnetics 183 (2000). 21. S. Ichioka, T. Nakatsuka, Y. Sato, M. Shibata, A. Kamiya and K. Harii, J. Surg. Res. 42 (1998). 22. E. Eriksson, J. V. Boykin and R. N. Pittman, Microvasc. Res. 374 (1980). 23. J. H. Barker, F. Hammersen, I. Bondar, E. Uhl, T. J. Galla, M. D. Menger and K. Messmer, Plast. Reconstr. Surg. 948 (1989). 24. D. Nolte, H. Zeintl, M. Steinbauer, S. Pickelmann and K. Messmer, Int. J. Microcirc. Clin. Exp. 244 (1995). 25. M. Shibata, S. Ichioka, J. Ando and A. Kamiya, J. Appl. Physiol. 321 (2001). 26. S. Ichioka, S. Kouraba, N. Sekiya, N. Ohura and T. Nakatsuka, Br. J. Plast. Surg. 1124 (2005). 27. J. Brunner, F. Krummenauer and H. A. Lehr, Microcirculation 103 (2000). 28. R. K. Saetzler, J. Jallo, H. A. Lehr, C. M. Philips, U. Vasthare, K. E. Arfors and R. F. Tuma, J. Histochem. Cytochem. 505 (1997). 29. H. A. Lehr, B. Vollmar, P. Vajkoczy and M. D. Menger, Meth. Enzymol. 462 (1999). 30. A. S. Golub, A. S. Popel, L. Zheng and R. N. Pittman, Biophys. J. 452 (1997). 31. N. R. Harris, S. P. Whitt, J. Zilberberg, J. S. Alexander and R. E. Rumbaut, Microcirculation 177 (2002). 32. I. Bondar, E. Uhl, J. H. Barker, T. J. Galla, F. Hammersen and K. Messmer, Res. Exp. Med. (Berl.) 379 (1991). 33. D. Kjolseth, M. K. Kim, L. H. Andresen, A. Morsing, J. M. Frank, D. Schuschke, G. L. Anderson, J. C. Banis Jr, G. R. Tobin and L. J. Weiner, Microsurgery 390 (1994).

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CHAPTER 3 IMPEDANCE CARDIOGRAPHY: DEVELOPMENT OF THE STROKE VOLUME EQUATIONS AND THEIR ELECTRODYNAMIC AND BIOPHYSICAL FOUNDATIONS DONALD PHILIP BERNSTEIN The Department of Anesthesia Palomar Medical Center, Escondido, California, USA HENDRIKUS J. M. LEMMENS The Department of Anesthesia The Stanford University School of Medicine Stanford, California, USA

Impedance cardiography is a branch of bioimpedance primarily concerned with the determination of left ventricular stroke volume. The technique involves applying a current field across a segment of the body, usually the thorax, by means of a constant magnitude, high frequency, low amplitude alternating current. By Ohm’s Law, the voltage measured within the current field is proportional to the electrical impedance Z. Upon ventricular ejection, a characteristic pulsatile impedance change occurs ∆Z(t), which, in early systole, represents both the radially oriented volumetric expansion of and longitudinal forward blood flow within the great thoracic arteries. Past methods assumed a volumetric origin for the peak systolic upslope of ∆Z(t), that is, dZ/dtmax . A new method assumes the rapid ejection of forward flowing blood in early systole causes significant changes in blood resistivity, and it is the peak rate of change of blood resistivity that is the origin of dZ/dtmax . As proposed, dZ/dtmax is an ohmic mean acceleration equivalent, and, as necessary for stroke volume calculation, must undergo square root transformation to yield an ohmic mean velocity equivalent. Further changes over older methods include a variable magnitude volume conductor based on body mass, rather than on current field segment length or body height. Keywords: Impedance cardiography; stroke volume equations; signal processing; dZ/dtmax ; square root acceleration step-down transformation; volume conductors; intrathoracic blood volume; blood resistivity.

1. Introduction Bioimpedance is a branch of biophysics concerned with the electrical hindrance offered by biological tissues to the flow of low amplitude, high frequency alternating current (AC). Bioimpedance encompasses a broad range of techniques, including impedance spectroscopy, single and multi-frequency bioimpedance body composition analysis, impedance tomography, impedance plethysmography and impedance cardiography (ICG). ICG is generally subdivided into the transthoracic and whole body techniques.1 This chapter will concern itself with the transthoracic technique, which is the one most often applied clinically and studied experimentally. 49

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Impedance cardiography, also known as transthoracic electrical bioimpedance cardiography (TEBC) is a clinically applicable, non-invasive, continuous, beatto-beat method for estimating left ventricular stroke volume (SV) and cardiac output (CO).2–6 The method is attractive, because it does not require a protracted learning curve on the part of the user, or the necessity of a skilled technician. Furthermore, data acquisition and waveform processing are not inordinately time consuming, and with the availability of user-friendly, menu-driven, programmable computers, data are displayed in a signal-averaged, real-time format. As compared to its original operational implementation, using circumferential bandelectrodes, the advent of the tetra-polar eight spot-electrode patient interface now renders the technique satisfactory for both patients and medical personnel (Fig. 1). While it is generally acknowledged that directional changes in impedance-derived SV follow changes of accepted standard reference methods, there is, unfortunately, no consensus as to the validity of absolute measurements, especially in patients with cardiopulmonary pathology.7,8 A review of the literature indicates that existing SV equations, originally modeled for simple extremity blood volume changes (impedance plethysmography),9 inadequately account for the much more complex and dynamic, vascular and hemorheologic interrelationships of pulsatile thoracic blood flow.10–13 Thus, a theoretically robust mathematical expression, linking impedance-derived SV to other standard methods has been suggested as necessary.7 As a consequence of these important issues, impedance cardiography has not yet found an established place in mainstream medicine for modulation of therapy in critically ill humans.

Fig. 1. Ohm’s Law as applied to the human thorax. An AC (∼) field is applied to the thorax generating a static voltage U0 and a cardiac-synchronous, time-dependent voltage drop ∆U (t). By Ohm’s Law, the ratio of the measured voltage (U ) to the applied current (I) is equal to the calculated impedance Z. Shown are the poorly conductive tissue impedances in the aggregate Zt and the highly conductive blood impedance Zb . L is the distance between the voltage sensing electrodes (Kubicek), approximated as 17% of overall body height for the Sramek–Bernstein method.

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2. Operational Implementation and Development of the Dual Compartment Parallel Conduction Model Described most simply, the impedance technique involves applying a current field across the thoracic volume over a defined segment L (cm). This is effected by means of a constant magnitude, high frequency (50–100 kHz) alternating current (AC) (I, mA) of small amplitude (1.0–4.0 mA rms). The AC is applied by means of a pair of current injecting electrodes; one electrode (or set of two) placed at the base of the neck, and the other electrode (or set of two) at the xiphisternal junction (Fig. 1). A potential difference is thus established, wherefrom a baseline thoracic voltage U0 (U, Volt) and a time-variable cardiovascular component ∆U (t) are obtained by means of a pair of voltage sensing electrodes. As shown in Fig. 1, the voltage sensing electrodes are placed proximal to the current injectors within the current field. The static, relatively non-conductive thoracic tissues, including bone, muscle, connective tissue, fat, vascular tissue, the thoracic gas volume, and the highly conductive fluids, including the intra-thoracic blood volume (ITBV, mL) and interstitial fluids, act as parallel impedances Z (Ω, Ohm) to AC flow. By virtue of Ohm’s Law, as applied to AC (Z = U/I), the measured quasi-static voltage U0 , and the voltage change ∆U (t), are proportional to the quasi-static impedance Z0 , and the cardiogenically-induced pulsatile impedance change ∆Z(t), respectively. Disregarding the impedance change caused by pulmonary ventilation ∆Zvent , the aforementioned relationships are best described by the following modification of Ohm’s Law: I · [Z0 + ∆Z(t)] = U0 + ∆U (t),

(1)

where Z0 comprises a static, relatively non-conductive tissue impedance Zt (Ω), the static component of the highly conductive blood resistance (impedance) Zb , and the normally present, very highly conductive interstitial extra-vascular lung water (EVLW) Ze (i.e. Z0 = Zt + Zb + Ze ) (Fig. 2). Since the combined tissue and pulsatile impedances are exposed to a field of AC, they are not summed directly, but rather, are added in parallel as their respective reciprocals (i.e. ). Thus, the following equation pertains to a two-compartment parallel conduction model in which all current is assumed to traverse the static DC (Zb ) and dynamic AC (∆Zb (t)) components of the blood resistance: I · [Z0 ∆Z(t)] = U0 ∆U (t).

(2)

Biophysically, the systolic portion of the cardiovascular component represents the simultaneously generated, serial composite of both the instantaneous volumetric changes of the great thoracic vessels ∆Zvol (t), principally from the aorta,14,15 and the instantaneous velocity (v) changes ∆Zv (t) of axially directed forward flowing blood contained therein.10–13,16 Thus, ∆Z(t) is a composite waveform, as depicted in Fig. 3, and can be expressed as follows: ∆Z(t) = ∆Zv (t) + ∆Zvol (t).

(3)

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Zt

Zb

Ze

Zo

∆Zb(t)

Z (t)

U

Fig. 2. Schematic of a multi-compartment parallel conduction model of the thorax. The transthoracic electrical impedance Z(t) to an applied AC field represents the parallel connection of a quasi-static base impedance Z0 and a dynamic, time-dependent component of the blood impedance ∆Zb (t). Z0 represents the parallel connection of all static tissue impedances Zt and the static component of the blood impedance Zb . In disease states characterized by excess EVLW, quasi-static Ze is added in parallel with Zt and Zb . Voltmeter (U ) and AC generator (∼) are shown.

Fig. 3. The components of ∆Z(t). The thoracic cardiogenic impedance change ∆Z(t) is composed of the blood velocity-induced impedance change ∆Zv (t), shown as v, and the pulsatile vessel volume impedance change ∆Zvol (t) shown as ∆V (t). The ∆Z(t) shown is an improvisation, superimposed on a velocity and pressure waveform. It is assumed that ∆V (t) = ∆P (t) · C. Velocity and pressure waveforms are from Van den Bos, G.C. et al. (1982) Circ. Res. 51, 479–485. Reproduced with permission.

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3. Hemodynamic Biomechanical Analogs and Bioelectric Components of the Transthoracic Cardiogenic Impedance Pulse Variation ∆Z(t) 3.1. The volumetric component: ∆Zvol (t) Biomechanically, the net aortic volume change ∆V (t)net over an ejection interval is a function of the aortic pulse pressure ∆P (t)(Psystolic −Pdiastolic ), modulated by vessel compliance (∆V /∆P ), minus the volume lost to simultaneous outflow.17,18 Because of the viscoelastic nature of the ascending aorta, in addition to distal vascular hindrance, the proximal aorta expands in systole. These factors cause the volume flow into an ascending aortic segment to be greater than the simultaneous outflow (i.e. Vin > Vout ) during the early portion of the ejection interval. The systolic volume lost to outflow is approximated as the quotient of the integral of the systolic portion of the pressure curve P (t) in (mmHg · s) and systemic vascular resistance (SVR) Rs . Rs is further defined as the mean systemic arterial blood pressure Pm divided by mean flow Qm (Rs = Pm /Qm ). Thus, the equation defining the mechanics of the net systolic volume change (mL) can be viewed most simply as conforming to that of a two-element Windkessel model, comprising a compliance chamber and a distal vascular resistance.17,18   tavc  P (t) dP (t) C− dt ∆V (t)net = dt Rs tavo (4) Vin − Vout . In practical operational terms equation (4) is unsolvable, because Rs and C, the Windkessel parameters, are derived values, requiring a priori knowledge of both the systemic mean flow and vessel compliance. Bioelectrically, the decrease in transthoracic impedance, corresponding to the pulsatile systolic expansion of the aorta, is due to the volumetric addition of blood of low static specific resistance (i.e. impedance) (ρb(stat) = 100−180 Ω cm in the physiologic range),19 ejected as SV, into a relatively fixed-volume, gas-filled thorax of much higher static specific impedance (ρ(gas) = 1020 Ω cm).16 The specific resistance ρ of a cylindrical electrical conductor is equal to its measured impedance Z over a defined segment, multiplied by its cross-sectional area (CSA, A, cm2 ) and divided by its length (L, cm) (i.e. ρ = ZA/L). By virtue of the simultaneous displacement of tracheobronchial gas by the ejected SV, there is also an absolute reduction of thoracic gas volume, provided that a positive pressure gradient does not exist between the upper airway and tracheobronchial tree. As determined by pneumocardiography, Wessale et al.20 showed that, although the displaced thoracic gas volume is several magnitudes smaller than measured SV, it is highly correlated with respect to tracking directional changes. Similar to the mechanical principle underlying pneumocardiography, the impedance change caused by volume displacement of bronchoalveolar gas by expanding vascular volumes, partially accounts for the original designation of the technology, impedance plethysmography. Thus, the plethysmographic component of the systolic time dependent decrease in transthoracic impedance ∆Zvol (t)

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is bioelectrically one of two components comprising the time dependent decrease in transthoracic specific impedance ∆ρT (t). The volumetric component ∆ρT vol (t) can be viewed as the oscillating, time variable ratio of the thoracic gas volume ∆Vg (t), simultaneously compressed and displaced by expanding vascular blood volumes ∆Vb (t). By magnitude, this ratio is defined as the relative change in volume of bronchoalveolar gas expelled from the thorax, [(Vg + Vb ) − ∆Vg (t)]/(Vg + Vb ), versus the relative change of the volume of blood simultaneously added during systole, [(Vg + Vb ) + ∆Vb (t)]/(Vg + Vb ). Simplifying and rearranging results in the following:   ∆ρT(vol) (t) (Vg + Vb ) − ∆Vg (t) , (5) ∝ 1− ∆Zvol (t) ∝ ρT (Vg + Vb ) + ∆Vb (t) where ρT is the static transthoracic specific resistance and Vg and Vb are the thoracic gas and blood volumes at end expiratory apnea and at end-diastole, respectively. 3.2. The blood velocity-induced “resistivity” component: ∆Zv (t) The contemporaneous blood velocity-induced transthoracic impedance decrease ∆Zv (t) is caused by dynamic, time-dependent changes in the specific impedance (“resistivity”) of flowing blood ∆ρb (t).10–13,16 These dynamic, temporally concordant changes are generated by phasic alterations in erythrocyte orientation and deformation during the cardiac cycle.6,10,21,22 At end-diastole, the state of highest specific impedance, poorly conductive erythrocytes are oriented randomly, causing the applied AC to take a circuitous route through the highly conductive plasma (ρ(plasma) = 60–70 Ω cm) (Fig. 4). By comparison, during the peak acceleratory and rapid ejection phase of early systole, erythrocytes are oriented with their long axes of symmetry parallel to the direction of flow. Parallel orientation establishes clear current pathways through the plasma, with an attendant steep decrease in transthoracic impedance.10,21 The impedance decrease, corresponding to the blood “resistivity” change, is known to be a function of the shear rate profile. When forward flowing blood is interrogated by AC, longitudinally, or parallel to the axis of flow, the shear rate profile is a function of the reduced average velocity. The reduced average velocity is defined as the mean velocity divided by the vessel radius, namely, vmean /R.21,22 In vitro, the magnitude of the relative resistivity change ∆ρb (t)/ρb(stat) is dependent on hematocrit and an exponential value of the reduced average blood velocity, (vmean /R)n . Thus, the dynamic, velocity-induced blood resistivity component of ∆ρT (t) (i.e. ∆ρT v (t)) can be viewed as the time variable, oscillating, relative resistivity change of flowing blood ∆ρb (t)/ρb . Thus, in relative terms, the sum of the volumetric and blood resistivity changes comprises the transthoracic cardiogenically-induced impedance pulse. Thus,    ∆ρT (Vel) (t) + ∆ρT (Vol ) (t) ∆ρb (t) (Vg + Vb ) − ∆Vg (t) . (6) ∆Z(t) ∝ ∝ + 1− ρT ρb (Vg + Vb ) + ∆Vb (t)

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| ∆ρ b(t)| >> 0 v >> 0 dA(t)

v

v=0 ∆ρb(t) = 0 End Diastole

Qin + Qout Systole

v=0 ∆ρb(t) = 0 Diastole

Fig. 4. Behavior of alternating current as applied to pulsatile blood flow — = AC flow; v = velocity of red cells; ∆ρb (t) = changing specific resistance of flowing blood; Qin = flow into aortic segment; Qout = simultaneous flow out of aortic segment; dA(t) = time-dependent change in vessel CSA.

The combined effects, corresponding to both vessel volume and blood velocity changes, contribute equivalently, producing a measured decrease in transthoracic impedance and transthoracic specific impedance to applied AC.10,13,21 Stated differently, these contemporaneous hemorheologic and plethysmographic changes produce enhanced electrical conductivity of the thorax. Despite the reported equivalency of the velocity and volume contributions to the impedance pulse,10,13 it is generally assumed that the peak magnitude of the first time-derivative of ∆Z(t), namely d∆Z(t)/dtmax , or simply dZ/dtmax (historically and empirically given the units Ω · s−1 ), represents the ohmic image of the peak rate of change of aortic volume or flow, namely dV /dtmax (Qmax , mL · s−1 ).1–6 Conceptually, this assumption is embodied in the two most widely used impedance SV models: the Kubicek and Sramek-Bernstein equations.23,24 For the remainder of the discussion, and the sake of simplicity, the extreme negative absolute impedance change, |∆Z(t)|min , and the extreme negative absolute value for its first time-derivative, |dZ/dt|min , will be referred to as ∆Z(t)max and dZ/dtmax , respectively. 4. Development of the Kubicek and Sramek-Bernstein SV Equations 4.1. Nyboer equation: Foundation and rationale for the plethysmographic hypothesis in impedance cardiography The Nyboer equation9 was originally proposed for determination of segmental blood volume changes in the upper and lower extremities. It is based upon the assumption that the arteries of the extremities are rigidly encased in muscle and connective

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tissue, and that both the arteries and surrounding tissue can be approximated as cylindrical electrical conductors placed in parallel alignment. Nyboer found that, by placing current injecting and voltage sensing electrodes on a limb segment, configured analogously as in Fig. 1, impedance changes proportional to strain gauge determined limb blood volume changes could be measured. He also determined that, in order to compensate for simultaneous volume flow out of the limb segment during arterial inflow, venous outflow occlusion of the limb was necessary. Using venous outflow obstruction, absolute arterial volume changes during each pressure pulse were accurately measured. Nyboer referred to this method, appropriately, as electrical impedance plethysmography.9 The foundation of Nyboer’s method is based on a companion equation to Ohm’s Law, relating to the electrical impedance offered by inanimate cylindrical conductors of static composition to the flow of alternating current. It is given as: Z=

ρL , A

(7)

where Z = the impedance in Ohms (Ω), ρ = the static specific resistance of the conductor (Ω cm), and in AC terminology, more correctly addressed as specific impedance (ζ, zeta), L = the length of the conductor (cm), and A = the cross sectional area (CSA) of the conductor (cm2 ). The numerically equivalent expression, modified for use as the developmental platform in both impedance plethysmography and impedance cardiography, is given as: Z=

ρb L 2 , Vb

(8)

where Vb = blood vessel volume (cm3 , mL), and ρb = the hematocrit dependent, static specific resistance of blood. If ρb and L are constants, and Vb is allowed to vary with time t, then the change in impedance of a conductor over time t, ∆Z(t), varies inversely with its change in volume over the same time interval ∆Vb (t). This is the basic premise in impedance cardiography, relating volumetric expansion of the ascending aorta over a fixed thoracic length L to a decrease in transthoracic impedance during the systolic ejection period.23 This concept is embodied in Sec. 3.1. Thus, rearranging Eq. (8) and solving for the time dependent net change in aortic volume: ∆Vb (t)net =

ρb L 2 . ∆Z(t)

(9)

Thus, according to Nyboer’s theory, Eq. (9) is the bioelectric equivalent of Eq. (4), relating ∆V (t)net to purely mechanical events. If the vessel segment described above is embedded in a larger thoracic encompassing cylinder of high static specific impedance and equivalent length L, but of much larger CSA, then the total parallel impedance of the thorax (Z(t)) can be expressed as: Z(t) = Z0 ∆Z(t),

(10)

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where Z0 = the base, or static transthoracic impedance of the thorax (males: 22 Ω– 33 Ω, females: 28 Ω–45 Ω). By virtue of the magnitude of ∆Z(t) normally being only 0.3%–0.5% of total thoracic impedance (i.e. ∆Z(t)max = 0.1 Ω–0.2 Ω), Z(t) and Z0 are approximated as equivalents. Thus, solving Eq. (10) for ∆Z(t) by the reciprocal rule for parallel impedances yields, ∆Z(t) =

Z02 Z0 · Z(t) ∼ = Z0 − Z(t) −∆Z(t)

(11)

Substituting the parallel equivalence of ∆Z(t) from Eq. (11) into Eq. (9), and solving for ∆V (t)net , yields a modification of the original Nyboer equation.9 Thus, ∆Vb (t)net = −

ρb L 2 ∆Z(t). Z02

(12)

The Nyboer equation, though originally proposed for extremity blood volume changes, is the plethysmographic foundation for existing SV methods. As directly applied to thoracic measurements, it implies that a measured change in transthoracic impedance ∆Z(t) is directly proportional to the measured net SV change ∆V (t)net . Clearly, Eq. (12) is also the bioimpedance equivalent of Eq. (4). As discussed above, the condition under which the change in impedance is representative of the absolute change in vessel volume is when complete outflow obstruction is imposed.25 This assumes that all other factors in Eq. (12) remain constant. As applied to thoracic applications, Nyboer professed to have solved the outflow problem by manually determining the maximum systolic down-slope of ∆Z(t) and extrapolating it backwards to the beginning of ejection. The maximum impedance change resulting from the backward extrapolation procedure was believed to be equivalent to the maximum impedance change attained as if no arterial runoff occurred during ejection (Fig. 5). Thus, the maximum volume change ∆Vmax , as a result of ventricular ejection (SV), was thought to be proportional to Nyboer’s ∆Zmax . SVNyboer = −

ρb L 2 ∆Zmax (down-slope extrapolation). Z02

(13)

Nyboer’s method was not widely accepted, because it was considered cumbersome, and, because of the difficulty in determining the true down-slope, inaccurate. 4.2. Kubicek equation In response to the objections of Nyboer’s down-slope extrapolation method, Kubicek et al.23 made the assumption that, if the maximum systolic up-slope of ∆Z(t) (i.e. ∆Z  = ∆Z · s−1 , Ω · s−1 ) is held constant throughout the ejection period, then compensation for outflow, before and after attainment of peak flow results.23,26 The theory underlying Kubicek’s forward extrapolation procedure assumes that little peripheral arterial runoff occurs during the rapid ejection phase of systolic ejection. While Nyboer’s ∆Zmax is directly measured, Kubicek’s method requires multiplying ∆Z  by left ventricular ejection time Tlve (i.e. ∆Z  × Tlve = ∆Zmax ).6 In the

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Fig. 5. ECG, ∆Z(t) and dZ/dt waveforms from a human subject TRR = the R-R interval, or the time for one cardiac cycle; Q = onset of ventricular depolarization; ——— = maximum systolic upslope extrapolation of ∆Z(t); B = aortic valve opening; C = peak rate of change of the thoracic cardiogenic impedance variation, dZ/dtmax ; X = aortic valve closure; Y = pulmonic valve closure; O = rapid ventricular filling wave; Q-B interval = pre-ejection period, TP E ; B-C interval = timeto-peak dZ/dt, TTP; B-X interval = left ventricular ejection period, Tlve . dZ/dt waveform to the right shows dZ/dtmax remaining constant throughout the ejection interval, Tlve , which represents outflow compensation. Nyboer’s down-slope backward extrapolation of ∆Z(t) is not shown.

original description of the technique, aortic valve opening and closure, and thus left ventricular ejection time (Tlve , s), were determined by phonocardiography. Finally, and without further explanation, this peak value, square wave extrapolation was assumed to produce the equivalent impedance change proportional to the mean flow velocity calculation required for SV determination. As opposed to manually extrapolating the maximum forward slope of ∆Z(t), Kubicek et al.23 found that it was obtained most accurately by electronically differentiating the ∆Z(t) waveform with respect to time (Fig. 5). This was especially true in the presence of spontaneous ventilation, where the breathing artifact ∆Zvent caused a wandering baseline and gross distortion of the cardiogenic component of Z(t), ∆Zcardiac (t). In modern impedance devices, however, the respiratory artifact is suppressed, assuring a stable baseline for ∆Z(t) and dZ/dt. The first derivative signal clearly identifies the maximum slope as a discreet landmark, point C (dZ/dtmax ), as well as defining Tlve as the interval from point B to point X. Differentiating both sides of Eq. (12) with respect to time purportedly yields the rate of change of volume, or flow (mL · s−1 ),

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its peak first time-derivative ostensibly related to its corresponding impedance analog by the following: dV (t) ρb L2 dZ(t) =− 2 , dtmax Z0 dtmax

(14)

where dZ/dtmax is the maximal systolic upslope, peak first time-derivative, or forward peak rate of change of the transthoracic cardiogenic impedance pulse variation d[∆Z(t)]/dt (Ω · .s−1 ). In an in vitro pulsatile open outflow system, using the Kubicek equation for SV, Yamakoshi et al.25 demonstrated that the outflow extrapolation procedure yielded close approximations of corresponding electromagnetic measurements of SV. As predicted, the Nyboer equation (Eq. 12), without outflow obstruction, systematically underestimated SV, thus tentatively validating the outflow hypothesis. Despite the rather sweeping assumptions justifying the outflow hypothesis, and its never having been directly confirmed experimentally, in vivo, it is still considered, by most, a theoretically plausible solution. In terms of basic hemodynamic theory, the outflow correction purportedly compensates for arterial runoff as defined in the Windkessel model for flow Q, which is given as follows:17,18 Qtotal =

dV (t) P (t) + , dt Rs

(15)

Qin + Qout where dV (t)/dt (i.e. (dP (t)/dt) × C, Qin ) equals the flow of blood entering segment L during ejection, and Qout , simultaneous aortic runoff (outflow). Thus, equating the differentiated Nyboer equation, containing Kubicek’s outflow correction, with the classical Windkessel expression for SV, found through integration of Eq. (15) over the ejection interval, yields:   tavc  ρb L2 dZ(t) dV (t) P (t) + dt , Tlve ≡ SVKubicek = − 2 Z0 dtmax dt Rs (16) tavo Vin + Vout Vin + Vout where the left hand side is known as the Kubicek equation and t0 and t1 represent aortic valve opening and closure, respectively. It is thus stipulated that impedance derived SV, employing dZ/dtmax , is expressly dependent on dV /dt, the components of which are the rate of change of ascending aortic pressure, Aortic dP/dt, and aortic compliance C. If compliance C of a vessel is equal to a given change of volume for a given change in pressure (pulse pressure) dV /dP , then dV /dt is equal to the following: dP dV dV = , dt dt dP

(17)

where, for a cylindrical conduit of cross-sectional area A, (πr2 , cm2 ), and length L (cm), the volume V (mL) is given as: V = πr2 L.

(18)

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Differentiating both sides of 18 with respect to the radius r yields, dV = 2πrdrL, (19) dr where dV /dr is the volume change of a cylinder with respect to the change in radius dr, 2πr the circumference, and 2πrL the internal surface area A (cm2 ) of the cylinder. It follows that the change in volume is directly proportional to the change in vessel internal surface area, dV /dA. Thus, dV = dA × L. (20) dA Hence, the rate of change of dV /dr (or, analogously, dV /dA) with respect to time is given as: dV /dr dV dr dA = = 2πr L ≡ L. (21) dt dt dt dt Substitution of the right hand side of Eq. (21) into the integral containing dV /dt in Eq. (16) shows that dZ/dtmax is explicitly dependent on the rate of change of internal and cross-sectional aortic area dA/dt, as a function of the rate of change of aortic radius, dr/dt. Equation (16) can therefore be rewritten as:   tavc  dr(t) P (t) ρb L2 dZ(t) + 2πrL dt (22) SV = T = lve Z02 dtmax dt Rs tavo Further inspection of Eqs. (16), (17) and (22) demonstrates that the magnitude of impedance-derived SV is expressly dependent on aortic dP/dt and pulse pressure dP , mean arterial blood pressure Pm , the static component of the ventricular afterload, Rs (systemic vascular resistance, SVR), and, using a three-element Windkessel model, the dynamic mechanical component of the ventricular afterload, characteristic aortic impedance.27 The mechanical impedance, also called Z0 , is a frequency dependent parameter, equal to the pulse pressure divided by peak aortic blood flow, (Ps –Pd )/Qmax . Furthermore, since ICG-derived SV is hypothesized as conforming to simple Windkessel theory as its foundation, it should also vary directly with C and Aortic distensibility (AoDistens), which are both age-dependent.28–31 Aortic cross-sectional distensibility is given in its various forms as: AoDistens =

2∆d (AoArea)max − (AoArea)min C ≡ , ≡ (AoArea)min (AoArea)min × ∆P d × ∆P (23)

where (AoArea)min is the aortic CSA area at end-diastolic pressure, (AoArea)max the aortic CSA at peak systolic pressure, ∆d the pulsatile change in aortic diameter measured over the pulse pressure, d the end diastolic aortic diameter, and ∆P the pulse pressure.30,31 However, studies have shown that, whereas reference-method t SV and stroke distance S (i.e. S = t01 v(t)dt, cm) decrease ∼ 20%–30% from age 20–60, or 5%–7% per decade, respectively,32,33 AoDistens and C decrease 80%–100% from age 20–60, or 20%–25% per decade.30,31 Thus, it appears highly unlikely that

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the theory behind present ICG equations adequately accounts and compensates for the mechanical changes in the aorta attendant with increasing age. Moreover, judging by the rather poor correlations between blood pressure levels and dZ/dtmax ,34 it is doubtful that ICG-derived SV is linearly correlated with changes in aortic diameter and cross-sectional area over a wide spectrum of mean, end-diastolic, and pulse pressures. Thus, while pulse pressure increases as C decreases with age, the changes in aortic diameter and CSA do not parallel the actual magnitude of SV. Given the complex relationship between Pm and Rs , it is rather doubtful that Kubicek’s outflow correction can compensate for this discrepancy. As verification of this assertion, Yamakoshi et al.35 demonstrated in an in vitro tube model that ICG-derived SV, using dZ/dtmax , shows little change, if any, over a wide range of vascular compliance C, systemic vascular resistance Rs , mean arterial pressure Pm , and vessel dP/dtmax . Clinically, Brown et al.36 showed that, despite progressive stiffening of the aorta over the full spectrum of age (20–80 years), ICG-derived CO is nonetheless highly correlated and in agreement with thermodilution CO. Upon reassessment of equations (16) through (23), these findings are antithetical to the very foundations upon which ICG SV is based. Specifically, a Windkessel model for impedance-derived SV seems improbable. Thus,   tavc  P (t) dP (t) ρb L2 dZ(t) C + dt (24) T =  lve Z02 dtmax dt Rs tavo Although the right hand sides of Eqs. (16) and (22) are considered a priori unsolvable, the bioimpedance expression for SV is assumed to produce a numeric output equivalent to the simple Windkessel models delineated above, as well as to those equations employed in other pulsatile methods; namely, electromagnetic and Doppler flowmetry. While both latter methods are prone to methodological error,37 they each employ the universally accepted, theoretically assumption-free statements defining SV, namely,  t1 2 v(t)dt ≡ πr2 vmean Tlve , (25) SV = πr t0

t where πr2 is the aortic valve cross-sectional area and t01 v(t)dt the time-velocity integral measured over the ejection interval (t0 → t1 ), which is equivalent to stroke distance, S (cm). In practice, the normalized peak value of the first time-derivative (dZ/dtmax )/Z0 is multiplied by the product of a static or quasi-static volumetric constant V (ρb L2 /Z0 ), hereafter known as the volume conductor Vc , and left ventricular ejection time Tlve to presumably yield SV. The Kubicek equation23 in modified form is thus given as: SV =

ρb L2 dZ/dtmax Tlve , Z0 Z0

(26)

where ρb equals the static specific resistance of blood, usually fixed at 135–150 Ω cm,2–5,23 L, the measured distance between the voltage sensing electrodes (cm)

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(Fig. 1), and Z0 , the quasi-static transthoracic base impedance (Ω) measured between the voltage sensing electrodes.

4.3. Assumptions implicit to the accuracy of Nyboer-Kubicek model (1) The transthoracic impedance is considered the parallel connection of an aggregate of static tissue impedances Zt , considered as one, and a dynamic blood resistance Rb , otherwise known as the blood impedance Zb . (2) The blood resistance is considered a homogeneously conducting tube of constant length L, or a parallel connection of an aggregate of tubes considered as one. (3) The current distribution in the blood resistance is uniform. (4) All current flows through the blood resistance. (5) The volume conductor Vc is homogeneously perfused with blood of specific resistance ρb . (6) The magnitude of SV is directly related to power functions of the measured distance L between the voltage sensing electrodes (Kubicek), or to heightbased (cm) thoracic length equivalents (Sramek, Sramek-Bernstein) (Fig. 1). (7) All pulsatile impedance changes ∆Z(t) are due to vessel volume changes ∆V (t), and, in the context of assumption 2, ∆Z(t) is due exclusively to changes in vessel radius dr and CSA dA (Eqs. (9), (12) and (22) and Fig. 4). (8) In the context of assumption 7, dZ/dtmax is the bioelectric equivalent of dV /dtmax . (9) Outflow, or runoff, during ventricular ejection can be compensated for by extrapolating the peak rate of change of ∆Z(t) over the ejection interval (i.e. dZ/dtmax × Tlve ) (Eq. (16) and Fig. 5). (10) The specific resistance (“resistivity”) of blood ρb is constant during ejection. (11) The specific resistance of the thorax ρT is constant.23 4.4. Sramek–Bernstein equation The Sramek–Bernstein24 equation and its assumptions are similar to Kubicek’s, except for the physical definition and magnitude of the Vc . In Sramek’s interpretation of Kubicek’s Vc , ρ, and a Z0 variable are eliminated by mathematical substitution. This simplification assumes that ρ is constant, equal to ZA/L, thus rendering the volume a constant for each individual. The resultant cylindrical model is constructed assuming the circumference of the thorax at the xiphoid process to be three times the thoracic length L (i.e. C = 3L). Solving for the CSA at C, and multiplying by L, yields a cylinder volume three times the magnitude required for physiologic levels of SV. Taking one third of the resultant volume produces a best-fit Vc . Sramek’s Vc is known as the volume of electrically participating thoracic tissue VEP T and is modeled conceptually as a frustum, rather than as a cylinder in the

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Kubicek approach. At variance with assumption 6 of Kubicek’s method, this equation assumes the magnitude of SV to be proportional to the third power of a fixed percentage of overall body height, and modified by Bernstein24 for deviation from ideal body weight. The Sramek–Bernstein equation is given as: SVS−B = δ

L3 dZ(t)/dtmax Tlve , 4.25 Z0

(27)

where δ = a dimensionless parameter which corrects for deviation from ideal body weight at any given height, and further modified for the indexed blood volume at that weight deviation, L = 17% of overall body height (cm), and 4.25 = a dimensionless, empirically-derived constant of proportionality. Thus, the purported equivalency of Eqs. (26) and (27) with (25) can be expressed as follows: dZ(t)/dtmax × Tlve (28) SV = Area × vmean × Tlve = V olume × Z0 where, (dZ/dtmax /Z0 ) × Tlve is assumed to be the dimensionless ohmic equivalent of stroke distance, vmean × Tlve . Thus, SV equivalency is assumed on the basis of dZ/dtmax being the ohmic analog of peak flow velocity (equivalent to dV /dtmax ), and the correctness of the outflow extrapolation procedure. Therefore, as governed by constraints of their ohmic origins, all other factors remaining constant, both ICG equations rigidly ascribe the magnitude of SV solely to the phasic peak rate of change of the transthoracic, volume-related, impedance variation; specifically, the relationship expressed in proportionality 5 and Sec. 3.1. 4.5. Validity of the plethysmographic techniques Despite the experimentally verified equivalence of impedance changes caused by red blood cell velocity, as discussed earlier, both equations obligatorily ignore this influence as a trivial contaminant. Indeed, some in vitro pulsatile and nonpulsatile models have shown that the resistivity of flowing blood contributes merely 5%–10%,38,39 or less,40 to the magnitude of ∆Z(t); they did not, however, study the effects on the peak first time-derivative, dZ/dtmax . While it is true that many studies, especially those of clinical nature, have shown good to excellent results using both the Kubicek and Sramek–Bernstein equations,1,2,7 there are unanswered questions regarding the theoretical underpinnings of the method. Although it has been proven that the parallel conduction model is valid for thoracic applications,41,42 there remain some unresolved issues, a few of which comprise the following: (1) The proper value, physiologic definition, and theoretical basis for ρ: blood resistivity ρb (hematocrit dependent or constant) versus thoracic resistivity ρT (a constant).43,44 (2) The proper methodology for determining thoracic length L in the Kubicek and Sramek–Bernstein equations and its physiologic relevance to SV.45

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(3) The validity of the outflow extrapolation procedure.46 (4) A coherent physiologic basis and correlate for the empirically-derived volume conductors defined above. (5) The relevance of δ in the Bernstein modification of the Sramek equation.45,47 (6) Lack of consideration for the changing transthoracic specific impedance,48 especially in critical illnesses characterized by increased thoracic liquids causing aberrant electrical conduction.49,50 (7) The effect of the blood velocity-induced change in transthoracic specific resistance ∆ρb (t). (8) Failure of the literature to validate a Windkessel model for ICG-derived SV.35 Therefore, it is the stated objective of this chapter to show how existing SV equations might be improved by exploring the following possibilities: (1) The general assumption of ρb L2 /Z02 × (dZ/dtmax ) or VEP T /Z0 × (dZ/dtmax ) being equivalent to dV /dtmax might be in error (Eq. (14)). (2) dZ/dtmax × Tlve might not be the ohmic equivalent of stroke distance (Eq. (28)). (3) The influence of the dynamic, blood velocity-induced resistivity change ∆ρb (t) is not trivial as concerns dZ/dtmax . (4) The magnitude of, and theoretical basis for existing electrical volume conductors might be incorrect. (5) Changing transthoracic specific impedance (“thoracic resistivity”) must be accounted for in states characterized by excess extra-vascular lung water.49 (6) Amendments to existing SV equations might be appropriate.3,4,7

5. Theory and Rationale for a New Stroke Volume Equation 5.1. Analytical methods Based on a literature search relevant to impedance-derived SV, a critical analysis of the biophysical and electrodynamic systolic origins of ∆Z(t) and dZ/dt was performed and extrapolative comparisons made with their respective mechanicallybased hemodynamic counterparts. Equations descriptive of the reported phenomena were then designed to elucidate the various factors responsible for the systolic components of dZ/dt, and specifically the origin of dZ/dtmax . To this end, a differential and time domain analysis was performed, and conclusions drawn with regard to the physical time domain in which dZ/dtmax resides. An analytically-based mathematical transformation, derived and extrapolated from the relevant bioimpedance and hemodynamic literature, was employed to find the ohmic mean velocity equivalent necessary for SV calculation. A new volume conductor Vc is also proposed, based on proven allometric relationships and accepted physiologic principles, as well as the constraints imposed by the magnitude of the resultant transformation. Additionally, a transthoracic electrical shunt index for aberrant conduction is introduced, rendering the magnitude of the volume conductor dynamic, rather than static, as in the

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Sramek–Bernstein approach. To test the above hypotheses, the results of a recently published study are presented, wherein the Kubicek, Sramek, Sramek–Bernstein equations were compared to the new equation, using thermodilution cardiac output as a reference standard.51 As a consequence of the theoretical discussion, empirical time domain analysis, equation derivations and results of the clinical study, a new SV equation is proposed. 5.2. Origin of dZ/dtmax 5.2.1. Resolution of origin by anatomic domain The systolic portion of ∆Z(t) has been shown to be comprised of elements from both the pulmonary artery and aorta.14,15 By magnitude, Saito et al.14 and Ito et al.15 demonstrated that about 30%–40% of the area beneath the ∆Z(t) waveform is generated by pulmonary blood flow and 60%–70% by aortic blood flow. Early correlative phonocardiographic studies of the dZ(t)/dt waveform by Lababidi et al.52 clearly show fiducial landmarks consistent with aortic valve opening and aortic and pulmonary valve closure (Fig. 5). When electrocardiographic signs of left bundle branch conduction delay are present, pulmonary valve opening can also be readily distinguished.53 Likewise, with right bundle branch block, the delay in pulmonary valve closure is also clearly apparent.53 As concerns the anatomic origin of dZ/dtmax , there is compelling evidence that it derives from the ascending aorta.2–4,54 Similar to Doppler velocimetry, in which the aortic peak velocities are proportional to the highest frequency shifts detected, the peak rate of change of aortic volume (or peak flow) is assumed to be proportional to the peak rate of change of impedance (Eq. (14)). In normal individuals, Gardin et al.55 reported mean values of 92 cm · s−1 , and 63 cm · s−1 , respectively, for the peak velocities of the aorta and pulmonary artery. Additionally, mean values for time to peak velocity (TTP), or acceleration time, were reported as 98 ms and 159 ms for the ascending aorta and pulmonary artery, respectively. Therefore, if one assumes a plethysmographic origin for dZ/dtmax , then it must be concluded that dZ/dtmax represents the highest aortic velocities detected, and that, by timing, aortic peak velocity would precede pulmonary artery peak velocity. Thus, the peak pulmonary artery component of dZ(t)/dt (dZ[pulmonic]/dtmax ) would be obscured and delayed in timing within the composite first derivative impedance envelope. Alternatively, if dZ/dtmax represents the peak time derivative of the resistivitybased velocity component of ∆Z(t) (Eq. (3) and Sec. 3.2) dρb (t)/dtmax (Ω cm · s−2 )(i.e. d[∆ρb (t)]/dtmax or d[∆Zv (t)]/dtmax ), then the aortic component would still predominate. Justification for this conclusion is based on the finding that average values for aortic and pulmonary artery mean blood flow acceleration are reported as 940 cm · s−2 and 396 cm · s−2 , respectively.55 Thus, if dZ/dtmax represents the greatest rate of change of the blood resistivity variation, which, in turn, is related to the peak acceleration of red blood cells, then dZ/dtmax would peak with aortic dv/dtmax and dQ/dtmax . In addition to purely mechanical considerations, dynamic changes

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in the specific resistance of flowing blood are characterized as being anisotropic.56 In concert with, and analogous to the magnitude of the ultrasonic frequency shift in Doppler velocimetry, this means that, in addition to red cell velocity, the measured magnitude of the resistivity change is related to the angle of interrogation between the applied AC and the direction of the forward flowing erythrocytes. The greatest negative resistivity changes of flowing blood occur when, from a state of random orientation in end-diastole, the long axis of symmetry of the red cells become oriented axially, aligned in parallel with the applied AC during systole (Fig. 4).21,22 This condition is fulfilled in the ascending and descending thoracic aorta, but not in the pulmonary artery, because only a short segment of the pulmonary artery trunk is aligned longitudinally, in parallel with the applied AC. Since the direction of the vast majority of pulmonary blood flow is via the left and right pulmonary arteries, and thus, perpendicular to the axis of the applied AC, the total pulmonary artery resistivity contribution is considered trivial as compared to that of the aorta.16

5.2.2. Resolution of origin by differential time domain analysis Since ∆Z(t) is a composite waveform, containing ohmic analogs of both velocity and volume changes (proportionality 6, and Fig. 3), dZ/dt should reflect their respective time-derivatives, linear acceleration and flow. Thus, by differentiating Eq. (3) with respect to time: dZv (t) dZvol (t) dZ(t) = + . dt dt dt

(29)

If a segment of aorta is considered a cylindrical, thin-walled blood-filled structure at end-diastole, its impedance Z to an applied AC field, across the measured segment, can be defined as given in Eq. (8). In this example, ρb is the static specific resistance of blood (Ω cm), L is the segment of aorta under electrical interrogation (cm), and Vb is the volume of blood (mL) in the aorta over segment L in enddiastole. If upon ventricular ejection, all variables in Eq. (8) become continuously differentiable functions of time, then, using the product and quotient rules, the rate of change of the cardiogenically-induced impedance variation dZ/dt is given as follows: ρb 2L dL(t) L2 dρb (t) ρb L2 dVb (t) dZ(t) = + − . dt Vb 1 dt Vb dt Vb2 dt (1) (2) (3)

(30)

If dL/dt → 0, then dZ/dtmax peaks in the time domain of either derivative 2 or 3. Conceptually, SV obtained by the Kubicek, Sramek and Sramek–Bernstein equations implement the peak first time-derivative of derivative 3, relating dZ/dtmax to

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the peak rate of change of the aortic volume variation, dV /dtmax (mL · s−1 ), the genesis of which has already been discussed. Thus, dZvol (t) ρb L2 dVb (t) =− 2 (Ω · s−1 ). dtmax Vb dtmax

(31)

Alternatively, dZ/dtmax might possibly peak with derivative 2, relating dZ/dtmax to the peak first time-derivative of the velocity-induced blood resistivity variation, d[∆ρb (t)]/dtmax , corresponding biophysically to, and in the time domain with peak aortic blood acceleration dv/dtmax . Therefore, dZv (t) L2 dρb (t) = (Ω · s−2 ). dtmax Vb dtmax

(32)

Analysis of Eqs. (31) and (32) demonstrate the two possible origins of dZ/dtmax , but unfortunately, this theoretical conundrum cannot be resolved by simple differential analysis. Although a temporal difference in time-to-peak (TTP) magnitude of the derivatives in Eqs. (31) and (32) is implied, there is insufficient information in either equation to establish a definitive biophysical origin for dZ/dtmax in the time domain. 5.2.3. Resolution of origin by comparative time domain analysis: time-to-peak (TTP) of dZ/dtmax While the blood resistivity change is necessarily deemed a trivial contaminant, using the plethysmographic approach, it has been demonstrated that the changing “resistivity” (specific resistance) of flowing blood contributes approximately 50% to the amplitude of ∆Z(t).10,13,57 Others11,12,16 have observed lesser, but still substantial contributions of the blood velocity induced resistivity change to the magnitude of ∆Z(t). Therefore, since dZ/dt is the first time-derivative of ∆Z(t) and acceleration is the derivative of velocity, it is suggested that dZ/dt might peak in the time domain of acceleration (dv/dt, dQ/dt), rather than flow (Q, dV /dt). Proof that dZ/dtmax peaks synchronously with dv/dtmax , is found to reside in its empirical correlation with known TTP durations (ms) of the time-derivatives of pressure and flow in the mammalian cardiovascular system. Physical measurements of the TTP from opening of the aortic valve to dZ/dtmax is reported to be 60 ± 20 ms in normal individuals at rest (Fig. 6),58–60 whereas peak velocity (vmax ), and in the time domain, equivalently maximum flow (Qmax , dV /dtmax ), peak at a mean of 98 ms, or at the end of the first third of systole.55 Concordant with the TTP of dZ/dtmax , it has been demonstrated that the TTP of peak aortic blood acceleration (dv/dtmax , dQ/dtmax ) occurs equivalently at about 50 ± 20 ms from valve opening, or in the first 10%–20% of systole.37,61,62 In Fig. 6 it is to be noted that, for an ejection time of ∼ 360 ms, dZ/dt peaks in ∼ 60 ms, or in the first 10%–20% (16.7%) of systole. Similarly, graphic evidence provided by Sheps et al.63 shows that, for an ejection time of approximately 250 ms, TTP for dZ/dtmax occupies 20% of Tlve , or about 50 ms. Corroboratively, close examination of the fiducial landmarks and time lines

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Fig. 6. dZ/dt peaks in the time domain of acceleration TTP = time-to-peak; LVET = left ventricular ejection time; % of systole = % LVET; (TTP/LVET) × 100 = % of systole.

supplied by Lababidi et al.52 show that dZ/dtmax peaks in the first 15–20% of the ejection phase of systole, as does peak acceleration.37,61,62 Matsuda et al.60 have shown that, while peaking out of phase, the maximum slope and TTP of LV dP/dtmax and dZ/dtmax are identical. It can also be demonstrated from their data that the time difference between the onset of the Q wave of the ECG to aortic valve opening, (i.e. Q wave to dP/dtmax ), and the time of onset from Q wave to dZ/dtmax , is approximately 60 ± 15 ms (i.e. [Q → dZ/dtmax (ms)] – Q → LV dP/dtmax (ms)] = 60±15 ms). Corroborative data from Lozano et al.,106 obtained by subtracting the R(ECG)–B(dZ/dt) interval (mean 68 ms, range 47–84 ms) from the R–dZ/dtmax interval (mean 120 ms, range 80–160 ms), shows that the TTP of dZ/dtmax from point B (rise time) is ∼ 52 ± 20 ms (i.e. [R–dZ/dtmax (ms)]– [R–B (ms)] = 52 ± 20 ms). In normal children, Barbacki et al.64 showed that the TTP for dZ/dtmax was appreciably less than for Doppler-derived peak ascending aortic blood flow. Figure 7 graphically demonstrates that dZ/dtmax (waveform G) precedes peak ascending aortic blood flow (waveform D), monotonically, during the period of peak acceleration. Kubicek54 and Mohapatra and Hill65 demonstrated that dZ/dtmax corresponds precisely in time with the “I” wave of the acceleration ballistocardiogram (BCG). The “I” wave of the acceleration BCG is known to represent the peak acceleratory component of the initial ventricular impulse during the inertial phase of very early systole.61 Kubicek et al.26 also showed that, when measured simultaneously, dZ/dtmax peaks substantially before ascending aortic Qmax . In this study,26 correct interpretation of both waveforms’ maxima requires measuring left

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Fig. 7. Relationship between esophageal ECG (A), aortic pressure (B), aortic expansion (C), aortic blood flow (D), pulmonic expansion (E), surface ∆Z(t) (F) and dZ/dt (G) waveforms. Calibration signals: aortic and pulmonic expansion (1 mm), aortic blood flow (10 L · min−1 ), ∆Z(t) (0.1 Ω), dZ/dt (1 Ω · s−1 ). Note that dZ/dtmax precedes peak aortic blood flow (Qmax ). Figure 7 with permission from John K. Hayes, PhD. Influence of simulated gastric reflux, body posture, aortic and pulmonic vascular expansion and erythrocytes on the impedance cardiogram recorded from an esophageal impedance probe. Doctoral Dissertation, University of Texas at Austin, p. 186, 1994.

ventricular ejection time on the dZ/dt waveform from point B, corresponding to aortic valve opening, which is above the baseline, to point X, corresponding to aortic valve closure on the ascending aortic flow tracing. Confirming this observation, image enlargement of the work of Ehlert and Schmidt66 subtly, but convincingly demonstrates that, over a wide range of SV, dZ/dtmax peaks prior to Qmax . Rubal et al.67 also showed that, when measured simultaneously, dZ/dtmax peaks appreciably before Qmax . Finally, Welham et al.68 showed that, over a full range of myocardial depression with halothane anesthesia, dZ/dt peaked simultaneously with dv/dtmax (acceleration), and very subtly before peak flow (Qmax ). As a result of this evidence-based, empirical TTP analysis, it is suggested that dZ/dtmax is the ohmic impedance analog of dv/dtmax , and not dV /dtmax , as generally believed. Based on this discussion, and the conclusions drawn therefrom, it is suggested that dZ/dtmax is best described by differential Eq. (32), relating dZ/dtmax to the peak rate of change of the erythrocyte velocity-induced resistivity change, dρb (t)/dtmax ; that is, peak aortic red blood cell acceleration. This implies that the requirements of Eq. (25) are unmet, insofar as the Kubicek and Sramek Bernstein equations are concerned, and Eqs. (14), (16), (22), and (26–28) do not represent equality. Clearly, SV is obtained through a mean flow (velocity) calculation, precluding the use of acceleration as a surrogate: Specifically,  t1 dv dZ(t)/dtmax dt ⇒ SV = Volume × × Tlve . (33) SV = Area × dt Z0 t0

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5.2.4. Hemodynamic and biomechanical origins of ∆Z(t) and dZ/dtmax : Analysis by means of a differentiable cylindrical model of ascending aortic blood flow Consider a cylinder of length S, and radius r, and assume that the cylinder at its maximum volume V is equal to SV . Hemodynamically, let the time-velocity integral, also known as stroke distance, or the displacement of blood over one ejection interval, equal S (cm). Also, assume that the mean radius of the aortic root over the ejection interval is a constant, equal to r (cm), as well as the aortic wall being visco-elastic and thin. Equation (25) then takes the form of, SV = V = πr2 S.

(34)

If r and S are allowed to vary with time, t, then V is a function of time. V (t) = πr(t)2 S(t).

(35)

If r(t) and S(t) are continuously differentiable functions, then, dS(t) dr(t) ∗ dV (t) = Q = 2πr S + πr2 , dt dt dt (1)

(36)

(2)

where, dV (t)/dt equals the rate of change of volume, or flow Q, and dr/dt is the time rate of change of the aortic radius (wall velocity, cm · s−1 ) from its end-diastolic nadir r. In this context, derivative 1 corresponds to (dA/dt) · S ∗ ,25 which is a function of, and equivalent to (dP/dt) · C (Eqs. (17) and (21)). For derivative 1, let S ∗ be a fixed-length aortic segment over which dr/dt is generated and analogous to thoracic segment length L over which ∆Z(t) and dZ/dtmax are measured. It is also assumed that vascular segment S ∗ in derivative 1 provides for simultaneous inflow and outflow during systole. In the currently used plethysmographic approaches, dS/dt (cm · s−1 ) in derivative 2, corresponding to axial blood velocity v in the Newtonian domain, and to ∆ρb (t) in the ohmic domain, is obligatorily considered trivial (vide supra). Thus, in the context of its composite architecture, ∆Z(t) contains the volume change from Eq. (35), S remaining constant, plus the velocity component of derivative 2 in Eq. (36), dS/dt, r (i.e. area) remaining constant (see Figs. 3 and 4). It is generally believed that dZ/dtmax peaks in the time domain of Eq. (36) and that the differential component best describing this biophysical phenomenon in impedance cardiography corresponds to derivative 1. Derivative 1 is in agreement with Kubicek assumption 7, which states that all volume changes are due only to changes in CSA. By contrast, it should be noted that in Doppler and electromagnetic flowmetry (EMF), both techniques implement derivative 2, where r(t) and dr/dt are assumed to be negligibly small (i.e. r(t) → 0, dr/dt → 0) as compared to dS/dt.37 License for this convention is considered valid, because, attendant with systole, the aortic annulus and ascending aortic radius in humans change by only ± 3.3% from their end-diastolic nadir (i.e. ∆r/r = ±3.3%).69 The small change in

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radius generates a maximum change in CSA of about 6–7%. Clearly, integrating differential 1 of Eq. (36) over the ejection interval is equivalent to dV /dr in Eq. (19). Thus, for an aortic segment length of 30 cm with open outflow, segment mean enddiastolic radius of 1.5 cm and a 3.3% change in radius over the pulse pressure (i.e. dr = 0.0495 cm), the net change in volume (i.e. net SV) would be approximately 14 mL. This value is approximately five to six times smaller than physiologic (i.e. 15%–20%), and thus, similar in magnitude to the expelled gas volume obtained and predicted from pneumocardiography (vide supra).20 As discussed earlier, this volume represents net volume inflow into the aortic segment, with the remainder of SV (i.e. 80%–85%) being simultaneously dissipated to the periphery as axially directed outflow. For this simple model of pulsatile blood flow, in contrast to classic Windkessel approaches, it thus follows that the outflow is represented by derivative 2 of Eq. (36). To compensate for radial displacement and volume inflow, Doppler-derived SV implements the measurement of aortic diameter at peak systolic displacement. Thus, by magnitude,  ACSA

t1 t0

dS(t) dt  AISA dt



t1

t0

dr(t) dt, dt

(37)

SVout  SVin , where ISA is the vessel internal surface area over segment S and CSA is the vessel cross-sectional area at the point of measurement of dS/dt. Differentiating flow (Q) with respect to time, yields radial wall and axial aortic blood acceleration dQ/dt (i.e. d2 V /dt2 ). Hence, dQ = 2π dt



dr(t) dt (1)

2 S + 2πr

d2 S(t) d2 r(t) dr(t) dS(t) + πr2 S + 4πr . 2 dt dt dt dt2 (2)

(3)

(38)

(4)

As concerns ventricular-vascular coupling, derivatives 1–3 are considered comparatively trivial at dQ/dtmax , because aortic wall velocity dr/dt, wall acceleration, d2 r/dt2 , and axial dS/dt approximate zero by comparison with d2 S/dt2max (cm · s−2 ) during the inertial phase of ventricular ejection. Thus, dZ/dt is a composite waveform, comprising ohmic analogs of derivative 1 of Eq. (36), and derivative 4 of Eq. (38). Therefore, based on this and preceding discussions, it is the thesis of this chapter that dZ/dtmax peaks in the time domain of derivative 4, the second derivative of stroke distance; that is, linear peak aortic blood acceleration, dv/dtmax (d2 S/dt2max ). Clearly, this conclusion is consistent with the theories of origin proposed in Eqs. (29)–(32) and the discussion in Sec. 5.2.3. Thus, in order to compute SV , the ohmic equivalent of derivative 4 in Eq. (38) (i.e. Eq. (32)) must be transformed into the ohmic equivalent of derivative 2 in Eq. (36). Ordinarily, velocity is obtained from acceleration through simple integration. Unfortunately, however, ohmic mean velocity cannot be directly extracted from dZ/dt using this method.

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6. Rationale for Application of the Square Root Acceleration Step-down Transformation in Impedance Cardiography: Conversion of dZ/dtmax Normalized by its Base Impedance Z0 (dZ/dtmax /Z0 ) to Normalized Ohmic Mean Velocity ∆Zv (t)max /Z0 Hypothetically assuming dZ/dt to be a mono-component, blood resistivity-based acceleration waveform (i.e. dZv (t)/dt), and conforming to derivative 2 in Eq. (30), simple integration would yield ∆Zv (t), its sole composition being the velocityinduced blood resistivity change ∆ρb (t). However, as discussed, ∆Z(t) is a composite waveform, its peak magnitude ∆Z(t)max probably representing a point temporally approximating peak pressure, which physically corresponds to the net volume change of the aorta in mid systole (Fig. 3, Eqs. (4) and (12)).35 This effectively voids integration as a solution for finding ∆ρb (t)max , and thus, ohmic mean velocity. Since it is suggested that dZ/dtmax possesses the dimensions of Ω · s−2 (Eq. (32)), and normalized by its base impedance Z0 to a dimensionless ohmic acceleration equivalent, s−2 , a square root transformation would seem mathematically plausible and intuitively appealing. When equation (32) is normalized to relative ohmic acceleration by Z0 , a dimensionless ohmic equivalent results, namely, [(L2 /Vb )dρb (t)/dtmax ]/Z0 , which reduces to s−2 . Operationally, the numerator would possess dimensionless magnitude  0 during early systolic ejection. Thus, it is suggested that dZ/dtmax /Z0 is related to the normalized differential in Eq. (32), as follows: dZv (t)/dtmax L2 dρb (t) 1 = = s−2 . Z0 Vb dtmax Z0

(39)

Since both sides of Eq. (39) are suggested to be dimensionless ohmic equivalents of acceleration, the square root of both sides should yield dimensionless ohmic equivalents of velocity. That is:    dZv (t)/dtmax L2 dρb (t) 1 ∆Zv (t)max ≡ = = s−1 . (40) Z0 Vb dtmax Z0 Z0 Thus,

 dZv (t)/dtmax = ohmic mean velocity, Z0

(41)

where Eq. (41) is to be known as the square root acceleration step-down transformation, or simply, ICG acceleration step-down transformation. 6.1. Peak aortic reduced average blood acceleration (PARABA): Hemodynamic analog of dZ/dtmax /Z0 The rationale for, and the correctness of the square root transformation is to be found in a modified version of theory originally propounded by Visser.21 Consider

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the following relationship for blood accelerating axisymmetrically through a circular aorta with radius R (cm):  R dv dQ = 2πrdr = mL · s−2 , (42) dt dt 0 where dQ/dt is the instantaneous acceleration. Integrating Eq. (42) by parts, and assuming the blood does not slip at the vessel wall, the following is ultimately obtained:  R  1 (dv/dt) 2 (dv/dt)  r 2  r dQ 3 = −π r dr = −πR . (43) d dt dr dr R R 0 0 Therefore, dQ/dt dv/dtmean = = πR3 R

 0

1



(dv/dt)  r 2  r = s−2 . d dr R R

(44)

The results obtained from Eq. (44) demonstrate that the blood flow acceleration resulting from integration over the vessel radius R is the mean acceleration divided by the aortic radius. At its aortic maximum, this is to be referred to as peak aortic reduced average blood acceleration (PARABA), [(dv/dtmean )/R]max , and is solely a function of the maximum shear acceleration, [−(dv/dt)/dr]max , and the ratio of r/R. Since r/R is dimensionless, the shear acceleration is purely a function of the reduced average acceleration. Thus, rearranging Eq. (44) and solving for volumetric mean acceleration:     dv/dtmean dQ 3 = mL · s−2 . = πR (45) dt mean R Concordant with this analysis, Visser21 showed that, in vitro, the peak relative resistivity change ∆ρ(t)max /ρstat is a function of the maximum reduced average blood velocity [(vmean /R)max ]exp . In vivo, peak aortic reduced average blood velocity (PARABV) is determined by peak aortic blood velocity, vmax . Analogously, an equivalent relationship exists between dZ/dtmax /Z0 , PARABA, and dv/dtmax . Thus, to obtain ohmic mean velocity by means of impedance cardiography, the square root of the impedance analog of PARABA must be taken, which √ is (dZ/dtmax )/Z0 . It can thus be stated that mean flow Q (mL · s−1 ) is equivalent to, 

 n  dZv (t)/dtmax dv/dtmean Qmean = K1 ≡ K2 Z0 R max  v

m ∆Zv (t)max mean ≡ K2 ≡ K1 , (46) R Z0 max where K1 and K2 , are volumetric constants of proportionality, and n and m are exponents equating the bioelectric and hemodynamic domains. Clearly, the relationship between dZv /dtmax /Z0 and ∆Zv (t)max /Z0 is parabolic in nature (y = x2 ),

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0.16

0.12

dZ/dtmax/Zo

0.08

0.04

0.0 0.0

0.1

0.2

0.3

0.4

(dZ/dtmax/Zo)

Fig. 8. Square root acceleration step-down transformation. Relationship between the bioelectric equivalent of peak aortic reduced average blood acceleration (PARABA), which is dZ/dtmax /Z0 (y√ axis), and the bioelectric equivalent of peak aortic reduced average blood velocity (dZ/dtmax /Z0 ) (x-axis).

as graphically demonstrated in Fig. 8. Shown on the x-axis is the linear extrapo√ lation of the square root transformation of dZ/dtmax /Z0 (x = y), which yields dimensionless ohmic equivalents of peak aortic reduced average blood velocity. Thus, it is suggested that Eqs. (16) and (26–28) do not represent equality, the suggested correct relationship between impedance-derived SV and classical expressions being,  t1 2 v(t)dt SV = Area × vmean × Tlve = πr t0  dZv (t)/dtmax Tlve . (47) = Volume Z0 Despite Visser’s experimental in vitro finding that the measured relative blood resistivity change is proportional to hematocrit,21,22 in vivo SV calculation using dZ/dtmax is known to be minimally affected (± 5%) over the full range of physiologically permissible hematocrit levels.44,70 Corroborative in vitro evidence reported by Visser et al.13 clearly shows that, while the measured peak magnitude of ∆Z(t) (i.e. ∆Z(t)max ) is dependent upon hematocrit, its maximal upslope, that is, dZ/dtmax is not (see Fig. 3 in Ref. 13). In as much as the ohmic mean velocity calculation is derived from the square root transformation of dZ/dtmax , and not from an in vivo measurement, impedance-derived SV , using the transformation, is predicted to be unaffected by hematocrit as well. By being mathematically and not physiologically coupled, this is surely the case. Thus, SV = Volume ×

∆Zv (t)max × Tlve , Z0

where ∆Zv (t)max /Z0 is the calculated ohmic equivalent of (vmean /R)max .21

(48)

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Thus, it is suggested that the mean value square root transformation renders, moot, the theoretical justification for the peak value outflow correction extrapolation procedure, and consequently, assumption 9 of Kubicek’s hypothesis. Evidence suggesting that the theory behind the maximal forward systolic upslope extrapolation of ∆Z(t) is flawed resides in the sweeping assumption that little outflow exists during the rapid ejection phase of systole. Johnson et al.71 showed that, by the end of the rapid ejection phase, or first third of systole, about 45% of total SV is already ejected in normal individuals. Thus, the extrapolation procedure would have to compensate for the roughly 50% of SV ejected before peak flow velocity occurs, plus the 50% of SV ejected in the final two-thirds of systole. The end of the first third of systole is precisely in time where dV /dtmax occurs in normal individuals and other primates,37,58,72 and where Kubicek54 and Kubicek et al.26 believed dZ/dt peaked. As per the discussion in 5.2.3., dZ/dtmax peaks in the very early, inertial phase of systolic ejection (i.e. peak acceleration), where only 5%–10% of SV is ejected.61,73 This, however, is irrelevant, insofar as Kubicek’s hypothesis is concerned, because dZ/dtmax is a priori a mean acceleration variable. Thus, while the outflow theory is flawed, the square wave integration is operationally correct, providing acceleration surrogates of mean flow over a narrow range of values (Fig. 8). Further evidence showing that dZ/dtmax is a mean value equivalent is demonstrated by the observation that, over a full range of Vc and dZ/dtmax /Z0 , both the Kubicek and the Sramek–Bernstein equations yield numeric outputs roughly 50% of those actually predicted for peak flow.72 Therefore, the direct proportionality of dV /dtmax and dZ/dtmax expressed in Eq. (14), the equivalencies expressed in Eqs. (16–22) justifying the plethysmographic hypothesis, and assumptions 7 and 8 of Kubicek’s method appear invalid. Interestingly, as compared to standard reference methods, the correlation of impedance derived “SV ”, using mean acceleration equivalents, is considered good (r2 = 0.67, range = 0.52–0.81 and, r = 0.82, range = 0.70–0.90).7 The closeness of association can be fully explained by the observation that both mean and peak acceleration, per se, are both highly correlated with peak velocity and SV (r = 0.75– 0.775),74,75 as well as with the systolic velocity integral (r = 0.75).76 Undoubtedly, the close association between acceleration with both SV and stroke distance, across a full range of values, has fortuitously kept the technology alive since its clinical introduction forty years ago.23 Thus, in conjunction with a Vc of appropriate magnitude, the “extrapolation procedure” mathematically forces the ohmic analog of PARABA through systole (Tlve ) producing a sometimes close facsimile of SV . Therefore, the well known overestimations of SV reported in both young normal individuals,77 and animals,78–80 using both the Kubicek and Sramek–Bernstein SV equations, can be fully explained on the basis of PARABA equivalents, proportionately disparate with their reduced average blood velocity counterparts. By inspection of Fig. 8, it becomes clear that there will be overestimations of mean velocity at higher levels of PARABA and underestimations at lower levels of PARABA. Indeed, there is evidence based data provided by Yamakoshi et al.35 showing that

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impedance-derived SV overestimates its EMF counterpart in healthy canines by as much as 70%, and correspondingly underestimates its EMF counterpart by as much as 25% when myocardial failure is induced. These findings are consistent with those of Ehlert and Schmidt,66 who reported that a linear relationship could not be found between impedance and EMF-derived SV over a wide range of hemodynamic perturbations. As a result of this discussion, it seems improbable that assumptions 7 and 8 of Kubicek’s hypothesis can be supported. Evidence suggesting that dZ/dtmax varies inversely with aortic valve CSA, or radius, as a function of PARABA, is inferentially demonstrated through the work of Sageman.81 He clearly showed that an inversely proportional and highly negatively correlated (r = −0.75) relationship exists between dZ/dtmax /Z0 and body mass in otherwise healthy individuals. The literature also shows that aortic valve CSA is highly and directly correlated with both body mass (kg) and body surface area (BSA).82 This would corroborate the experimental results and hypotheses of Visser,21 and explain the findings of Sageman,81 showing progressively diminished values of dZ/dtmax /Z0 as weight, aortic valve radius, and CSA increase. This is in stark contrast to Doppler or electromagnetically derived peak velocities and systolic velocity integrals, which are totally independent of body mass.83 Thus, peak Newtonian velocities of equal magnitude, measured between age-matched normal individuals of different body mass and aortic valve CSAs, will produce correspondingly disparate values of dZ/dtmax /Z0 . However, while there is no clear direct proportionality between hemodynamically-based, and impedance-derived systolic velocity inte√ grals within or between individuals (i.e. vmean × Tlve versus dZ/dtmax /Z0 × Tlve ), a linear equivalence does exist between Newtonian-based mean flow and impedancederived mean flow. Specifically,  dZ(t)/dtmax . (49) Area × vmean = Volume × Z0 Taken to its logical conclusion, using the relationships established in Eqs. (42–47), the right hand side of Eq. (49) can also be equated with the hemodynamic domain through a convoluted abstraction of the equation of continuity:

 n  dv/dt mean Qmean = πR3 R max  2  t1 πr dZ/dtmax = v(t)dt = Vc , (50) t1 − t0 t0 Z0 where exponent n has been experimentally determined to be between 1.20– 1.25 (authors’ unpublished results). Because of the absolute dependency of (dZ/dtmax )/Z0 upon PARABA, as established earlier, this means that, for any given value of mean acceleration, the magnitude of dZ/dtmax is related to, and explicitly dependent on aortic root CSA by its dependency on R. Since aortic root CSA is a function of body mass,82 age,33 and gender, dZ/dtmax will also vary accordingly

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with these parameters.84 Thus, the magnitude of dZ/dtmax is multi-factorial and not explicitly dependent on the respective levels of myocardial contractility or Z0 alone.35,68,85 Specifically, as a first-order approximation, the input variables determining the magnitude of dZ/dtmax are given as follows:

 n 2  πR3 dZ(t) dv/dtmean = Z0 (Ω · s−2 ). (51) dtmax Vc R max 7. Stroke Volume Equation Implementing the Square Root Acceleration Step-down Transformation 7.1. The volume conductor As a consequence of the square root acceleration step-down transformation of (dZ/dtmax )/Z0 , the resultant dimensionless values are numerically greater by several magnitudes (Fig. 8). Thus, the volume conductors of the Kubicek and Sramek– Bernstein methods are approximately four to five magnitudes larger than required, and therefore, as utilized with the square root transformation, inappropriate. As an alternative to present approaches, it is suggested that the thoracic Vc , or VEP T would then best approximate the intra-thoracic blood volume (ITBV). The rationale for the new hypothesis derives from studies correlating both body mass and ITBV with SV. It is well known that TBV is a major input for venous return and ITBV, which, in turn, are major input variables for ventricular preload, and thus, SV.86 Furthermore, SV is highly correlated with (r = 0.97), and virtually a linear allometric function of body mass (0.66 W(kg)1.05 ).87 It is also known that body mass is linearly and highly correlated (r2 = 0.997) with TBV by the allometric relationship, 73 W(kg)1.02 .88 The allometric relationships linking body mass to both SV and TBV89 extend to ITBV and global end-diastolic volume (GEDV).90,91 These indices are now recognized as the measurements most closely correlated with left ventricular preload, and, as such, the major determinants of SV, CO, and changes thereof.92–96 As reported by Feldschuh and Enson,97 indexing TBV for body height shows poor correlation (r = 0.35–0.37) as compared to BSA (r = 0.80–0.87), weight (0.77–0.88), and deviation from ideal body weight (r = 0.90). The relatively poor correlation of overall body height with indexed values of blood volume is concordant with its poor correlation with SV.98 Indeed, allometrically, SV is most highly correlated with body mass, and fat-free body mass by univariate statistical analysis.98 Thus, the high correlation of body mass with ITBV, and both body mass and ITBV with SV, suggests that the most appropriate input variable for the Vc would be body mass (kg), allometrically related to the ITBV. Therefore, in the otherwise normal individual, and as a first order approximation, Vc is suggested to be equivalent in magnitude to the ITBV (VITBV ), which approximates 17–20 mL/kg,91 or about 25% of TBV. For purposes of SV determination, VITBV is approximated as 16 W(kg)1.02 (or 0.25 × TBV). Thus, in otherwise healthy individuals of ideal body mass, the Vc in impedance cardiography is linked with the aortic radius and

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CSA by the relationship established in Eq. (50). In defense of the new hypothesis, an alternative explanation supporting the biophysical appropriateness of the thoracic length or height-based volume conductors has never been proposed, much less proven.99 Thus, in retrospect, given the range of magnitude of dZ/dtmax /Z0 , it is suggested that the empirically derived “best-fit” volume conductors of both the Kubicek and Sramek–Bernstein methods were unknowing searches for close approximations of TBV. Supporting this contention, consider the following argument: For the Kubicek equation, using 17% of overall height as a surrogate45 for measured L in healthy young persons of 150–180 cm, where ρ = 135 Ω cm, and Z0 = 24–30 Ω, produces volume conductors in the range of 3,500–5,500 mL. Correspondingly, indexing blood volume at 65–75 mL/kg in humans of ideal body mass index (BMI(ideal) = 22 kg/m2 ) from 50–80 kg, produces TBV closely approximating the volume conductors constructed above. As concerns the appropriate value for ρ, Quail et al.44 rearranged the Kubicek equation, solved for ρ as the dependent variable, and measured SV by EMF. By means of normovolemic exchange transfusion, they showed that over a wide range of hematocrit from 26%–66%, the value of ρ remained virtually constant about a mean of approximately 135 Ω cm. Accordingly, they changed the designation of the hematocrit dependent specific resistance of blood ρb in the Kubicek equation to the assumed constant specific resistance of the thorax ρT . These findings were seminal in generating the Sramek equation,24 wherein he substituted ZA/L for ρT in the Kubicek equation. Meijer et al.,48 however, measured the transthoracic specific impedance and found the mean value for ρ approximated 330 Ω cm ±40 Ω cm, which is about 2.5 times greater than the value predicted by Quail et al.44 They also found that ρT varied with age, the extremes being 290 Ω cm at age ten to 430 Ω cm at age fifty, probably reflecting the increased ratio of thoracic gas volume to ITBV. At variance with Quail et al.,44 they also demonstrated that ρT was non-constant intra-individually and varied directionally with Z0 . As predicted by Eq. (7), and solving for ρ, this would be the expected result with varying levels of hydration, intravascular blood volume and interstitial extra-vascular lung water (EVLW). Therefore, as suggested, it is not ρT that is constant, but rather, the monotonic constancy of the ratio ρT /Z0 . Thus, the nonconstant magnitude of Kubicek’s volume conductor, based on changing values of Z0 and a constant value for ρT , appears somewhat untenable. Sramek’s conductor is somewhat more compatible with the findings of Meijer et al.,48 because it is operationally more consistent with the concept that ρT varies directly with Z0 ; that is, VEP T is constant over a full range of ρT and Z0 . The concept of the ITBV being the appropriate Vc is consistent with Kubicek assumption 4, stipulating that all electrical conduction flow through the blood resistance. As implicitly mandated and proposed herein, the ITBV is, in fact, the equivalent physical embodiment of the ohmic blood resistance Rb . Furthermore, except for errors imposed by its allometrically defined magnitude, ITBV is anatomically and biophysically assumption-free, inherently unambiguous in meaning, and intuitively understood. By comparison, existing volume conductors are modeled as

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simple geometric abstractions, which are firmly rooted, uniquely, in basic electrical theory.99 By virtue of their “best-fit” mathematical construction, they bare little relevance to, and have virtually no association with other commonly acknowledged physiologic, anatomic or hemodynamic parameters. 7.2. Index of transthoracic aberrant conduction ζ (zeta): Genesis of the three-compartment parallel conduction model Analysis of the appropriate volume conductor does not address the issue of changing thoracic resistivity in pathologic states, especially those characterized by excess intra-thoracic liquids. Using the Sramek–Bernstein method, it is well known that SV is systematically underestimated in sepsis and other conditions characterized by increased thoracic liquids, such as pulmonary edema.49,100–102 Because quasistatic Z0 appears as a squared function in the Kubicek equation, with a constant value for ρ (Eqs. (16) and (26)), the underestimation of SV is less marked.102 The underestimation is most likely due to a bypassing or shunting of AC, circumventing the VITBV to the more highly conductive edema fluid.49,50,103 In its extreme form, a virtual short circuit is produced. Since the specific resistance of plasma is about 60–70 Ω cm,104 and that of blood 100–180 Ω cm, within the physiologic range,19,44 it is likely that the AC takes the path of least resistance (i.e. lowest impedance), which is the plasma-like edema fluid. Thus, the magnitude of the Vc increases as a function of an enlarging edema volume Ve , causing pathologic parallel conduction between the blood and edema fluid compartments. In this scenario, Vc = VIT BV + Ve . Thus, in effect, a three-compartment parallel conduction model is generated. According to Ohm’s Law, this produces an appropriate, but pathologic transthoracic voltage drop, and a progressive decrease in AC flow through the dynamic VIT BV as Ve increases. Consequently, a smaller than predicted pulsatile change in voltage occurs, rendering it no longer proportional to its mechanical hemodynamic counterpart.35 If Eqs. (1) and (2) represent normal conduction, wherein all current flows through the static Zb and dynamic blood resistance ∆Zb (t), then the following interpretation of Ohm’s Law characterizes the discussion above, if Z is defined in terms of Eq. (8).      ρb L 2  ρe L 2  ∆ρb (t)L2 ρb L 2 ρt L 2     = U0 ∆U (t) , + (52) I·  Vt  Vb  Ve Vb ∆Vb (t) (1) (2) (3) (4) where subscripts t, b, and e represent thorax, blood, and edema, respectively (Fig. 2). Compartment 1 represents the non-conductive static thoracic tissue impedance Zt ; compartment 2, the highly conductive, static component of the blood resistance, or impedance Zb ; compartment 3, the most highly conductive, variable magnitude static edema impedance Ze ; and compartment 4, the pulsatile, dynamic components of the blood resistance, comprising ∆ρb (t) and ∆Vb (t). It should be noted that compartments 1 through 3 represent the quasi-static base impedance Z0 . Clearly, as compartment 3 increases in volume, expanding within a fixed volume thoracic

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cage, the total transthoracic impedance Z(t) decreases as a function of displacement, compression and reduction of the thoracic gas volume. The result of this is an appropriate, but pathologic drop in the measured static transthoracic voltage U0 . Moreover, since a progressively increasing proportion of AC flows through compartment 3, drawing current at the expense of compartments 2 and 4, an inappropriate electrical damping of the peak magnitude of ∆Z(t) (∆Z(t)max ), and reduction of its maximum systolic upslope dZ/dtmax ensue.35,50 Specifically, this sequence of events causes spuriously low values of dZ/dtmax /Z0 , which, by absolute magnitude, no longer parallel PARABA or proportional changes thereof.49,50 The diminishing magnitude of dZ/dtmax , as the volume conductor increases in magnitude, PARABA remaining constant, is predicted by Eq. (51). Thus, the equality established in Eq. (50) no longer exists, the final result being underestimation of ICG-derived SV . In conjunction with reduced levels of myocardial contractility, as in congestive cardiomyopathy, it is suggested that this is the major cause of underestimation of SV and CO in sepsis and other pulmonary edematous states using present impedance techniques. While it is beyond the scope of this chapter to fully elucidate the bioelectric mechanisms operative in states of excess thoracic liquids, it has been found that the magnitude of the volume conductor must be corrected for by a dimensionless AC shunt index. This parameter is to be known as the Index of Transthoracic Aberrant Conduction, ζ (zeta), and becomes operative when pathologic parallel conduction evolves. ζ is mathematically equivalent to: ζ=

Zc2 − Z0 · Zc + K 2Zc2 + Z02 − 3Zc · Z0 + K

(Dimensionless),

(53)

where Zc = critical level of base impedance = 20 Ω, Z0 = measured transthoracic base impedance ≤Zc , and K = a trivial constant → 0. It has been empirically established that pulmonary edema with pathologic conduction supervenes at critical values of Z0 (Zc ) between 20 Ω ± 5 Ω.49,103 The precise value for Zc depends on premorbid Z0 , age, gender, body habitus, and the frequency of the applied AC.105 Until being further quantifiable, Zc is ascribed a nominal default value of 20 Ω at 2.5 mA (rms) and 70 kHz, using a tetra-polar spotelectrode array. Operationally, ζ is a dimensionless index, mirroring the magnitude of the AC shunt, and decreases progressively from a value of unity at levels below Zc (i.e. If Z0 < 20 Ω, 0 < ζ < 1.0). Consistent with a cylindrical model of constant length, but expanding CSA, ζ is implemented as an exponential function, reflecting a change in edema volume. At all levels of Z0 ≥ 20 Ω, ζ = 1.0. As the denominator of the VIT BV , diminishing values of ζ produce progressively larger values of Vc , thus accounting for AC lost from normal conduction through the VITBV to pathologic conduction through the Ve . Thus, a new SV equation is proposed51,107 :  VITBV SV = ζ2

dZ(t)/dtmax Tlve , Z0

(54)

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where: (1) VITBV = K · 16 W 1.02 (2) ζ = index of transthoracic aberrant conduction, (0 < ζ ≤ 1.0) ζ = (Zc2 − Zc Z0 + K)/(2Zc2 + Z02 − 3Zc Z0 + K), where Zc is the critical level of Z0 , nominally ascribed a default value = 20 Ω, and K is a trivial constant → 0 (3) dZ/dtmax = the peak rate of change of the transthoracic cardiogenic impedance pulse variation, (Ω · s−2 ) (4) Z0 = the transthoracic quasi-static base impedance (Ω) (5) Tlve = left ventricular ejection time (s)

8. Proof of Hypothesis: The New Equation versus the Kubicek, Sramek, and Sramek–Bernstein Equations using Thermodilution as the Standard Reference Technique To show the superior behavior of Eq. (54) over the Kubicek, Sramek, and Sramek– Bernstein equations, 106 adult cardiac surgery patients were studied within 24 hours of cardiopulmonary bypass.51 Thermodilution cardiac output (TDCO) was

0

0

S (mL/min) 4000 8000

Sramek Rs = 0.51 (P 0. For u = 0 the diffusion without drift model must be used. As for the diffusion without drift case, (33) is substituted into (26), which is now expressed as  t l(µ, t − τ )p(0, τ )dτ. (34) p(µ, t) = 0

Once again, (34) can be solved in the Laplace domain. The resulting L(µ, s) represents the FPT model in the Laplace domain when drift is included. L(µ, s) is expressed as given in (35): L(µ, s) = e−

µ(1−



1−4sK 2 ) 2K 2

.

(35)

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Taking the inverse Laplace transform, (35) is expressed in the time domain as (µ−t)2 µ l(µ, t) = √ e− 4K 2 t . (36) 4πK 2 t3 l(µ, t) represents the FPT model with drift in the time domain. Since L(µ, 0) = 1, the integral of l(µ, t) in the interval (0, ∞) equals 1 and l(µ, t) may be considered as a statistical distribution. With the substitution λ = (u2 µ)/(2D), it follows that K 2 = µ/2λ and (36) is expressed as given in (24), which is the FPT model definition as reported by Bogaard, Reth, and Wise.6,7,17 The first moment of the model, as proven by Sheppard, is equal to µ.10

4. Cardiovascular Quantification by Indicator Dilution Principles The application of the indicator dilution principles for the measurement of several cardiovascular parameters is explored. The link with specific medical technologies is limited because it is the object of the next section. 4.1. Cardiac output Several different techniques are clinically used for Cardiac Output (CO) measurements. They are based on different principles with different reproducibility and accuracy.24 In general, the CO is defined as the volume of blood that is ejected by the left ventricle (LV) into the aorta in one minute, and it is expressed in liters per minute. Due to the cyclic contraction-expansion of the ventricles, the CO, as well as the blood pressure, is a periodic function of time. The first harmonic of these functions is the Pulse Rate (PR), which represents the number of ventricular systoles (or diastoles) per minute. Despite the fact that the CO is a periodic function, it is usually represented by a single value, which is a measure of the average flow in liters per minute. The CO is related to the ventricular volume variations during one cardiac cycle. If Ved is the end diastolic volume (maximum volume) and Ves is the end systolic volume (minimum volume), then the Stroke Volume (SV), which is the volume of blood ejected from the ventricle to the artery in one cardiac cycle, equals the difference Ved − Ves . Therefore, the CO can be defined as given in (37): CO = SV · PR.

(37)

The CO definition in (37) is correct only if no regurgitation (valve insufficiency) is present, otherwise SV = Ved − Ves . The CO for an average size adult (70 kg) at rest is about 5 L/min. During severe exercise it can increase to over 30 L/min. Miguel Indurain (who won the Tour de France in five successive years) had a resting PR of 28 beats per minute and could increase his CO to 50 L/min and his PR to 220 beats per minute. The CO is often divided by the Body Surface Area (BSA) to normalize the value with respect to the size of the subject. In this case, it is referred to as Cardiac Index

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(CI). Several formulas can be adopted to determine the BSA based on the weight and the height of the patient. One of the most common is the Dubois and Dubois formula (1917), which estimates the BSA as BSA(cm2 ) = 71.84 · weight0.425 · height0.725 ,

(38)

where weight and height are given in kg and cm, respectively. The indicator dilution theory is based on the following concept: If the concentration of an indicator that is uniformly dispersed in an unknown volume V is determined, and the volume of the indicator (dose) is known, then the unknown volume can be determined too. Since Φ(t) = dV (t)/dt (Φ(t) and V (t) are respectively the instantaneous flow and the volume of the carrier) and C(t) = dm/dV (m and C(t) are respectively the mass of the tracer and its concentration at time t), the following differential equation can be derived: Φ(t) =

1 dm dV (t) = . dt C(t) dt

(39)

A typical approach for CO measurements makes use of a rapid injection of an indicator dose. This permits to relax the constraints about the nature of the indicator, since only a small bolus is injected. However, also methods based on continuous indicator infusion are used (see Secs. 5.1.1 and 5.1.2). In rapid injection methods, C(t) in (39) is not constant. In practice, as shown in Fig. 6, the indicator is rapidly injected into a fluid dynamic system where a carrier fluid (in our specific case blood) is flowing, and the indicator concentration C(t) is measured versus time for the registration of an IDC. The IDC contains all the information to estimate the flow, whose value is derived from (39) by an integration over time as shown in (40). The flow Φ is assumed to be constant, so that it can be moved out of the integration. The resulting formula is the Stewart-Hamilton equation,  ∞  ∞  ∞ dm m dt = m =⇒ Φ = ∞ , ΦC(t)dt = Φ C(t)dt = dt C(t)dt 0 0 0 0 (40) which provides the measurement of the mean flow Φ.25,26 Therefore, the injection and subsequent detection of an indicator allows the measurement of the mean flow. The calculation of the integral in (40) is not trivial. Since the circulatory system is a closed system, the recirculation of the indicator produces rises of the concentration that mask the tail of the IDC related to the fist passage of the contrast (see Fig. 7). In addition, the IDC is often very noisy. Therefore, the estimation of the integral of the first passage IDC in (40) requires the employment of one of the models discussed in Sec 3. 4.2. Mean transit time and blood volumes Blood volume measurements provide valuable information on the circulatory system functionality. In particular, the Pulmonary Blood Volume (PBV, blood volume

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Fig. 6. Measurement of the indicator concentration versus time (IDC) in an infinite tube model. The indicator is injected at distance 0 and detected at distance x0 from the injection site.

between the pulmonary artery and the LA), the Central Blood Volume (CBV, blood volume between the pulmonary artery and the LV) and the Intra-Thoracic Blood Volume (ITBV, blood volume between the right atrium and the LV) are important parameters in anaesthesiology, intensive care, and cardiology to evaluate the cardiac preload and the symmetry of the cardiac efficiency. For instance, LV EF and SV are closely related to PBV and CBV.27–29 More in general, asymmetries in the cardiac efficiency due to LV malfunction or heart failure lead to increased intrapulmonary pressures (cardiac preload) and, therefore, volume.27,29–33 The persistency of such condition results in the increase of pulmonary interstitial fluid and, eventually, in the formation of pulmonary edema, e.g. extravasations of fluid into the alveoli.34 The interstitial fluid is often referred to as Extra-Vascular Lung Water (EVLW). Since cardiogenic pulmonary edema is usually combined with increased cardiac preload, invasive pulmonary pressure measurements can serve as useful diagnostic means. Nevertheless, non-cardiogenic pulmonary edema can also occur, due for instance to infections, kidney failure,

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Fig. 7. The dots represent a measured IDC while the continuous line shows the real (theoretical) first passage IDC. The second rise due to the recirculation and the noise due to the measurement system are evident.

Fig. 8. Scheme of the blood volume measurements: Pulmonary Blood Volume (PBV), Central Blood Volume (CBV), and Intra-Thoracic Blood Volume (ITBV). The volumes of the compartments that are included in the measurement are filled with gray color.

and injuries. The fluid accumulated in the alveoli may become a barrier to normal oxygen exchange, resulting in dangerous hypoxia. A scheme of the standard blood volume measurements is shown in Fig. 8. In order to make the volumes independent on the size of the subject, they are usually normalized (indexed) as already explained for the CI (Sec. 4.1). PBV measurements are based on trans-pulmonary indicator dilution techniques, which make use of the

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injection of an indicator bolus before the lungs and its subsequent detection after the lungs.27,28,35 The measured IDC is analyzed for the estimation of the MTT that the indicator takes to cover the distance between the injection site (pulmonary artery or right ventricle) and the detection site (ascending aorta or LV). The estimated MTT is then multiplied times the CO as given in (41) for the measurement of the blood volume (V ) between injection and detection site: V = MTT · CO.

(41)

Nowadays, trans-pulmonary indicator dilution techniques are very invasive due to the need for a double catheterization. In fact, a catheter for thermodilution (or dye dilution) must be inserted through the femoral artery up to the aorta, where the IDC is measured. Moreover, since the indicator must be injected into a central vein, the insertion of a second catheter is necessary to reach the injection site. Common applications make use of a Pulmonary Artery Catheter (PAC) in order to inject the contrast in the pulmonary artery and measure the CBV.36 The measurement of the MTT between the first and the second passage of the indicator in the same site allows the assessment of the Total Circulating Blood Volume (TCBV). In this context, it is important to make the distinction between intravascular and extravascular indicators. Quantification of EVLW is possible if an intravascular indicator (e.g. a dye bolus) and an extravascular indicator (e.g. a cold saline bolus) are injected. This technique is referred to as double indicator transpulmonary dilution.37,38 Two IDCs, one for each indicator, are then registered and analyzed to assess the MTT of the indicator between the injection and detection site. The intravascular PBV can be estimated from the MTT of the intravascular indicator, while the total (intravascular and extravascular) fluid can be estimated by the MTT of the extravascular indicator, under the assumption of rapid extravascular diffusion. The difference between the two estimated MTTs permits the quantification of the EVLW. Based on a number of assumptions, the estimation of the EVLW can also be derived by the injection of a single extravascular indicator.38,39 The time constant of the exponential decay of the IDC is adopted as the estimate for the MTT of the largest compartment along the transpulmonary circulation, i.e. the total PBV.40 The difference between the ITBV and the total PBV provides the global enddiastolic volume (GEDV). In Ref. 38 the GEDV is reported to be linearly correlated to the intravascular PBV (measured by an intravascular dye). In particular, PBV = 1.25 · GEDV.

(42)

Therefore, the EVLW can be derived as the difference between the total PBV, derived from the time constant of the IDC exponential decay, and the intravascular PBV, measured by (42). Here below a more accurate derivation of the volume estimates based on the dilution of an indicator bolus is provided. The obtained general results are discussed with respect to the FPT and the LDRW models.

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The infinite tube model in Fig. 6 is adopted to derive a formula for the volume measurement. A carrier fluid flows through the tube with a steady flow Φ. An indicator bolus is injected (fast injection) at time t = 0 into the tube. The volume to measure is defined as the tube segment between the indicator injection and detection sections. We define f (t)dt as the fraction of injected indicator that leaves the tube segment in the time interval [t, t + dt]. It is assumed that the indicator may not pass more than once through the detection section (FPT model hypothesis). Due to the single passage hypothesis, the fraction of leaving particles corresponds to the fraction of particles that appear at the detection section. Therefore, f (t) equals the normalized indicator concentration that is measured at the detection site and it is given as C(t) . f (t) = ∞ C(τ )dτ 0

(43)

The fraction of indicator that has left the segment by time t is determined by F (t) as  t f (τ )dτ. (44) F (t) = 0

The volume of fluid that enters the tube segment in the time interval [0, dt] is Φ · dt and the fraction that leaves the segment by time t is Φ · dt · F (t). Therefore, the volume of fluid that enters and leaves the tube segment in the time interval [0, t] is given as  t F (τ )dτ. (45) Φ 0

The difference between the entering and the leaving fluid volume in the time interval [0, t] is then given as  t F (τ )dτ. (46) Φt − Φ 0

Therefore, (46) expresses the volume of fluid that enters the segment in the time interval [0, t] and is still in the segment at time t. For time t → ∞ all the fluid in the segment is replaced by fluid that has entered for t ≥ 0. Therefore, the total volume V of the segment is given as    t F (τ )dτ . (47) V = lim Φ t − t→∞

0 b

(47) can also be formulated as

 t  t V = lim Φ t − [τ F (τ )]0 + τ f (τ )dτ = Φ t→∞

0



τ f (τ )dτ.

(48)

0

R R R b The integration per parts of t τ f (τ )dτ allows replacing t F (τ )dτ in (47). In fact, t τ f (τ )dτ = 0 0 0 Rt Rt t 0 τ dF (τ ) = [τ F (τ )]0 − 0 F (τ )dτ .

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Due to the definition of f (t) and the FPT hypothesis, the right term of (48) represents the multiplication of the flow Φ times the MTT of the indicator, i.e. the average time that the indicator takes to cover the distance between injection and detection section. Moreover, due to the FPT hypothesis and (43), f (t) may be represented by (24) (except for the coefficient m/Φ), which proves the convergence of the integral in (48). In conclusion, the volume is given as in (41), where the MTT of the indicator, which equals the first moment of the IDC, is given as ∞  ∞ τ C(τ )dτ . (49) τ f (τ )dτ = 0 ∞ MTT = C(τ )dτ 0 0 The definition of the indicator MTT as the first moment of the IDC is appropriate only under single passage hypothesis. In this case, the MTT corresponds to the MRT of the indicator in the defined segment and equals µ in (24).17 In fact, the indicator appearance time at the detection section also corresponds to the disappearance time from the segment. The LDRW model is more general and does not assume a single passage of the indicator. As a consequence, the first moment of the model, which still represents the MRT of the indicator in the tube segment, differs from the MTT.9,16 The MTT, which is defined as the average time that the indicator takes to go from the injection to the detection site, is by definition equal to µ in (21). In fact, in the LDRW model µ equals the time that elapses to cover the distance between the injection and detection site at the carrier fluid velocity, i.e. the MTT of the indicator. Instead, the first moment of the model, which corresponds to the MRT, equals µ(1+1/λ) (see also (23)).23 Therefore, the MRT exceeds the MTT by the term µ/λ=2D/u2, i.e. twice the ratio between indicator diffusion constant D and squared velocity of the carrier fluid u2 . Large diffusion constants lead to increased numbers of indicator particle passages through the detection site and, therefore, to large differences between MRT and MTT. In conclusion, when either the LDRW model or the FPT model is adopted to interpolate the IDC, (41) corresponds to V = Φ · µ.

(50)

4.3. Ejection fraction The measurement of the Ejection Fraction (EF) is a common clinical practice to evaluate the cardiac condition. In particular, it is a measure of the efficiency of the myocardial contraction. If the end-diastolic (Ved ) and the end-systolic (Ves ) ventricular volumes are measured, the percent EF is defined as given in (51): EF% =

Ved − Ves · 100. Ved

(51)

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The EF can be estimated by any clinical imaging technique that allows a geometric estimation of Ved and Ves , such as magnetic resonance (MR), ultrasound, X-ray, and nuclear imaging techniques.24,41–44 Three-dimensional techniques, such as MR or three-dimensional ultrasound imaging, are particularly attractive for these applications. The acquired images are analyzed by manual or automatic segmentation. Due to the complex geometry of the right ventricle (RV), the estimation of the RV EF is not commonly performed in clinical practice. In general, geometric EF measurements are time consuming due to the need for segmentation, which in clinical practice is usually performed manually. In fact, the reliability of automatic border detection algorithms is sometimes very limited and cardiologists prefer a manual delineation of the endocardial contours. As a consequence, the EF assessment not only slows down the clinical practice, but also requires the employment of specialized personnel, such as radiologists or cardiologists. Also magnetic resonance imaging (MRI), despite the better image quality, requires a long procedure both for patient scanning and for data analysis. Moreover, patients that are claustrophobic or have an implanted pace-maker cannot undergo an MRI scan. The EF assessment can be also performed by means of indicator dilution techniques. A cold saline (thermodilution) or a dye (dye dilution) bolus is injected for the measurement. The injected indicator bolus is detected either in the LV or in the ascending aorta. A mathematical interpretation of the measured IDC allows the EF assessment. A ventricle can be modeled as a mono-compartment system, whose volume changes as a quasi-periodic function of time. This can be represented by a cylinderpiston system as shown in Fig. 9. The system is filled with an incompressible fluid. Two valves are used for the fluid input and output and are driven by pressure variations. If an indicator bolus is rapidly injected within a diastolic phase and the mixing is complete, then the contrast concentration at the nth end-diastolic phase is given by Cn and it equals the indicator mass m in the ventricle divided by Ved . During the following systole, part of the contrast mass (∆m) is ejected out

Fig. 9. Mono-compartment cylinder-piston LV model. An input and an output valve are included and represent the mitral and aortic valve, respectively.

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of the cylinder. The concentration Cn+1 at the subsequent end-diastole is given as in (52):   m − ∆m Ved − Ves . (52) = Cn · 1 − Cn+1 = Ved Ved Combining (51) and (52), the percent EF can be expressed in terms of Cn and Cn+1 as given in (53):   Cn+1 · 100. (53) EF% = 1 − Cn The sampling timing of Cn and Cn+1 is usually controlled by an electrocardiographic (ECG) trigger, so that the sampling rate is based on the cardiac electrical activity. In order to ensure that the indicator recirculation (see Fig. 7) does not influence the measurement, the samples Cn and Cn+1 should be measured before the recirculation time. The EF estimate in (53) only considers the fluid that is ejected through the output valve (aortic valve in the LV). If the input valve is insufficient, i.e. it leaks, part of the contrast is ejected back through the input valve (mitral valve in the LV). However, this fraction of contrast comes again into the ventricle during the subsequent diastole and, therefore, does not contribute to ∆m. As a consequence, the EF definition in (53) is better referred to as Forward EF (FEF). A similar reasoning may be also applied to aortic valve insufficiency. If the SNR of the measured IDC is low, the definition of the right samples to estimate Cn and Cn+1 is difficult, resulting in very inaccurate measurements. A better approach makes use of an IDC model interpolation. The LV is well approximated by a simple mono-compartment model, whose impulse response equals an exponential function (see Sec. 3). If C0 is the concentration after a sudden indicator injection in the compartment at time t = 0, then the IDC C(t) is represented as C(t) = C0 · e

−t τ

,

(54)

where τ is the time constant.3 The exponential function in (54) can be used to fit the IDC as measured from the model that is shown in Fig. 9. The ripple due to the pulsatile flow does not disturb the measurement because it is averaged over a large number of cycles. Once the exponential model is fitted to the IDC down-slope, the FEF is measured by (53) as given in (55), where ∆t is the cardiac period: −(t+∆t)

FEF = 1 −

e τ Cn+1 =1− −t Cn eτ

=1−e

−∆t τ

.

(55)

The FEF measurement by (55) is valid only when the contrast bolus is rapidly injected into the ventricle within a diastolic phase, which is the reason why a correct FEF estimation requires a ventricular injection (Holt method).45 In fact, the measurement must be performed in the LV (or ascending aorta) during contrast

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wash-out with no incoming contrast.3 Therefore, catheterization is needed and the clinical application of the method is very limited due to its invasiveness. An invasiveness reduction is accomplished by use of radio-opaque contrast or radionuclides for X-ray or nuclear angiography (see Secs. 5.2.2 and 5.2.3), which allow a non-invasive detection of the indicator. In fact, the FEF can be assessed by videodensitometry of cine-loops.46 –48 However, despite a non-invasive contrast detection, contrast injection still needs cardiac catheterization (invasiveness issue) and, as discussed in Secs 5.2.2 and 5.2.3, the use of X-rays or radionuclides is not recommended in several cases. In healthy people the EF is usually bounded between 50% and 85%. However, in heart failure patients the EF can even drop down to 10%. In order to keep a sufficient value of CO, the decrease of EF is compensated by an increase of heart rate and both Ved and Ves (dilated ventricle), so that the SV can still guarantee a sufficient CO. 4.4. Myocardial perfusion The perfusion of the myocardium is an important indicator for the presence of coronary artery diseases (CAD) and ischemia. Ischemia is an inadequate blood supply to the myocardium, usually due to stenoses or blocked arteries (clots), which often results in hypokinesis (reduction of wall motion and thickening) and asynchrony of the cardiac wall motion. The tests to evaluate the myocardial perfusion are usually qualitative, and an absolute quantification of the perfusion defects is not feasible. By X-ray angiography the perfusion of single arteries can be analyzed and evaluated. This technique, however, requires the insertion of a catheter through a peripheral artery (e.g. the femoral artery) up to the LV coronary arteries for the indicator injection. The level of perfusion in the myocardium can be also evaluated by looking at the contrast enhancement of the myocardial region by nuclear imaging and, more recently, also by contrast ultrasound and MR imaging. These techniques, however, are limited in time (MRI) or spatial (ultrasound and nuclear imaging) resolution with respect to X-ray angiography. A common test that is performed to have a better assessment of the blood supply to the myocardium is referred to as stress test. A stress test is performed during or immediately after exercise. An indicator is then injected and detected under stress condition. The stress test is commonly performed by means of ultrasound or nuclear imaging investigations.49 In patients that are unable to exercise, the stress test is performed by the injection of vasodilator drugs, such as adenosine or dipyridamole, which increase the level of perfusion. Other drugs that induce heart rate increase, such as dobutamine, may also be used. However, these drugs are preferably adopted for wall motion analysis. Recently, also semi-quantitative information on myocardial perfusion can be obtained. The myocardium is considered as a mono-compartment dilution system whose input is represented by the indicator concentration in the ascending aorta. If

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perfusion intensity curves are measured by imaging techniques in several locations in the myocardium, then the mono-compartment system between the aorta and the measured output curve can be identified. Usually the input function can be assumed to be well represented by a step function C0 u1 (t − t0 ), where u1 (t − t0 ) is the unitary step function at time t0 and C0 is the concentration after the step rise. The system response for t ≥ t0 (causal system) is given by the convolution between the step function (C0 u1 (t−t0 )) and the mono-compartment system impulse t response (τ −1 e− τ ) as   t−t0 . (56) C(t) = C0 1 − e− τ The time constant τ of the compartment as well as the steady state signal (concentration C0 ) are then associated to the level of perfusion and flow. An interesting application of this method is the replenishment technique, which can be performed by means of contrast ultrasound measurements as discussed in the end of Sec. 5.2.1. An important parameter that can be estimated by means of the same methods is the coronary flow reserve, i.e. the increase of myocardial blood flow in stress condition. It corresponds to the ability of the cardiovascular system to increase coronary blood flow in response to vasoactive mechanisms. Perfusion techniques in the context of cardiovascular applications have recently extended to novel applications. Of particular interest is the perfusion of vasa vasorum and plaques in the carotid arteries.50 The presence of angiogenesis in plaques might be very interesting to distinguish between stable and unstable plaques.51 A plaque is unstable when the risk that it detaches and causes a stroke (in the brain as well as in the myocardium) is high. Up until now, a qualitative evaluation of plaque perfusion is mainly performed by contrast ultrasound or MRI.52 Semi-quantitative analysis may be expected in the future. For this studies, the use of intravascular ultrasound probes can be very useful.53

5. Indicator Dilution Techniques for Cardiovascular Quantification The most common methods for cardiovascular quantification by means of indicator dilution are discussed. A distinction is made between invasive techniques (Sec. 5.1 and Refs. 54, 55, 56), where the indicator detection requires contact between sensor and indicator, and indicator imaging (minimally invasive) techniques (Sec. 5.2), where the injected contrast is detected by means of non-invasive medical imaging techniques. Particular attention is dedicated to echocardiographic and MR methods (Secs. 5.2.1 and 5.2.4). 5.1. Invasive techniques 5.1.1. Fick method These techniques uses a continuous indicator infusion. Due to the injection of a large amount of contrast, the adopted contrast must be absolutely inert, harmless, and

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Fig. 10.

Scheme of the Fick method for CO measurement.

non-toxic. There are at least two tracers that fulfill these requirements: Oxygen, which is used in the Fick technique (Fick, 1870), and heat, which is used in the continuous thermodilution technique (Fegler, 1954).57 In the Fick method two sites for the indicator concentration measurement (see Fig. 10) are fixed: the first site a is located before the injection point while the second one b is located after it. Thus, assuming constant concentrations of the tracer Ca and Cb , steady flow Φa = Φb = dV /dt, and using (39), (57) can be derived:     d(m −m ) Cb − Ca =

dmb dt dVb dt



dma dt dVa dt

b

=

dt

Φ

a

.

(57)

Since d(mb −ma )/dt represents the tracer injection dm/dt between the sampling points, (57) can be expressed as given in (58), which is the basic equation of the Fick method: dm dt

. (58) Cb − Ca The “trick” of the method is that the injection is naturally made by the lungs, and Ca and Cb are respectively the venous and the arterial concentration of oxygen (O2 ). Since the concentration of O2 is different in the different venous returns, the sampling point a is placed in the pulmonary artery, after the venous blood has been mixed by the RV. The blood samples of the so called “mixed venous blood” are drawn by a catheter inserted through the jugular vein (or the subclavian vein) across the right atrium (RA) and RV up to the pulmonary artery. Then the blood samples are analyzed by a gas analyzer device for the measurement of the O2 concentration. The choice for the arterial sampling site is not critical, since the blood from the lung capillaries is well mixed. An arm or a leg artery is usually used. The measurement of the injected tracer (i.e. the inhaled oxygen) is performed by making the patient breath pure oxygen from a spirometer.54 The exhaled CO2 is absorbed by a soda-lime absorber, so that the oxygen injection rate dm/dt (or consumption) can be directly measured by the net gas-flow. This method had been considered as the standard technique until the thermodilution replaced it. Φ=

5.1.2. Thermodilution The thermodilution technique was first introduced by Fegler in 1954.57 This technique can be performed in a continuous mode, in which the indicator consists of the

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heat diffused through a resistor (warm thermodilution), or in a single mode, in which the indicator consists of a cold saline bolus (cold thermodilution). In both cases a Swan-Ganz catheter is inserted via a central vein (usually the internal jugular or subclavian) through the RA and RV, so that its tip lies in the pulmonary artery.58 This catheter is carried in the correct position by the dragging force of the flowing blood thanks to a doughnut-shaped air-filled balloon on the tip of the catheter. In the warm thermodilution,59 the catheter (described in Fig. 11) includes the circuitry for the thermistor that measures the temperature in the pulmonary artery and the wires for the heating coil (resistor), which lies in the RA. A thermistor is a semiconductor thermometer, which uses the relation between temperature and material resistivity for temperature measurement.54 Like oxygen, heat is non-toxic and naturally cleared. Therefore, it is a perfect indicator to perform a continuous infusion technique. Heat may also be dissipated through the walls of the blood vessels (extravascular indicator) between the injection and the sampling site, therefore, this distance should be as short as possible. Unfortunately, in order to have an adequate mixing this distance should be long. The compromise that is usually adopted consists of the indicator infusion in the RA and its sampling in the pulmonary artery. The warm thermodilution is still based on (58). The term dm/dt is given by the heat derivative q˙ (expressed in watt), and the term Cb − Ca is given by the temperature difference Tb − Ta (expressed in kelvin) times the specific heat of the blood cb (expressed in J · kg −1 · K −1 ) times the density of the blood ρb (expressed in kg · m−3 ). In conclusion, the formulation of (58) for the thermodilution technique is given as Φ=

q˙ . cb ρb (Tb − Ta )

(59)

The mass of the tracer is now expressed by the amount of supplied energy q (in joule) and its concentration is given in J · m−3 by the term cb ρb T . The thermistors are usually placed in a Wheatstone configuration.54 This technique has become very common in the clinical practice and everything is integrated in one single catheter. The advantage of these continuous methods consists in the possibility of a continuous CO monitoring.

Fig. 11.

Swan-Ganz catheter for warm thermodilution. Tb represents the reference temperature.

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Fig. 12. Swan-Ganz catheter for cold thermodilution. The IDC is measured by the temperature fall ∆T .

A different application of (39), i.e. the cold thermodilution, makes use of an injection of a cold indicator bolus, which can be either cold dextrose or saline (NaCl blood-isotonic solution). In this case, the resistor is replaced by a port for the bolus injection as shown in Fig. 12, which is performed by a syringe. The cold solution mixes with blood in the RA and RV before passing into the pulmonary artery, where the temperature fall is sensed by a thermistor on the side of the catheter. The CO is then calculated from the temperature-time curve. With the same interpretation of heat and temperature in (59), (40) can be written as Φ=

cb ρ b

∞ t0

q , −∆T (t)dt

(60)

where q is the total injected heat, ∆T is the temperature fall, and cb and ρb are the same as in (59). This method does not allow continuous CO monitoring. Apart for the PAC, sometimes a transpulmonary thermodilution is performed. In this case, the bolus is injected in the RA or pulmonary artery and detected in the aorta by a second catheter usually inserted through the femoral artery. The transpulmonary thermodilution between RA and aorta can be performed in children to assess the CO when cardiac catheterization is complicated by the small size of the heart. This application is also necessary for the measurement of the ITBV or the CBV based on the methods explained in Sec. 4.2. Cold saline is an extravascular indicator, since the heat is lost (diffused) through the vessel walls. Therefore, these methods also allow the estimation of the EVLW based on the models reported in Sec. 4.2. The same measurement can be performed when an intravascular indicator, such as a dye (following section), is injected and detected together with the cold saline bolus (double indicator method). In fact, the MTT difference between the indicators is related to the EVLW. 5.1.3. Dye dilution Also a colored dye such as indocyanine green,c usually referred to as cardiogreen, meets all the necessary requirements of a “good” indicator: It is inert, non-toxic, measurable, and even economic. Using the principle of absorption photometry, the c Also

other dye indicators are used, such as Evans Blue (absorption peak at 640 nm and 50% clearance in 5 days) and Coomassie Blue (absorption peak around 590 nm and 50% clearance in 15–20 minutes).

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concentration of cardiogreen, which is usually injected into the pulmonary artery, can be detected by the light absorption peak at the wave-length of 805 nm. In the past, blood samples had to be drawn by a catheter placed in the femoral or brachial artery and analyzed by an external photometry device. Nowadays, the use of optical fibers allows in situ measurements. About 50% of the dye is cleared by the kidneys in the first 10 minutes, so that repeated measurements are possible too. Once the system is calibrated, meaning that the peak absorption is related to the concentration C(t) of the dye, the flow is directly given by (40).

5.1.4. Lithium dilution Lithium is a fluid that can be injected and detected by a lithium-selective electrode in a flow-through cell.60–63 Two Ag-AgCl electrodes measure the potential across a lithium selective membrane. According to the Nernst equation (more details are reported in Ref. 54), the electric potential E (volt) across a membrane is given as E=

RT ln(Cext ) , nF ln(Cint )

(61)

where Cext and Cint are the external and the internal (with respect to the membrane) lithium activities, which correspond to the lithium ionic concentrations, R is the gas constant (8.314 J · mol−1 · K−1 ), T is the absolute temperature (kelvin), F is the Faraday constant (96485 C · mol−1 ), and n is the valence of the ions, which for lithium ions (Li+ ) is 1. As a result, the transducer measures a voltage that is logarithmically related to the lithium concentration. The sensor is connected to a three-way tap on the arterial line and a small peristaltic pump draws blood with a flow of few mL per minute. After calibration, the lithium concentration is determined and a lithium IDC generated. The time integral of the measured IDC is used as given in (40) for the CO assessment. Moreover, based on the IDC CO measurement and the establishment of the pressure-volume relation (compliance), also a continuous CO monitoring can be performed.

5.2. Indicator imaging techniques for cardiovascular quantification 5.2.1. Echocardiography Ultrasounds are elastic waves characterized by a frequency f that is higher than the audibility threshold (20 kHz). Assuming a non-viscous medium, ultrasound waves are longitudinal and are described by the wave equation. The real part of the wave equation solution, Re[A(t, z)] = A0 cos (k (vt − z)) ,

(62)

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is commonly used for the wave representation: A(t, z) represents the medium displacement as a function of time (t ) and position (z ), A0 is the maximum displacement, k is the wave number, which is equal to 2πλ−1 (λ is the wave length and it is equal to v · f −1 ), and v is the ultrasound velocity.64 The ultrasound propagation velocity v is 343 ms−1 in air and 1480 ms−1 in water at 20◦ C. The velocity of the sound in the soft biological tissues is about 1540 ms−1 . It is similar to the velocity in water, since soft biological tissues are made by 80%  −1 of water. The ultrasound propagation velocity is equal to Bρ , where ρ is the density of the medium (kg · m−3 ) and B is the bulk modulus, which is measured in pascal (1 Pa = 1 N · m−2 ) and it is the measure of the stiffness of the material. The energy carried by an acoustic wave is defined by its intensity I, which represents the power across an unitary surface (watt · m−2 ). It is given as I=

1 Z(2πf A)2 , 2

(63)

where Z is the acoustic impedance, which for a planar wave equals P/A˙ = ρv (P is the ultrasound pressure and A˙ the oscillation velocity).64 The physical principle of echography is the reflection of the ultrasound waves at the discontinuities of the medium. The discontinuity is described in terms of acoustic impedance Z, which is usually expressed in rayls: 1 rayl = 1 N · s · m−3 . For water Z = 1.49 M rayls. For larger discontinuities a larger fraction √ of the energy is reflected. The amplitude P of the pressure wave is given as P = 2ZI (i.e. P = 2πf AZ), therefore, the reflection can be described in terms of pressure, intensity, or displacement.64,65 An important aspect for a complete characterization of sound propagation is the acoustic attenuation. The intensity of the sound wave decays with an exponential law asd I = Ii e−2az ,

(64)

where Ii is the initial intensity, z is the covered distance, and a is the absorption coefficient in neper per cm.64,65 The coefficient a depends on the material as well as on the frequency of the ultrasound wave. It increases as the frequency increases. The relation between a (usually given in cm−1 · MHz−1 ) and frequency for soft tissues is nearly linear. The absorbtion coefficient for soft tissues is a linear function of f b , where b is slightly larger than 1 (b 1.15) and f ∈ [0.1 MHz, 50 MHz].64 Very often the attenuation a is measured in dB · cm−1 · MHz−1 . As a consequence, (64) is given ase I = Ii e−0.23adB zf .

(65)

d Since

the intensity is related to the squared value of the pressure amplitude, for pressure (64) becomes P = Pi e−az . e It is common to find a expressed in dB, i.e. a dB = 20 log10 (P/Pi ). As a consequence, P = Pi 10− adB .

adB f z 20

= Pi e− ln(10)

adB f z 20

− Pi e−0.115adB f z . The conversion is then a =

ln 10 adB 20

= 0.115 ·

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Ultrasound transducers are made of piezoelectric crystals that are placed on the tip of a probe. The crystals are the vibrating material that converts electrical energy into acoustic energy and vice versa. Each crystal makes an approximately linear conversion between mechanical pressure and electrical voltage and shows a specific resonance frequency that maximizes the energy transfer.55,65 The acoustic adaptation between crystals and skin tissue in order to transmit with the maximum bandwidth and the minimum loss is made by a matching layer designed with a thickness of λ/4 (λ is the length of the acoustic wave in the layer at the resonance frequency of the transducer) and an acoustic impedance between those of the crystal and the external medium (the acoustic impedance of the human skin).64,65 Echography, as the name suggests, is based on the analysis of the echoes: when a pulse (usually few ultrasonic cycles) is transmitted inside the body and the resulting echoes (reflections) received back, it is possible to derive the position of the intercepted discontinuities that generated the echoes. In fact, since the propagation velocity of ultrasonic waves through biological tissues is known, the delays of the received echoes can be interpreted in terms of distance. The distance d between the transducer and the discontinuity is given as d=

v∆t , 2

(66)

where v is the ultrasound velocity in tissue and ∆t is the time interval between the transmission and reception of the ultrasonic pulse. An example is given in Fig. 13. This is the basic Mode of an ultrasound scanner, and it is referred to as A-Mode (“A” stands for “Amplitude”). A pulse is transmitted and received back in order to reconstruct the discontinuities along a line. The received signal is demodulated in

Fig. 13. Distance estimation by means of echography. The discontinuity is represented by a plastic layer inserted in a water-filled basin. Notice that the border of the basin represents a discontinuity too, and that the resulting echo is more attenuated due to the longer path.

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order to suppress the frequency (usually between 2.5 MHz and 7 MHz for clinical applications) of the ultrasound pulse. This signal is referred to as A-line (Amplitude line). The A-line that is obtained after demodulation represents the amplitude of the ultrasound echoes as a function of the ultrasound depth, i.e. the profile of the acoustic impedance discontinuities. An important characteristic of the system is the resolution. Two different resolutions can be distinguished: the axial resolution and the lateral resolution.64 The axial resolution depends on the length of the pulse envelope (e.g. 5 cycles at 5 MHz result in a resolution distance in tissue d 0.8 mm). The lateral resolution is a measure or the narrowness of the ultrasound beam. It can be derived from the Huygens principle.55,64,66,67 For a circular transducer of radius r the lateral resolution is maximal (approximately equal to r) for a distance corresponding to r2 /λ, i.e. the transition between Fresnel and Fraunhofer zone. In the Fraunhofer zone the main lobe of the beam can be described by the opening angle θ0 as   0.61λ . (67) θ0 = arcsin r Two points covered by the main lobe cannot be distinguished, therefore, θ0 describes the angular lateral resolution of the system, which decreases as the distance of the discontinuities from the transducer increases. Although the beam pattern depends on the shape of the transducer, in every transducer the lateral resolution is proportional to the ultrasound frequency and crystal size. The same principles that are used to reconstruct the distribution of the acoustic impedance discontinuities along one line (A-line) can be used to obtain the discontinuity distribution on one plane.55,68 In fact, a plane can be spanned by translating the transducer in one direction and taking measurements for a series of lines. The voltage of the multiple A-lines (dynamic range) is mapped into gray-level by using a logarithmic compression, or, more in general, a non-linear compression. The result is a bi-dimensional image that represents a slice of the body. The bidimensional application of echography is referred to as B-Mode, where “B” stands for “Brightness”. The first simple B-Mode implementation was based on the mechanical translation of the transducer. Current solutions are mainly based on electronic array processing, which is implemented in the array transducers and allows dynamic focalization and steering of the ultrasound beam to different directions (beam steering). An array transducer is made of an array of crystals. Each crystal can be activated separately with a specific delay. By choosing the appropriate delays it is possible either to focus the ultrasound beam (like an optical lens) or change (steer) its orientation to span a wide angle. The crystals are therefore “phased” along one direction. The use of linear phased arrays is especially desirable in cardiology for transthoracic imaging, since the transducer must be sufficiently small to fit between the ribs and, at the same time, it must be able to span a large plane inside the body to image the heart. Linear phased arrays can be made in 1.5 dimensions, i.e. the array

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also contains a few elements (3 to 9) along the second dimension (elevation plane), which are used to improve the resolution along this second dimension (elevation focus). Obviously, this solution can be extended to a fully bi-dimensional array, which allows spanning complete volumes (3D imaging) inside the body.69 The most common echocardiographic test is referred to as transthoracic echocardiography (TTE). The name derives from the fact that the ultrasound transducer is positioned on the chest (thorax) of the patient. From that position, several windows can be found between the ribs in order to obtain different views (slices) of the heart.70 Miniaturized transducers are also available. They can be introduced through the mouth into the esophagus or the stomach as shown in Fig. 14. Since the heart is analyzed from the esophagus, this technique is known as transesophageal echocardiography (TEE) and has a significant advantage over classical TTE. Using TTE the ultrasound beam has to pass through the ribs and the lungs, resulting in lower quality images. The lungs may cause problems especially in patients with lung emphysema. Furthermore, added difficulties arise from the female breast and the fat tissue when dealing with obese patients. Instead, the esophagus is directly behind the heart, separated only by a thin layer of tissue. That is the reason why TEE produces higher Signal-to-Noise Ratio (SNR) images, which are also suitable for contrast detection and quantification.71 However, a disadvantage of this technique consists of its application procedure, which is unpleasant for the patient due to the probe insertion through the mouth. Nevertheless, for applications during surgery or in intensive care unit, TEE is becoming widely used. Related complications, such as gastrointestinal tissue damage, are very low, but they are not completely negligible.72 Figure 14 shows a TEE four chamber view, which permits the simultaneous observation of all the four cardiac chambers.

Fig. 14.

Transesophageal echocardiography: example of a four-chamber view.

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Ultrasound contrast agents Ultrasound Contrast Agents (UCAs) are made of a solution of micro-bubbles (diameter from 1 µm to 10 µm). They are smaller than red blood cells (diameter from 6 µm to 8 µm) and, therefore, suitable for passage through the capillaries and, in particular, the transpulmonary circulation. As for all indicators, ultrasound contrast agents have to be inert and non-toxic. Commercial agents can be distinguished in three generations.73 The first generation includes the early air micro-bubbles that were not stabilized by a shell, such as Echovist (Schering, Berlin, Germany), which was approved in Europe in the early 90s. Second generation agents were introduced in the mid-90s and included the first encapsuled air bubbles, such as Albunex (Molecular Biosystems, San Diego, CA, USA) and Levovist (Schering), whose shell was made of albumin and galactose, respectively. The latest third generation micro-bubbles for ultrasound detection are composed of air, SF6 , C3 F8 , or other perfluorocarbons encapsuled in a phospholipid, albumin, or polymer shell.74–77 The optimized use of a shell creates a strain that opposes to the Laplace pressure and stabilizes bubbles against dissolution. In fact, according to the Laplace law, the pressure inside a gas sphere, referred to as Laplace pressure, equals 2σ/R, where σ and R are the surface tension (N · m−1 ) and the sphere radius, respectively.78 The high internal pressure due to small radius is held by the strain that is provided by the shell. Examples of third-generation agents that are currently used in clinical practice are SonoVue (Bracco, Italy) and Definity (Bristol-Myers Squibb, USA). Once injected into blood, the effect of the bubbles (see Fig. 15) is a significant increase of the ultrasonic energy backscatter, which at first analysis could be simply explained by the reflection across the blood-air acoustic-impedance discontinuity. Due to the natural oscillations (contraction-expansion) of the bubbles when invested by a pressure input, the interaction between contrast agents and ultrasound is a nonlinear process, which adds several harmonics to the backscattered ultrasound. The oscillations of a single bubble are commonly characterized by the model developed by Rayleigh and Plesset to describe the motion of vibrating spheres.78,79 The bubble is represented as a sphere of radius R and its motion is considered spherically symmetric as shown in Fig. 16. Therefore, bubble oscillations are described by radius variations. The surrounding fluid is assumed to be Newtonian (incompressible with constant viscosity). The equation that is generally used to describe the relation between velocity and pressure in a fluid-dynamic system is the Navier-Stokes equation.80 If the steady external forces (e.g. the gravitational force) are not considered, the Navier-Stokes equation can be written as given in (68), where ρ and µf are the density and the viscosity of the fluid, respectively:  ρ

v2 ∂v + ∂t 2



4 + µf ( ×  × v) − µf ( · v) = −P. 3

(68)

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Fig. 15. TTE using SonoVue contrast agent. The contrast is recognizable in the left side of the heart. Courtesy of the department of cardiology of the Catharina Hospital in Eindhoven, The Netherlands.

Fig. 16.

Schematic drawing of an oscillating bubble in an ultrasound pressure field.

For a Newtonian incompressible fluid  · v = 0. If the vorticity is negligible, the velocity field is irrotational ( × v = 0) and the viscosity terms in (68) disappears.78,80 Therefore, we can define a potential Ψ so that v = Ψ and replace (68) by the Bernoulli equation as given in (69)78,80 :  ρ

∂Ψ v 2 + ∂t 2

 = −P.

(69)

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Due to the radial symmetry hypothesis, the fluid velocity field has a radial symmetry and can be described as a function v(r, t) of the radial distance r from ˙ where R the bubble center and the time t. As a direct consequence, v(R, t) = R, is the bubble radius. Moreover, Ψ = 0 for r → ∞. Another condition concerns the pressure field for r → ∞, which is defined as P∞ and equals the sum of two contributes: the hydrostatic pressure P0 and the ultrasound driving pressure P (t). Using these conditions, the integration of (69) over r allows defining the relation between the pressure field P (r, t) and the bubble radius R(t), which for r = R is given as ¨ + 3 ρR˙ 2 . P (R) − P∞ = ρRR 2

(70)

P (R) can be related to the internal pressure Pi of the gas bubble as given in (71), where Pv is the vapor pressure and σ is the surface tension coefficient (N · m−1 )79 : 2σ + Pv . (71) R Since the gas expansion and contraction can be considered respectively isothermal and adiabatic, the whole process is described by a polytropic transformation with exponent k, i.e. P · V k is constant.81 For a driving pressure that does not cause bubble collapse we may assume the internal gas to be ideal. In this case Pi is given as   3k  R0 2σ , (72) Pi = P0 − Pv + R0 R P (R) = Pi −

where R0 is the bubble radius at the equilibrium condition. Combining (72) and (71) with (70), we obtain (73), which describes the nonlinear motion of an ideal bubble and is referred to as Rayleigh-Plesset equation 78 :    3k  R0 3 ˙2 2σ ¨ ρRR + ρR = + P0 − Pv − 1 − P (t). (73) 2 R0 R A modified expression of the Rayleigh-Plesset equation also adds to the second ˙ −1 , which defines the pressure drop that is caused by member of (73) the term 4µf RR the viscous damping of the bubble-fluid system and is related to the fluid viscosity µf (see Ref. 78 pp. 189–190 and Ref. 82 pp. 68–70). The resulting equation is given as in (74). It is referred to as the modified Rayleigh-Plesset equation,83,84 which is the result of the work of Noltingk, Neppiras, and Poritsky:    3k  R 2σ 4µf R˙ 3 0 2 ¨ + ρR˙ = − P (t). (74) + P0 − Pv −1 + ρRR 2 R0 R R Since modern contrast agents (second and third generation) are made of encapsuled bubbles, the shell properties must be included in the bubble motion equation. Starting from the Rayleigh-Plesset equation, several authors have made modifications and added different terms according to different shell characterizations. The

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major contributions were proposed by de Jong (1993), Church (1995), Frinking-de Jong (1998), and Hoff (2000).73,79,82,85–88 Also other models have been derived to remove some of the assumptions of the Rayleigh-Plesset equation. They are based on the Herring (1941) and Gilmore (1952) equations,89,90 which consider the enthalpy energy and a compressible medium (non Newtonian) in order to give a better prediction of large oscillations.78,91,92 Based on these equations, Flynn (1975) developed a model that also included the thermal effects inside the bubble.93 Still based on the Herring equation, the Trilling (1952) and the Keller-Kolodner (1952) models were derived.82 The Trilling model was then extended to a population of bubbles by Chin and Burns (1997),94 while the Keller-Kolodner model was modified first by Keller-Miksis (1980) to introduce the driving pressure field and then by Prosperetti (1988).95,96 Another model — still based on a modified Herring equation — was also introduced by Morgan and Hallen (2000).97 A complete overview of all the proposed bubble dynamics models is beyond the purpose of this chapter. Due to its simplicity, we limit our discussion to the model proposed by de Jong in 1993.79 With respect to (74), the model does not consider only the damping due to the fluid viscosity µf , but also other contributes related to re-radiation and heat conduction, together with the contributes to pressure due to the shell-properties. The pressure that is related to the presence of a shell is mainly caused by the shell viscous damping and elasticity. After collecting all these terms together, the resulting equation is given as ¨ + 3 ρR˙ 2 = ρRR 2



2σ + P0 − Pv R0

 

R0 R



3k

− 1 − Sp

− 2πf δt ρRR˙ − P (t) .



1 1 − R0 R

 + (75)

The pressure due to the shell elasticity (restoring force) is defined by the shell elasticity parameter Sp , which is measured in N · m−1 and can be derived from the application of the Hooke law (first order approximation) under hypothesis of homogeneous, thin, and perfectly elastic shell.79,98 Notice that in this model the shell thickness is assumed to be constant during the oscillations.f The term δt in (75) is the total damping factorg and it is the sum of four terms as given in (76): δt = δrad + δvis + δth + δf .

f Other

(76)

models, such as the Hoff model, assume a constant shell volume and, therefore, the shell thickness becomes a function of the bubble radius R. g In a linear system of the second order, such as a mass-spring damped system, the damping factor (non-dimensional) is defined as the inverse of the quality factor and equals b/2πfn m, where b is the damping coefficient, fn is the natural frequency of the system, and m is the mass.

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The term δrad represents the re-radiation damping, δvis the fluid viscosity damping (already included in (74)), δth the damping due to thermal losses, and δf the damping due to the shell friction.78,82,99 The expression for the four dimensionless terms, usually considered at the natural frequency fn (a derivation of fn for a linearized system is given in (80)), is given below, where µs is the shell viscosity, v is the ultrasound velocity in the medium, and the expression for B(f, R) can be found in Ref. 99 (pp. 296–299): 2πfn R0 , (77) δrad = v 2µf δvis = , πfn ρR02 f2 δth = B(f, R0 ) n2 , f 6µs Ts . δf = πfn ρR03 All the damping factors are function of the frequency f and the bubble radius R0 . For instance, the thermal damping δth is only effective for intermediate frequencies (near the natural frequency fn ). In fact, for high frequencies the process is adiabatic (no heat transfer due to the short period of the oscillations), while for low frequencies the process is isothermal (the oscillations are sufficiently slow to allow the heat transfer to keep the bubble temperature constant). Instead, for intermediate frequencies, like for instance around the natural frequency fn , the temperature oscillates in the bubble and causes pressure variations that are out of phase with respect to the driving pressure. The bubble dynamic model is used to study and predict the response of the bubble system to a pressure input (driving pressure). The major interest is the definition of a frequency range that maximizes the bubble oscillations and, therefore, the backscattered ultrasound signal. In order to derive the frequency response of the bubble, the system is analyzed for small oscillations and a Taylor first order approximation of (75) is considered. The resulting system is a typical second order linear system (like a damped mass-spring system, see Ref. 82, p. 109) as given in (78), where represents the small radius oscillation R − R0 for a driving pressure P (t) (see Fig. 16): m¨ + b ˙ + s = −4πR02 P (t), m = 4πρR03 , b = 2πf mδ t ,  

2σ 2σ 2Sp − . + pv + s = 4πR0 3k P0 − Pv + R0 R0 R0

(78)

The frequency response | (f )| · |P (f )|−1 of the system (see also Refs. 82, 78) is given as | (f )| 1  = , (79) |P (f )| R0 ρ (fn2 − f 2 )2 + (f fn δt )2

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where fn is the natural frequency of the bubble without damping and is given as 1 fn = 2π



 s 1 = √ m 2π ρR0

  2σ 2σ 2Sp − 3k P0 − Pv + + Pv + . R0 R0 R0

(80)

Based on (79) and (80), the peak frequency response (resonance frequency f0 ) of the damped bubble is given as in (81):  f0 = fn

1−



δt 2

2 .

(81)

As a consequence, the maximum oscillation, when for instance the viscous damping is included, is generated for a driving frequency f0 given as in (82) (see Ref. 78, pp. 305–306):  1 f0 = √ 2π ρR0

  4µ2f 2σ 2σ 2Sp − 3k P0 − Pv + + Pv + − . R0 R0 R0 ρR02

(82)

As already mentioned, the de Jong model, as well as all the models in literature, are based on a series of assumptions that are not always realistic. In particular, in a real clinical application a large number of bubbles may interact with each other with multiple thermal and radiation energy exchanges. The interaction forces are not considered in the presented model. Moreover, as recently observed by de Jong and Versluis using an ultra-fast camera, the assumption of spherical symmetry of the bubble oscillations is not realistic.100,101 Bubbles oscillate using several different geometrical modes, so that more sophisticated models are needed. Also a large variability of the maximum bubble expansion for same driving pressures and same bubble original diameters is recognizable. This phenomenon is complex to explain and might be the result of different elastic properties of the bubble shell.102 In this case, the elasticity parameter Sp could be substituted with a statistical distribution. However, especially for small pressure and low concentrations (negligible interaction between bubbles), the available models can already predict with sufficient accuracy the interaction between ultrasound and bubbles and they are widely adopted for contrast quantification.8,73,103–108 In particular, contrast quantification involves the estimation of the agent echogenecity, which is commonly determined by the measurement of the ultrasound backscatter. The ultrasound backscatter is defined by the backscatter coefficient β, which is the scattering cross-section (cm2 ) per unit volume (cm3 ) and per scattering angle (sr). The scattering cross-section of a bubble is the ratio between the power scattered in all directions and the incident acoustic intensity. The scattered power equals the energy that is dissipated because of the radiation damping.

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In general, if the damping coefficient equals b, then the force F that is necessary to compensate for it equals ˙b. For (t) = 0 cos(2πf t), the average power W (energy per cycle) that is dissipated is given as in (83), where T = f −1 :   1 T 1 T 1 W = (F · (t))dt ˙ = ( (t)b ˙ · (t))dt ˙ = 4π 2 f 2 20 b. (83) T 0 T 0 2 The re-radiation average power is given as in (83) for radiation damping coefficient brad = 2πf mδrad = 16π 2 ρR04 f 2 v −1 . Combining (83) with (79), the scattering cross-section Σ for a single bubble, which is defined as the ratio between average scattered power and ultrasound intensity, is a function of the radius R of the bubble and the ultrasound frequency f as given in (84), where W is the average scattered power, Ii is the amplitude of the incident intensity (Ii = Pi2 /2Z), and fn is the natural frequency of the bubble:78,79,82,88,108,109 Σ(R0 , f ) =

1 4π 2 f 2 20 b W 4πR02 . = 22 =  2 −1 Ii Pi (2Z) 2 (fn (R0 )/f ) − 1 + δt (R0 , f )

(84)

The term δt (R0 , f ) summarizes all the damping factors. Since the adopted model represents a second order system, the scattering cross-section shows a resonance frequency where the system gives the strongest response in terms of scattered power. For f fn ⇒ Σ (R0 , f ) 4πR02 , which is the physical cross-section, i.e. the bubble surface.104,108 The resonance frequency is inversely proportional to the radius R0 of the bubbles at the equilibrium. Therefore, the total scattering cross-section Σtot (f ) depends on the normalized radius distribution n(R) of the bubbles as given in (85)110,111 :  Rmax Σtot (f ) = n(R)Σ(R, f )dR. (85) Rmin

n(R) is a characteristic of the specific contrast. Assuming an isotropic scattering and a low concentration of bubbles, the backscatter coefficient β (expressed in cm−1 · sr−1 ) is given as in (86), where ρn is the number of bubbles per unit volume (concentration)110 : ρn Σtot (f ) . (86) 4π Therefore, the backscatter coefficient is a linear function of the UCA concentration and the backscattered acoustic intensity I that is measured by the transducer can be approximated by8,105,106,110 –113 β(f ) =

dV dV ρn Σtot (f ) Ii , β(f )Ii = 2 (87) 2 z z 4π where Ii is the acoustic intensity insonating the contrast, dV = dz ·dA is the volume of the insonated contrast, and z is the distance between dV and the transducer.h I=

h An

homogeneous scattering with a spherical symmetry of the backscattered pressure is assumed (see Ref. 114, pp. 357–361).

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When β(f ) is averaged over the frequency spectrum of the ultrasound transducer it is referred to as Integrated Backscatter Index (IBI). The interaction between ultrasound and UCA is not only described by the backscatter coefficient, but also by the attenuation coefficient, which represents the loss of acoustic pressure in neper per cm. It occurs along the distance that the ultrasound beam covers through the contrast solution, and it is related to the micro-bubble decay (ad ) due to both chemical decay (dissolution) and dispersion through the capillaries, the scattering of the acoustic energy in multiple directions (as ), and the viscous, thermal, and friction damping (δvis , δth , and δf in (76)) of the ultrasound waves (aδ ). The total increase of attenuation ∆a is given as in (88), where ad is proportional to the chemical constituents of the contrast while (as + aδ ) is proportional to the concentration of the contrast: ∆a = ad + as + aδ ,

(88)

The attenuation that is expressed by the term as +aδ is referred to as extinction. It is possible to define an extinction cross section Σe as the sum of the backscatter cross section Σ given in (84) and the absorption cross section Σa . Similarly to (84), the absorption cross section is defined as the ratio between the loss of power and the incident intensity Wloss /Ii . The power can be always expressed as a linear function of the damping coefficient b as given in (83). Since b = 2πδmfn by definition, the ratio between the loss of power caused by damping and the acoustic scattering Wloss /W equals the ratio between the respective damping factors. This concept can be formulated as given in (89) (see Ref. 82, pp. 25–28 and Ref. 99, pp. 302–304):   δvis + δth + δf Σ. (89) Σa = δrad Since Σe = Σa + Σ, Σe is also given as   δt Σ. Σe = δrad

(90)

The extinction coefficient ae = as + aδ can be derived from the extinction cross section. In fact, from the definition of extinction cross section, the differential loss of power dWloss can be written as given in (91), where dV = dA · dz is a volume sample as shown in Fig. 17: dWloss = Iρn Σetot dV = Iρn Σetot dAdz.

(91)

Therefore, the loss of intensity dI is described by the differential dI = Iρn Σetot dz,

(92)

I(z) = Ii e−ρn Σetot z .

(93)

whose integral is given as

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Fig. 17. Schematic passage of an ultrasound beam of intensity I through a volume sample dV of bubbles. The volume length is dz while the cross-section area is dA. The intensity loss through the sample equals dI.

The total extinction cross-section Σetot is derived from Σe as given in (85) for the scattering cross-section Σtot :  Rmin Σetot = n(R)Σe (R, f )dR. (94) Rmax

As a consequence, the extinction coefficient ae is given as in (95):  ρn ρn Rmin Σe = ae = n(R)Σe (R, f )dR. 2 tot 2 Rmax

(95)

The division by a factor 2 allows deriving (64) from (93), and it is due to the fact that the attenuation coefficient is usually defined in terms of pressure loss, which is related to the intensity by the quadratic relation I = P 2 (2Z)−1 . For shell encapsuled bubbles, the first term ad in (88) can be neglected if the measurement is executed in a short time (i.e. few minutes). When only the extinction is considered, ae in (95) corresponds to the attenuation coefficient a introduced in (64). As a result, the attenuation is linearly proportional to the concentration of the contrast ρn , as also shown in (86) for the backscatter coefficient β.79,105,110,111,115,116 Because of the linear relation between total attenuation and contrast concentration, some authors have considered the opportunity of measuring the attenuation-time curve rather than the backscatter-time curve for the estimation of fluid-dynamic parameters.116 According to (64), (87), (88), and (95), the received echo intensity from a contrast perfused organ can be rewritten as dV (β + ∆β)Ii e−4a(z0 +∆z)−4∆a∆z , (96) z2 where β is the backscatter coefficient in the examined organ (without contrast), a is the attenuation coefficient of tissue, z0 is the depth of the organ, and ∆z is the depth of the insonated volume sample dV within the contrast perfused organ. ∆β I=

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and ∆a are the backscatter and attenuation increase due to the presence of contrast in the investigated organ. The return distance that is covered by ultrasonic waves (after reflection) is taken into account by the factor 4 at the exponent (instead of 2). This is the model that is usually adopted to interpret the backscattered intensity. In fact, both ∆β and ∆a are linearly proportional to the contrast concentration. A good measure of the efficacy of UCAs in terms of ultrasound detection, which considers both backscattering and attenuation, is represented by the Scattering-To2π∆β 73,110,111 Attenuation Ratio (STAR) ΣΣetot = δrad δt = ∆ae . tot

This characterization of UCA is largely based on the assumption of small oscillations, i.e. small ultrasonic driving pressures. For higher pressures the bubble cannot hold the inner gas and the bubble collapses. The violent collapse of bubbles is referred to as cavitation. The standard indicator to determine the mechanical interaction or impact of ultrasonic waves on bubbles is the mechanical index (MI), which is defined as P [MPa] , MI =  f [MHz]

(97)

where P is the peak rarefactive pressure measured in MPa and f is the central frequency of the ultrasound pulses expressed in MHz. For MI > 0.6, a large fraction of the bubbles collapses. It is worth mentioning that bubble cavitation phenomena find important applications in the field of drug and genes delivery.75,117–120 In these applications, bubbles are loaded with either drugs or genes. A targeting ligand in the bubble shell makes the injected bubbles settle and attach to specific sites.121,122 Once the bubble is located where the drug is intended to be delivered, high MI ultrasound pulses make the bubbles collapse and release the drug. This technique allows both targeting the delivery and enhancing the drug uptake through the increased porosity of the cell membranes due to bubble cavitation. In cardiology, this technique is very promising for adenosine release during cardiac reflow therapy (i.e. when the flow through the coronary arteries is reestablished). Also applications for lysis of clots in the coronary arteries are very interesting and promising.123 The nonlinear behavior of bubble oscillations has given the opportunity to develop specific imaging modes to preferentially detect the echoes from the agent while suppressing those from other linear structures, such as tissue.76,77,105,124–126 The nonlinear behavior of the bubbles is mainly evident for driving pressures above 50 kPa.104 Above this threshold, the backscatter cross-section Σ differs from the physical value of 4πR02 according to (84). We refer to as fundamental mode when the ultrasound scanner is set to transmit and receive at the same frequency (fundamental frequency). Nowadays, also contrast detection modes, referred to as harmonic modes, are widely available in ultrasound scanners. The objective of these harmonic imaging techniques is to maximize the Contrast-to-Tissue Ratio (CTR).127–129 In harmonic mode, the system transmits at one frequency (resonance frequency of the transducer), but is tuned to receive echoes

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Fig. 18.

Frequency response spectrum of SonoVue.

preferentially at a different frequency. Therefore, the bandwidth of the transducer must be broad. As shown in Fig. 18, the frequencies that are commonly used for contrast detection are distinguished in sub-harmonic (about half of the fundamental harmonic), ultra-harmonic (between the fundamental and the second harmonic), second and higher harmonics, and super-harmonic (between higher harmonics).76,77,128,130,131 For these frequencies, the signal coming from contrast shows larger amplitudes with respect to that coming from tissue. In fact, due to the linear response of tissue, tissue echoes contain approximately the same frequency components as the pressure waves that are generated by the transducer (fundamental harmonic). Figure 18 shows an example of frequency spectrum of UCA backscatter. The implementation of filters in order to detect specific frequencies is not the only solution to enhance the UCA detection. Alternative techniques involve the modulation of amplitude and phase of the transmitted ultrasound. They are referred to as power modulation imaging and phase modulation imaging, respectively. These techniques are economically more convenient, since they do not require the employment of expensive broad-band transducers. The most common phase modulation technique is referred to as pulse inversion. Two pulses p1 (t) and p2 (t) = −p1 (t) are transmitted in sequence into the tissue. The sum of the received echoes results in the cancellation of the echoes from linear structures (i.e. tissues), while the echoes from the bubbles do not cancel, resulting in a selective detection of UCAs. Amplitude modulation (or power modulation) techniques quantify directly the nonlinearity of the ultrasound transmission-detection channel (i.e. the medium), which is related to the presence of bubbles. If h(P ) represents the channel for pressure P and the channel is nonlinear, then 2h(P ) = h(2P ). Several implementations combine Power Modulation and Pulse Inversion (PMPI) with a number of different pulse sequence schemes. The use of high MI (MI > 1) produces a high-rate bubble disruption.104,120,132 Some techniques exploit the bubble disruption for contrast imaging.133,134 Two

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pulses are transmitted in fast sequence. The first one is reflected and destroys the bubble. After the bubble destruction, the second pulse is not reflected. By taking the difference between the reflections from the two pulses it is possible to distinguish bubbles from tissue, unless the fast motion of tissue is confused with the bubble destruction (clutter noise). More sophisticated implementations of the same concepts are possible and are referred to as release-burst imaging.76 ,126,134 Cardiovascular quantification Nowadays, the main application of UCA dilution in cardiology aims at the improvement of the endocardial and epicardial visualization and segmentation.43 In particular, an accurate segmentation of the endocardium is fundamental for accurate assessments of LV end-diastolic and end-systolic volumes, which also permit the EF derivation. These applications are referred to as LV opacification (LVO) and require UCA infusion rather than the injection of a single bolus. The use of LVO also allows the detection of ventricular obstructions that are difficult to recognize otherwise. Apart from LVO, which is not directly a quantitative application of UCA dilution, a technique that is implemented in all recent ultrasound scanners is referred to as replenishment. It consists of the destruction of micro-bubbles followed by a low MI detection phase. This technique is used for myocardial flow quantification. During perfusion, the contrast in the myocardium is destroyed by a high MI ultrasound burst (MI ≥ 1). The following replenishment curve is detected by recording the acoustic intensity-versus-time curves in defined regions of interest of the myocardium. The interpolation of these curves by a specific model allows the quantification of the myocardial flow.135,136 Also this application requires UCA infusion rather than a bolus injection. In practice, a simple mono-compartment model is adopted to represent the dilution system of the myocardium as shown in Fig. 19. Therefore, the replenishment curve C(t) is modelled as the step response of a mono-compartment model as given in (56). The flow is proportional to C0 · τ −1 and an image of the myocardium can be generated such that the gray level (or color coding) corresponds to the perfusion level.49,135,136 As discussed in Sec. 4.4, the myocardial flow can be estimated before and after inducing stress (stress test) in order to assess the coronary flow reserve. An interesting emerging application of UCA dilution concerns the measurement of the PBV. A region of interest (ROI) is placed on the RV and a second ROI is placed on the LA after the injection of a small bolus of UCA. Two IDCs can be measured (one for each ROI) and interpolated by the distributed models presented in Sec. 4.2, so that the MTT of the transpulmonary circulation can be estimated and multiplied times the CO for the assessment of the PBV.137 Another novel technique consists of the EF assessment by analysis of two IDCs in the LA and LV after a peripheral bolus injection. A deconvolution technique is adopted to estimate the impulse response of the LV. Due to low SNR, the deconvolution can be implemented by means of squared error minimization techniques, such as for instance a Wiener filter.21 The LV is modelled by a mono-compartment

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Fig. 19.

Replenishment curves for lower (Φ ∝ C1 · τ1−1 ) and higher (Φ ∝ C2 · τ2−1 ) flow.

model as described in Sec. 4.3, so that the EF can be measured by the Holt method described in (55) in a minimally invasive manner.138

5.2.2. X-ray angiography X-ray imaging is based on the detection of X-rays that pass through the body. The different absorbtion of X-rays associated to different tissues permits to use the X-ray projection to reconstruct a medical image of the inner body. The X-ray angiography is the standard technique to diagnose blood-vessel stenosis and aneurysm, but it can be also adopted to estimate CO and MTT, especially in the myocardium.139 Stenosis is an abnormal narrowing of an artery or vein (it can be also referred to the cardiac valves) while aneurysm is an abnormal widening of it. Cardiac catheterization is a combined hemodynamic and angiographic procedure, which can be done for diagnostic and therapeutic purposes. It is an invasive procedure, therefore, the decision to perform cardiac catheterization must be based on a balance of the risk of the procedure against the possible benefit to the patient. Catheterization is performed mainly by percutaneous approach, through femoral or radial arteries, transseptal catheterization, or apical left ventricular puncture. While it is technically possible to perform a catheterization in a freestanding facility, it is still strongly advised to have it performed in the hospital. For the operating personnel, cardiac catheterization requires a thorough training, which includes study of cardiac anatomy, physiology, and instruction about radiologic equipment, radiation safety, and the proper use of contrast agents. During angiography, the X-ray scanner is used in fluoroscopy-mode so that a continuous monitoring can be performed.140 This is realized by means of a BI

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(Brightness Intensifier ) or flat panels made of TFTs (Thin-Film Transistors) for real-time imaging. To permanently acquire a movie of the catheterized anatomic districts, 35 mm cine-cameras have been utilized so far in every cardiac catheterization facility although, more recently, these devices have been substituted by digital encoders which operate with the “filmless” technology. Because of historical reasons, this operating mode is still referred to as cineangiography. During a high resolution image recording performed with the cineangiographic technique, the X-ray beam intensity can be 100 times higher than it is during fluoroscopy, when the images are acquired by a standard television chain; this has to be taken into high consideration for the calculation of the X-ray dose absorbed by the patient and by the operator. A standard angiographic procedure requires the insertion of a catheter through the femoral artery up to the heart. The injection of a radiopaque medium is performed through this catheter. A radiopaque medium is a substance that absorbs X-radiation (e.g. iodine). As a consequence, the blood mixed with the radiopaque absorbs the X-rays and the blood vessels can be detected by the X-ray scanner. Figure 20 shows an example of angiographic image. Early generations of intravascular contrast agents contained heavy metals (e.g. bismuth and barium). However, all modern contrast agents are based only on iodine, which has a high atomic number and, therefore, it is proved to be a very good intravascular opacification agent. Inorganic iodine, however, cause toxic reactions. Ratio 1.5 ionic compounds with a sodium concentration around that of blood, a pH between 6.0 and 7.0, and a low concentration of calcium disodium ethylenediaminetetraacetate (i.e. chelated with ethylenediaminetetraacetic acid) are the traditional high osmolar ionic contrast media (HOCM): Renografin (Bracco, Milan, Italy), Angiovist (Berlex-Schering), Hypaque (Nycomed). Ratio 3 or low osmolarity ionic and non-ionic contrast media (LOCM), which reduce hypertonicity side effects, have been introduced in the end of the 1980s. Hexabrix (Mallinckrodt, Hazelwood, MO) is the only ionic LOCM while several examples of non-ionic LOCM are available on the market. Both HOCM and LOCM have advantages and drawbacks, however, LOCM have not yet completely substituted HOCM only because of the cost of LOCM, still 2 to 3 times higher than that of HOCM.140 A new class of

Fig. 20. Example of X-ray coronary angiography with evident stenosis of the middle right coronary artery (indicated by an arrow).

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isomolar contrast medium (IOCM), which is a ratio 6 non-ionic dimmer, has been recently made available on the market: Visipaque (Nycomed). The accuracy of X-ray analysis is improved by the employment of threedimensional imaging techniques. The technology that allows a three-dimensional image reconstruction is the Computerized Axial Tomography (CAT). It is based on the Radon transform (Johann Radon, 1917) of multiple X-ray projections.141,142 Since an angiographic procedure takes about 30 minutes, the dose of X-rays absorbed by the patient should be seriously considered. This dose is defined as the energy that is absorbed per unit mass of material invested by X-rays. In the International System it is measured in gray [Gy], 1 Gy = 1 Jkg−1 . In case of biological tissues it is measured by the equivalent dose, expressed in sievert [Sv]. The equivalent dose takes into account many factors that determine the biological interaction. The limit dose for common people is 1.7 mSv/year, while the dose given by a fluoroscopy is about 1 cGy 1 mSv. 5.2.3. Nuclear imaging In the radionuclide angiography a small amount of radioisotopes, normally indicated as radiopharmaceuticals, is peripherally injected. The radioactive decay of radioisotopes leads to the emission of α and β particles as well as γ- and x-radiation. Hence, the detection of the indicator is performed by radioactivity measurements. It is performed by a scintillation camera (Anger, 1959), usually referred to as gammacamera. The gamma-camera is a photon counter (minimal photon energy 50 keV).i It consists of a scintillator crystal (NaI or NaTl) that covers a matrix of photomultiplier tubes. Only γ- or x-radiation can be detected with detectors that are outside the body. The information detected and recorded by these scanners is analyzed and processed to generate images of the target anatomical structures. The emission tomography systems are divided in two main groups, depending on the type of radiation emitted by the adopted radiopharmaceutical. • The Single Photon Emission Computed Tomography (SPECT) system makes use of routine single photon gamma emitters such as 99m Tc, 131 I, 123 I, 67 Ga, and 201 Tl. It is generally designed to collect data from different angles. • The Positron Emission Tomography (PET) system detects annihilation radiation from positron emitters such as 11 C, 13 N, 15 O, 18 F, and 68 Ga. It consists of two or more opposed detectors that permit the detection of the two 511 keV gamma photons that are emitted simultaneously in opposite directions by the annihilation process. Therefore, the line that intercepts the emission point can be determined. is the energy acquired by an electron (1.602 × 10−19 C) when inserted in an electrical field of 1 V. Hence, 1 eV = 1.602×10−19 J. For instance, in case of X-radiation, we can calculate the energy by the formula E =  λc , with c equal to the light speed, λ equal to the radiation wave-length, and  equal to the Planck’s constant (6.626 × 10−34 Js). The average wave-length of the X-radiation

i 1 eV

is 10−1 nm, resulting in an energy E =

6.626·10−34 Js·3·108 ms−1 10−10 m·1.602·10−19 eV−1 J

= 12.5 keV.

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The most advanced radionuclide technique for dynamic image analysis is the Multi-Gate imaging (MUGA, also known as ventriculogram), where the gammacamera takes more images triggered by the ECG signal. This technique permits the geometrical estimation of EF as well as the application of the indicator dilution theory for blood flow, perfusion, and volume measurements according to the models described in Sec. 3.29,33,143–145 In fact, since the images represent the concentration of the radiopharmaceuticals, it is always possible to place a ROI and record an IDC based, for instance, on the average video-intensity in the ROI. Since radionuclides cannot be considered as inert, patients in special condition, such as for instance women who are pregnant or breast-feeding, are not allowed to undergo nuclear imaging.

5.2.4. Magnetic resonance imaging Principles of magnetic resonance imaging Magnetic Resonance Imaging (MRI) is an imaging technique used to produce high quality images of the inside of the human body. To this end, the MRI scanner uses high magnetic fields (several tesla). A detailed discussion of the MRI physics and technology is beyond the purpose of this chapter. Here we provide a brief description of the basic principles. A complete description of MRI can be found in Refs. 142, 146–149. The MRI technique is based on the principles of Nuclear Magnetic Resonance (NMR), a spectroscopic technique that allows the distinction between different materials depending on the magnetic properties of their atoms. In particular, biological tissues can be analyzed depending on the hydrogen concentration. Hydrogen-1 (mass number = 1) is the most abundant element in the human body in its association in water molecules. Due to an uneven atomic number, each hydrogen nucleus (proton) shows an angular momentum (spin), which is due to the nucleus rotation around its own axis. Since the proton is electrically charged and rotates, it generates a magnetic field, which can be represented by a magnetic dipole moment. Therefore, each proton can be considered as a small magnet with separate poles as shown in Fig. 21. Its magnetic moment is parallel and proportional to the spin by a physical constant, the gyromagnetic ratio γ, which is a property characterized by quantum mechanics, and it is equal to 2.685 × 108 rad s−1 T−1 . → − Thus, the relationship between the spin (angular momentum J ) and the magnetic → − moment µ is given as → − − → µ =γJ.

(98)

In an external magnetic field, the spinning protons tend to align themselves in the direction of field. According to the quantum theory, only two orientations of the nuclear magnetic moment relative to the field direction are possible. Both

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Fig. 21.

Spinning nuclei.

orientations are at an angle to the external field and are labeled parallel and antiparallel. These two states correspond to two allowed energy levels. At equilibrium, the number of spins at the lower energy level exceeds that at the higher level. According to classical mechanics, the magnetic moment experiences a torque → − → − τ from the external magnetic field B 0 , which is equal to the rate of change of → − its angular momentum J , and it is given by the equation of motion for isolated spins: → − ∂J → − → → − =− µ × B 0. τ = (99) ∂t Combining (98) and (99), we can derive → ∂− µ → − → = γ− µ × B 0. ∂t

(100)

As a result, the magnetic moment precesses around the axis of the external magnetic field (Fig. 22) at a particular frequency that is referred to as Larmor frequency and is given as fL =

Fig. 22.

γ B0 . 2π

Magnetic moment precession.

(101)

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This is the frequency at which the nuclei can receive the RF energy to change their states and exhibit nuclear magnetic resonance. In a macroscopic sense, the sum of the spin vectors leads to a net magnetization in the direction of the field, which is the vectorial sum of all the magnetic moments of the nuclei in the considered object, and it is referred to as longitudinal mag→ − netization. If an additional alternate field B 1 perpendicular to the z-axis with a frequency equal to the Larmor frequency is applied, then the spin system absorbs energy. The population at the upper energy level increases while that at the lower → − level decreases. The alternate field B 1 is referred to as RF pulse and the following process is referred to as resonant process. As a result, the net magnetization vector is no longer in the direction of the external magnetic field. It moves away from the z-direction and it is flipped of an angle — referred to as flip angle — equal to θ = γB1 ∆t, where ∆t is the duration of the RF pulse. Once the RF pulse is finished, the net magnetization vector returns to its original equilibrium (steady) state through a process that is referred to as relaxation. The previously absorbed energy is emitted at the Larmor frequency and it is detected by receiving coils perpendicular to the z axis (see Fig. 22). The MR signal (transient response) after a RF pulse excitation is referred to as free induction decay (FID) signal. The induced voltage V (t) in the receiving coil can be derived as follows, applying the principle of reciprocity (102) and the Faraday law (103):  →− − − →→ → B (→ r , t) · M (− r , t)d− r, (102) Φ(t) = obj

∂ ∂Φ =− V (t) = − ∂t ∂t



− − → − →→ → B (→ r , t) · M (− r , t)d− r.

(103)

obj

− → → B is the magnetic field generated by the coil at location − r and Φ is the magnetic flux through the coil. The longitudinal (z direction) component varies much slower than the transversal (xy-plane) component and can be neglected. As a result, the voltage detected by the receiving coil can be approximated as  ∂− → → → − − → B xy (→ r , t) · M xy (− r , t)d− r. (104) − V (t) ∼ = ∂t obj Magnetic relaxation The relaxation expresses the recovery towards equilibrium of nuclear dipoles that have been perturbed by RF excitations. The time-dependent behavior of the net − → → − magnetization vector M in the presence of an applied magnetic field B 1 is described by the Bloch equation as given in (105), − → k Mxi + My j ∂M (Mz − M0 ) − → → − = γM × B − − , (105) ∂t T2 T1 → − where M0 is the equilibrium Mz magnetization in the presence of B 0 only. T1 and T2 are, respectively, the longitudinal and transverse relaxation times. There are relaxation parameters unique for each tissue. The solution of (105) is given as t

Mx (t) = M0 e− T2 cos(γB0 t),

(106)

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My (t) = M0 e− T2 sin(γB0 t),   − t Mz (t) = M0 1 − e T1 , which represents the time dependent behavior of the magnetization vector. √ Representing the xy-plane by means of the complex plane with i = −1, the transverse magnetization can be expressed as −t t − → M xy (t) = Mxy (cos(γB0 t) − i sin(γB0 t))e T2 = M0 e− T2 e−i(γB0 t) . (107) From the definition of Mz in (106) it is clear that the longitudinal or spin-lattice relaxation time T1 refers to the time that is necessary for the tissue magnetization to return to its steady state, i.e. parallel to the external magnetic field, after the RF pulse. Since an exchange of energy between protons and environment takes place, the T1 relaxation time depends on the nature of the surrounding molecules. For instance, the magnetization associated with lipids relaxes faster than that associated with much smaller, such as water, or much larger molecules, such as proteins. The second relaxivity property of tissue is the transverse or spin-spin relaxation, referred to as T2 relaxation. In this relaxation process the magnetic moments of the spins turn out of phase as a result of their mutual interaction. Each of the spins experiences a slightly different magnetic field and rotates at its own Larmor frequency, leading to loss of spin phase coherence (dephasing). As a result, the transverse magnetization Mxy decays. The longer the elapsed time, the larger the phase difference and the transverse magnetization decay. Unlike the T1 relaxation, no energy is transferred from the nuclei to the environment during the T2 relaxation. Each tissue has a characteristic T2 relaxation time that is referred to as the decay of the transverse magnetization. However, the actual rate of signal decay is faster than the prediction based on T2 . In fact, the actual observed transverse relaxation time, referred to as T2 *, is affected by local magnetic field inhomogeneities, which cause the precessional rates of the individual spins to differ from each other and dephase. An important parameter in MRI is the spin density ρ. The spin density is proportional to the effective number of hydrogen nuclei per unit volume contributing to the MR signal. Thus, the amplitude of M0 in (106) is proportional to ρ. The density ρ can be used together with T1 and T2 to distinguish different tissues. In general, the T2 * relaxation does not provide reliable information on the specific tissue. In order to obtain information related to the T2 relaxation time of tissue, a 180o RF pulse is used to invert the dephasing process and rephase the spins at a later moment as an echo. When the MR signal decay is evaluated for refocused spins, then the signal depends on T2 rather than T2 *. An example of FID refocusing is shown in Fig. 23. This is a typical Spin-Echo (SE) sequence, which begins with a 90o RF pulse that initiates the relaxation precession and is followed after TE/2 (TE is the echo time) by a rephasing pulse. This sequence can be repeated several times in order to obtain all the information that is necessary to allow reconstruction of

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Fig. 23.

139

T2 relaxation and echo formation.

an image from a slice of tissue inside the body. Spatial localization is fundamental for the reconstruction of an image. Spatial localization The spatial localization of the signal contribution per voxel is implemented by means →→ − → − of a spatial gradients G (− r ). These are added to the magnetic field B 0 . After a number of approximations, the expression of the MR signal S(t) detected by the → receiving coil from an object with spin density ρ(− r ) can be derived from (104) and 148 (107) as given in (108) :  → − → − → → ρ(− r )e−iγ( G · r )t d− r. (108) S(t) ∝ obj

After demodulation (i.e. removal of the carrier signal e−iγB0 t by a mixer) (108) can be written as:  → − → − → → ρ(− r )e−iγ( G · r )t d− r. (109) S(t) ∝ obj

− → If we define a real vector k as − G γt→ − → k = , 2π

then (109) can be written as → − S( k ) ∝



→ − → − → → ρ(− r )e−i2π K · r d− r.

(110)

(111)

obj

Expression (111) corresponds to a three-dimensional Fourier transformation of → − → the space − r into the space k , which is commonly defined as k-space. From (111) → it is clear that the reconstruction of an image ρ(− r ) consists of the Fourier inverse → − transformation of S( k ) (see Fig. 24). Notice that both the k-space and the output image are discrete. Since the origins of MRI, extensive developments have been implemented to reduce the time required to fill the samples of the k-space. The selection of a slice

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Fig. 24. LV short axis view (courtesy of dr. vd Bosch, Catharina Hospital Eindhoven, The Netherlands) in k-space and in real space after Fourier inverse transformation.

perpendicular to the z direction is usually performed by choosing the frequency of the RF pulse. Only the slice of which the Larmor frequency γ(B0 + Gz z)/2π corresponds to the frequency of the RF pulse is excited. Once a slice is selected, the space (kx , ky ) must be filled. Typically, the bi-dimensional k-space is filled line by line. This is accomplished by the combination of phase and frequency encoding. A phase encoding gradient ∆Gy is activated for a time Tpe before the signal readout in order to select one line of the k-space, while a frequency gradient Gx is activated during the readout to span the selected line of the k-space. For ∆t equal to the time sampling period during readout, the discrete k-space in Fig. 24 shows the following sampling distances: γ Gx ∆t, ∆kx = 2π (112) γ ∆Gy Tpe . ∆ky = 2π As for most discrete systems, the Nyquist threshold must be fulfilled in order to avoid aliasing. For an image size (Wx × Wy ), the Nyquist theorem imposes the following limits: 2π , γGx Wx 2π . ∆Gy = γTpeWy ∆t =

(113)

Apart from the Cartesian structure of the k-space given in Fig. 24, alternative strategies to fill the k-space exist. For instance, one can make use of polar coordinates, resulting in a polar sampling of the k-space made of concentric circles or a spiral trajectory. When polar coordinates are used, the image reconstruction is based on the Inverse Radon Transform.141,142 As an example, the description of a SE sequence is provided (see Fig. 25). A SE sequence starts with the slice selection gradient, which spreads out the Larmor frequency over a broad range, so that the frequencies contained in the RF pulse affect only one slice perpendicular to the z direction. Then the phase-encoding gradient

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Fig. 25.

141

Spin echo timing sequence.

Gy is pulsed briefly (time Tpe) after the first RF excitation pulse has been turned off, making the precession frequency depend on spatial position along the phaseencoding direction (y). Simultaneously, a frequency encoding gradient Gx is turned on to prepare the initial position to span the k-space line. After time TE/2 a 180◦ pulse is transmitted, so that an echo centered at time TE is generated and recorded during the readout time with the frequency encoding gradient Gx turned on. The following pulse is sent after time TR (repetition time) since the initial 90◦ RF pulse. Each sequence permits the registration of one line of the k-space. The MR signal SSE measured by a SE sequence depends on the settings TR and TE as well as on the tissue properties T1 , T2 , and ρ as given in (114):   ( 1 T E−T R) −T R −T E 2 T1 T1 SSE = ρ 1 − 2e e T2 . +e (114) Expression (114) can be derived from (105) in case of a SE sequence. It is clear that the dependency of the image contrast on T1 or T2 can be varied by choice of the scan parameters TE and TR. In that way, the contrast in the resulting image can be made depend strongly on either T1 or T2 , and the image contrast is referred to as T1 or T2 weighted, respectively. The SE sequence is one of the most common sequences that have been introduced in MRI. Another common technique is to leave out the RF refocusing pulse. This method is referred to as Gradient Echo (GE) or Field Echo (FE).146,148–150 An image is still generated by a number of excitation pulses that is equal to the number of lines. Parameters like the TR and TE can be used to control the image contrast.151

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The characteristics of the image and its relation with different tissues can also be controlled by use of pre-pulses (magnetic preparation), like for instance in Inversion Recovery (IR) pulse sequences. In general, even though the use of smaller flip angles (< 90◦ ) can increase the time resolution of GE scans, these techniques are too slow for real-time imaging of moving structures as well as for perfusion imaging. Especially for rapid processes, the need for a fast scanning technique becomes a fundamental requirement. Recent technical developments have increased the time resolution, which is now approaching that of X-ray CT scan and ultrasound echography. These improvements mainly consist of advanced strategies to fill the k-space with a reduced number of excitation pulses. Developments in this direction of SE and GE techniques result in the Fast or Turbo SE (FSE or TSE) and in the Echo Planar Imaging (EPI) techniques.146,150 By EPI, a single excitation pulse can lead to the reconstruction of the entire image (single shot technique). Combinations of EPI and FSE are also used and referred to as Gradient Echo And Spin Echo (GRASE) or Turbo Gradient Spin Echo (TGSE).150,152,153 Especially for GE techniques, the scan time can be reduced by using short flip angles and TR. These techniques are often referred to as Fast Field Echo (FFE).146,150,153 In this family of techniques, a surprisingly large number of members exists and the associated names differ per manufacturer. The signal is usually read during magnetic steady state, i.e. when the magnetization has reached a stable value depending on the flip angle and the sequence parameters TR and TE. Sometimes the readout is performed prior steady state to reduce the scanning time. This technique is referred to as Turbo Field Echo (TFE). In this case, the degree of T1 -weighing can be increased by use of inversion or saturation pre-pulses. A survey of MRI nomenclature for different manufacturers is presented in Ref. 154. Alternative techniques to boost the time resolution of MRI based on smart k-space construction strategies consist of exploiting the symmetry of the Fourier transform of a real signal.150,153 An example is the Half Fourier Single-Shot Turbo Spin Echo (HASTE) pulse sequence.155 As already mentioned, the k-space can be also filled by a radial or spiral trajectory.150,153 This permits to fill the center of the k-space in the initial part of the readout, resulting in better contrast and reduced sensitivity to motion. In fact, the external part of the Fourier spectrum contains limited information, and it could be filled by zeros (zero padding) to further decrease the scan time. Recent developments to increase speed of image production consist of the combined use of arrays of receiving coils and frequency-temporal interpolation of the k-space. These go by the names of SiMultaneous Acquisition of Spatial Harmonics (SMASH) or SENSitivity Encoding (SENSE) method.156,157 In sequential scanning of the dynamics of events, in addition to all techniques already mentioned, spatio-temporal k-space interpolation can often be used. A modern version of this technique is the k-t BLAST (or k-t SENSE if combined with the SENSE technique)

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method, where a least squares approach is used to treat aliasing problems due to simultaneous down-sampling in frequency and time.158 MRI contrast agents MRI contrast agents often are molecular constructs containing paramagnetic atoms such as gadolinium (Gd3+ ), iron (Fe2+ , Fe3+ ), and manganese (Mn2+ ) ions. Their influence on the MR signal is caused by the induced change in the relaxation times of nearby protons. By proper choice of the molecular structure, this effect is maximized. Since the contrast-agent enhancement is based on the alteration of the two relaxivity parameters, it can be categorized according to the shortening it produces on either T1 or T2 . Shortening of T1 is the result of dipole-dipole interactions, which occur when the frequency of time-varying magnetic fields produced by contrast agents (as a result of molecular rotation and tumbling) is similar to the resonance (Larmor) frequency of hydrogen nuclei. Increases in local magnetic field inhomogeneities enhance dephasing of non-stationary nuclei, resulting in decreased T2 or T2 *. A contrast agent that predominantly affects T1 relaxation is referred to as a positive relaxation agent because the enhanced shortening of T1 relaxation results in increased signal intensity on a T1 -weighted image. Instead, a contrast agent that predominantly affects T2 or T2 * relaxation is referred to as a negative relaxation agent because the reduced T2 * results in decreased signal intensity on a T2 -weighted image.159 The influence on the T1 and T2 * values of tissue can be formalized as given in (115) and (116), respectively146,160–163 : 1 (observed) = T1 1 (observed) = T2

1 + R1 C, T1 1 + R2 C. T2

(115) (116)

Here R is the relaxivity of the contrast agent and C is its concentration. At low concentration, the dominant effect often is the reduction of T1 so that T1 -weighted imaging is preferred in most contrast enhanced MRI applications. Especially gadolinium is currently used in several MRI studies. Its toxicity is reduced by chelation to molecules such as diethylene triamine pentaacetic acid (DPTA). Cardiovascular quantification Applications of contrast MRI in cardiovascular diagnostics are mainly qualitative. Contrast agent infusion is performed either to delineate the cardiovascular system (major arteries and cardiac lumina) or to assess the level of perfusion of the myocardium. The use of contrast agents improves the detection of stenoses and aneurisms in the vessels as well as abnormal structures in the cardiac cavities. The assessment of regional perfusion in the myocardium is an important procedure for the detection of infarcted areas, i.e. areas in which the tissue viability is reduced due to lack of oxygenation.164,165 In general, perfusion defects are related to stenosis

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or strokes in the coronary arteries, i.e. the myocardial perfusion arteries. Figure 26 shows a cardiac opacification by gadolinium infusion. As a consequence of (115) and (116), several recent studies confirm that an approximately linear relationship between MR signal and gadolinium concentration can be found for low concentrations.161,166,167 This finding has opened the way to novel perfusion imaging applications. Therefore, although the use of contrast MRI mainly aims at qualitative diagnostics, some studies are available that provide more quantitative measurements of blood perfusion and volumes. Perfusion studies in the myocardium, brain, kidney, and liver have been reported.167–170 Despite a linear relationship between contrast concentration and MR signal, an absolute relationship is difficult to establish because it is flow dependent.171 Due to their small size, gadolinium molecules can diffuse through the capillary wall into the interstitial space. Several multi-compartment models (see also Sec. 3.1) have been tested to describe this diffusion process.167,168,172–174 For some cardiovascular studies, the use of blood pool contrast is more suitable. Novel opportunities for perfusion can be provided by novel blood pool agents under development, such as Gd labeled albumin (e.g. MS-325, Epix Pharmaceutical, Cambridge, MA) and Gd-DPTA labelled dextran (e.g., Gadomer, Schering, Berlin, Germany), whose characteristic is a longer intravascular life time (slow clearance).167,175–177 Extravasations are avoided by albumin binding or larger molecular size for the first and the second class of contrasts, respectively. An interesting novel application of quantitative perfusion imaging relates to the assessment of the blood flow through the myocardium. To this end, a bolus of contrast agent is injected in a peripheral vein. The MR signal due to the bolus passage is registered versus time in the LV or the ascending aorta (input IDC) and in several regions of the myocardium (output IDC). The impulse response of the dilution system between the input and output IDC is estimated by a deconvolution technique (see also the end of Sec. 5.2.1).178,179 The dilution system is usually assumed to be represented by a mono-compartment model and the estimated impulse response is interpolated by an exponential decay (see Sec. 3.1). The time

Fig. 26. Four chamber view MRI before (left side) and after (right side) gadolinium infusion. Courtesy of dr. vd Bosch, Catharina Hospital Eindhoven, The Netherlands.

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constant of the fitted model, which corresponds to the MTT of the contrast between the detection sites, is used to estimate flow and perfusion. If calculated pixel by pixel, a new image that corresponds to perfusion timing (and therefore flow) can be constructed.

6. Conclusions and Outlook This chapter concerns the quantification of cardiovascular parameters by means of indicator dilution methods. The interpretation of an indicator dilution curve permits to characterize the dilution system where the indicator has passed. The mathematical background that is necessary for the interpretation of an indicator dilution curve is provided and tailored to the estimation of the cardiovascular parameters of interest. The main focus is on the quantification of cardiac output, blood volumes in the pulmonary circulation, ejection fraction of the ventricles, and perfusion of the myocardium. The clinical methods for the measurement of the indicator and the assessment of the clinical parameters of interest are introduced. A distinction is made between invasive methods that require catheterization and minimally invasive methods that use medical imaging techniques. The invasive methods are usually employed in anesthesiology, both in the operating rooms and in the intensive care units. The need for catheterization does not represent an obstacle for these applications and the derived parameters are considered important for monitoring the condition of the patient. Instead, the indicator imaging methods, with the exception of the X-ray angiography, are minimally invasive as only the injection of the indicator in a peripheral vein is necessary. Large part of the chapter is dedicated to contrast agent imaging techniques and, in particular, ultrasound and magnetic resonance imaging. The use of these minimally invasive methods for the measurement of indicator dilutions have two major advantages. The first one concerns the applicability of these methods without the need for hospitalization. This makes the methods available to cardiologists, adding diagnostic possibilities to standard cardiologic investigations made for instance by ultrasound or magnetic resonance imaging. The second one, still under development, is related to the possibility of measuring multiple indicator dilution curves, so that several central dilution systems can be characterized by means of system identification methods (deconvolution) without catheterization and central injection of the indicator. This adds diagnostic potential to these methods and represents a major advantage over invasive methods. Indicator dilution methods are under continuous development due to the introduction of new types of indicators. Novel technologies that extend the use of indicators from purely diagnostic applications to therapeutic ones are also growing. A clear example is given by the use of ultrasound contrast agents for myocardial perfusion. Apart from the assessment of perfusion and viability of the myocardial tissue, contrast agents can also be loaded with adenosine and used to locally release it for cardiac reflow therapy.

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149. V. Kuperman, Magnetic Resonance Imaging. Fisical Principles and Applications (Academic Press, San Diego, 2000). 150. M. A. Bernstein, K. F. King and X. J. Zhou, Handbook of MRI Pulse Sequences (Elsevier Academic Press, 2004). 151. W. R. Nitz and P. Reimer, Contrast mechanisms in MR imaging, European Radiology 9 (1999) 1032–1046. 152. K. Oshio and D. A. Feinberg, GRASE (gradient-and spin-echo) imaging: A novel fast MRI technique, Magnetic Resonance in Medicine 20(2) (1991) 344–349. 153. W. R. Nitz, Fast and ultrafast non-echo-planar MR imaging techniques, European Radiology 12(12) (2002) 2866–2882. 154. M. A. Brown and R. C. Semelka, MR imaging abbreviations, definitions and descriptions: A review, Radiology 213 (1999) 647–662. 155. R. C. Semelka, N. L. Kelekis, D. Thomasson, M. A. Brown and G. A. Laub, HASTE MR imaging: Description of technique and preliminary results in the abdomen, J. of Magn. Reson. Imag. 6, 4 (1996) 698–699. 156. K. P. Pruessmann, M. Weiger, M. B. Scheidegger and P. Boesiger, SENSE: Sensivity encoding for fast MRI, Magnetic Resonance in Medicine 42 (1999) 952–962. 157. M. Weiger, K. P. Pruessmann and P. Boesiger, Cardiac real-time imaging using sense, Magnetic Resonance in Medicine 43 (2000) 177–184. 158. J. Tsao, P. Boesiger and K. L. Pruessmann, k-t blast and k-t sense: Dynamic MRI with high frame rate exploiting spatiotemporal correlations, Magnetic Resonance in Medicine 50 (2003) 1031–1042. 159. D. D. Stark and W. G. Bradely, Magnetic Resonance Imaging (Mosby, London, third edition, 1999). 160. V. Dedieu, P. Fau, P. Otal, J. P. Renou, V. Emerit, F. Joffre and D. Vicensini, Rapid relaxation times measurements by MRI: An in vivo application to contrast agent modeling for muscle fiber types characterization, Magnetic Resonance Imaging 18 (2000) 1221–1233. 161. J. Morkenborg, M. Pederson, F. T. Jensen, H. S. Jorgensen, J. C. Djurhuus and J. Frokier, Quantitative assessment of Gd-DTPA contrast agent from signal enhancement: An in-vitro study, Magnetic Resonance Imaging 21 (2003) 637–643. 162. J. Zheng, R. Venkatesan, E. M. Haacke, F. M. Cavagna, P. J. Finn and D. Li, Accurancy of T1 measurements at high temporal resolution: Feasibility of dynamic measurement of blood T1 after contrast administration, J. Magn. Reson. Imag. 10 (1999) 576–581. 163. E. Toth, L. Helm and A. E. Merbach, Relaxivity of MRI Contrast Agents (SpringerVerlag, Berlin, 2002). 164. J. P. Vallee, F. Lazeyras, L. Kasuboski, P. Chatelain, N. Howarth, A. Righetti and D. Didier, Quantification of myocardial perfusion with fast sequence and Gd bolus in patients with normal cardiac function, J. of Magn. Reson. Imag. 9 (1999) 197–203. 165. T. Nakajima, N. Oriuchi, Y. Tsushima, S. Funabasama, J. Aoki and K. Endo, Noninvasive determination of regional myocardial perfusion with first-pass MR imaging, Academic Radiology 11 (2004) 802–808. 166. M. K. Ivancevic, I. Zimine, X. Montet, J. N. Hyacinthe, F. Lazeyras, D. Foxall and J. P. Vallee, Inflow effect correction in fast gradient-echo perfusion imaging, Magnetic Resonance in Medicine 50 (2003) 885–891. 167. E. Henderson, J. Sykes, D. Drost, H. J. Weinmann, B. K. Rutt and T. Y. Lee, Simultaneous MRI measurement of blood flow, blood volume, and capillary permeability in mammary tumors using two different contrast agent, J. of Magn. Reson. Imag. 12 (2000) 991–1003.

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168. E. P. Ruediger, M. V. Knopp, U. Hoffmann, S. Milker-Zabel and G. Brix, Multicompartment analysis of gadolinium chelate kinetics: Blood-tissue exchange in mammary tumors by dynamic MR imaging, J. of Magn. Reson. Imag. 10 (1999) 233–241. 169. L. Ostergaard, A. Gregory Sorensen, K. K. Kwong, R. M. Weisskoff, C. Gyldensted and B. R. Rosen, High resolution measurement of cerebral blood flow using intravascular tracer bolus passages. part II: Experimental comparison and preliminary results, Magnetic Resonance in Medicine 36, 5 (1996) 726–736. 170. J. A. Detre, J. S. Leighand D. S. Williams and A. P. Koretsky, Perfusion imaging, Magnetic Resonance in Medicine 23, 1 (1992) 37–45. 171. R. L. Perrin, M. K. Ivancevic, S. Kozerke and J. P. Vallee, Comparative study of fast gradient echo MRI sequences: Phantom study, J. of Magn. Reson. Imag. 20 (2004) 1030–1038. 172. P. S. Tofts, Modelling tracer kinetics in dynamic Gd-DPTA MR imaging, J. of Magn. Reson. Imag. 7(1) (1997) 91–101. 173. P. S. Tofts, G. Brix, D. L. Buckley, J. L. Evelhoch, E. Henderson, M. V. Knopp, H. B. W. Larsson, T. Y. Lee, N. A. Mayr, G. J. M. Parker, R. E. Port, J. Taylor and R. M. Weisskoff, Estimating kinetic parameters from dynamic contrast-enhanced T1-weighted MRI of a diffusible tracer: Standardized quantities and symbols, J. of Magn. Reson. Imag. 10 (1999) 223–232. 174. Y. Cao, S. L. Brown, R. A. Knight, J. D. Fenstermacher and J. R. Ewing, Effect of intravascular-to-extravascular water exchange on the determination of blood-totissue transfer constant by magnetic resonance imaging, Magnetic Resonance in Medicine 53 (2005) 282–293. 175. D. L. Kraitchman, B. B. Chin, A. W. Heldman, M. Solaiyappan and D. A. Bluemke, MRI detection of myocardial perfusion defects due to coronary artery stenosis with MS-325, J. of Magn. Reson. Imag. 15, 2 (2002) 149–158. 176. J. S. Troughton, M. T. Greenfield, J. M. Greenwood, S. Dumas, A. J. Wiethoff, J. Wang, M. Spiller and P. Caravan, Synthesis and evaluation of a high relaxivity manganese(II)-based MRI contrast agent, Inorganic Chemistry 43, 20 (2004) 6313– 6323. 177. M. Jerosch-Herold, X. Hu, N. S. Murthy, C. Rickers and A. E. Stillman, Magneticresonance imaging of myocardial contrast enhancement with MS-325 and its relation to myocardial blood flow and perfusion reserve, J. of Magnetic Resonance Imaging 18, 5 (2003) 544–554. 178. I. K. Anderesen, A. Szymowiak, C. E. Rasmussen, L. G. Hanson, J. R. Marstrand, H. B. W. Larsson and L. K. Hansen, Perfusion quantification using gaussian process deconvolution, Magnetic Resonance in Medicine 48 (2002) 351–361. 179. R. Wirestam, L. Andersson, L. Ostergaard, M. Bolling, J. P. Aunola, A. Lindgren, B. Geijer, S. Holtas and F. Stahlberg, Assessment of regional cerebral blood flow by dynamic suscettibility contrast MRI using different deconvolution techniques, Magnetic Resonance in Medicine 43 (2000) 691–700.

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CHAPTER 5 ANALYZING CARDIAC BIOMECHANICS BY HEART SOUND ANDREAS VOSS∗ , R. SCHROEDER† , A. SEECK‡ and T. HUEBNER§ ∗,†,‡ University of Applied Sciences Jena, Department of Medical Engineering and Biotechnology, Carl-Zeiss-Promenade 2, Jena D-07745, Germany ‡,§ Medtrans

GmbH, Tatzendpromenade 2 Jena D-07745, Germany

Diseases of the heart have become the Number One cause of death in the industrialized nations of the world. Every heart disease affects the biomechanics of the heart in a direct or indirect way. These effects can primarily be analyzed by the signals of the heart sounds and cardiac murmurs using techniques such as auscultation and phonocardiography. But these methods are very sophisticated and require a high degree of specialization. During the last years electronic stethoscopes and commercial PC techniques have improved essentially so an automatic analysis of heart sound has become a potential supporting tool for physicians in particular as a screening method for heart diseases by the general practitioner. This paper introduces a new automatic system to diagnose heart valve diseases based on time and frequency analyzing methods and feature extraction. It also describes an multivariate approach for an enhanced risk stratification in patients with heart failure considering the time interval between the ECG signal, especially the R-wave, and the first heart sound. In conclusion we could demonstrate, that analysis of the heart sound is a suitable method to evaluate the state of the heart and to detect changes on the biomechanics at an early stage. Keywords: Heart sound; heart valves; heart failure; frequency analysis; signal processing.

1. Introduction The incidence of heart diseases has increased dramatically during the last few decades, especially in the industrialized nations of the world. According to mortality statistics, heart diseases have become the Number One cause of death in these countries. For example, 27.3% of all deaths in the USA in 2004 were due to heart diseases (NCHS).1 The American Heart Association estimates for 2006 that the direct and indirect cost of cardiovascular heart diseases amount to more than $142.5 billion a year in the US alone.2 Every heart disease affects the biomechanics of the heart. Some of these diseases affect the biomechanical system directly (heart defects, diseases of the myocardium, heart valve disease and destruction of heart muscular cells by myocardial infarction), others have an indirect effect on it because of insufficient supply of the myocardial tissue or pathological changes of the generation or conduction of electrical stimuli (coronary heart disease, cardiac arrhythmia). 157

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Heart valve diseases, congenital heart diseases and myocardial diseases cause typical cardiac murmurs. Listening to the sound of the murmur and its different characteristics, the trained medical doctor is able to identify not only the type of a heart defect but often also the severity. Such characteristics include, for example, the frequency of the murmur (high or low frequency) or the way in which a murmur develops in a course of a heart beat, e.g. whether it starts faintly and becomes louder or it starts loud and then decreases. The relation of the cardiac murmur with the preceding and the following heart sounds is also important. Conventional stethoscopes in the market transmit the heart sound and murmur to the human ear via a diaphragm and airborne sound conduction. Auscultation is a purely manual method without any possibility of obtaining an objective measurement and analysis. This method depends entirely on the routine and experience of the examiner. Not infrequently, physicians of advanced age suffer from impaired hearing or are even hard of hearing. They can no longer sufficiently detect the very faint heart sounds and even more faint cardiac murmurs. For this reasons, efforts to develop electronic stethoscopes have been reinforced especially during the last five years. These stethoscopes record sound signals, for example, by small encapsulated microphones or vibration sensors and amplify them. Then the signals are output by small encapsulated headphones in different filter steps. In most cases, these devices do not improve the quality of the signals significantly, they only amplify them (and along with the signals also the noise). At present, there are only a few electronic stethoscopes which also permit recording the heart sound data in a PC or similar device. By recording the heart sound in a PC, phonocardiography is generally possible and the heart sounds and cardiac murmurs can be visualized. With the patient in supine position, the physician places a stethoscope (formerly a heart sound microphone, today an electronic stethoscope) at several points of the thorax (similar to established auscultation sites). The microphone/stethoscope is connected to the phonocardiograph (formerly) or the PC (today) which records the heart sound and displays it graphically. Because the microphone records every sound, including noise, the examination must take place in a low-noise environment. Besides, there should be neither breathing nor speaking for the time of the examination. The traditional phonocardiography is extremely demanding and as a rule is or was applied only by specialized cardiologists. Because echocardiography, as an imaging method, is a more reliable method for identifying congenital or acquired heart defects, phonocardiography is rarely used today. As in the industrialized countries of the world, ultrasonic examination equipment for echocardiographic examination has become standard for cardiologists in hospitals as well as in private practices, these phonocardiographic methods have been replaced, by and large. Phonocardiography was a purely visualizing method for manual diagnosis, without any automatic measurement or interpretation. In view of the fact that phonocardiography was pushed in the background, there is very little recent research in the area of heart sound analysis anywhere in the world. Automatic interpretation algorithms (similar to automatic ECG interpretation) are not known in daily medicine.

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The approaches discussed below are not intended to revive phonocardiography for the cardiologist; their objective is to transfer a modified form of phonocardiography as a generalized method of examination to the level of the general practitioner (GP). The interpretation of difficult to understand heart sound tracings can be automated and assist the family doctor in diagnosing heart diseases. Heart sounds are no longer recorded by a phonocardiographic sensor but a newly developed electronic stethoscope with low-noise high amplification of the signal and minimization of noise signals and external disturbances e.g. contact pressure, tremble or temperature influences. Electrodes integrated in the head of the stethoscope provide an ECG recording at the same time. The standardized digitization of ECG and heart sound, the control of electronic auscultation program and PC control and data transmission are handled by a small manual control. The newly developed methods of heart sound analysis in time and frequency domains will be discussed in detail. The gold standard for the evaluation and classification, especially of subjects and patients with heart valve disease, were parameters from echocardiographic reference examinations. It is a main objective to obtain information via heart sound analysis comparable to that obtained by echocardiographic examination for early detection of pathological changes of biomechanical processes as heart valve diseases. To reduce the number of surgeries on heart valves it is necessary to identify the defect as early as possible. If a correct diagnosis is available in time, surgical operation could be avoided by the reduction of risk factors by the patient and/or treatment with drugs which might still be effective at that stage or at least could be delayed significantly. There may be cases in which a heart valve can be corrected surgically instead of replacing it, which would avoid other consequential factors significantly impairing the quality of a patient’s life, such as life-long anticoagulation in case of implanted artificial mechanical heart valve. A patient with complaints will at first consult his family doctor or nearest GP. The further efficient medical treatment depends on the family doctor’s or GP’s ability of assessing the patient’s complaints and refer him or her to the right specialist in time. Hence, the GP or family doctor should already be in a position to diagnose and interpret pathological biomechanical changes, e.g. heart valve defects, by a simple and reliable method.

2. Background 2.1. The biomechanics of the healthy heart The primary function of the heart is to pump blood through blood vessels to the body’s cells. To make the supply as efficient and effective as possible, high-precision valve systems are needed. This function is served in the body especially by the heart valves. In fact, the heart valves control the blood flow and its direction in the heart and are opened and closed by the pressures generated in the heart.3

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Fig. 1. The heart and the heart valves (T Tricuspidal valve, P Pulmonary valve, M Mitral valve, A Aortic valve).43

The heart consists of four chambers, the right and left atrium and the right and left ventricle. The atrium and ventricle of each side of the heart are separated by heart valves (atrioventricular valves). These are the tricuspid valve in the right half of the heart and the mitral valve in the left half (Fig. 1). Two other valves located at the two points where blood exits the heart: these are the pulmonary valve (at the exit of the right ventricle into the pulmonary artery) and the aortic valve (exit of the left ventricle into the aorta). Both these valves are also known as semilunar valves (from Latin semi = half, luna = moon).3,4 A cardiac cycle is divided in two stages, the systole and the diastole. The systole is the contraction phase of the heart, the diastole is the phase during which the heart relaxes. The systole is again divided into two phases. During the tension phase (approx. 50 ms), the contraction of the myocardium causes the intraventricular pressure to rise quickly and the atrioventricular valves (AV valves) to close. At this time, the pulmonary and aortic valves are also closed because the pressure in the ventricle is still below the pressure in the outgoing blood vessels (Fig. 2; phase I). The tension of the ventricular muscles around the incompressible blood volume now leads to a pressure increase. If the pressure in the ventricle exceeds the pressure in the vessels, the semilunar valves open and the ejection phase (approx. 210 ms) begins (Fig. 2; Phase II). The pulmonary valve opens at a pressure of about 10 mmHg, whereas a much higher pressure of about 80 mmHg is needed for opening the aortic valve. The reason why the pressure in the right ventricle is much lower than in the left is the

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very much weaker resistance of the vessels in the pulmonary circuit. The ejected blood volume is the same and is about 70 ml at rest. This volume is referred to as ejection fraction and amounts to approx. 50% of the blood volume in the ventricle. In this phase, the atria also fill with new blood from the large body veins. Following the systole with tension and ejection phase is the diastole, which is also divided in two phases. The relaxation phase (approx. 60 ms) begins when the ventricle pressure drops below the arterial pressure. This causes the pulmonary and aortic valves to close so that all heart valves are closed at that time (Fig. 2; phase III). The pressure in the ventricle drops very quickly to below the pressure level in the atria. The AV valves open and the blood volume in the atria can flow in the ventricles. This phase is the filling phase (approx. 500 ms) (Fig. 2; phase IV). If the needed blood volume has flown in the ventricles, the next tension phase begins.4,5 Figure 2 illustrates the time relationship between ECG, heart sound and the corresponding mechanical action of the heart during the four phases of cardiac action. 2.2. Heart sounds and cardiac murmur Heart sounds and cardiac murmurs6,7 can be identified during a heart action (Fig. 3). Heart sounds include vibrations up to 0.1 s caused by the vibration of structures capable of oscillation. Cardiac murmurs are vibrations of longer than 0.1 s duration and are generated by turbulent flow of blood within the heart. Murmurs are also transmitted to structures capable of vibration. 2.2.1. Heart sounds During a cardiac cycle, a healthy heart produces two heart sounds which are rather intensive compared to the other heart sounds. These heart sounds differ in amplitude and frequency. The first heart sound is associated with the closing of the AV valves, i.e. the tricuspidal and mitral valves, at the beginning of ventricular contraction, or systole and ejection of the ventricle volume. The origin of the second heart sound is the closing of the semilunar valves, i.e. the aortic and pulmonary valves, at the end of ventricular systole. As the right/left ventricle empties, its pressure falls below the pressure in the aorta/pulmonary artery, and the respective valve closes. 2.2.1.1. The first heart sound The first heart sound begins about 0.01 to 0.03 s after the Q wave of the electrocardiogram (electromechanical latency). It can be divided into three sections with different frequencies and physiological origins. The pre-segment with lower frequencies of up to about 50 Hz is caused by the first movements of the ventricles. The main-segment starts approximately 0.06 s after the beginning of the QRS complex and comprises frequencies between 60 and 100 Hz. It is produced by the sudden tension of the cardiac wall after closure of the AV valves, when the ventricle

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Fig. 2. ECG, heart sound and heart mechanics during a cardiac cycle. Pressures and volumes and valve states during the different action phases of the heart.

contracts around the non-compressible blood volume and due to this starts oscillating, including the valve mechanism. The post-segment with frequencies up to about 30 Hz already marks the beginning of the ejection phase. The semilunar valves have opened under the ventricular pressures and the ventricle volume flows into the aorta

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Fig. 3.

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Heart sound, heart murmur and ECG [5].

and the pulmonary artery, respectively, which also start oscillating due to the pressure rise in the vessels. 2.2.1.2. The second heart sound The second heart sound comprises higher frequencies of up to 150 Hz. It is of shorter duration than the first heart sound and of higher pitch. It is generated by pressure changes and vibration of valves and contiguous structures induced by motion of the aortic and pulmonic leaflets toward their respective ventricles. The second heart sound can be divided into an aortal and a pulmonary section, with the aortal section generally preceding the pulmonary section. 2.2.1.3. The third and fourth heart sounds Occasionally, a third heart sound occurs during diastolic rapid inflow of the blood into the ventricle. This can be heard only in children or young adults due to their more favorable conditions of sound conduction. The third heart sound is low in frequency and intensity. A fourth heart sound can occasionally be noted between the P wave and the Q wave in the ECG. This heart sound is caused by the ejection phase of the atrium just before the first heart sound. Under normal auscultation conditions,a this sound is not audible. In addition to the third and fourth heart sounds, further extra sounds, such as the opening sound of the mitral valve or tricuspidal valve, may occur. Extra sounds are also caused by artificial heart valves.

a Auscultation — Listening, mostly with a stethoscope, to the murmurs and sounds produced by the body.8

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The intensity of the heart sounds depends on a multitude of cardiac and extracardiac factors, such as: • the intensity of the heart action or the force and speed of contraction of the heart muscles, • the oscillation capability of the heart valve apparatus, which can be impaired by calcification or scarring, • the ventricular mass; the larger the ventricular mass, the lower the frequency and amplitudes of the oscillations, • the blood viscosity. 2.2.2. Cardiac murmurs Cardiac murmurs are caused most frequently by blood flow turbulences. In cases where the flow velocity of fluid within a pipe exceeds a certain value, turbulence develops and energy is dissipated generating audible vibrations. The determining factors in this process are the viscosity of the fluid, the lumen and the flow rate. Practically, the sound level of the murmur depends on the rate of the blood flow, the viscosity of the blood, the oscillation capability of the cardiac wall and the valve apparatus and the big vessels leading away from the heart, the diameter of the opened valve and the width of the vessels downstream the valves. Thus, the sound level of the murmur is not necessarily an indicator of the severity of a valve disease. 2.3. The pathological heart Basically, there are five different groups of heart diseases: (1) (2) (3) (4) (5)

Arrhythmia Coronary heart diseases Myocardial diseases Heart defects Valve diseases

These five groups of heart diseases can have significant effects on the biomechanical function of the heart and can be diagnosed by medical specialists/cardiologists using the usual cardiological diagnostic methods, such as heart catheter (HC), echocardiography (Echo), 12-channel electrocardiography (ECG), ergometry (ERGO) and long term Holter ECG (LECG). Opposed to these is the very exact (but expensive) method of computer tomography (CT). Phonocardiography (Phono) can take over part of the diagnostic action (Table 1). 2.3.1. Cardiac arrhythmias Cardiac arrhythmias are abnormalities in rate, regularity or sequence of cardiac activation. In a healthy person, the cardiac rhythm is generated by the sinus node

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165

Methods for diagnosing heart diseases.

Arrhythmia (blocks, tachycardia, extra-systole) Coronary heart disease (ischemia, infarction) Myocardial diseases (cardiomyopathy) Heart defect Atrial and ventricular septal disease Valve diseases (insufficiency, stenosis)

ECG +LECG

ECG +ERGO

x

x

Echo

x

HC

CT

x

x

Phono

x

x

x

x

x

x

x

x

x

generating electrical impulses (about 60–100 beats per minute) which are conducted via the AV node and along specific pathways to the ventricles, where they cause ventricular contraction. Arrhythmias are caused by malfunctions of the heart beat generation centers or the electrical conduction system. There are different types of arrhythmias: • Tachycardia: heart rate that is faster than the normal range (from 100 up to 200 heartbeats per minute). This reduces the pumping efficiency and increases the oxygen consumption of the heart. • Bradycardia: heart rate that is lower than the normal range (less than 60 heartbeats per minute) at rest (threshold). As a consequence of this, the body is not supplied with enough oxygen and nutrients. The threshold heartbeat is variable; for example, the heart rate at rest can be substantially lower in trained sportsmen. • Atrial fibrillation or atrial flutter: the electrical signals that coordinate the muscle of the upper atria of the heart become rapid and disorganized, typically causing the atria to beat faster than 300 beats per minute. The myocardium has insufficient time for coordinated blood filling of the ventricles. This impairs the pumping performance and leads to a reduced blood flow to the body. • Ventricular fibrillation: The frequency of the ventricle contractions rises up to 300 beats per minute. This prevents further pumping of the blood. This state is also known as functional cardiac arrest. • Heart block: This is a delay or inhibition of the pathway of the stimulus. The following blocks are known: sinoatrial (SA) block (block between sinus node and atrium), atrioventricular (AV) block (block between atrium and ventricle), and bundle-branch block (block in ventricular pathway). • Premature beats: Sudden heartbeats outside the normal baseline rhythm (if originating in the ventricle: ventricular premature beats; if originating in the atrium: premature atrial contractions).

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2.3.2. Coronary heart disease A coronary heart disease (CHD) is a constriction or occlusion (narrowing) of one or several coronary arteries. As a consequence the blood supply of the heart muscle cells is reduced. The most frequent cause of CHD is an arteriosclerosis of the coronary arteries. Fat deposits on the walls of the vessels, which together with calcium form so called plaques. The vessel loses the ability to expand and constricts gradually. If a vessel is occluded, the person suffers a myocardial infarction. The risk of coronary heart disease is increased by having a family history of coronary heart disease before age 50, older age, stress, smoking, high blood pressure, high cholesterol, diabetes, lack of exercise, and overweight. Treatment includes the elimination of the risk factors and a number of methods of vessel expansion and drug therapy. Constricted vessels can be bridged by implants inserted in a by-pass operation. 2.3.3. Myocardial diseases A myocardial disease is a structural change of the myocardium caused by the changes or degeneration of individual muscle cells. Myocardial inflammation is a frequent cause, others are drug or alcohol abuse or hereditary disposition. In all such cases, the heart becomes weak, and its capacity to pump is diminished. Symptoms of cardiac insufficiency (heart failure) start to develop. This change is seen as a thickening, enlargement or stiffening of the ventricular muscles. There are three different types: • Dilatative cardiomyopathy: This disease implies enlargement of the left or right or of both ventricles with restriction of the force of contraction. The ventricular systolic pump function of the heart is impaired, leading to progressive cardiac enlargement and hypertrophy (remodeling). Possible consequences are arrhythmia, cardiac insufficiency or sudden cardiac death. • Hypertrophic cardiomyopathy: Typical of this disease is a massive muscle thickening and the ventricular cavities are small without an obvious cause. In addition, microscopic examination of the heart muscle in HCM shows an abnormal alignment of muscle cells. The walls are stiff, and relax poorly during diastole, leading to increased end-diastolic pressure and resulting in pulmonary congestion and dyspnea. • Restrictive cardiomyopathy: While the heart rhythm and the pumping action seem to be healthy, the stiff walls of the heart chambers keep them from filling normally. So blood flow is reduced, and blood that would normally enter the heart is backed up in the circulatory system. 2.3.4. Heart defect A heart defect is a congenital or acquired specific structural phenomenon of the heart or the adjacent vessels impairing the function of the cardiovascular system.

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Most frequent are atrial or ventricular septum diseases and valve anomalies (e.g. Ebstein anomaly, aortal stenosis, and mitral valve prolaps).

2.3.5. Heart valve diseases Most heart valve diseases are acquired defects. Only about 1% of heart valve diseases are congenital. Acquired heart valve diseases can have several causes. About one in three of all acquired heart valve diseases is due to calcification and degeneration. These diseases are often due to the typical ways of life in the industrialized nations. Unhealthy food, overweight and lack of physical exercise mostly lead to arteriosclerosis and high blood pressure. As people become older, calcific degeneration of the valves increases and the valves lose mobility and become stiff. As a consequence of this, heart valves fail to open and close properly, possibly hampering the heart’s ability to pump blood adequately through the body. Due to the high mechanical load of the aortic valve, this is affected most often. The most frequent cause of aortic valve stenosis is arteriosclerosis. Compared with other valve diseases, aortic valve diseases have become most frequent.8 Another cause of heart valve disease is endocarditis, an inflammation of the endocardium. It can be caused by bacteria or after an infection with rheumatic fever. The valve cusps and flaps are part of the endocardium. During endocarditis, they are also affected by bacteria. This leads to swelling and conglutination of the cusp or flaps leading to valvular stenosis and/or insufficiency. Until 1970, mitral stenosis was the most frequently diagnosed heart valve disease. The combined mitral valve disease (stenosis and insufficiency) occurred second most frequently. Mitral valve diseases almost always occur in connection with rheumatic fever. Advances in the treatment with antibiotics have dramatically reduced the incidence of these diseases. Today, aortic valve diseases (aortic valve stenosis, in particular), account for 65% of all heart valve diseases (Table 2) whereas the mitral valve defects has dropped down to about 30%.9 Acquired heart valve diseases can also occur after a preceding myocardial infarction. The AV valves are most at risk because a myocardial infarction usually damages the muscles which prevent disruption of the cusps when the ventricle becomes tense. These muscles develop small fissures and become necrotic with time causing valve insufficiency. Often, mitral valve insufficiency is a consequence. Right-side heart valve disease is very rare and mostly congenital or the consequence of severe left-sided insufficiency of the heart. Table 2.

Frequency of acquired heart valve diseases.

Valves disease left-sided right-sided

Aortic valve disease Mitral valve disease Pulmonary and tricuspid valve disease

Frequency 65% 30% 5%

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Heart valves can have one of two malfunctions leading to cardiac murmur: • Regurgitation The valve does not close completely, causing the blood to flow backward instead of forward through the valve. • Stenosis The valve opening becomes narrowed or does not fit properly, inhibiting the ability of the heart to pump blood to the body due to the increased force required to pump blood through the stiff (stenotic) valve. (Besides, shunts, can cause turbulence in the blood accompanied by cardiac murmur. As they are rare, they will not be considered in the following.) Stenosis or insufficiency can occur alone or in combination with each other at each of the four heart valves. It is important for diagnosis, in which phase of the heart action the murmur occurs: • if the AV valves are insufficient, blood flows back into the atria during contraction of the ventricles. Therefore, murmur is in the systole of the heart action, directly after the first heart sound. • if the AV valves are stenotic, the murmurs are caused by the flow of blood from the atrium in the ventricle, i.e., during the diastole after the second heart sound. • if the semilunar valves are insufficient, the blood, after ejection from the heart, flows back from the vessels into the ventricles. The murmur occurs during the diastole, immediately after the second heart sound. • if the semilunar valves are stenotic, turbulences occur during ejection of the blood from the ventricles into the large vessels during the systole after the first heart sound. In simplified terms: At the AV valves, stenoses cause diastolic murmur, insufficiencies cause systolic murmurs. At the valves to the large vessels, it is vice versa.6,7 In this way, typical heart sound curves are obtained for each heart valve disease; which are illustrated in Fig. 4. In the following, the examined heart valve diseases will be described in greater detail as to their origin and specific change of the heart sound. 2.3.5.1. Aortic valve stenosis Normally, the opening area of the aortic valve is about 2.5 to 3 cm2 . Blood flows through it with a velocity of up to 0.2 l/s during the systole. If the aortic valve is stenotic (narrowing greater 50%), the pressure in the left ventricle increases (up to 300 mmHg) and a pressure difference occurs between the left ventricle and the aorta. To compensate for the increasing resistance at the aortic valve, the muscles of the left ventricle thicken to maintain pump function and appropriate cardiac output. This muscle thickening causes a stiffer heart muscle which requires higher pressures in the left atrium and the blood vessels of the lungs to fill the left ventricle.

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Typical heart sound characteristics of different heart diseases.10

Even though these patients may be able to maintain adequate and normal cardiac output at rest, the ability of the heart to increase output with exercise is limited by these high pressures. This constantly higher pressure causes hypertrophy (pathological enlargement of an organ in response to the extra load) of the left heart, which again results in reduced coronary blood circulation and increased oxygen demand of the heart. Affected patients complain of pain in the chest, loss of efficiency, respiratory distress and sudden unconsciousness, mainly under physical strain.11 The changes of the heart sound curve are typical and, as a rule, easy to find by auscultation in patients with higher severity of the disease. Typical of it is a spindle-like murmur during the ejection phase, i.e. between the first and second heart sounds. This murmur reaches its maximum the later the stronger the stenosis. Its intensity increases if the stroke volume increases, e.g. under physical strain. Another change of the heart sound curve occurs at the heart tones. As the mass of the left ventricle increases, the oscillation frequency is lower and the attenuation stronger, letting the first heart sound appear weaker. In cases of very severe aortic valve stenosis, the sequence of the aortic and the pulmonary valve portions can occur inversely at the second heart sound because the systole at the aortic valve is extended. The hardening of the aortic valve and the related decrease in the oscillation capability also make the second heart sound weaker.6,7,12

2.3.5.2. Aortic valve insufficiency After ejection of the blood from the left ventricle, the pressure in the ventricle drops below the pressure in the aorta and the aortic valve closes. This causes a pressure gradient towards the ventricle to be established. If the aortic valve is insufficient, i.e. leaks, blood flows back into the heart. Aortic valve insufficiency can be congenital

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or caused by inflammatory changes of the valves (endocarditis after rheumatic fever or bacterial infection). The consequences of aortic valve insufficiency depend on the amount of backflow (regurgitant blood flow) which, in turn, is determined by the size of the leaky opening area and the pressure difference and the duration of the diastole. To ensure that a sufficient amount of blood reaches the periphery, the stroke volume must be increased. Constant pumping of a higher blood volume finally causes dilatation and hypertrophy of the left ventricle. Unlike the aortic valve stenosis, patients are not necessarily incapacitated by high physical strain because when the heart rate is higher, less time is available for the diastole, during which blood can flow back. Dizziness, respiratory distress and physical capability at rest and during light physical activity are observed.7,11 The main indication for aortic valve insufficiency in the heart sound curve is a decrescendo murmur which directly follows the second heart sound. The duration of the murmur depends on the severity of the insufficiency. In severe cases, it can extend into the presystole. Another typical indication is a fainter second heart sound in severe cases of the disease. The reason for this is the reduced oscillation capability of the aortic root and the aortic valve.6,12 2.3.5.3. Mitral valve insufficiency The closing ability of the mitral valve depends on the function of the cusps and the mitral ring, also on the papillary muscles and tendons which tension the cusps. If the valve cusps do not close completely, blood flows back from the left ventricle into the atrium during the systole. Hence, an additional regurgitant blood flow between the atrium and the ventricle must be managed without being available for circulation. In the long term, this extra muscle work also causes hypertrophy of the left heart. Patients suffering from mitral valve insufficiency cannot perform normally and fatigue more easily due to insufficient supply of blood. The number of mitral valve insufficiency after myocardial infarction has increased especially in recent years. But also endocarditis or a Marfan’s syndrome (hereditary disease of the connective tissue) are potential triggers of the disease. Endocarditis causes the cusps and tendinous cords to shrink, become thick or stiff, which impairs valve closure. Another frequent cause of mitral valve insufficiency is the Barlow’s syndrome (also mitral valve prolaps syndrome), which is characterized by too long tendinous cords. The cusps bulge into the atrium far enough as to open again. This effect is also caused by a necrosis of the papillary muscles (tensor muscles) after myocardial infarction.7,11 By auscultation, mitral valve insufficiency is identified as a murmur directly after the first heart sound. This murmur is heard loudest at the apex of the heart. The murmur decrescends towards the second heart sound but ends before the second heart sound starts because the contraction of the ventricle causes the leaky spot to become smaller. Only in very severe cases will the murmur persist at the same level

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until the second heart sound. The difference to aortic valve stenosis is in the onset of the murmur. Whereas the murmur of mitral valve insufficiency follows the first heart sound directly, a pause until the actual expulsion phase can be considered in the case of aortic valve stenosis. Another typical feature of the heart sound curve in the presence of mitral valve insufficiency is the fainter first heart sound. Changes of the valve (e.g. due to calcification) impair the latter’s capacity for vibration. Besides, less tension energy is available. In most cases, the third heart sound is also more pronounced. As blood flows back into the atrium, the latter is filled strongly at the end of a systole. This large volume very quickly flows into the ventricle as a wave at the beginning of the diastole. In addition, a click murmur can occur as third heart sound in patients with Barlow’s syndrome; it is caused by the reversal of the cusps in the atrium due to overlength of the tendinous cords.6,12 2.3.5.4. Tricuspid valve insufficiency Typical of a tricuspid valve insufficiency is the backflow of blood from the right ventricle in the right atrium. The most frequent cause is preceding rheumatic fever, which is often accompanied by mitral valve disease. In this case also, the cusps and the tendinous chords shrink, deform or become thicker. As with mitral valve insufficiency, auscultation will note a systolic decrescendo murmur after the first heart sound. Frequently, the level of the murmur is constant. Compared with murmurs of the left heart, the loudness of the murmurs with tricuspid valve insufficiency increase during inspiration. Occasionally, there is an inverse splitting of the second heart sound due to the shorter systole in the right half of the heart. Often, the third heart sound is more pronounced because of the large volume flowing into the right ventricle.6,7 2.3.5.5. Pulmonary valve insufficiency Pulmonary valve insufficiency occurs if the blood flows back from the pulmonary artery into the right ventricle during the diastole. Pulmonary valve insufficiency alone is a very rare disease. Frequently the causes are dilatation of the valve ring in case of pulmonary hypertension, but also connective tissue disorders, for example, the Marfan’s syndrome. Inflammatory processes or malformation are involved rarely. Pulmonary valve insufficiency leads to extra volume strain on the right ventricle. Because it usually occurs together with pulmonary hypertension, the consequences for the patient are normally determined by the hypertension. As with aortic valve insufficiency, the principal acoustic feature is a diastolic decrescendo murmur. The difference is in the beginning of the murmur. The earliest possible onset is after the pulmonary section of the second heart sound, but mostly after a short pause. The frequency and duration of the murmur depend on the pressure in the pulmonary artery.6,7 Table 3 summarizes the important heart sound indicators of heart valve diseases.

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Typical changes of the heart sound caused by heart valve diseases.

Heart valve disease Aortic valve stenosis Aortic valve insufficiency Mitral valve insufficiency Tricuspid valve insufficiency Pulmonary valve insufficiency

Heart sounds First and second heart sounds fainter Second heart sound fainter First heart sound fainter Frequent third heart sound No change

Cardiac murmurs Spindle murmur between first and second heart sound Decrescendo murmur directly after the second heart sound Decrescendo murmur after the first heart sound Decrescendo murmur after the first heart sound Decrescendo murmur after a short pause following the second heart sound

2.3.6. Biomechanical replacement systems 2.3.6.1. Replacement of the heart valves The target of heart valve surgery is to recover the impaired pumping capacity of the heart caused by pathological changes of the valves. Because natural human heart valves are unique in life and functionality, the first method of choice is to repair the diseased heart valve whenever possible (valve reconstruction) and spare the patient possible disadvantages of an artificial heart valve. The repair option of the heart valves is limited by marked calcification and severe changes of the valve cusp tissue. In these cases, replacement of the heart valve is the only possibility. The ideal heart valve replacement should meet the following requirements: • unlimited durability • normal blood flow conditions in the vessel (there should be no pressure difference before or after the implantation of the heart valve replacement and the blood flow through the heart valve replacement should be normal and without obstruction) • no complications due to the heart valve replacement, e.g. increased thrombogenicity (clot formation), susceptibility to endocarditis (inflammation of the endocardium) • no complications due to the heart valve replacement, e.g. cradle fracture • simple to implant • low-murmur and comfortable • constant availability None of the present valves meets all these criteria. Generally, there are two different valve types: biological heart valves, which are divided further into biological prostheses (xerografts) and human valves (homografts), and mechanical artificial heart valves. All types have specific advantages and disadvantages. Which valve type is used depends on age, medication, other diseases and the personal conditions of the patient.

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Bioprostheses (xerografts) Bioprostheses are heart valves made of animal material. Some prostheses are made from pig heart valves (porcine prostheses) and some prostheses from bovine pericardial tissue (bovine prostheses). The prostheses are prepared by special chemical and physical methods. Their function and appearance is very similar to those of human heart valves. Some tissue heart valves are sewn onto a plastic valve frame to which an external ring of dacron or teflon tissue is applied, which ensures easier implantation. Other tissue heart valves have no frame. The advantage of biological heart valves prostheses is that their hemodynamic behavior is very similar to that of the natural human heart valve. Besides, anticoagulation treatment is not normally required. The essential disadvantage of bioprostheses is their limited durability; about 5% of all modern bioprostheses in 65-year old patients show marked changes after about 10 years. Degenerative changes, calcification, in particular, occur the earlier the younger the patients. Another problem is sudden avulsion or tearing of the cusp, which can lead to massive cardiac insufficiency within very short time. Human valve replacement (homografts) These valve replacements are heart valves obtained either during a post mortem or a heart transplantation. They are prepared and preserved for storage in an organ bank. Generally, the advantages and drawbacks are the same as those of animal heart valves. The durability here is between 15 and 20 years. Mechanical heart valves prostheses Mechanical heart valves prostheses are heart valves made of extremely durable materials, such as metal or plastic. They have an outer ring of synthetic tissue (dacron or teflon). The purpose of the ring is to sew the heart valve in the tissue of the patient’s heart. The large number of different mechanical heart valve prostheses at present in clinical use indicates that the ideal prosthesis has still not been found. The big advantage of mechanical heart valves in comparison with bioprostheses is in their virtual unlimited durability and constant availability. However, extensive anticoagulation therapy is required, in spite of which thrombo-embolic complications are frequent. The risk of endocarditis is also higher in case of a mechanical heart valve replacement. 2.3.6.2. Replacement systems for cardiac arrhythmia If cardiac arrhythmia cannot be controlled by drug therapy, a replacement system may have to be implanted. There are the following systems: • Cardiac pacemakers, which are implanted in cases of low heart frequency, sinus node disorder or a heart block. If the heart frequency is low, the pacemaker sends electrical signals to the heart through electrodes which cause the heart to contract.

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• Implantable cardioverter/defibrillators (ICD), which are implanted preferably in patients with ventricular tachycardia. If a life-threatening arrhythmia (tachycardia) occurs, the device releases an electric shock which normalizes the heart frequency. 2.3.6.3. Cardiac assist device and artificial heart A cardiac assist device is usually used in addition to the natural heart in case of heart insufficiency, to bridge the time until a donor organ is available. Another use is as temporary relief of the heart if the heart muscles can regenerate, e.g. after a severe myocarditis. The term artificial heart describes an electromechanical pumping system within (intracorporeal) or outside (extracorporeal) the body which unlike the cardiac assist device takes over the complete function of the natural heart.

3. Methods of Analyzing Cardiac Biomechanics 3.1. Computer tomography (CT) and magnetic resonance tomography (MRT) Until a few years ago, computer tomography and magnetic resonance tomography were only used as imaging methods for the anatomy and morphology of the heart. The main uses were and still are congenital heart diseases, heart tumors and the diseases of the thoracic aorta. Rapid progress in science and technology in the last few years has opened new applications for both methods in hospital routine examinations of heart function. For example, functional MRT can be used to assess function, perfusion and vitality by a single examination. MRT is an imaging process for anatomy and morphology, malformation of coronary arteries, quantification of valvular defects and shunt defects, the examination of myocardial diseases and the assessment of the metabolism of the heart. The focus of CT is on the examination of the coronary arteries, in particular, the quantification of coronary calcium.13 3.2. Ultrasound — echocardiography In addition to electrocardiography (ECG), echocardiography is one of the most important non-invasive examination methods of the heart and by now an indispensable tool in cardiological diagnostics. It allows to assess the function of the ventricles, the atria and the heart valves. Quantification of defects by measuring certain parameters is also possible. Congenital heart defects can also be diagnosed reliably. In particular, the examiner obtains information on the structure of cardiac walls and heart valves as well as their movements, the wall thickness of heart atria and ventricles, the size of heart chambers and the ejection capacity of he heart.

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3.3. Heart sound — auscultation and phonocardiography Many heart diseases, most of all the heart valve diseases and the congenital heart defects, produce typical cardiac murmurs due to a sometimes massive change of the biomechanics of the heart (see Chapter 2). Mostly, cardiac murmurs are the first symptoms of a pathological process at the heart valves. For a long time, auscultation was the only method by which heart valve diseases could be detected.14 Auscultation is listening with a stethoscope to sounds produced by the heart, lungs, and blood. The cardiac murmurs which the physician hears are diagnosed directly. Phonocardiography was introduced by EINTHOVEN as late as in 1907.15 Phonocardiography makes the murmurs visible as a curve and records them. From these curves, several parameters can be selected for diagnosis. Mostly, the curves are referenced to the heart curves of healthy or other pathological heart curves. Despite its high diagnostic potential, phonocardiography never prevailed in everyday hospital work. One reason for this is the sensitivity of this method and, in relation with it, the difficulty in obtaining noise-free records whose quality is sufficient for exploitation.16,17 Likewise, the interpretation of heart sound curves requires high medical qualification and clinical experience on the part of the physician.6,7 An important step ahead in phonocardiography was made by MAASS and WEBER in 1952. They introduced a standardized recording technology (amplifiers, filters, etc.) allowing more exact comparisons between different heart sound curves.18 The introduction of ultrasonic diagnostics and especially of echocardiography (first examinations were made by KEIDEL around 1950) almost completely displaced phonocardiography. This imaging method is a safe and exact diagnostic tool but requires substantial technical equipment and related high costs and is, as a rule, used only by cardiologists and internal specialists but not by the general practitioner.19 Only the development of electronic stethoscopes gives diagnosis by means of heart sounds a new lease of life.

4. Heart Sound Evaluation for Analyzing Cardiac Biomechanics 4.1. Stethoscopes and heart sound sensors Whereas, in earlier times, a difference used to be made between purely acoustic stethoscopes and sensors for phonocardiography, today these functions mostly merge to a single product, the electronic stethoscope, it “hears”and “records” at the same time. 4.1.1. State of the art The present stethoscopes and sensors can generally be grouped in three categories (Fig. 5): • conventional, purely acoustic stethoscopes,

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Fig. 5.

Overview of stethoscopes and sensor principles.

• electronic stethoscopes, airborne sound microphone principle, • electronic stethoscopes, structure-borne sound or vibration sensor principle. Conventional stethoscopes with and/or without diaphragm are for the purely acoustic hearing of heart sound phenomena and subjective diagnostic findings, they are not able to record heart sound signals. They are comparatively low-cost and still regarded as “status quo” in the hospital and doctor’s practice. At least, these stethoscopes support two mechanical frequency ranges (heart sound and lung sound). Electronic stethoscopes on the principle of airborne sound microphones can essentially be compared with conventional stethoscopes with the difference that an encapsulated microphone for recording the sound is integrated in the acoustic path, and an electronic amplification and filter circuit with output via loudspeaker or encapsulated headphone is available. The two frequency ranges for heart sound and lung sound are controlled electronically. The almost only advantage of this first generation of electronic stethoscopes is the better amplification of the sound events, in particular for users whose hearing ability has suffered with age. The younger generation of electronic stethoscopes no longer records the (airborne) sound of the heart or lung activities but the vibrations these activities produce at the surface of the thorax are recorded by a structure-borne sound or vibration sensor. At present, stethoscopes of this type supply the best signal quality

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and signal-noise ratio. Today, vibration sensors can be contact film sensors based on a piezo-polymer film. The common features of electronic stethoscopes in comparison with conventional acoustic microphones are the following: • Signal amplification (e.g. for hearing impaired due to age) or volume control, • Electronic filtering, selectable frequency ranges for heart and lung sounds, • Recording the signals in the stethoscope or the PC, with the possibility of playback and analysis. 4.1.2. Design of a new electronic stethoscope The “Audira” system (Fig. 6) developed by our group is an electronic stethoscope of the latest generation with all above-named features and standard functionalities. Essential component is a newly developed, high-sensitivity vibration sensor based on a bimorph rod. The design provides a very clear and highly amplified signal with attenuation of noise and jitter on the basis of floating and pressure-balancing sensors. This design ensures essentially that the device is independent of the contact pressure as well as skin and ambient temperatures. The Audira stethoscope simultaneously records a single-channel ECG by electrodes integrated in the head of the stethoscope. A handheld module digitizes the ECG and the sound in a standard format, electrically isolates and transmits them to the PC via USB or Bluetooth for further processing. The selection of the point of auscultation and the entire program and process control of electronic auscultation are also made with the manual control that the examining physician is not required to operate the PC. 4.2. Heart sound analysis for detection of biomechanical disorders of the heart valves A main objective of this work was to develop an automatic heart sound analysis system for diagnosing acquired heart valve diseases which with simple handling and

Fig. 6.

“AUDIRA” stethoscope, inclusive of hand-held control module.

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at reasonable cost, can be used by the general practitioner and the family doctor. The algorithms for the classification of the different diseases of heart valves are based on the feature extraction from heart sound curves.

4.2.1. State of the art The last few years have originated several research works focusing on the use of neuronal networks for the automatic diagnosis of heart disease. The differentiation of functional and pathological heart murmurs in children for diagnosing congenital heart defects, in particular, has archived high levels of sensitivity and specificity. However, no commercial analytical systems for the diagnosis of all heart valve diseases based on neuronal networks are available in the market so far.20–22 Another approach is that of correlation analyses, which bases on the similarity of patterns of heart sound curves of identical diseases. For this, a new signal is set in correlation with all diagnosed signals, for example, in a database. Signals of high similarity to each other have a high correlation coefficient. The disease with which the signal has the highest average correlation coefficient is diagnosed. For example, if a signal has a higher average correlation coefficient to signals of the group of aortic stenoses than to signals of the tricuspid valve insufficiency group it is more similar to the aortic stenoses than the tricuspidal insufficiencies), the patient suffers from aortic valve stenosis with higher probability than from tricuspid valve insufficiency. First internal studies showed 23 that the correlation analysis for the detection of moderate and severe aortic valve stenosis achieves good results (sensitivity 91%, specificity 94% in comparison with a healthy reference group; sensitivity 94%, specificity 97% in comparison with a group with other heart valve diseases). Both approaches, i.e. neuronal networks and correlation analysis, require a lot more computer hardware and computing time than parameter extraction does. For some time, several studies have applied different methods of signal analysis for the interpretation of heart sounds. For example, high sensitivity and specificity levels were achieved in the detection of degenerated biological replacement valves (aortic and mitral valves). For this, different methods of frequency analysis were combined with traditional classification methods (Bayes classificator, nearest neighbor) and neuronal networks.24–26 Much more recently, a device for home monitoring of mechanical replacement valves based on frequency analysis (Fast Fourier Transform) of the heart sounds is available in the market. The purpose of the device is the early detection of disorders of the mechanical heart valve to enable timely intervention.27 Pavlopoulos et al.28 presented a method of differential diagnosis of aortic valve stenosis and mitral valve insufficiency based on parameter extraction followed by a decision-tree method. This method although of high accuracy is unsuitable as

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screening method on GP level because a valve disease must already have been diagnosed. Another study tried, by different methods of frequency analysis (CWT, FFT), to determine the dominating frequency from the heart sound signal and use it as a parameter for the degree of severity of aortic valve stenosis.29 Kim et al. proved in a study that the duration of certain frequency ranges in the spectrum correlate with the pressure gradient derived from the Doppler signal and that these are useful indicators of an aortic valve stenosis.30 Most recent studies also prove that the Wavelet Transformation, which we also apply, is especially useful for a detailed analysis of the heart sound components.31 Important research work on automatic interpretation using parameters of the heart sound has been done by Barschdorff.17,46 However, the signal analytic methods by which signals are analyzed (envelope formation and pattern recognition) and parameters are defined (e.g. mean value calculation, variance, envelope function form factor, flattening coefficient in different segments), are completely different, classification is also based on neuronal networks.32,33 So far, there have been no algorithms for the automatic interpretation of heart sounds based on feature extraction using methods of the time and frequency domain which are also applicable as screening method for all heart valve diseases. 4.2.2. Heart sound analysis based on feature extraction 4.2.2.1. Signal recording and preliminary processing For this study, the heart sound signals of patients were recorded using an electronic stethoscope on five different auscultation areas for 15 seconds each. Auscultation areas 1 to 5 were those which the physician normally checks with the acoustic stethoscope. Their locations can be seen in Fig. 7. Simultaneous with the heart sound, a single-channel ECG (Einthoven II lead) was recorded to obtain a time relation of the heart sound signal. The cardiologist examined every patient’s heart extensively by echocardiography. For this, a special appraisal form was developed which the cardiologist had to complete. R wave detection and noise eliminating With reference to the detected R waves in the ECG, the heart sound was divided into single heart periods. Each of these heart periods was replayed acoustically and inspected visually for noise. Heart periods in which noise was detected were excluded from the following examinations. Wavelet decomposition The heart sound signals were filtered by multi-scale analysis using wavelets. It is very suitable for discriminating background noise from biosignals.34,35 The method which

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Fig. 7.

Fig. 8.

Auscultation areas AP1 to AP5.

Scheme of stepwise decomposition of the signal by multi-scale analysis.

we applied splits a signal successively into a low-frequency segment and a highfrequency segment. The low-frequency segment is split again for the next step.36 In Fig. 8, c0 is the input signal. Signal c1 is the first low-frequency band generated by low pass H and d1 the first high-frequency band generated by high pass G. The filtering continued up to m = 10 so that totally 20 frequency bands (10 low-frequency bands, 10 high-frequency bands) were obtained.

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Representative heart period To reduce the time and computing effort, an attempt was made to carry out the examinations only with one heart period for each auscultation area. For this, the period in one recording was determined automatically which was most similar to all other heart periods of this recording. For the purposes of the study, this heart period was declared to be the representative heart period whose sum of correlation coefficients was the highest in relation to all other heart periods. The calculations were not made with the original signals because this would have consumed too much time due to the high sample frequency of 44100 Hz. Instead, this analytical step was carried out in the wavelets within the frequency band from 0 to 2756 Hz, which saved computing time. The resulting information loss was tolerable because all examinations were limited to the frequency bands of up to 2756 Hz because only relevant information to the analyzed diseases were expected in the range of up to approximately 2000 Hz. Heart sound detection The temporal occurrence of the heart sound phenomena is essential for the assignment of the affected heart valve. Therefore, it is necessary to divide the heart period in functional sections according to the mechanical phases of the heart action. The maximums of the first and second heart sounds can serve as triggers. They were detected by an algorithm and thus the heart period was divided. For detection, two independent methods were developed and their results were compared with each other to improve the degree of accuracy. At first, the time progression of the Shannon energy of the wavelet filtered signal in the range from 0 to 172 Hz was calculated. The analysis of the energy characteristic served as reference method. For this, the frequency-time spectrum was determined by Short Time Fourier Transform (STFT) and all coefficients within a time window were summated. In the two obtained time series, the two highest peaks within defined time windows in which the first and second heart sounds were expected, were detected (first heart sound — 0 to 150 ms, second heart sound — 350 to 600 ms). If the difference of the detected positions in both methods did not exceed 70 ms, the mean of the two methods was calculated and the maxima declared to be the first and second heart sound. Figure 9 illustrates the method. The first graph shows the original heart sound signal. The graphs shown are those of the STFT, the energy progression obtained from it and the progression of the Shannon energy. The maximum values found in graph three and four are back-projected onto the original signal. Finally, they are compared and declared to be valid if the difference is sufficiently small after the mean has been calculated. The problem of this method also becomes apparent with patients of strong heart murmurs and faint heart sounds, e.g. in presence of severe aortic valve stenosis. In

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Fig. 9. Heart sound detection in a patient with healthy valves on the left and a patient with severe aortic valve stenosis on the right (in each case original signal, STFT spectrum, squared sum of frequency components from STFT and Shannon energy; maxima are marked, amplitudes normalized, frequency in Hz, time in samples and window steps).

both detection methods, the heart murmurs have higher maximum peaks than the heart sounds. For this reason, the maximum values were sought in limited time windows in which the heart sounds with the utmost probability are situated. These time windows were defined empirically. Signal segmentation Starting from the positions of the first and second heart sound, the signal was split into 12 segments, S1 to S12 (Fig. 10). For this, the distance between the heart sounds was determined and divided by five. This value, x, described the length of segments S2 to S11, beginning from the first heart sound minus x/2. Segment S1 consists of all sample points from the R wave until that time. The last segment, S12, consisted of all sample points after S11 until the end of the heart period. In

Fig. 10. Segmentation of a heart sound signal from segment S1 to S12 (amplitude normalized, time in samples).

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this way a dynamic segmentation was obtained which guarantees that functionally equivalent segments during a heart cycle among patients were studied independently of the length of a heart period or the position of the heart sounds. 4.2.2.2. Parameter extraction Global parameters from the Fast Fourier Transform The Fast Fourier Transform (FFT) splits a measured signal into its sinusoidal and cosinusoidal frequency parts and displays them as a spectrum. This provides information on the frequencies in the signal and their respective parts, but not on the time of their occurrence. Stenosis or insufficiencies at a heart valve cause turbulences in the blood flow. These flow murmurs lead to higher frequency portions in the heart sound signal. Theoretically, sound signals from persons without heart diseases only contain frequencies in the range of the heart sounds up to approximately 150 Hz. If a heart valve disease exists, much higher frequencies (partly over 1000 Hz) can be contained in the spectrum. For the parameter extraction, the amplitude spectrum of the heart sound signals was calculated. For this, the original signals were weighted with the Blackman-Harris window function to avoid the occurrence of high frequencies due to amplitude jumps at the edges of the window. The window length was 4096 sample points. This yielded a spectral resolution of 10 Hz. Now the frequency parts of different bands between 15 and 700 Hz were summed as parameters. For this, the frequency range was divided in bands of 25 Hz width each and the sum of all parts in each band was calculated. Each of these values was one parameter which was checked for significance using the Mann-Whitney U test, e.g. the parameter sum of all frequency components between 15 and 40 Hz, the parameter sum of all frequency components between 40 and 65 Hz, etc. After this, the same method was applied to frequency bands of 50 Hz, 75 Hz, 100 Hz, 125 Hz, 150 Hz, 175 Hz, and 200 Hz width. Figure 11 illustrates the method by the example of the 50 Hz frequency band width. Graph A is the original heart period, graph B the associated FFT spectrum from 15 to 700 Hz and its division into sections of 50 Hz width each. The sum of all frequency components in each section is then calculated. Global parameters from the time relation between ECG and heart sounds Changes at the heart valves can shift the time of occurrence of the heart sounds relative to the R-peak in the ECG (Fig. 12) because, for example, the pressure conditions among the vessels change.4 Therefore, the following four parameters were analyzed: (1) Time between the R-peak and the first heart sound, (2) Time between the first and second heart sound,

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Fig. 11. Example of FFT parameter extraction. (A) Original signal with heart sounds; amplitude normalized, time in samples, (B) Fourier spectrum up to 700 Hz, divided into sections of 50 Hz, each, in each of which the area was determined as parameter; amplitude normalized, frequency in Hz.

(3) Time between the R-peak and the second heart sound, (4) Time between the second heart sound and the end of the heart period. These parameters were also analyzed regarding to their significance for each disease. In Fig. 12 an example is given for this parameter extraction method. Graph A shows the ECG (Einthoven II lead), Graph B the associated heart sound signal. Segmental parameters from time and frequency domain analysis For diagnosing a heart valve disease, it is extremely important to know in which section of the heart period the heart sound changes. In most cases, this is the only way to find the origin of the defect. Therefore, it was necessary, in addition to the information on the frequency content, to analyze the temporal behavior of the signal. For this, analytical methods from the time and frequency domain were applied. In all the representations and curves of the signal derived from these methods, the following parameter classes were extracted from each of the 12 segments: A total area of the segment B proportion of the area of the segment to the total area

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Fig. 12. Parameter extraction from the time relationship between ECG and heart sound. (A) ECG with R-peak marked, (B) original heart sound with marked heart tones; amplitudes normalized; time in samples.

C sum of differences between two points D sum of the absolute values of the differences between two points. Each parameter is composed of its class, the segment and the analytical presentation of the signal. Segmental parameters from time-domain analysis The analysis in the time-domain was not carried out with the original signal but with the wavelet-filtered signals. The frequency components in the range from 0 to 2756 Hz were included in the analysis. The following parameters were extracted: • • • •

Intensity (absolute value) Energy Shannon energy Shannon entropy

Figure 13 illustrates this method by an example. Graph A is a wavelet-filtered heart period at level 6.0 (0 to 172 Hz), B represents the intensity, C the energy, D the Shannon energy, and E the Shannon entropy of the signal. Segmental parameters from time/frequency methods Heart murmurs are classified by high frequencies within the sound signal. Therefore, the examination of the frequencies in connection with their time-related occurrence

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Fig. 13. Example of parameter extraction from time domain. (A) wavelet-filtered heart period 0 to 172 Hz, (B) Intensity, (C) Energy, (D) Shannon energy, (E) Shannon entropy; extracted from each curve were in each segment the parameters: area, area proportion, sum total of differences between two points and sum of absolute values of the differences between two points; the cursors denote the heart sounds; amplitudes normalized, time in samples.

promised to yield the most significant information. Only the time/frequency methods can reveal information which frequencies occur and with what intensity at what time during a heart cycle. Parameters from STFT The STFT was calculated for each heart period with the following parameters: (1) Window length of 4096 samples; this yields a frequency resolution of 10 Hz and a time resolution of 100 ms,

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(2) Shift of the window by 100 samples each; this somewhat impairs olution but this had to be tolerated to reduce computing time capacity, (3) Windowing of the signal by Blackman-Harris window function occurrence of high frequencies due to amplitude jumps at the window.

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the time resand memory to avoid the edges of the

From the calculated 3-dimensional transformations, new signal descriptions were generated by calculating different sums of all frequency sections in each time window. (1) (2) (3) (4) (5) (6)

sum of sum of sum of sum of scaling scaling

all frequency components all frequency components and squaring of sum, all logarithmized frequency components, all logarithmized frequency components and squaring of sum, of 3. between 0 and 1, of 4. between 0 and 1.

The method is illustrated by an example in Fig. 14. Graph A is the original heart sound signal, graph B the associated STFT spectrum. Graph C to H show the generated signal presentations in the above-named order. Extracted from each curve in each segment were the four parameters absolute area, area proportion of the total signal, sum of differences between two points and sum of absolute values of the differences between two points. Parameters from Wavelet Transform The Wavelet Transform (WT) also supplies information on the time and frequency dependence. Instead of a window of fixed width, such as for STFT, a changing window of a constant number of oscillations is slided over the signal and the similarity with the current signal section is calculated. The variation of the analytical function, the so called wavelet, is obtained from the compression or elongation defined by the so called scaling function. The basic form of the wavelet does not change. For parameter extraction, a WT with the following parameters was calculated from every representative heart period: • “Mexican hat” mother wavelet because this shape resembles that of the heart sounds, • elongation of the wavelet from 20 to 4410 sample points in 100 steps after a cubic root function; in this way, the lower frequency ranges up to 500 Hz were analyzed in narrower steps than were the higher frequencies from 500 to 2000 Hz, because more information was expected to be obtained from the lower frequencies, • stepwise shifting of the wavelet by 10 sample points (to save computing time and memory capacity).

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Fig. 14. Example of parameter extraction from STFT. (A) Original signal, (B) STFT, (C to H) from STFT generated different signal representations (see text); extracted from each curve were in each segment the parameters: area, area proportion, sum total of differences between two points and sum of absolute values of the differences between two points; the cursors denote the heart sounds; amplitudes normalized, frequency in Hz, time in samples and window steps.

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From the calculated transformations, new signal descriptions were generated by summation of the frequency sections of each time window similar to the approach with STFT. (1) (2) (3) (4) (5) (6) (7) (8)

sum of sum of sum of sum of scaling scaling sum of sum of

all frequency components, all frequency components and squaring of sum, all logarithmized frequency components, all logarithmized frequency components and squaring of sum, of 3. between 0 and 1, of 4. between 0 and 1, absolute values of all frequency components, absolute values of all frequency components and squaring of sum.

Figure 15 illustrates this method graphically. The figure starts with the presentation of the original signal (A) and the associated WT spectrum (B). Graph (C) to (J) show the generated signal presentations in the above-named order. Extracted from each curve in each segment were the parameters absolute area, area proportion of the total signal, sum of differences between two points and sum of absolute values of the differences between two points. All segmental parameters were now tested for significance to the tested item by means of the Mann-Whitney U Test. 4.2.2.3. Using the methods for analysis of heart sounds in patients with heart valve disease

Fig. 15. Example of parameter extraction from WT. (A) Original heart sound, (B) WT spectrum, (C to J) signals derived from WT (see text); extracted from each curve were in each segment the parameters: area, area proportion, sum of differences between two points and sum of absolute values of the differences between two points; the two cursors denote the heart sounds; amplitudes normalized, frequency in Hz, time in samples and window steps.

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Patients Data of 402 patients were included in this retrospective randomized study. Not included were patients with at least one non-native cardiac valve, i.e. a replacement valve or a surgically corrected valve. Another condition for the usability of the examination was that at least one heart period includes noise-free sound signals from all auscultation areas. These 402 patients were divided in two groups, one training group (TrG) for the determination of significant parameters and the resulting discriminant functions, and one test group (TeG) for verification of the classification methods. Patients were classified into these groups by a random procedure. The training group comprised of 206 patients, the test group of 196 patients. Table 4 shows the group arrangement concerning sex, age and body weight. A classification was made in patients with healthy heart valves (REF) and patients with heart valve disease (VD). The parameter sets and the discriminant functions were determined from the training group by Mann-Whitney U test and discriminant function analysis. The obtained test methods were validated on the test group. The higher severity levels of the analyzed diseases were partly underrepresented so that groups were summarized (Table 5). Classification methods Classification method 1 The simplest possibility was to separate the patients with the analyzed disease (without considering the severity level) from those patients who were not affected by that disease. This required only one parameter set with one discriminant function (Fig. 16). CS 1 : (ASl + ASms) versus (REF + OV D) Classification method 2 This method considers the possibility that mild and higher severity levels of a disease do not necessarily have the same significant parameters and as such are not, of

Table 4.

Patient data (BMI-Body mass index).

Group

Sex

Age

BMI

Male

Female

TrG REF VD

44 50

33 79

58 ± 16 66 ± 14

27 ± 4 26 ± 4

TeG REF VD

51 58

25 62

57 ± 15 66 ± 13

27 ± 5 26 ± 4

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Table 5. Arrangement of the patient groups for the analysis of valve diseases (AS — Aortic valve stenosis, AI — Aortic valve insufficiency, MI — Mitral valve insufficiency, TI — Tricuspid valve insufficiency, PI — Pulmonary valve insufficiency). Heart valve disease REF AS AI MI TI PI

Fig. 16.

mild moderate–severe mild moderate–severe mild moderate–severe mild moderate–severe mild–severe

TrG

TeG

Total

77 20 27 42 20 53 23 40 23 22

76 12 30 39 21 50 16 45 16 17

153 32 57 81 41 103 39 85 39 39

Classification method 1 — aortic valve stenosis.

necessity, placed in the same group. In the first step of this method, only higherlevel patients are identified and in the second step, patients with mild levels of the disease are identified (Fig. 17).

For every classification step, the respective significance of each parameters was to be determined in terms of a possible separation of the currently examined groups. Due to the large number of parameters, the level of significance had to be reduced

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Fig. 17.

Classification method 2 — aortic valve stenosis.

according to the Bonferroni criterion: Local significancelevel α =

global significancelevel α . N umber testparameters

So the significance level for each parameter was α < 3∗ 10−6 . In addition, it was arranged that parameters of not more than three auscultation areas in each case were included in the discriminant analysis. A linear stepwise discriminant analysis with maximum four of the significant parameters was carried out. Evaluation Example: Aortic valve stenosis For examination of the aortic valve stenosis, the patients were divided into four groups: (I) (II) (III) (IV)

REF – OVD –

patients without valve disease patients with a valve disease other then aortic valve stenosis ASl – patients with mild aortic valve stenosis (and possibly other valve diseases) ASms – patients with moderate or severe aortic valve stenosis (and possibly other valve diseases)

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In Table 6 the following group arrangement was obtained: The following concrete test situations and results were obtained for the two classification methods: (1) CS 1: Discriminant function for separation of all patients with aortic valve stenosis from patients without Aortic valve stenosis (Table 7) (REF + OVD) versus (ASl + ASms) (2) CS 1: Discriminant function for separation of all patients with moderate or severe aortic valve stenosis from all other patients (REF + OVD + ASl) versus (ASms) CS 2: Discriminant function for separation of the patients with mild aortic valve stenosis from patients without aortic valve stenosis (Table 8) (REF + OVD) versus (ASl) The primary aim was to achieve high sensitivity and specificity not less than 80%. For this reason, test method 2 was selected for aortic valve stenosis. This method will be explained in detail. Selected classification method for aortic valve stenosis The purpose of the first stage of the test was to separate all patients with moderate or severe aortic valve stenosis from all other patients. The results in Table 9 were obtained for the training group with the developed parameter set PAS CS1. Table 6.

Composition of patient groups.

Group arrangement REF OVD ASl ASms

TrG

TeG

Total

77 82 20 27

76 78 12 30

153 160 32 57

Table 7. Results of classification method 1 — Aortic valve stenosis.

Sensitivity Specificity Correct diagnosis

TrG

TeG

Total

85.1% 93.1% 91.3%

71.4% 89.6% 85.7%

78.7% 91.4% 88.6%

Table 8. Results of classification method 2 — Aortic valve stenosis.

Sensitivity Specificity Correct diagnosis

TrG

TeG

Total

89.4% 93.7% 92.7%

81% 85.1% 84.2%

85.4% 89.5% 88.6%

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194 Table 9.

Results of CSI — aortic valve stenosis (TrG).

AS CS 1

REF + OVD + ASl

ASms

Total Correct In %

77 + 82 + 20 76 + 78 + 14 93.9%

27 26 96.3%

Table 10.

Result of CS2 — aortic valve stenosis (TrG).

AS CS 2

REF + OVD

ASl

Total Correct In %

77 + 82 74 + 77 95%

20 15 75%

Parameter set PAS CS2 was to find patients with mild aortic valve stenosis (Table 10). For the evaluation of the total result, the two test stages had to be considered one after the other. For this, every patient who is not identified in CS1 as suffering from moderate or severe aortic valve stenosis, is again examined at CS2 (which includes undetected wrongly negative patients). This method was also evaluated with the test group (Table 11). As expected, the test group performed less good than the training group. Altogether, 94.7% of moderate or severe aortic stenoses and 68.8% of mild aortic stenoses were detected. Total sensitivity for detection of aortic valve stenosis is 85.4%. Of this, 4 of the detected 54 moderate or severe aortic valve stenosis cases (7%) were only detected in the second test step and were therefore diagnosed as mild aortic stenosis. In contrast 12 of 22 detected mild cases of aortic valve stenosis (54.6%) were already detected at the first test step and therefore diagnosed as being moderate or severe aortic valve stenosis than they actually are. The specificity for patients without valve disease is 93.5%, for patients with valve diseases other than aortic valve stenosis it is 85.6%. Thus, a total specificity of 89.5% is obtained. Finally, 88.6% of all patients are classified correctly concerning the question whether an aortic valve stenosis does or does not exist. The used parameters are listed in Table 12. Table 11.

Overall result of the classification of aortic valve stenosis. TrG

True positive Sensitivity True negative Specificity

TeG

Total

ASms

ASl

ASms

ASl

ASms

ASl

27/27 100% REF 74/77 96.1%

15/20 75% OVD 75/82 91.5%

27/30 90% REF 69/76 90.8%

7/12 58.3% OVD 62/78 79.5%

54/57 94.7% REF 143/153 93.5%

22/32 68.8% OVD 137/160 85.6%

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Explanation of parameter sets PAS CS1 and PAS CS2.

Parameter

Description AA

PCS 1 Par1 1

Parameter extraction method

Frequency range (Hz)

REF + OVD + AVS l vs. AVS ms 3 STFT 15–172

Par1 2

2

Par1 3 Par1 4 PCS 2 Par2 1

195

3

WT decomposition STFT

172–689 15–689

1

STFT

172–345

REF + OVD vs. AVS l 1 WT

639–2205

Par2 2

1

WT

79–269

Par2 3 Par2 4

2 2

FFT FFT

115–140 115–165

Time series

Extracted parameter

power-timeSTFT Shannon entropy power-timeSTFT power-timeSTFT

area S4

power-timeWT power-timeWT spectrum spectrum

ratio of area of S2 to total area ratio of area of S4 to total area total power total power

ratio S5 ratio S5 ratio S4

of area of to total area of area of to total area of area of to total area

The associated discriminant functions are: CS1: CS2:

d = −2.827 + 0.073*Par1 1 + 7.273*Par1 2 + 9.975*Par1 3 + 4.190*Par1 4. d = −1.730 − 2.283*Par1 1 +2 4.818*Par1 2 − 0.008*Par1 3 + 0.006*Par1 4.

Now, this method was also applied to the other four heart valve diseases: aortic valve insufficiency, mitral valve insufficiency, tricuspid valve insufficiency and pulmonary valve insufficiency. In contrast to aortic valve stenosis, classification method 1 supplied the best results in each of these cases. The overall results of the training and test groups analyses are presented in the following tables (Tables 13–16). Aortic valve insufficiency Table 13.

Overall results of the classification of aortic valve insufficiency. TrG

True positive Sensitivity True negative Specificity

TeG

Total

AImms

AIg

AImms

AIg

AImms

AIg

15/20 75%

27/42 64.3%

14/21 66.7%

23/39 59%

29/41 70.7%

50/81 61.7%

REF

OVD

REF

OVD

REF

OVD

70/77 90.9%

43/67 64.2%

59/76 77.6%

40/60 66.7%

129/153 84.3%

83/127 65.4%

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Mitral valve insufficiency Table 14.

Overall results of the classification of mitral valve insufficiency. TrG

True positive Sensitivity True negative Specificity

TeG

Total

MImms

MIg

MImms

MIg

MImms

MIg

19/23 82.6%

36/53 67.9%

10/16 62.5%

34/50 68%

29/39 74.4%

70/103 68%

REF

OVD

REF

OVD

REF

OVD

64/77 83.1%

30/53 56.6%

58/76 76.3%

16/54 29.6%

122/153 79.7%

46/107 43%

Tricuspid valve insufficiency Table 15.

Overall results of the classification of tricuspid valve insufficiency. TrG

True positive Sensitivity True negative Specificity

TeG

Total

TIms

TIg

TIms

TIg

TIms

TIg

17/23 73.9%

24/40 60%

7/16 43.8%

34/59 57.6%

24/39 61.5%

49/85 57.7%

REF

OVD

REF

OVD

REF

OVD

67/77 87%

47/66 71.2%

64/76 84.2%

34/59 57.6%

131/153 85.6%

81/125 64.8%

Pulmonary valve insufficiency Table 16.

Overall results of the classification of pulmonary valve insufficiency.

True positive Sensitivity

True negative Specificity

TrG

TeG

Total

PI

PI

PI

18/22

12/17

30/39

81.8%

70.6%

76.9%

REF

OVD

REF

OVD

REF

OVD

72/77 93.5%

90/107 84.1%

68/76 89.5%

89/103 86.4%

140/153 91.5%

179/210 85.2%

4.2.2.4. Discussion Very high values of sensitivity and specificity for the classification of aortic valve stenosis were achieved. In particular, the detection of moderate and severe grades of aortic valve stenosis is nearly 100% and even about 70% for mild forms. Besides, this classification method provides information as to the severity of the disease. On the other hand, most cases of mild aortic valve stenosis are classified higher whereas the

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classification of medium and severe levels of aortic valve stenosis is correct but for a few exceptions. Most of the wrongly positive findings are not concerning patients with healthy valves but patients suffering from valve diseases other than aortic valve stenosis. In view of this, the result of specificity should also be rated highly. The parameters Par 1 1 to Par 1 4 of the parameter set PAS CS1 concern all segments 4 and 5, i.e. the segments between the first and second heart sound. This is the area at which the heart murmur of aortic valve stenosis typically occurs. Parameter Par 2 1 of the parameter set PAS CS2 describes a change of the second segment in which the first heart sound is located. This reflects the established diagnostic criteria of the fainter heart sound. The changes in the frequency domain caused by the occurrence of the heart murmur also appear, in particular, in parameters Par 2 3 and Par 2 4. The values for sensitivity and specificity of the test for aortic valve insufficiency do not meet the expected target values. For the test, all aortic insufficiencies had to be combined in one group. Mild, moderate or severe aortic insufficiencies are detected at 70.7% with the result for the test group being about 8% below that of the training group. The aortic insufficiencies of more severe aortic valve insufficiency only represent for about 30% of the total number so that their impact on parameter optimization is not strong enough. Two-thirds of these patients suffer only from a mild form whose characteristic changes apparently are not important enough for better detection. Possibly these changes are masked by additional other valve diseases. Thus, the sensitivity for mild aortic insufficiencies was only 61.7%. The specificity also does not confirm the characteristic changes accompanying aortic valve insufficiency or a mingling with other valve diseases. The specificity in comparison with other valve diseases is only 65.4%, whereas it is a sufficient 84.3% in comparison with patients with healthy valves. The classification of mild, moderate and severe mitral insufficiencies was successful with a sensitivity of totally 74.4%. In this, the test group performed poorly of 20% than the training group. The patient population in that group was rather low giving the wrong classification of a single patient a massive impact on the percentage result. The detection of mild mitral insufficiencies reaches 68% sensitivity. This result is satisfactory considering the mild grade of disease. The biggest problem with the method chosen for mitral valve insufficiency is the very low specificity in connection with patients with valve diseases other than mitral valve insufficiency, in particular, in the test group, where it is below 50%. This means that more than half of all patients who are suffering from a valve disease that is not mitral valve insufficiency are diagnosed as being mitral insufficient. Also the number of patients with wrongly positive detection who actually have healthy valves is fairly high when compared with the other tests. One reason for this is the very high proportion of mild mitral insufficiencies (over 70%). Less than 30% of all mitral insufficiencies are of higher severity as a result of which the clear symptoms of this disease cannot sufficiently be detected.

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Typically, the tricuspid insufficiencies as diseases of the right heart very commonly occur together with left-side valve diseases. For example, 90% of patients with higher-degree tricuspid valve insufficiency suffer from at least one other valve disease, 80% even from at least 2 additional valve diseases. The detection of the mild tricuspid valve insufficiency was not as high as expected. One should consider that the pressure conditions in the right half of the heart are very much lower than on the left side which explains that changes at the valves will only be noted at very minor degree. The results of the pulmonary valve insufficiency test method are very good. Sensitivity, however, is just short of 80% but it should be considered that with the exception of one patient all patients have only mild valve insufficiencies at the right half of the heart. The generated algorithms achieved a correct detection of patients with heart valve disease of 76.3%, i.e. three of four patients are diagnosed correctly. This result meets the expected target values, in particular, as most of the analyzed diseases are at the initial stage. The specificities are a problem of the developed test method. If a patient suffers from a cardiovascular disease other than a heart valve defect, there is a high probability for that patient still to be diagnosed among the patients with heart valve disease. In this case, it does make sense that the GP refers that patient to the specialist despite the initially wrong suspected diagnosis. For a final validation of the method as a whole, a prospective study with entirely new patient data will be carried out. At the next step, it will be necessary to optimize the preprocessing procedures, for one, to make this process fully automatic (e.g. develop a system of automatic noise detection) and, for another, to further reduce computing time (e.g. by calculating the representative heart period in a narrower frequency band). Another aim is the improvement of the classification results. For this, several approaches can be combined. For example, one possibility is to include the correlation coefficient as an additional parameter. Unfortunately, this would make the time for computing even longer and necessary to reduce signal preprocessing which might lead to worse quality of other parameters. Besides, the developed parameters could be used as inputs for a neuronal network which can use non-linear relations among them for classification. Finally, other approaches can be applied, such as the development of a classification method on fuzzy basis. Furthermore, it is planned to apply these principles to classify other heart diseases. These will mainly be congenital heart defects (e.g. septum diseases) and myocardial diseases (cardiomyopathies) as well as vascular diseases as these lead to changes in the heart and vessel sounds. Another area of application where the developed methods and principles could be used is in follow-ups after heart valve operation or other cardiological interventions, e.g., after implantation of a stent. Summarizing, it can be said that the developed algorithms are suitable for an early detection of patients with heart valve disease. They require inexpensive technical equipment (commercial medical PC).

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4.3. Heart sound analysis for risk stratification in patients with heart failure Heart failure is one of the most common diseases worldwide and has a high prevalence especially in the industrialized countries. In America, about 5 million people suffer from heart failure37 and approximately 10 million in Europe.38 Every year, further 550000 patients contract the disease, both in America and in Europe (incidence). The direct and indirect costs for the treatment of heart failure are estimated to amount to about USD 29.6 billion in America. According to the National Center for Health Statistics (NCHS)1 57218 patients died of the direct consequences of heart failure in America in 2003, the 1-year mortality rate is about 30%.39 The term “heart failure” denotes the weakening of the right and/or left myocardium, which changes the biomechanics and, in particular, restricts the ejection fraction of the heart. As a result of this, the organism is not supplied with enough blood and nutrients. The development of heart failure can be triggered by a large number of adverse factors and can occur quickly or take longer to develop (months/years). The most frequent causes of heart failure are40 : • Excessive pressure load on the heart, e.g. due to heart valve diseases (aortic valve stenoses) or high blood pressure causes myocardial hypertrophy (Fig. 18), • Excessive volume load on the heart, e.g. due to heart valve diseases (aortic or mitral valve insufficiency) leads to a dilatation of the left and/or right ventricle (Fig. 18), • Disturbed filling of the left ventricle, e.g. due to mitral stenosis, restrictive cardiomyopathy, • Diminished pumping efficiency due to primary myocardial disease, e.g. cardiomyopathy, coronary heart disease, • Indirect diseases, such as diabetes, thyroid disease. At the initial stage of heart failure (compensated heart failure), the organism can compensate the reduced cardiac output by appropriate compensation mechanisms so

Fig. 18. Schemes of healthy heart, hypertrophic heart (myocardium is hypertrophied, increased wall thickness especially of the left ventricle) and a dilated heart (enlarged heart, dilatation of the left and/or right ventricles, reduced wall thickness).

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that none or only minor discomfort appears during the patients normal course of life. On a continuing basis, compensation mechanisms damage the circulatory system, the organs and also the tissue and lead to a progressive weakening of the heart. The progressive heart failure cannot constantly be compensated (decompensated heart failure) and causes symptoms such as shortness of breath, fatigue, fluid retention (edema) in the lung, tissue and organs, and discomfort during mild physical activity. The diagnosis of heart failure can be based on resting or stress ECG, echocardiography, X-ray examination, heart catheter and laboratory values. The severity of heart failure is assessed by means of the subjective NYHA (New York Heart Association) index which classifies the functional capacity (NYHA I–IV) of a patient. Heart failure can be treated as follows: By wholesome diet, adequate physical activity, medication (ACE-blocker, beta-blocker, digitalis, diuretics) and when in advanced stage by implantation of a cardiac pacemaker or a defibrillator or heart transplantation.41 Until now an early detection of high-risk patients with heart failure has not been solved satisfactorily. In a first pilot study we investigated how far different methods for the analysis of cardiovascular variability and heart sound provide additional diagnostic information for a prognostic individual risk stratification of heart failure.42 Under standardized resting conditions, 30-minutes of ECG and non-invasive continuous blood pressure (Portapres M2, TNO-TPD, Amsterdam, Netherlands) were recorded from 43 patients with heart failure (NYHA ≥ 2). Afterwards, heart sound r Master Elite Plus stethoscope) and on 9 defined auscultation areas (Welch Allyn synchronized ECG and blood pressure were acquired over five heartbeats each. To eliminate respiratory sounds, the acoustic signals were recorded during a short respiratory pause. Use of an USB sound card (Maya EX, Audiotrak, sample frequency: 44.1 kHz, resolution: 16 bit) allowed digitization of the heart sounds (Fig. 19). During data preprocessing the tachograms from ECG (time distance between successive R waves) as well as systograms and diastograms (succession of systolic and diastolic blood pressure values) from the blood pressure were extracted. Following, artifacts and extrasystolic beats were replaced by interpolated beats. Heart rate variability was quantified by defined parameters from time and frequency domain according to the Task Force of the European Society for Cardiology.43 For the classification of the dynamic, biomechanical interactions between ECG and blood pressure, additional nonlinear parameters of the Joint Symbolic Dynamics (JSD) were calculated.44 The analysis of interactions by means of JSD is based on the transformation of the tachograms and systograms in symbol sequences consisting of “0” (heart frequency and blood pressure decrease) and “1” (heart frequency and blood pressure increase). Thereafter, words are formed from the symbol sequences. The maximum word length is determined by a statistically required probability of the occurrence of the words and thus the length of the symbol sequences. If a uniform distribution and a required minimum frequency of occurrence of 30 events per bin are assumed, a word length of three is obtained for the 30-minute standardized recordings made as part of the analysis at an estimated mean heart frequency of

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Fig. 19.

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Measuring equipment.

75 beats per minute. Consequently, a 3 × 3 matrix W is spanned from the word type [000,000] to the word type [111,111]. To ensure the comparability of data records of different length the total occurrence is normalized to 1. Due to this simplification of time series, some detailed information of the time series is lost whereas robust, invariate information will be stayed. In addition to the analysis of variability within a time series and the interaction analysis between the recorded signals, the examination of the blood pressure morphology supplies important information.45 For this, parameters which describe the amplitudes and the temporal characterization (intervals, slopes) of the blood pressure morphology were estimated (Fig. 20). The analysis of heart sound as an important indirect method to determine the biomechanical characteristics of the heart is based on the determination of parameters describing the first heart sound (frequency range: 5–100 Hz) and the second heart sound (100–150 Hz) and the temporal coupling with ECG and blood pressure. The heart sounds were detected by a wavelet method.23 At first, RR interval related heart sound sections were generated for every auscultation point and decomposed in the frequency range 0–172 Hz by wavelet decomposition.36 The time points of the first and second heart sounds were detected applying the methods of Shannon energy and STFT.46 In case of the Shannon energy method the considered filtered heart sound interval was divided into 40 segments. From the mean values of the

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Fig. 20. Extracted parameters from blood pressure morphology. SYS — end-systolic blood pressure; DIA — end-diastolic blood pressure; BPA — blood pressure amplitude; SSS — maximum systolic slope; MSS — mean systolic slope; SDS — maximum diastolic slope; IC — incisure; DP — dicrotic peak; T1-T4 — time intervals; A1-A6 — areas below the blood pressure curve.

Shannon energy of each segment, the envelope of the heart sound interval was constructed. Within this envelope, the maxima for the first and second heart sounds were searched within certain time ranges. The time window for the first heart sound is 0–200 ms (= 1st to 7th sample point of the envelope) after the R wave and for the second heart sound it is 250–700 ms (= 15th to 25th sample points of the envelope) after the R wave. In addition, the short-time Fourier transform (Blackman-Harris window, window length = 4096 samples, time shift = 100 samples) was calculated for the analyzed heart sound interval. Within the transform, the maximum of the first heart sound was also determined in the range of 0–200 ms (STFT section from 0 to 60) and of the second heart sound in the range of 250–700 ms (STFT section from 145 to 250) after the R wave. As a next step, the difference of the two time points (from both methods) for the first heart sound was calculated. If the difference was less than 3000 sample points, the mean of both time points was determined and used as position of the first heart sound, otherwise the time point of the first heart sound was discarded. The position of the second heart sound was defined in the same way. During a 6-month follow-up, 7 of the total of 43 patients died due to a cardiac event, in 15 patients the state of health was deteriorated in comparison with the time of data acquisition (group of high-risk patients). The state of the remaining 21 patients did not deteriorate (group of low-risk patients). The statistical evaluation for determination of univariate differences between high-risk and lowrisk patients was based on the Mann-Whitney U test. Considering the Bonferroni

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criterion for multiple tests, the significance level of univariate statistics was corrected. An optimized parameter set for risk stratification of heart failure patients was determined applying the Cox regression model. The statistical analysis of the calculated parameters of different domains showed that the standardized parameters of heart rate variability did not contribute to the assessment of the individual risk in case of heart failure. On the other hand, parameters of blood pressure morphology and non-linear JSD as well as from the heart sound analysis showed significant differences between high-risk and low-risk patients. Using one high-significant parameter from each of the three analysis methods, an optimized parameter set for risk stratification of heart failure was developed. By application of this parameter set, 93.3% of survivors (specificity) and 91.7% of high-risk patients (sensitivity) were classed correctly. The parameter HS1 in the set corresponds with the time shift from the R wave to the first heart sound and is increased in high-risk patients. The time delay between the QRS complex and the first heart sound is based on the stimulus conduction in the ventricles. After electrical excitation of the septum (Q wave), the excitation reaches the apex of the heart (R wave) and causes increased tension of the ventricles (beginning of the first heart sound). The maximum of the first heart sound is reached when the electrical excitation has been conducted through the ventricular walls causing contraction of the entire ventricular myocardium. The parameter HS1 describes the time delay between R wave and maximum of the first heart sound. Another parameter BBI011 of JSD describes the probability of occurrence of the symbol sequence “011” within the RR interval time series and is decreased in high-risk patients. In addition, a less steep negative slope was found between the dicrotic wave and the diastolic minima in high-risk patients (parameter SDS of the blood pressure morphology analysis). Several studies proved the occurrence of a third heart sound47,48 in patients with heart failure. In patients with progressive heart failure, the third heart sound occurs almost obligatory in 96% of all patients.49 The third heart sound indicates a reduced left ventricular function and is generated by the transmitral inflow as an oscillation signal of the intracavitary blood volume, the ventricular walls and the surrounding structures.50 Patients with limited ejection fraction of less than 50% were identified by the third heart sound with a sensitivity of 51% and a specificity of 90%.51

5. Discussion In the industrialized nations, the incidence of cardiac diseases with partly massive impairment of the biomechanics of the heart is constantly increasing and is now the Number One in the mortal statistics in many of these countries. This points to the need of detecting cardiac diseases as early as possible to start treatment with promise of success. Because a patient usually consults the family doctor first, it is very important that the latter can rely on a simple-to-use, non-invasive and low-cost system for screening.

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The heart allows blood to flow from the venous to the arterial vascular system and in this way supplies all organs of the body with sufficient amounts of oxygen. This requires an intricate and accurate interplay of cardiac ventricles and muscles and the heart valves. If this system is disordered, a heart disease develops as a result of which certain parts of the body receive less oxygen than they need. The disease or disorder can involve the biomechanical system as such (heart defect, myocardial diseases, heart valve disease) or other areas of the heart as drivers or suppliers of the system (coronary heart disease, arrhythmia). The biomechanics of the heart can primarily by analyzed by the signals of the heart sound. The analysis of the heart sounds and cardiac murmurs can provide information on the state of the heart. The position of cardiac murmurs in relation to the heart sounds is a decisive criterion for the diagnosis of a cardiac disease. Heart defects, myocardial diseases and heart valve diseases can be detected by the heart sounds, in particular, because the disordering of the biomechanical system causes typical changes. These changes can be analyzed and diagnosed by auscultation and phonocardiography. Both methods are very sophisticated and require a high degree of specialization if a disease is to be detected early. Another problem is the susceptibility of these methods to patient movements and murmurs, e.g. breathing, action of the bowels, and external noise. In the last few years modern imaging methods, such as computer tomography and magnetic resonance tomography have been making inroads in the clinical arena. These methods are not available to the general practitioner or the family doctor due to the high level of technical equipment and costs. The purpose of these studies was, on the one hand, the development of a fully automatic system for diagnosing heart valves diseases which relies on an electronic stethoscope and a commercial PC. This should assist the GP in a first finding after which the patient can be referred to the required specialist, in particular, a cardiologist or internist. For this, the heart sound should be studied by methods of time and frequency analysis. The parameters extracted from these methods were statistically evaluated in a real patient population. On the other hand, the suitability of heart sound analysis for characterizing basic cardiac diseases (no heart valve diseases) was to be established. In this respect, it was shown that the time delay between the R wave and the maximum of the first heart sound is an important measure for the risk assessment in patients with heart failure. Proceeding from these very encouraging results from the author’s work and the recent results of other groups, the analysis of the biomechanics of the heart based on the heart sound analysis can be regarded as a very promising method, in particular, for screening of heart diseases by the general practitioner.

References 1. National Center for Health Statistics, Deaths: Preliminary Data for 2004, http://www.cdc.gov/nchs/data/hestat/preliminarydeaths04 tables.pdf (2004).

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2. American Heart Association, http://circ.ahajournals.org/cgi/content/short/ 113/6/e85, Circulation 113 (2006) e85–e151. 3. P. Deetjen and E. J. Speckmann, Physiologie (Auflage, Urban & Schwarzenberg 1994). 4. R. F. Schmidt, G. Thews and F. Lang, Physiologie des Menschen (Auflage, Springer Verlag 2000) 28. 5. S. Silbernagel and A. Despopoulos, Taschenatlas der Physiologie (Auflage, Dt. Taschenbuchverlag 1991) 4. 6. K. Holldack and K. Gahl, Auskultation und Perkussion. Inspektion und Palpation (Auflage, Georg Thieme Verlag 1991) 11. 7. K. Holldack and H. W. Rautenburg, Phonokardiographie (Georg Thieme Verlag 1979). 8. http://blick-in-den-op.de/herzklappen/klappenfehler/ursachen.html. 9. http://www.die-herzklappe.de/anders1.htm. 10. http://www.gesundheit.de/roche/pics/p29967.000-1.html. 11. S. Silbernagel and F. Lang, Taschenatlas der Pathophysiologie (George Thieme Verlag 1998). 12. D. Michel, Internist 6 (1965) 515–529. 13. J. Sandstede, K. F. Kreitner, D. Kivelitz, S. Miller, B. Wintersperger, M. Gutberlet, ¨ C. Becker, M. Beer, T. Pabst, A. Kopp and D. Hahn, Deutsches Arzteblatt 27 (2002) 1892–1897. 14. R. M. Rangayyan and R. J. Lehner, Critical Reviews TM in Biomedical Engineering 15, 3 (1987) 211–236. 15. W. Einthoven, Arch ges Physiol. 117 (1907) 461–478. 16. L. F. Barker, Bull Johns Hopkins Hosp 21 (1910) 358–389. 17. E. G. Dimond and A. Benchimoi, Ann. Intern. Med. 1 52 (1960) 145–153. 18. H. Maass and A. Weber, Cardiologica 21 (1952) 773. 19. M. Tavel, Circulation 93 (1996) 1250–1253. 20. I. Cathers, Artif. Intell. Med. 7 (1995) 53–66. 21. T. Oelmez and Z. Dokur, Pattern Recognition Letters 24 (2002) 617–629. 22. C. De Groff, S. Bhatikar, J. Hertzberg, R. Shandas, L. Valdes-Cruz and R. Mahajan, Circulation 103 (2001) 2711–2716. 23. J. Herold, R. Schroeder, F. Nasticzky, V. Baier, A. Mix, T. Huebner and A. Voss, Med. Biol. Eng. Comput. 43, 4 (2005) 451–456. 24. L. G. Durand, M. Blanchard, G. Cloutier, H. N. Sabbah and P. D. Stein, IEEE. Trans. Biomed. Eng. 37, 12 (1990) 1121–1129. 25. L. G. Durand, Z. Guo, H. N. Sabbah and P. D. Stein, Med. Biol. Eng. Comput. 31, 3 (1993) 229–236. 26. Z. Guo, L. G. Durand, H. C. Lee, L. Allard, M. C. Grenier and P. D. Stein, Med. Biol. Eng. Comput. 32, 3 (1994) 311–316. 27. D. Fritzsche, T. Eitz, K. Minami, D. Reber, A. Laczkovics, U. Mehlhorn, D. Horstkotte and R. Korfer, J. Heart Valve Dis. 14, 5 (2005) 657–663. 28. S. A. Pavlopoulos, A. C. Stasis and E. N. Loukis, Biomed. Eng. Online 3, 1 (2004) 21. 29. Z. Sun, K. K. Poh, L. H. Ling, G. S. Hong and H. Chew, J. Heart Valve Dis. 14, 2 (2005) 186–194. 30. D. Kim and M. E. Tavel, Chest 124, 5 (2003) 1638–1644. 31. S. M. Debbal and F. Bereksi-Reguig, Med. Phys. 32, 9 (2005) 2911–1917. 32. D. Barschdorff, S. Ester, T. Dorsel and E. Most, Biomedizinische Technik 35 (1990) 271–279. 33. Patent DE3722122C2: ‘Vorrichtung zur Bewertung einer periodischen Folge von Herzger¨ auschsignalen’, Inhaber Prof. Barschdorff. 34. S. Messer, J. Agzarian and D. Abbott, Microelectronics Journal 32 (2001) 931–941.

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35. L. Hall, J. Maple, J. Agzarian and D. Abbott, Microelectronics Journal 31 (1999) 583–592. 36. S. G. Mallat, IEEE Trans. Pattern Anal. Mach. Intell. 11, 7 (1989) 674–693. 37. Heart Disease and Stroke Statistics — 2006 Update. At-a-Glance, American Heart Association 21, (2006). 38. U. C. Hoppe, M. B¨ ohm, R. Dietz, P. Hanrath, H. K. Kroemer, A. Osterspey, A. A. Schmaltz and E. Erdmann, Z Kardiol 94 (2005) 488–509. 39. D. Levy, S. Kenchaiah, M. G. Larson, E. J. Benjamin, M. J. Kupka, K. K. Ho, J. M. Murabito and R. S. Vasan, N Engl J. Med. 347, 18 (2002) 1397–1402. 40. C. Thomas, Spezielle Pathologie (Special Pathology), Schattauer Verlag, ISBN 3-79451713-X (1996) 167–169. 41. R. Gross, P. Sch¨ olmerich and W. Gerok, Die Innere Medizin (Internal medicine), 9 (Auflage, Schattauer Verlag, ISBN: 3794516990, 1986) 157–185. 42. A. Voss, R. Schroeder, M. Baumert, S. Truebner, M. Goernig, A. Hagenow and H. R. Figulla, Comput. Cardiol. 32 (2005) 275–278. 43. Task Force of the European Society of Cardiology and the North American Society of Pacing and Electrophysiology, Circulation 93, 5 (1996) 1043–1065. 44. M. Baumert, T. Walther, J. Hopfe, H. Stepan, R. Faber and A. Voss, Med. Biol. Eng. Comput. 40, 2 (2002) 241–245. 45. R. Faber, H. Stepan, M. Baumert, A. Voss and T. Walther, J. Hum. Hypertens. 18 (2004) 135–137. 46. A. Voss, A. Mix and T. Huebner, Ann. Biomed. Eng. 33, 9 (2005) 1184–1191. 47. M. H. Drazner, J. E. Rame, L. W. Stevenson and D. L. Dries, NEJM 345 (2001) 612–614. 48. S. M. Butman, G. A. Ewy, J. R. Standen, K. B. Kern and E. Hahn, J. Am. Coll. Cardiol. 22, 4 (1993) 968–974. 49. W. L. Stevenson and J. K. Perloff, JAMA 261 (1989) 884–888. 50. P. M. Shah and P. N. Yu, Am. Heart J 78, 6 (1969) 823–828. 51. R. Patel, D. L. Bushnell and P. A. Sobotka, West J. Med. 158, 6 (1993) 606–609.

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CHAPTER 6 METHODS IN THE ANALYSIS OF THE EFFECTS OF GRAVITY AND WALL PROPERTIES IN BLOOD FLOW THROUGH VASCULAR SYSTEMS S. J. PAYNE∗ and S. UZEL Department of Engineering Science, University of Oxford Parks Road, Oxford OX1 3PJ, UK ∗ [email protected]

Simple one-dimensional models of blood flow are widely used in simulating the transport of blood around the human vasculature. However, the effects of gravity have only been previously examined briefly and the effects of changes in wall properties and their interaction with gravitational forces have not been investigated. Here the effects of both gravitational forces and local changes in wall stiffness on the one-dimensional flow through axisymmetric vessels are studied. The governing fluid dynamic equations are derived from the Navier-Stokes equations for an incompressible fluid and linked to a simple model of the vessel wall, derived here from an exponential stress-strain relationship. A closed form of the hyperbolic partial differential equations is found. The flow behavior is examined in both the steady state and for wave reflection at a junction between two sections of different wall stiffness. A significant change in wave reflection coefficient is found under the influence of gravity, particularly at low values of baseline non-dimensional wall stiffness. Keywords: Gravity; wall properties; blood flow; vascular systems.

1. Introduction The flow of blood around the circulatory system is the fundamental basis of the human anatomy. The correct understanding of the behavior of blood flow in the vasculature is thus crucial if any realistic physiological model of the human anatomy is to be accurate and representative. The difficulties involved in modeling the flow of blood in the vessels are well-known: There is considerable interaction between the fluid flow and the vessel wall, which makes modeling this flow a non-trivial matter. The governing fluid dynamics equations, the Navier-Stokes equations, are known, but to solve them the coupling between the fluid and vessel wall behaviors must be known. Since the vessel wall has a very complex structure, comprising three layers of different structure and function, it is frequently characterized with a simple pressurearea relationship, although more advanced mechanical models have also been used, as outlined below. This enables the fluid-structure equations to be reduced to two area-averaged equations, as shown below. Under certain assumptions, these equations can be approximated by a set of equations that is of identical form to an equivalent electrical circuit: The use of this analogy is extremely widespread and forms the basis for most models of the whole vasculature, although the precise nature of the equivalent circuit varies widely from author to author. Units, each comprising resistors, inductors and capacitors, are 207

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generally joined together to form a complete circuit. Essentially resistance, inductance and capacitance are included to model friction, inertia and wall compliance respectively. With the advent of powerful computers and efficient Computational Fluid Dynamics (CFD) solvers, many problems that were not previously amenable to analysis can now be solved numerically. However, one of the major drawbacks of using detailed numerical solvers is that only small regions of the vasculature can be examined at any given time. The setting of suitable boundary conditions is crucial and lumped parameter models are still frequently used. However, one-dimensional models of blood flow generally assume that the wall properties are invariant with distance along the vessel. It is well known that vessel walls exhibit local regions of increased stiffness, for example stenoses, or decreased stiffness, for example aneurysms. CFD solvers have been widely used to examine the details of the flow fields around such local variations, but the equations governing one-dimensional flow around such features have not received equivalent attention. Clearly, an assumption has to be made that the wall properties are invariant with circumferential position, but this should still provide a good first approximation to realistic situations. In addition, the effects of gravity on such features have not been considered. In this chapter, the equations governing flow in an axi-symmetric tube with an elastic wall are thus derived, including gravitational forces and variable wall stiffness. The theory is presented and some numerical simulations are shown to illustrate the effects of gravitational forces and variable wall stiffness on the flow field.

2. Theory 2.1. Fluid dynamic equations The approach set out in Canic and Kim1 is followed here, starting with the incompressible axi-symmetric form of the Navier-Stokes equations in cylindrical co-ordinates (x, r, θ). The momentum equations are:  2  ∂ux ∂ux 1 ∂p ∂ ux ∂ux 1 ∂ux ∂ 2 ux + ur + ux + =ν + − gS, (1) + ∂t ∂r ∂x ρ ∂x ∂r2 r ∂r ∂x2  2  ∂ur ∂ur ∂ur 1 ∂p ∂ ur ur 1 ∂ur ∂ 2 ur + ur + ux + =ν − , + + ∂t ∂r ∂x ρ ∂r ∂r2 r ∂r r2 ∂x2

(2)

and the continuity equation is: ∂ux 1 ∂(rur ) + = 0, ∂x r ∂r

(3)

where the velocity components are given by (ux , ur , uθ ) and the slope of the vessel is denoted by S. The value of slope varies between 1 and −1 for vessels inclined vertically upwards and downwards respectively in the direction of flow. It is equal

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to the sine of the angle of the vessel axis with the horizontal. The pressure is p and the fluid has density ρ and viscosity ν. It is assumed that the vessel radius is small in comparison with its length, so the gravitational force component in the radial direction can be neglected. 2.1.1. One-dimensional flow Using order of magnitude arguments, the penultimate term in Eq. (1) is neglected and the radial pressure gradient can be assumed small. Equations (1) and (3) are then integrated over the cross-sectional area to give: ∂(R2 ) ∂(R2 U ) + = 0, ∂t ∂x

  ∂(R2 U ) ∂(αR2 U 2 ) R2 ∂p ∂ux + + + gSR2 = 2νR , ∂t ∂x ρ ∂x ∂r r=R where the following non-dimensional parameters are used:  R 1 U= 2 2rux dr, R r=0  R 1 α= 2 2 2u2 rdr, R U r=0 x

(4) (5)

(6) (7)

where U is the area-averaged axial velocity and α is the correction term, also known as the Coriolis coefficient, which compensates for the fact that the final equation will be based on conservation of the area-averaged momentum, rather than the actual momentum. The wall inner radius is R, which varies with both longitudinal distance and time. To calculate the viscous term on the right-hand side of Eq. (5), the velocity profile is often assumed to be of the form of Hagen-Poiseuille flow:  r γ  γ+2  , (8) U 1− ux = γ R where γ is a constant that defines the shape of the velocity profile: A value of 2 corresponds to a Newtonian fluid, although much larger values have been reported to be a good compromise fit to experimental data, reflecting the fact that blood is not quite a Newtonian fluid.2 Equations (4) and (5) are then expressed in terms of flow rate, Q, and area, A: ∂A ∂Q + = 0, ∂t ∂x  ∂Q ∂ Q2 A ∂p παν Q + α + + gSA = −2 . ∂t ∂x A ρ ∂x α−1A

(9) (10)

Although area and flow rate are used as the primary variables of interest, these equations can be reformulated in terms of the area and area-averaged axial

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velocity3 : ∂A ∂U ∂A +A +U = 0, ∂t ∂x ∂x

(11)

U 2 ∂A ∂U 1 ∂p παν U ∂U + (α − 1) + (2α − 1)U + + gS = −2 . ∂t A ∂x ∂x ρ ∂x α−1 A

(12)

The viscosity of the fluid is well known to vary across the vessel cross-sectional area. Near to the wall, the fluid comprises a plasma layer, the red blood cells being confined to a core layer in the center of the vessel. Although two-phase models of the flow have been used,4 for a one-dimensional model of fluid flow only the correction factor and wall shear stress are required, so such models are not widely used here. The velocity profile, although well fitted by a Hagen-Poiseuille profile with a high value of γ, thus still has a low shear stress, since the plasma viscosity is much lower than that of the core flow. The right-hand side of equations 10 and 12 is thus often neglected, even though α is close to 1, due to the high value of γ. 2.1.2. Two-dimensional flow Under certain conditions, the flow cannot be assumed to be one-dimensional, for example where there are local changes in wall stiffness. If the two-dimensional flow field is to be solved, assuming again that the flow field is axi-symmetric, the governing equations can be written in the vorticity/stream-function form,5 for example. Since this approach is much less amenable to simple analysis, it is not examined further here. In addition, the two-phase nature of the fluid means that, except in larger blood vessels, the viscosity is not constant over the cross-sectional area and some means must be employed to determine the boundary between the flow layers. 2.2. Vessel wall equations Since the fluid dynamic equations are expressed in terms of three variables, but there are only two equations, the vessel wall equations play a very important role in determining the fluid behavior. This fluid-structure interaction is of great interest for a number of reasons, including the fact that wall shear stress is implicated in the behavior of the vessel wall. In particular, regions of low shear stress are thought to be implicated in the onset of atherosclerosis,6 although the precise mechanisms for this remain contested.7 The development of a physiologically realistic model for the vessel wall is complicated by the fact that the vessel wall comprises three layers, each of which has a different structure and function and the relative fractions of which vary widely around the vasculature. However, it is generally assumed that the wall has a small thickness and thus has negligible inertia in comparison to the fluid. The equilibrium equations are thus very widely used. The stress field will be assumed uniform over the wall thickness

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here. We briefly examine a number of different models, considering first models that consider both axial and radial deformations before examining those that include only radial deformation. Since such models can be expressed in terms of a pressure-area relationship, such relationships are finally reviewed. 2.2.1. Two-dimensional deformation The general equations for an axi-symmetric membrane with finite deformation can be expressed in terms of the principal deformation ratios8 :

2  2  ∂R ∂S , (13) λ1 = 1 + ∂x ∂x λ2 =

R , Ru

(14)

where R, S and Ru are the wall radius, axial position and undeformed radius respectively. The equilibrium equations in the tangential and normal directions are:

2   ∂R ∂R ∂T1 (T1 − T2 ) + R = τR 1+ , (15) ∂x ∂x ∂x −

T1 ∂ 2R T2 + = p, 2 2 3/2 ∂x (1 + (∂R/∂x) ) R(1 + (∂R/∂x)2 )1/2

(16)

in terms of the applied wall shear stress, τ , and pressure. T1 and T2 refer to wall tensions in the longitudinal and circumferential directions respectively. Since the membrane is assumed thin, it is frequently assumed to have negligible bending stiffness, as in the analysis presented here. The relationship between the deformation ratios and the wall tensions is the final link in the model. A linear visco-elastic model3,5,9 can be used, which characterizes the wall behavior by a non-dimensional elasticity coefficient, K, and viscosity coefficient, C:   3 ∂λ1 1 ∂λ2 λ2 − +C + , (17) T1 = K λ1 + 2 2 ∂t 2 ∂t   3 ∂λ2 1 ∂λ1 λ1 − +C + . (18) T2 = K λ2 + 2 2 ∂t 2 ∂t Although K can be determined in terms of the Young’s modulus and wall thickness, the value of C is much less well characterized. Its value, however, has a significant effect on the fluid flow patterns.9 The linearized version of these equations3 can be written in the form: ∆p =

3 Eh ∆R , 4 Ro Ro

where Ro is the characteristic value of radius.

(19)

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The inclusion of axial deformation makes the analysis highly complex: however, since the axial displacements are some orders of magnitude smaller than the radial displacements,3 these are frequently neglected. The vessel deformation is thus considered purely in radial terms, which means that the model reduces to a simple pressure-area relationship. This approach is thus now examined in detail as it forms the basis for a considerable proportion of the fluid-structure interaction models. 2.2.2. One-dimensional deformation The model for one-dimensional deformation is often referred to as the ‘independent ring model’. It has been shown that this gives the leading-order term in the approximation of the pressure in terms of the displacement.10 The equilibrium equation reduces to11 : pR = pext (R + h) + σθ h,

(20)

where the vessel wall has thickness h, the circumferential stress component is σθ and the external pressure is pext . Note that the external pressure is frequently neglected or it is assumed that pressure refers directly to trans-mural pressure, which is only the case for negligible wall thickness. This equation can be re-arranged into the form:  h h + σθ . (21) p = pext 1 + R R If it is assumed that the vessel wall is incompressible: (R + h)2 − R2 = (Ro + ho )2 − Ro2 , neglecting any longitudinal strain, the wall thickness to inner radius is:   2 Ao ho ho Ao h = −1 + 1 + 2 + , R Ro A Ro A

(22)

(23)

which, for thin-walled vessels, is approximately: h ho Ao = . R Ro A

(24)

The circumferential strain can be directly related to the change in area: dεθ =

1 dA , 2 A

which gives: 1 εθ = ln 2



A Ao

(25)

.

(26)

The relationship between circumferential stress and strain is thus the final step in determining the pressure-area relationship, linking Eqs. (21) to (26). The simplest

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relationship is that which assumes a linear isotropic material with constant Young’s modulus and Poisson’s ratio, i.e. Hooke’s law: εθ =

1 (σθ − νσz ), E

(27)

radial stress being much smaller than circumferential or longitudinal stresses for thin walled vessels. However, since the vessel wall increases in stiffness with increased strain, a nonlinear relationship is more accurate. One such possible relationship is of the form:  k 1 (28) εθ = ln 1 + (σθ − νσz ) , k E where k is a constant that quantifies the non-linearity of the stress-strain relationship, which assumes that the stress is related to the exponential of the strain. As k tends to zero, Eq. (28) tends towards Eq. (27). Another, similar, relationship between stress and strain that can be used is derived from the strain energy relationship proposed by Zhou and Fung12 : W =

K (exp(a1 Eθ2 ) − 1), 2

(29)

where K is a measure of the stiffness and a1 a measure of the non-linearity of the material, W being the strain energy. Longitudinal strain is assumed zero here, hence this expression is a simplified version of that given by Zhou and Fung.12 Circumferential stress is then: σθ = Ka1 Eθ (1 + 2Eθ ) exp(a1 Eθ2 ),

(30)

where the strain component is related to the change in cross-sectional area by:  1 A −1 . (31) Eθ = 2 Ao The relationship between pressure and area can be directly calculated using either Eq. (28) (Eq. (27) being taken as a special case of Eq. (28), where k tends to zero) or Eq. (30), using the equilibrium conditions for circumferential stress (Eq. 21) and longitudinal stress: 2   h h h 2+ . (32) + σz p = pext 1 + R R R The pressure-area relationship for the logarithmic model (Eq. 28) is thus:   E ho 1 1 1 ho (k/2)−1 + , (33) a − p = pext 1 + Ro a k (1 − ν/2) Ro a where: a=

A . Ao

(34)

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Since the wall thickness is assumed small, only the leading terms have been considered. In the limit as k → 0, Eq. (33) tends towards:  E ho ln (a) ho 1 + , (35) p = pext 1 + Ro a 2(1 − ν/2) Ro a which is the response for a linear elastic material. The corresponding relationship for the strain energy model (Eq. 30) is:  a  Ka1 ho ho 1 1 1 p = pext 1 + + (a − 1) exp (a − 1)2 . (36) Ro a 2 Ro (1 − ν/2) 4 For convenience, we define the baseline stiffness of the vessel as:  ∂p  Go = , ∂a a=1

(37)

which gives the logarithmic and strain energy pressure-area relationships as:   ho 1 1 2 (k/2)−1 + Go a , (38) p = pext 1 + − Ro a k a   a ho 1 1 + Go (a − 1) exp (a − 1)2 . p = pext 1 + (39) Ro a 4 The resulting pressure-area characteristics are shown in non-dimensional form in Figs. 1 and 2 for varying values of k and a1 . External pressure is taken to be zero here. Large values of k are required to give a significant non-linear effect in the

Fig. 1.

Pressure-area relationship for varying values of k for logarithmic and linear function.

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215

Pressure-area relationship for varying values of a1 for strain energy function.

logarithmic model, since a linear material automatically gives a reduction in stiffness with pressure. This indicates that the vessel wall material is very strongly nonlinear. However, the stiffness increases with raised pressure for any positive value of a1 , although it should be noted that the value of a1 quoted by Zhou and Fung12 is 0.264, for which the area increases very rapidly with pressure, which is not physiologically realistic. It is also convenient to consider the compliance of the vessel, since this is known to exhibit certain characteristics. This is defined here in non-dimensional terms as: C = Go

∂a , ∂p

(40)

and for the logarithmic and strain energy relationships is, respectively: a2 k  , k 2 − 1 ak/2 + 1 2  a  1 1 2  (a − 1) exp − . C=  a1 4 1 + (a − 1)2 2 C=

(41)

(42)

The resulting compliance against area ratio for both relationships is shown in Figs. 3 and 4. Note that for the definition of compliance used here, the value is always equal to one at zero pressure. Both Eqs. (41) and (42) show a peak in compliance, which is exhibited by experimental data.13 However, this peak is found at

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Fig. 3.

Compliance for varying values of k for logarithmic and linear function.

Fig. 4.

Compliance for varying values of a1 for strain energy function.

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an area ratio that is dependent upon the non-linearity parameter for the logarithmic model:  apc =

8 (k − 2)(k − 4)

k2 ,

(43)

whereas it is always found at baseline area for the strain energy model. The former is more physiologically realistic.13 For the logarithmic model, the area ratio at peak compliance is a monotonically decreasing function for medium values of k, as shown in Fig. 5. Note that a maximum is only found in the range k ≥ 4. As the value of k increases past 6, the peak compliance is found below an area fraction of 1. As k tends towards a very large value, the area at peak compliance tends back towards 1, having a minimum value of 0.679 at a value of k of 13.4. The logarithmic model thus has two advantages: Peak compliance that occurs at different area ratios, although, since there is only one degree of freedom in the model, this is wholly dependent upon the nonlinearity parameter, and algebraic simplicity in comparison with the strain energy relationship, which, although based on experimental data, results in a complex pressure-area relationship that is not amenable to inclusion within the one-dimensional flow equations. Before considering this coupling of the fluid and vessel equations, we briefly survey some of the existing pressure-area relationships available in the literature.

Fig. 5.

Area at peak compliance as a function of k for logarithmic and linear function.

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2.2.3. Existing pressure-area relationships Probably the most popular form of the pressure-area relationship used in the fluidstructure interaction equations is the power law relationship2 :

  β/2 A −1 , (44) p − pext = Go Ao which is a generalized version of the expression quoted by Formaggia,14 where Go is given by: Go =

ho E . Ro (2 − ν)

(45)

This power law relationship is clearly very similar to that derived above, based on the logarithmic model for stress-strain (Eq. (38)), although Eq. (38) includes a finite wall thickness. Equation (44) also has a monotonically decreasing compliance with area, which is not found in Eq. (38) and which is found in the experimental data. Olufsen15 quotes a similar relationship:

 1/2  A 4 Eho 1− , (46) p − pext = 3 Ro Ao Eho = k1 exp(k2 Ro ) + k3 , Ro

(47)

based on the experimental results of several studies.16–18 A quadratic relationship was proposed by Stergiopoulos17 : A = Aref (1 + Co (p − pref ) + C1 (p − pref )2 ), an exponential relationship by Hayashi19:

 p = po exp β

A −1 Ao

(48)

 ,

and a tangential relationship by Langewouters13:   A 1 , − p = po + p1 tan π Am 2

(49)

(50)

where po is the pressure at maximum compliance and p1 the ‘half-width’ pressure, i.e. the change in pressure from the pressure at maximum compliance required to halve the compliance. Am is the maximum area of the vessel at high pressure, beyond which the pressure cannot increase. All other parameters in Eqs. (48)–(50) are constants. The data presented by Langewouters13 is of particular interest as the compliance shows a clear peak at low area ratios. From the above survey, it is clear that there is no generally agreed form of the pressure-area relationship for use in the fluid-structure interaction equations. Although they all exhibit an increasing stiffness with pressure, they differ in their algebraic complexity and in their ability to model the peak in compliance found

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in experimental data. The optimal relationship, however, should be of a form that it models the variation in compliance accurately and that it can be incorporated within the fluid equations simply. The only such relationship is the one that we have derived here (Eq. (38)). This is directly based on the material properties, assuming small, but not negligible wall thickness, and predicts a peak in compliance, whilst being algebraically simple. It will thus be incorporated within the fluid-structure equations, as now outlined below. 2.3. Coupled form of equations The pressure-area relationship in Eq. (38) is substituted into Eq. (10). The vessel wall properties are assumed all to be invariant with time, although the wall stiffness is allowed to vary with axial distance. This latter addition is used to model the fact that there may be local variations in the wall properties, for example at an aneurysm or a stent. The external pressure is determined by static equilibrium: pext = pext,o − ρgSx,

(51)

relative to a reference pressure, defined at x = 0, which is taken to be the vessel inlet. The external pressure is thus only constant for a vessel inclined along the horizontal. Substitution of Eqs. (38) and (51) into Eq. (10) gives:     (k/2)−1 ∂Q ∂ Q2 2Go k A Ao ∂A + α + −1 + ∂t ∂x A kρ 2 Ao A ∂x   k/2 ho A ho Ao pext ∂A παν Q 2Ao dGo = gSAo − , − −2 −1 Ro A ρ ∂x Ro α−1A kρ Ao dx (52) where variations in wall stiffness only are assumed, the non-linearity parameter being taken as constant. The non-zero wall thickness, which leads to a further term in the equilibrium equation, thus contributes two terms in Eq. (52), one related to external pressure and one to gravitational forces. The relative magnitude of these terms, even for small values of the wall thickness to radius ratio, has been investigated in detail.20 The term relating directly to the external pressure on the left-hand side of Eq. (52) is generally negligible, except for walls with very low stiffness, and will thus be neglected here. However, the term due to the gravitational force (the first term on the right-hand side of Eq. (52)) has been shown to be significant, even for thin-walled vessels, when the vessel is not horizontal.20 As a fraction of the viscous term, the gravitational term is: Sα gho Ro , (53) ψ= 2(α − 1) νU

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where U is the flow velocity. The non-dimensional parameter ψ can be used to determine the importance of the gravitational forces relative to the viscous forces in a similar way to the Reynolds number being used to determine the relative importance of momentum and viscous terms. For a Newtonian fluid with viscosity ν = 3.2 × 10−6 m2 /s and a flow velocity of 0.5 m/s in a vessel of radius 5 mm and wall thickness 0.5 mm, this ratio is approximately equal to twice the slope of the vessel and is thus large even for a thin-walled vessel. A second non-dimensional parameter can be defined that compares the gravitational term to the momentum term: Sα gho ψ = . (54) φ= Re 2(α − 1) U 2 This is similar in form to the Froude number and interestingly is primarily dependent upon only the wall thickness and the flow velocity. It has been shown that gravitational forces are highly significant in comparison with either the viscous forces or the momentum forces for every type of vessel.20 ψ is large in the larger vessels since both the wall thickness and inner radius become small in the microvasculature, whereas φ is larger in the smaller vessels, since the velocity becomes very small. It is suggested that, for any vessel, both non-dimensional parameters are calculated to ascertain the importance of gravitational forces, since gravitational forces should not in general be neglected unless the vessel under consideration has low values of both ψ and φ. Before considering the solutions to the governing equations, we re-express them in non-dimensional form. The following non-dimensional parameters are used: a = A/Ao , q = Q/Uo Ao , go = Go /ρUo2 , z = x/L, τ = t/T , where Uo , L and T denote characteristic velocity, length and time scales respectively. The resulting continuity and momentum equations are thus: ∂a ∂q + = 0, ∂τ ∂z  2   ∂ q 2go 1 ∂a ∂q k + α + − 1 a(k/2)−1 + St ∂τ ∂x a k 2 a ∂z St

= gr S − 2

1 q 2 α dgo − [ak/2 − 1] , α − 1 εRe a k dz

(55)

(56)

where: L , Uo T ho gL , gr = Ro Uo2 St =

(57) (58)

Ro Uo Ro . (59) L ν In non-dimensional form, the equations are thus dependent upon four nondimensional parameters (St, gr , εRe, α) together with the wall stiffness parameters εRe =

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(the variation of stiffness along the vessel wall and the degree of non-linearity). The first parameter determines the relative importance of the time-varying behavior, the second the importance of gravitational forces, the third the viscous forces and the last characterizes the velocity profile. In matrix form the equations become: ∂θ ∂θ + H(θ) = B(θ), (60) ∂τ ∂z where:   a θ= , (61) q   0 1 1     2 H (θ) = (62)  k q , 1 (k/2)−1 St −α q + 2go −1 a 2α + 2 a k 2 a a   0 1   dg  B (θ) = (63) .  1 q 2  k/2 o St g S − 2 α − a −1 r α − 1 εRe a k dz The eigenvalues of the matrix H(θ) are:      q 2 k q 2go 1 − 1 a(k/2)−1 + . (64) (α − 1) + λ1,2 = α ± α a a k 2 a Since the parameter α is approximately one, they are thus:    √ k 2go 1 λ1,2 ≈ ± − 1 a(k/2)−1 + ≈ ± go . k 2 a

(65)

The orientation of the vessel has no effect on the eigenvalues; we have neglected the external pressure, which will tend to reduce their magnitude, but this is a very small effect. Having presented the governing equations, the solutions are now examined, both the steady state solution and the wave behavior where there are changes in wall stiffness under the influence of gravitational forces. 3. Model Predictions 3.1. Steady state behavior In the steady state the flow rate is invariant with axial position (Eq. (55)) and the steady state area varies according to:   dg  1 q 2  k/2 α o − gr S − 2 a −1 ∂a α − 1 εRe a k dz = (66)   ,  2 ∂z q k 2go 1 (k/2)−1 − 1 (a) −α + + a k 2 a

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where the over-bar denotes time-averaged values. This result is a more general version of that presented previously.2,20 Although Eq. (66) is somewhat complicated, some simplifications can be made to interpret its behavior. Firstly, it can be noted that the numerator consists of three separate terms, each of which contributes separately to changes in cross-sectional area: The first two correspond to gravitational and frictional forces respectively, whereas the third relates to the local change of wall stiffness. The importance of this term will be examined in more detail below. Secondly, it can be shown that the momentum term in the denominator of Eq. (66) is much smaller than the wall stiffness term under most physiologically realistic conditions and that the local area is normally close to the unstressed area.20 Equation (66) thus reduces, when wall stiffness is constant, approximately to:   ∂a 1 1 α = gr S − 2 q . (67) ∂z go α − 1 εRe For short lengths, the area variation with distance is approximately linear. A positive change in area is induced by a positive slope and a reduction in area is caused by a negative slope and by frictional forces. Changes in the wall stiffness adjust the rate of change of area with axial distance. The area change due to frictional forces is relatively small in physiologically realistic vessels.20 However, that due to gravitational forces alone is somewhat more significant: ∂a gr 3ρgL = S= S, ∂z go 2E

(68)

where Poisson’s ratio is assumed to be 0.5 for an incompressible material. For a Young’s modulus E = 300 kN/m2 and blood density ρ = 1050 kg/m3 , this is equal to 5.15% per meter per unit slope. Note that it is not dependent upon either the flow field or the cross-sectional geometry of the vessel. In the longer blood vessels, the gravitational force will thus have a significant effect on the flow field, even though the change in cross-sectional area is only a few percent: it has been shown elsewhere1 that a small tapering in the vessel cross-sectional area has a significant effect on the flow field, particularly the shock formation behavior. The wave behavior will thus now be examined in more detail, in particular the response to local changes in wall stiffness under gravitational forces. 3.2. Wave behavior Since the changes in wall stiffness are likely to occur over short sections, such that the distance over which the change occurs is of comparable length to the vessel radius and less than the wavelength of pulsatile flow (of order 0.1–1 m), any local change in wall stiffness can be considered as a first approximation as a step change. The behavior of waves approaching such sharp changes in wall stiffness is thus investigated here.

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To consider such behavior, the governing equations are linearized about the baseline values: a = a + a ,

(69)

q = q + q ,

(70)

where the overbar is used to denote steady state conditions, as determined by Eq. (67) for area and shown to be constant for flow rate, and the prime is used to denote the fluctuating component. Substitution of these into Eqs. (55) and (56) and subtraction of the steady state condition gives: ∂q  ∂a + = 0, ∂τ ∂z     2 ∂q  ∂ q a 2go k 1 ∂a a (k/2)−1 St +α 2 − + −1 a − ∂z ∂z q a k 2 a ∂z a      2go α 1 a k q 1 ∂a (k/2)−1 + −1 a = −2 − , + k 2 a ∂z α − 1 εRe q a St

(71)

(72)

where the rate of change of stiffness term is now neglected, since it is assumed that there is a step change in wall stiffness. It was previously shown that the rate of change of area in the steady state is small under most conditions of physiological interest. It can thus be assumed that both the steady state flow and area are approximately one. Substitution of Eq. (67) into Eq. (72) thus gives: St

∂q  α 1  ∂a ∂q  + 2α + (go − α) +2 q ∂τ ∂z ∂z α − 1 εRe    1 k k α − 2 gr S − 2 − 1 a = 0. + 2 α − 1 εRe 2

(73)

To analyze the behavior of this system, we consider a vessel with two sections: the first section with wall stiffness goL and the second with wall stiffness goR . In the first section, there are forward and backward waves, but there is only a forward wave in the second section:     z z  + DaL exp jω τ + , (74) aL = CaL exp jω τ − cL cL     z z  + DqL exp jω τ + , (75) qL = CqL exp jω τ − cL cL   z  , (76) aR = CaR exp jω τ − cR   z  , (77) qR = CqR exp jω τ − cR

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where the coefficients C and D determine the relative magnitudes of the forward and backward waves respectively and cL and cR are the wave speeds in the two sections. Substitution of Eqs. (74)–(77) into the linearised continuity and momentum equations gives the following equation for wave speed in either section: c2 (jω(St)2 + Kq St) + c(Ka − 2αjωSt) − jω(go − α) = 0, giving the following solution:    1 (go − α)  2 c= α + α + , St (1 + Kq /jωSt)

(78)

(79)

where: α =  Ka =

α − Ka /2jωSt , 1 + Kq /jωSt

 1 k k α − 2 gr S − 2 −1 , 2 α − 1 εRe 2

(80) (81)

1 α . (82) α − 1 εRe To calculate the reflection coefficient, continuity of area and its first differential at the boundary between the two sections is used to relate the wave coefficients: Kq = 2

CaL = DaL + CaR , ω ω ω −j CaL = j DaL + j CaR , cL cL cR

(83) (84)

which yields a reflection coefficient: ρR =

c L − cR . c L + cR

(85)

This reflection coefficient is a function of the two values of wall stiffness, the wall nonlinearity parameter and the four non-dimensional parameters (St, gr , εRe, α) that govern the flow field. In the limiting case where frictional and gravitational forces are negligible (i.e. gr is small and εRe is large) and α is approximately one, the wave speed is directly proportional to the square root of the wall stiffness. The reflection coefficient thus tends towards:  1 − goR /goL  , (86) ρR = 1 + goR /goL which is only dependent upon the ratio of the wall stiffness values in the two sections. This is shown in Fig. 6 for a range of values of this ratio: there is significant reflection at both low and high values of this ratio. However, since gravitational forces have been shown previously to have a significant impact on the flow behavior, the effects of non-zero gravitational forces will now be examined.

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Fig. 6. Reflection coefficient for changes in wall stiffness with stiff walls and zero friction and gravitational forces. Table 1. Typical values of physiological parameters for human cardiovascular system.21

Aorta Arteries Arterioles Capillaries Venules Veins Vena cavae

Wall thickness ho (mm)

Inner radius ro (mm)

Length L (mm)

Number of vessels

2.5 0.5 0.025 0.001 0.010 0.5 1.5

10 1 0.050 0.005 0.050 2 30

0.5 0.01 0.001 0.001 0.0005 0.01 0.3

1 3450 18600000 16100000000 160000000 512 1

To estimate typical values of the non-dimensional parameters, characteristic vessel dimensions and numbers are used, Table 1. The wall is taken to have Young’s modulus E = 300 kN/m2 .2 Blood is assumed to have density ρ = 1050 kg/m3 and viscosity ν = 3.2 × 10−6 m2 /s. The resulting non-dimensional parameters are shown in Table 2, using a characteristic time period of 1 second. The characteristic velocity is calculated using an assumed average flow velocity in the aorta of 0.5 m/s: The flow in the remaining vessels is calculated by dividing the corresponding flow rate amongst the vessels equally. There is a very wide range of values for most parameters, over a number of orders of magnitude, apart from St, which ranges only between 0.4 and 8 over the entire cardiovascular system. The gravitational parameter is largest in the smallest vessels, as is the frictional term, since εRe is

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Table 2. Typical values of non-dimensional parameters for human cardiovascular system.

Aorta Arteries Arterioles Capillaries Venules Veins Vena cavae

St

gr

ε.Re

go

1 0.69 0.93 8.05 4 0.41 5.4

4.9 233 4240 127,000 62,800 41.1 47.7

31.3 0.453 8.40e-4 9.70e-7 1.95e-4 3.05 52.1

381 9.07e5 1.65e8 4.94e9 4.88e9 1.60e5 6170

very small. The wall stiffness term is large in all vessels, but largest in the smallest vessels: this is due to the flow velocity being very small. The value for k can be approximately derived by comparing Eqs. (38) and (44) and using estimated values for β from the literature: values quoted for the arterial network are in the range 3.23 to 3.53 and 2.15 for the venous network.2 This gives values of k in the approximate range 4 to 9. The resulting variation in reflection coefficient with wall stiffness ratio (taken throughout as the value in the second section as a fraction of the value in the first section) is shown in Fig. 7 for the parameters for flow in the arteries. Throughout, we plot the magnitude of the reflection coefficient, the phase being of less interest. Although the reflection coefficient is predominantly determined by the wall stiffness ratio, there is also an influence due to the gravitational force, with the reflection

Fig. 7. Reflection coefficient for changes in wall stiffness for flow in the arteries, using parameters from Table 2.

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coefficient having a larger magnitude for positive slopes and a smaller magnitude for negative slopes. This effect is, however, not symmetrical, as the effects of gravity are much more significant at wall stiffness ratios less than one than at values greater than one. The solution for zero slope is almost identical to the ‘ideal’ solution shown in Fig. 6, showing that frictional forces have a negligible effect on the wave behavior, as expected. Since the wave speed derived in Eq. (79) is complex, it should strictly be split into real and imaginary components: the real part corresponding to the actual wave speed and the imaginary part relating to the wave attenuation: 1 1 1 =  + j  . c c c

(87)

In the absence of frictional and gravitational forces, the attenuation is zero. The wave speed and attenuation coefficient for flow in the arteries is shown in Fig. 8. There is only a small change in either parameter as the slope varies, with the wave speed increasing as slope increases (either positively or negatively), and the attenuation coefficient, which is very small, increasing with slope in a near linear manner. For small values of the attenuation coefficient, this coefficient is equivalent to the fractional reduction in the wave amplitude over the length of the vessel in which the wave is traveling. The reflection coefficient as a fraction of the value at zero slope is shown in Fig. 9. There is a significant change in reflection coefficient

Fig. 8. Non-dimensional wave speed and attenuation coefficient for flow in the arteries, using parameters from Table 2.

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Fig. 9. Reflection coefficient for changes in wall stiffness for flow in the arteries, using parameters from Table 2, as a fraction of the values at zero slope.

at low wall stiffness ratios with slope: even at a ratio of 0.1, there is a change of ± 4% with slope changing from −1 to +1. However, if the wall stiffness reduces, thus reducing the corresponding nondimensional parameter, the effect of gravitational forces becomes progressively more significant. The reflection coefficient is shown in Fig. 10 for a baseline wall stiffness reduced by a factor of 10 and the corresponding wave speed and attenuation coefficient in Fig. 11. The reflection coefficient as a fraction of the value at zero slope is shown in Fig. 12. There is a more significant change in reflection coefficient at low wall stiffness ratios with slope: Even at a ratio of 0.1, there is a change of ± 4% with slope changing from −1 to +1. At ratios higher than one, the change is much smaller: Reflection coefficient is thus predominantly affected by gravitational forces when incurred by a reduction, rather than an increase, in wall stiffness. For small changes in reflection coefficient, the percentage change with slope is proportional to the gravitational parameter gr . This scaling drops off for high values, as the reflection coefficient saturates at one. The gravitational parameter is dependent upon the vessel wall thickness to inner radius ratio and length and the flow velocity: In thicker walled vessels, the gravitational parameter thus becomes more significant. This also increases the non-dimensional wall thickness, the ratio of the two being independent of both the wall thickness to inner radius ratio and the flow velocity.

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Fig. 10. Reflection coefficient for changes in wall stiffness for flow in the arteries, using parameters from Table 2, but with baseline wall stiffness reduced by a factor of 10.

Fig. 11. Non-dimensional wave speed and attenuation coefficient for flow in the arteries, using parameters from Table 2, but with baseline wall stiffness reduced by a factor of 10.

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Fig. 12. Reflection coefficient for changes in wall stiffness for flow in the arteries, using parameters from Table 2, but with baseline wall stiffness reduced by a factor of 10, as a fraction of the values at zero slope.

From the numerical simulations, it is found that, at a fixed value of wall stiffness ratio, the reflection coefficient is scaled under gravitational forces in an approximately linear manner with the gravitational parameter and the square root of the baseline wall stiffness.

4. Discussion It should be noted that the analysis presented here only considers a single boundary between two regions of uniform stiffness. In reality, there are likely to be more than one change in wall stiffness, as there may be regions of decreased stiffness (for example an aneurysm) or increased stiffness (for example a stent). There will then be multiple reflections, which will complicate the flow field, although the analysis here could easily be extended to such situations. In earlier work,20 velocity waveforms propagating along the vessel were found to exhibit similar behavior to that shown previously1: The waveform appears to ‘pile up’ with the front edge becomes steeper. These solutions were obtained using the Lax-Wendroff numerical solver.22 This ‘piling up’ behavior has not been examined here as the focus has been on the behavior of waveforms at sharp discontinuities in wall stiffness under the influence of gravity. The waveform amplitude increases under positive slope and vice versa, relative to the behavior at zero slope. The amplitude increase with distance down the vessel is almost linear and is strongly dependent on the slope. For a vertically inclined

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vessel, the amplitude of the pulse increases by approximately 4% per meter for flow downwards and decreases by approximately 5% per meter for flow upwards. The front of the waveform is also steeper for positive slopes, growing more rapidly and thus promoting shock formation. It was shown using numerical simulations20 that the rate of shock formation is increased for downwards flow and decreased for upwards flow by a factor that is approximately 1.2 times the wall thickness to inner radius ratio times the slope, i.e.: xs (S) ho ≈ 1.2 S. xs (S = 0) Ro

(88)

Thus even for thin-walled vessels, a slope causes a significant change in the rate of shock formation. A linear tapering in area, simulated by allowing the baseline area to decrease linearly with distance, was found to postpone shock formation.1 Both of these effects thus have an effect on the shock behavior, it being postponed by both a linear tapering in baseline area and an upwards slope.

5. Conclusions We have presented here an investigation into the effects of gravity on the flow field in compliant axi-symmetric vessels. The particular focus has been on the cardiovascular system, although the results here are applicable to any similar fluid-structure interaction system. We have outlined the theory governing this flow in detail, covering the mechanics of both the fluid and the vessel wall and illustrating the complex nature of this fluid-structure interaction. After presenting a survey of models for the vessel wall, we have presented a new model that captures the characteristics of the vessel wall behavior well. This has been coupled to the fluid dynamics equations and the governing equations thus derived. The steady state and wave behavior have then been examined to investigate the difference in behavior of the flow field under the influence of gravitational forces. It is found that gravitational forces induce significant changes in the steady state area variation of such vessels: Even though this change is only a few percent, this is known to have a significant effect on the flow field, particularly the shock wave formation. The wave behavior at the junction between two sections of different wall stiffness is also found to be affected by gravitational forces, particularly where the wave encounters a region of reduced wall stiffness. The reflection coefficient is increased in magnitude for positive slopes and decreased for negative slopes, whereas the wave speed is increased slightly for either positive or negative slopes and the attenuation coefficient increases near linearly with slope. This effect is most significant when the non-dimensional gravitational parameter is large and the nondimensional stiffness parameter is small. The analysis presented here, however, is only a part of the answer: There remains much work to be done to improve our understanding of the fluid-structure interaction found in the human vasculature. At each stage of modeling, rigorous mathematical analysis of the impact of different effects is vital. Although a number of

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assumptions that are widely made have been relaxed in this paper, there are still several issues to be considered: In particular, the models of the wall behavior are very simplistic, neglecting any damping or inertial effects. For better understanding of the flow field around changes in wall stiffness, more detailed models will be required, probably using two-dimensional models, where the radial velocity is also considered and the velocity profile is not assumed to be of a fixed shape. This is the subject of current research and will be very important if the wall shear stresses are to be predicted accurately: this will be crucial in improving our understanding of how the flow field influences the wall behavior in both normal and diseased states.

Acknowledgments I would like to thank Professor Paul Taylor and members of my research group for helpful comments.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.

D. Canic and E. H. Kim, Math. Meth. Appl. Sci. 61 (2003) 1161–1186. N. P. Smith, A. J. Pullan and P. J. Hunter, SIAM J. Appl. Math. 62 (2002) 990–1018. G. Pontrelli and E. Rossoni, Int. J. Num. Methods Fluids 43 (2003) 651–671. M. Sharan and A. S. Popel, Biorheology 38 (2001) 415–428. G. Pedrizetti, J. Fluid Mech. 375 (1998) 36–64. A. M. Malek, S. L. Alper and S. Izumo, J. Am. Med. Assoc. 282 (1999) 2035–2042. N. Resnick, H. Yahav, A. Shay-Salit, M. Shushy, S. Schubert, L. C. M. Zilberman and E. Wofowitz, Prog. Biophys. Mol. Biol. 81 (2003) 177–199. A. E. Green and J. E. Adkins, Large Elastic Deformations (Clarendon Press, Oxford, 1960). G. Pontrelli, Med. Biol. Eng. Comp. 40 (2002) 550–556. S. Canic and A. Mikelic, Comptes Rendus de l’Academie des Sciences, Serie I (2002) 661–666. M. Ursino and C. A. Lodi, Am. J. Physiol. 274 (1998) H1715–H1728. J. Zhou and Y. C. Fung, Proc. Natl. Acad. Sci. 94 (1997) 14255–14260. G. Langewouters, K. Wesseling and W. Goedhard, J. Biomech. 17 (1984) 425–435. L. Formaggia, F. Nobile, A. Quarteroni and A. Veneziani, Comput. Visual. Sci. 2 (1999) 75–83. M. S. Olufsen, Am. J. Physiol. 276 (1999) H257–H268. N. Westerhof, F. Bosman, C. Vries and A. Noordergraf, J. Biomech. 2 (1969) 121–143. N. Stergiopoulos, D. F. Young and T. R. Rogge, J. Biomech. 25 (1992) 1477–1488. P. Segers, F. Dubois, D. De Wachter and P. Verdonck, J. Cardiovasc. Eng. 3 (1998) 48–56. K. Hayashi, K. Handa, S. Nagasawa and A. Okumura, J. Biomech. 13 (1980) 175–184. S. J. Payne, Med. Biol. Eng. Comp. 42 (2004) 799–806. D. J. Schneck, The Biomedical Engineering Handbook, ed. J. D. Bronzino (CRC Press, 2000). R. J. Leveque, Finite Volume Methods for Hyperbolic Problems (Cambridge University Press, 2002).

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CHAPTER 7 NUMERICAL AND EXPERIMENTAL TECHNIQUES FOR THE STUDY OF BIOMECHANICS IN THE ARTERIAL SYSTEM THOMAS P. O’BRIEN, MICHAEL T. WALSH, LIAM MORRIS, PIERCE A. GRACE, EAMON G. KAVANAGH and TIM M. McGLOUGHLIN∗ Centre for Applied Biomedical Engineering Research MSSI, University of Limerick, Limerick, Ireland ∗ [email protected]

Cardiovascular disease is a major cause of death in Western society. Complications arise throughout the cardiovascular system with the arterial sub-system being prone to atherosclerosis and aneurysm formation. Atherosclerosis is a disease characterized by the deposition of lipoproteins in the arterial wall while an aneurysm is characterized by an abnormal swelling of the wall itself. Medical therapy, comprising life-style changes and drug administration is the mainstay of treatment for the majority of people with arterial vascular disease. However, a variety of surgical interventions are available to treat diseases associated with the arterial system and the primary objective of such interventions is to restore normal arterial function. These treatments may be performed using open surgery or minimally invasive techniques and are biomechanical in nature. A range of numerical and experimental techniques has been developed to quantify and qualify disease-influencing biomechanical factors in the arterial system. These techniques may be used to perform basic research on disease forming mechanisms, to diagnose disease in vivo and to assess or develop new biomedical treatments. The techniques presented in this chapter are applied for various purposes at different locations in the arterial system using an integrated research and development approach. The chapter concludes with a discussion on new medical devices currently under development in order to demonstrate how numerical and experimental methods may be applied in the search for new and improved treatments for arterial disease. Keywords: Cardiovascular; artery; hemodynamics; biomechanical modeling; computational fluid dynamics.

1. Introduction Numerical and experimental techniques have been widely used for the study of biomechanics in the human vascular system.1–4 At the macroscopic or systemic level, application of these techniques have yielded extensive knowledge on how blood interacts with the blood vessels through which it flows, and, if complications in the form of disease or injury arise, how the mechanical function of the vascular system may be restored.5,6 At present, numerical and experimental techniques are being used to qualify and quantify the biomechanical environment and to hypothesize, design and develop new biomedical treatments for vascular disease.7–9 The function of the cardiovascular system is to transport nutrients and oxygen around the body and to remove waste products. In mechanical terms, it comprises a pump, in the form of the heart, and a distribution network, consisting of arteries, 233

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capillaries and veins.10 While complications can occur throughout the vasculature, a major focus of research to date has been investigating the principal diseases associated with the arterial system.11 The arterial system consists of major arteries such as the aorta and the iliac arteries, and minor arteries such as the femoral, carotid and coronary arteries. The major disease associated with the arterial system is atherosclerosis characterized by the deposition of low-density lipoproteins onto the endothelial cell layer that lines the lumen of the artery.12 These depositions may accumulate over time causing narrowing of the artery lumen, which can result in hypertension, stroke, limb ischemia or heart attack, some of the major causes of death in western society today.13 In a simple analysis, a segment of the arterial system may be modeled as a Newtonian fluid moving through a constant diameter rigid-walled pipe at constant velocity.14 A constant pressure gradient acts between the inlet and outlet of the pipe and the blood develops fully according to Poiseuille’s flow. This simplification has been widely used in biomechanical models to estimate the mean shear stresses acting on the endothelial cells, often cited as being a major disease-influencing factor.15–18 Previous work has assumed that the Newtonian assumption is valid in larger diameter arteries such as the aorta and the iliac arteries.19 However, in smaller diameter arteries, such as the coronary and femoral arteries, blood is more accurately modeled as a non-Newtonian fluid, with a viscosity/shear characteristic that may be determined experimentally.10 At the microscopic level, blood is comprised of red blood cells, white blood cells, platelets and plasma, but it is the red blood cells which coalesce to form Rouleux chains, which result in non-Newtonian characteristics in smaller diameter arteries. A number of non-Newtonian models have been reported previously.19 A further assumption, which limits the applicability of this simple model, is that of the blood velocity boundary condition. The heart does not provide steady flow of blood through the vascular system; rather, it is an oscillatory pump which pulsates blood through the vasculature.20,21 The resulting transient flow rates vary throughout the system depending on location, with the large diameter aortic flow rates being far higher compared to those associated with the small diameter coronary and femoral arteries.22 This creates a need to model arterial biomechanics with three-dimensional transient blood flow according to the continuity and NavierStokes equations.23 Transient blood flow differs significantly from steady flow and is characterized by temporally varying velocity profiles, wall shears and flow separation/recirculation regions.24 The model may be improved by modeling the compliant nature of the artery wall. Arteries vary in mechanical stiffness; as a person ages, their arteries become less compliant.25 However, the elasticity is such that, in younger people, it can affect the velocity profiles through the vasculature. In addition, the elasticity is responsible, in part, for the reversal in direction of blood flow over the pulsatile cycle, caused by vessel contraction and wave reflections.26 Though arteries are heterogenous and

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anisotropic, their non-linear elastic behaviour may be modeled to an acceptable degree by assuming linear elasticity.26,27 The original simplified model has thus evolved to represent non-Newtonian, pulsatile flow through a constant diameter complaint pipe. At this point, the model captures the main biomechanical characteristics of an idealized model of an artery. However, the foremost variable of interest throughout this chapter is that of geometry. Geometrical attributes have been cited as being the one of the most important biomechanical variables contributing to the initiation and development of atherosclerosis.28–30 In addition, for the treatment of disease, geometry is often the single most influential factor in designing surgical interventions.31,32 This chapter presents a range of numerical and experimental techniques used in the study of the biomechanics of the arterial system. Medical imaging techniques may be used to qualify and quantify biomechanical phenomena in the arterial system and to measure quantities suitable for analysis using in vitro biomechanical methods. Data acquired from medical images may be used to create models for use in numerical and experimental studies. Digital image processing techniques are discussed together with computer-aided methods to design and manufacture test models. Numerical methods, which are used to investigate the biomechanics of arteries and associated biomedical devices, are described with particular attention to the techniques of computational fluid dynamics and finite element analysis. Next, experimental validation and testing techniques are described. Flow modeling methods and measurement systems are presented with emphasis on laser Doppler anemometry, video extensometry and photoelasticity. Finally, an application of the aforementioned techniques is used to demonstrate the hypothesis, design and development process used to create novel medical devices. These techniques may be combined to describe an integrated investigative approach for arterial biomechanics (Fig. 1).

2. Medical Imaging Techniques Imaging methodologies are widely applied in the clinical environment for the diagnosis of disease and for follow-up evaluation of surgically implanted treatments.33,34 Methods such as magnetic resonance imaging (MRI) and computerized tomography (CT) have been used to assess the geometrical nature of aneurysms, to locate lesions and occlusions and to follow-up the incorporation and performance of vascular bypass grafts.35–37 What makes modern medical imaging techniques so powerful is the portability of the resulting data. While traditional X-ray systems were limited in the portability of the photographic plate, modern MRI and CT systems store their image data digitally, enabling rapid data transportation and processing. Additional techniques, such as Doppler ultrasound, enable researchers to obtain qualitative data as to the patency of bypass grafts and arteries and quantitative data in the form of flow rates through blood vessels.38 Other techniques such as

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Fig. 1.

The integrated approach to performing biomechanical investigations of the arterial system.

positron emission tomography (PET) and electrical impedence tomography (EIT) are less widely used but provide additional capabilities and applications. 2.1. Magnetic resonance imaging Magnetic resonance imaging has attained widespread use in recent times for clinical imaging and diagnostics. It is a non-invasive imaging technique posing minimal risk to the patient due to its non-ionizing radiation. It is capable of measuring blood velocity and acquiring geometrical information and is widely used for investigative biomechanics studies.39–42 2.1.1. Operating principles The core operating principle underlying MRI is that certain atomic nuclei, when placed into a magnetic field, undergo realignment as a result of the effect that the

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magnetic field has on their spin.43 The realigned nuclei, mainly hydrogen protons in the case of MRI, may then be periodically ‘flipped’ from the alignment or longitudinal plane towards the direction of the transverse plane by targeting a radio frequency (RF) pulse at them. Net magnetization is then in the transverse plane. This results in a voltage being induced in a coil placed parallel to the transverse plane thereby enabling data acquisition. Following the RF pulse, the protons realign with the longitudinal magnetic field. The time it takes for realignment is termed the relaxation time (T1). Different tissues have different relaxation times, resulting in a difference between the transverse magnetization being received for different tissues.37 These different tissues may then be distinguished on the basis of the induced voltage in the receiver coil. Different RF signals have been developed for imaging different tissues. The simplest is termed the saturation recovery (SR) signal. This involves a single excitation pulse with a fixed time between each pulse, termed the repetition time (TR). The relaxation time must be different from the repetition time; otherwise there will be artifacts present in the images. This method is not capable of varying the contrast between different tissues. A more widely used method incorporates an echo pulse following the excitation pulse, termed the spin echo (SE) method. The echo pulse is used to gradually restore phase coherency of the protons, which gradually decays after the RF pulse, as the gradual loss of phase results in a gradual loss of signal. The echo time (TE) is the time between the RF pulse and the echo pulse. TE may be varied in order to give different contrasts between different tissues. Multi-spin echo imaging used several echo pulses, following the initial RF pulse, to give images that have different tissue contrasts. Other variations on the echo pulse include turbo spin echo (TSE)44 and gradient echo (GE), a rapid acquisition method, which is used for imaging blood. There are a number of other parameters associated with the MR imaging of tissue. The field of view (FOV) is the anatomical area being imaged and the matrix (m × n) contains the 2D pixel information.29 MRI scanners consist of a number of basic elements. There is a magnet, which produces the main magnetic field, shim coils, which reduce inhomogenities in the main field; gradient coils, which produce linear gradient magnetic fields along three orthogonal planes to enable spatial location of the MR image; RF transmitter and receiver coils; an MRI computer and surface coils. Image quality is quantified by the signal-to-noise ratio (SNR) and the contrastto-noise ratio (CNR). It is desirable to increase these ratios when possible for a given image. There are a number of factors that affect these, the main ones being voxel size, slice thickness, matrix resolution, field-of-view, number of signal averages, pulse sequence, interslice gap, magnetic field strength and contrast media. Contrast agents include gadolinium-based compounds. These agents help to reduce the relaxation time of certain tissue thereby reducing TR, TE and the scan time. Image artifacts are prevalent in MRI arising from a number of different sources.45 Magnetic field artifacts are due to inhomogenities in the magnetic field and

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distortion due to nearby magnetic materials. Hardware related artifacts are caused by misalignment of gradient coils, receiver coils, image resolution and computergenerated artifacts. Motion related artifacts are critical in imaging blood flow and arterial walls and are due to the pulsatile nature of blood flow. Since MRI scans take a certain length of time, there may be a blurring of boundaries which means synchronizing the scan with the low velocity region following diastole. Magnetic resonance angiography (MRA) is the imaging of the cardiovascular systems using an MR machine.29,43 MRA may be used to determine both the velocity of the blood as well as the cardiovascular geometry. There are a variety of different methods used to acquire this information. These methods have several common features including small flip angles, and short TR and TE.29,39,46,47 Since blood flows, it is possible to capture the wall image by having such a time lag between the RF pulse and the signal acquisition that the blood has already left the imaging plane and so appears as a void on the image yielding detailed information of the wall. This is termed time-of-flight angiography. In addition, blood velocity may be determined by saturating the surrounding tissues with the RF pulse. Since the blood is flowing through the image plane, the net magnetization shows the unsaturated blood surrounded by saturated tissue. 2.1.2. Application This example illustrates how MRI may be used to screen for disease. The velocity and geometry image files of the healthy human abdominal aorta from above the renal arteries to below the aortic bifurcations were acquired using a 1.5T MR imager (Philips Electronics N.V.). Time-of-flight angiography was used to acquire the geometrical information. No contrast agent was used. A total of 50 geometry slices of spacing 4 mm were acquired. Repetition time was 16.52 ms, echo time was 6.9 ms, a 70◦ flip angle was used and the field of view was 300 mm. The image information was recorded into a 256×256 matrix. The velocity conditions at the inlet of the descending aorta, at the outlets of each renal artery, and, at the outlets below the aortic bifurcations were determined in the temporal and spatial domain for a total of 16 time-intervals per pulse. The total time for scanner configuration and data acquisition was approximately 1 hour. The 178 files were saved in DICOM format and split into separate header and raw image files and were made available for processing. An example of an MRI scan is shown in Fig. 2 (inset) together with models resulting from MRI and CT scans. While the resolution is low, the aorta (circular white region) is clearly evident in the image. The scans were each examined in order to reveal that no obvious disease was present and that the aorta appeared to be functioning as normal. Figure 2 also shows the inlet velocity profile at a certain time in the pulse and illustrates the nature of the skewed velocity profile. 2.2. Computerized tomography Computerized tomography (CT) is a non-invasive imaging technique that utilizes non-ionizing radiation to generate a two-dimensional image of the tissue.

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Fig. 2. Models of healthy and diseased abdominal aortas generated from MRI and CT data. The inset shows the peak velocity profile occurring at the inlet to the healthy aorta as determined using MRA together with an MRI scan. It is possible to apply this velocity profile to the inlet of the model using CFD.

It offers high-resolution images and is capable of differentiating between a wide range of different tissues. It may be used dynamically enabling surgeons to perform minimally invasive procedures using CT to guide instruments and catheters around the patient’s internal organs. CT is widely used for imaging of the arterial system.48–50

2.2.1. Operating principles A CT scanner utilizes non-ionizing radiation to capture two-dimensional images of internal organs.51 The patient may receive a dose of contrast agent such as iodine or barium sulphate. The contrast agent absorbs some of the incident radiation enhancing the contrast on the resulting image. The radiation source is mounted on a gantry, which rotates about the patient, in the case of spiral CT, acquiring a single image for each rotation. The gantry is then indexed a certain distance in the axial direction, and further gantry rotation produces the next image. The process is repeated until a series of contiguous images is acquired which may then be processed to produce a 3D image of the internal organ under investigation. Each image is separated by a certain distance, and each image has associated with it a certain thickness. The image will be stored in digital format and the resolution, a

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measure of the size of each pixel in the digital image, may be determined by dividing the field-of-view by the dimensions of the m×n pixel array. 2.2.2. Application CT is generally used in disease diagnosis and follow-up. The CT scan shown in Fig. 2 illustrates a complication associated with the descending abdominal aorta. Abdominal aortic aneurysm (AAA) is a potentially fatal swelling of the aortic wall and is attributed to hereditary and lifestyle factors.52 A localized weakening in the artery wall causes the wall to expand. As it expands, the thickness of the wall decreases and an increase in wall stress results, leading to further expansion. Eventually, the stress reaches a critical level causing the wall to rupture. The model shown was generated from a set of contiguous CT data acquired using a spiral CT imager (Siemens A. G.). A spiral acquisition was performed with an interslice spacing of 4 mm. One hundred slices in total were taken with a 422 × 422 mm2 FOV stored in a 512 × 512 pixel array. The model was generated using an image processing software (Scion Image, Scion Corp.). 2.3. Ultrasound Ultrasound is a flexible medical imaging technique capable of acquiring approximate geometrical and flow-rate data rapidly for use in disease diagnosis and post-operative follow-up.53,54 While it does not produce accurate quantitative data, the qualitative data is useful as it is easily acquired and poses no risk to the patient. Ultrasound and its variant, Doppler ultrasound, offer effective techniques in obtaining approximate in vivo flow-rates and geometrical information. 2.3.1. Operating principle When high frequency sound waves are emitted from one medium into another, some of the soundwaves are reflected at the interface. It is possible to measure the time delay from the emission of the waves to the detection of the reflected waves, thus providing a measure of the depth of the interface from the emitter. Ultrasound is a based on this concept, in that the geometrical nature of certain tissues can be determined by measuring their depth in a body. 2D and 3D ultrasound machines are available, the latter of which is essentially an extension of the 2D machine, capable of creating approximate 3D models of tissue. Ultrasound is suitable for obtaining a snapshot of a particular tissue and is widely used in gynecology for identifying the development of the foetus. Doppler ultrasound operates on the principle that there is a change in the frequency of a wave reflected from a moving body. In the case of angiography, the moving body is the red blood cell, and the change in the frequency of the emitted and received sound waves is dependant on the insonating frequency, the velocity of the moving blood and the angle incident between the emitted wave and direction

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of blood flow. There are several different methods of representing the velocity for the end-user, the principal ones being colour, pulsed and power Doppler ultrasound. Colour Doppler uses a colour code to identify regions of different mean flow velocity while pulsed Doppler represents a spectrum or graph of temporally varying flow velocity at a specific location. Power Doppler is a more sensitive technique than colour Doppler. Several significant artifacts may occur with Doppler ultrasound. The first, termed aliasing, happens when the frequency shift to be measured is over twice the pulse repetition frequency. The result of this is a wraparound of the Doppler spectrum in pulsed or colour Doppler ultrasound. On occasion, aliasing may indicate high velocity flow resulting from stenoses. The second artifact, inherent to colour Doppler, is termed bleeding. This occurs when the colour gain is set too high and blood regions may appear larger than they actually are. This may veil certain attributes of the flow volume, including aneurysms and stenoses. 2.3.2. Application Doppler ultrasound may be used to acquire a snapshot of the patency of blood vessels. The model (Fig. 3) illustrates a carotid bifurcation showing a patent junction. 2.4. Angiography Angiography is an imaging technique, which operates using similar principles to CT. It is used exclusively for imaging the vascular system. A contrast medium is injected into the bloodstream proximal to the artery under investigation and the blood vessel shows up as a white image when using standard angiography techniques and a dark image when using digital subtraction angiography.55 As an example, coronary angiography was used to image the left coronary artery during a screening process for coronary artery disease (Fig. 4). Stenoses and plaques may be located using angiography and regions necessary for stenting may be identified. As it is a dynamic process, angiography is used by cardiologists during the implantation of coronary stents.

Fig. 3.

Doppler ultrasound of a carotid artery bifurcation. The sinus of the carotid is visible.

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Fig. 4.

A left coronary artery generated from an angiogram.

3. Computer Aided Design and Manufacture Concurrent engineering principles may be applied in the biomechanical study of the arterial system by performing numerical and experimental investigations simultaneously. While numerical techniques are particularly useful in performing design, research and optimization studies, there remains the need for experimental validation of numerically derived results and techniques.56,57 Advances in computer technology means that models designed and developed using computer aided design (CAD) systems for use in computational simulations can be manufactured using computer aided manufacturing (CAM) systems. Therefore, as research and optimization studies are being performed using numerical methods, experimental validations of these studies can be carried out concurrently. While there is a wide range of different CAM techniques available and under development, two techniques of particular interest in the study of arterial biomechanics are discussed here. 3.1. Digital image processing The process of converting the two-dimensional digital medical images into threedimensional solid models for use in numerical and experimental investigations is termed digital image processing (DIP). It encompasses a broad field of mathematical processes, which convert the unsmoothed, pixel images into smoothed vector models. The essence of DIP applied to medical imaging involves acquiring geometrical co-ordinates of an ambiguous object from a pixel array and assembling these co-ordinates into a well defined 2 or 3 dimensional model.58 Images acquired using CT or MRI are represented in digital form. Each image scan results in the creation of one file that contains the information associated with that particular image. The general file format used in medical imaging is the digital imaging and communications in medicine (DICOM) format. A single DICOM file consists of the image header, which contains information on the scan process

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parameters, positioning and patient data, as well as the ASCII representations of the m × n pixel array. Each pixel is represented by a certain number of bits depending on how many levels of gray or colour was used. Typically a digital medical image is represented by either 8 or 16 bit pixels. A 16 bit pixel results in 256 levels of gray together with addressing and is suitable for most medical applications. When opening DICOM image data it is necessary to know a certain number of parameters. The dimensions of the pixel matrix, m × n, must be known and from this the total number of pixels may be calculated. Typical pixel arrays are 256 × 256 or 512 × 512. The pixel size may be determined by dividing the x and y FOV lengths by their corresponding m and n pixel array values. The gray-scale or colour levels and distance between slices must also be known. This information is stored in the DICOM header. Once the files have been opened and all parameters correctly specified, processing may occur. The quality of the images is determined by the resolution of the scanner. Resolution refers to the size of an individual pixel in the image. All digital images have a certain resolution and it is this property that limits their accuracy. Tissues with a relatively large cross-sectional area will be represented by a large number of pixels, resulting in clear boundary definition. However, in small arteries, low resolution means that a circular cross-section may be represented by a cluster of 4 or 5 pixels. Therefore, as the cross-sectional area of the tissue being scanned decreases, so too does the accuracy. Recent developments in scanner technology means that high-resolution images for major and minor arteries are more readily obtained.

3.1.1. Process application This example is concerned with the human abdominal aorta screened for disease (Sec. 2.1.2). This data was acquired using time of flight angiography. 178 DICOM data files were available for processing. There are a number of different methods available for processing data of the type described above. Commercially available software, such as Mimics (Materialise Inc.), provides an efficient and comprehensive method for processing data. Mimics is capable of importing and exporting a wide variety of different file formats. Image files may be thresholded, edited and assembled into 3D models suitable for export for numerical study or manufacture. Mimics forms an integral part of computer aided engineering in the medical domain. Curve processing and smoothing is performed using automatic curve and surface processing algorithms. A solid model of the abdominal aorta was generated from the DICOM image data using Mimics and is shown in Fig. 2. A second method involves using image processing software such as Scion Image (Scion Corp.). This software is capable of reading raw image data and DICOM format, and provides a range of common image processing and analysis tools. However, it is capable of 2D image processing only, and does not support 3D model generation. Therefore, once the 2D boundary configuration is determined, further processing is

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needed to create a 3D model. In addition, controlled curve processing and surface smoothing must be performed manually, according to a defined algorithm. There are numerous curve-smoothing algorithms available and new algorithms may be written and implemented using a high-level language. Algorithms may record the spatial co-ordinates of pixels located at the boundary and fit polynomial curves of a certain order to these spatial coordinates. Axial smoothing, or smoothing between adjacent slices, is also necessary and may be accomplished using different mathematical techniques. The main objective of curve and surface smoothing applied to arterial biomechanics is to create an artery lumen wall, which approximates the wall of the actual patient (Fig. 5).

3.1.2. Velocity profile processing MRA can be used to determine the velocity in an arterial system.29 In this example, the velocity and phase information for the various inlet and outlets were recorded at 16 times in the cardiac cycle for the abdominal aorta scan. The volunteer’s heart rate was approximately 60 beats per minute. The velocity data was represented as a series of images, the gray-level of the blood pixels being proportional to the blood velocity at that point. Depending on the cross-section of interest, the velocity was encoded from 0 to 4095 (grayscale level). In the case of the inlet of the aorta, the phase velocity encoding spanned –100 cm/s to +100 cm/s. Therefore a pixel value of 0 (black) corresponded to a velocity of –100 cm/s and a pixel value of 4095 (white) corresponded to a velocity of 100 cm/s. By reading the coordinates of the pixels,

Fig. 5. The stages involved in generating a solid model from a set of CT scans. The set of scans is thresholded individually (a), and the boundaries of the arteries are identified according to a certain image processing procedure. Assembly of these boundaries results in the wire-frame geometry (b). Further curve smoothing and surface fitting results in a surface model (c) which can then be processed further to produce a solid model suitable for CFD or FEA investigations (d).

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together with their gray-level value, a 3D velocity profile could be established. This procedure was tested for the maximum velocity step at the inlet slice. Positions and levels were recorded for all blood pixels and a 3D surface was created resulting in the profile (Fig. 2).

3.1.3. Vector transformations A pair of images acquired using angiography can also be used to create a 3D model of an artery using vector transformation provided that the angle and scale between the two images is known.59 Figure 6 illustrates the vector transformation process applied to a right coronary artery. Two images were acquired and were used to construct a 3D model.

3.2. Rapid prototyping Rapid prototyping (RP) is a CAM technique that manufactures a solid homogenous prototype directly from a CAD model. There is a range of different techniques available capable of producing prototypes using a variety of different manufacturing processes and materials. Stereolithography, selective laser sintering, 3D printing

Fig. 6. Ortohogonal angiograms of the right coronary artery. These images were processed using vector transformations to produce a realistic representation of the geometrical nature of the right coronary artery.

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and fused deposition modeling are all popular RP processes enabling fast and efficient manufacture of prototypes of computer solid models. While RP has not been widely used to date in investigating arterial biomechanics, it has been used extensively for orthopaedic applications. Studies60,61 have manufactured components for reconstructive surgery, while rapid prototyping has also been used to develop custom endoprostheses.62 3.2.1. Application In this application of RP, a model of a rigid graft was manufactured from fused deposition modeling (FDM). The dimensions of a commercially available abdominal aortic graft was measured and used to generate a 3D computer model of the part. This part was then split so the two halves modeled could be clamped together (Fig. 7). As the halves were symmetrical one model was exported as an STL file to the FDM processing software Insight (Stratasys Inc.). This software mathematically slices and orientates the part. Fused deposition modeling operates by extruding a filament of semi-molten material from an extrusion head in a prescribed pattern onto a platform. This generates a layer or slice of the geometry. The material layer then cures, the platform drops in the z direction according to the slice thickness, and the next layer is generated. Successive layers of the part are then built up on top of each other until the part is completed. In addition to the design of the part, the user must specify the material choice, the tool path, the layer thickness and part orientation. At present, FDM uses only thermoplastic materials. For the graft model discussed here, acrylonitrile butadiene styrene (ABS) was used. Due to the nature of the manufacturing process, accuracy and surface finish are the main limitations of FDM.

Fig. 7. Abdominal aortic graft lumen model generated using fused deposition modelling. The lumen computer model was split in two through its centreplane, a wall thickness was declared and a clamping flange and holes were added to the model.

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3.3. Casting Casting is an additional method available to create both rigid and complaint models of elements of the arterial system.63 There are a variety of different techniques available, but common to all, is the need to manufacture a solid part, the outer surface of which corresponds to the artery lumen wall. This male lumen cast defines the geometry of the artery inner wall and so accurate casting of this is critical to capturing the internal geometry of the artery for biofluid mechanics investigations. This cast may then be placed into a female mould which is used to define the outer wall of the artery, and pouring or injecting a curable material into the void between the male cast and the female mould results in a solid model of the artery with a certain wall thickness.20,64 Casts of the arterial system may be used in flow visualization studies, wall deformation analyses, medical device implantation tests and photoelasticity investigations.65–67 A general technique is described in the following section for creating compliant casts of human arteries. 3.3.1. Application A solid model is generated from medical images using the techniques described in Sec. 3.1. In this example, a solid model of an aortic arch is developed using a CAD package. A rendered model of the arch is shown in Fig. 8. This model is then split along its centerline into two halves, each one corresponding to one mould half. Each half is then defined by a curved surface representing the lumen wall, bounded by two edges defining the split lines. These split lines are swept orthogonally away from the model thereby defining two additional surfaces. A similar process is performed for the other mould half. These two mould halves may then be brought together to ensure that the swept surfaces are concurrent and that no interference or gaps occur. The halves are then processed for machining into two blocks of material such as aluminium or acrylic. Solid modeling packages are capable of defining the computer numerical control (CNC) machining codes required for machining surfaces defined in a computer solid model onto a workpiece. The workpiece is then mounted onto the table of a CNC machining center, a datum is set, and the surfaces are machined. The two mould halves are manufactured using this technique. Gates, vents, supports and guide-pins may be added during the solid modeling procedure or after the surfaces have been machined. The completed mould may then be prepared for casting. Prior to casting, a light coat of mould release agent is applied to the walls of the lumen mould. The halves are then joined together. A low viscosity casting wax is then melted and poured into the mould via the gate. Upon solidification of the wax, the male component is then removed from the mould. A lumen cast, representing the blood volume whose surface corresponds to the lumen wall, has been created (Fig. 9). The principal difference between the female mould and the male mould is that the female mould has a wall thickness associated with the artery wall included into

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Fig. 8. A computer solid model of the aorta generated from a set of CT images. Two sample images and their location are also shown. A cast of the aortic arch section was manufactured using an injection moulding procedure.

Fig. 9. A female wall mould manufactured using a CNC machine, a lumen cast which is inserted into the mould and the resulting silicone rubber cast of the aortic arch.

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the diameter of the solid model derived from the medical images. In addition to this, supports must be added to the female mould to support the male cast rigidly in place during material injection. The process for manufacturing the female mould is similar to that of the male mould. The female mould is coated in mould release agent and the male cast is placed on supports in the female mould. The two mould halves are then joined together, and casting material is injected into the injection port, filling the gap between the male lumen cast and the female mould wall. The mould is filled once the casting material begins escaping from the vents. Different types of material may be injected depending on the application. Latex and silicone rubber have been used to manufacture rigid and compliant casts of the arterial system.68,69

4. Numerical Investigations Numerical studies are arguably the most powerful tools available to researchers seeking to study the biomechanics of the arterial system. As was discussed in earlier sections, the main areas of biomechanical study in the human artery involve investigations of wall stresses in the artery and blood flow dynamics.70–73 Computational fluid dynamics (CFD) has been used to investigate artery hemodynamics, mass transfer and disease formation mechanisms while finite element analysis has been widely used in studying artery wall stresses, deformations and load responses.

4.1. Computational fluid dynamics Computational Fluid Dynamics is a numerical technique capable of modeling the hemodynamics of the arterial system.74 The technique involves taking a geometrical description of the volume the fluid occupies and dividing or discretizing this into a finite number of volumes. There are a number of different numerical algorithms available for CFD and their operation depends on the discretization scheme used. The classical method for CFD is called the finite volume method, whereby a fluid volume is discretized into a finite number of control volumes over which the governing equations are imposed.75,76 The finite element method, originally developed for structural analyses, may also be implemented in fluid analyses, and it has the added advantage that it enables both fluid-structure investigations to be performed using one integrated package.77,78 Additional methods applicable include the finite difference method and the boundary element method. CFD operates on the principle that it is possible to take a complex fluid continuum and divide it into a finite number of simple fluid elements. The intial process in CFD involves creating a geometrical model of the fluid volume using a solid modeler or a purpose build CFD pre-processor. Once the edges, surfaces and volumes defining the fluid volume have been created, it is possible to subdivide or discretize the model into a finite number of elements. During this pre-processing stage, the

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names of the boundary entities are defined and edges and surfaces of interest in analysis may also be defined. The next stage in CFD is to define the material properties, boundary conditions and solution method and procedure. This is done using a specialist software termed the ‘processor’. Numerous commercially available processors are available and some specialized purpose built codes have been developed by research institutions and organizations.70,75,78–81 The processor imports the discretized geometry and applies the governing equations across it. By defining boundary conditions, such as flow-rates and pressure waveforms, at the inlets or outlets of arterial systems, the processor is able to calculate the entire flow-field. During processing, the material properties, boundary conditions, species concentrations and temperatures may be defined while additional sub-routine libraries are able to model many other flow phenomena. The final stage of CFD involves post-processing the results. This stage involves selecting areas of interest in the mesh, recording the associated data, and arranging it to provide meaningful graphical methods of interpreting the flow.

4.1.1. Application The following simulation was performed in order to investigate the nature of the blood flow around the aortic arch and down into the abdominal aorta. The model was generated from a set of CT images acquired using a spiral CT imager (Siemens A. G.). A total of 198 images were acquired with an interslice spacing of 2 mm. The model was processed according to a smoothing process outlined previously.52 The model was then exported to the CFD pre-processor Gambit (Fluent Europe) where a paved quadrilateral mesh was specified. A total of 250000 first order elements were used to mesh the model. The resulting mesh was then exported to the finite volume processor Fluent (Fluent Europe) for solution. Models were solved on a Dell Dimension 4100 1Ghz personal computer with 512 Mb RAM. A laminar flow, segregated solver incorporating the QUICK momentum scheme, and, PISO pressure-velocity coupling was used to perform the analysis. A fully developed profile inlet boundary condition used was described elsewhere.82 The convergence criterion was set to 0.001 for the residuals of the continuity equation and of the X, Y and Z momentum equations. The model was run, found to converge and the peak velocity flowfield was recorded. Velocity vector and contour plots are shown in Fig. 10. It is evident from the results that significant skewness of the flow occurs upon passing around the arch. Slices 3 and 4 illustrate how significant that skewness is. A second model illustrates how CFD can be used to measure how fluids interact with artery walls. In this study, a model of a right coronary artery was generated from a cine-angiogram. Using a similar numerical procedure outlined in the previous model, the wall shear stresses of the coronary artery were recorded at a number of critical locations in the geometry during the peak flow phase. Figure 11 illustrates

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Fig. 10. Hemodynamic flow patterns through a rigid model of an aortic arch. The model was generated from a CT image set. Pulsatile, Newtonian flow was applied at the inlet (the ascending aorta). Note the skewed flow through the arch which develops again down through the descending aorta.

the nature of the flow through the geometry by means of velocity contours and also how different levels of wall shear stress are evidenced at different locations.

4.2. Finite element analysis Finite element analysis is a powerful numerical technique developed initially to solve complex structural engineering problems, which could not be solved empirically or analytically. From the geometrical definition of a physical body it is possible to sub-divide or discretize the body into a finite number of individual geometrical elements.83–85 The constitutive laws or equations governing the system are then applied to individual geometrical entities.86 By assembling a global stiffness matrix for the governing equations, creating geometrical relationships between each element and its neighbour, it is possible to define a single matrix representing the entire system. This may then be solved to yield an approximate solution for the boundary value problem. A particular advantage of the finite element method over the finite volume method is its capability in modeling both fluids and structures. While efficient solvers that can capture physiologically realistic fluid-structure interactions are currently available and under continuous development this section will concentrate on

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Fig. 11. Hemodynamic flow patterns through a rigid model of a coronary artery. The model was generated from cine-angiography. Pulsatile, Newtonian flow was applied at the inlet. The flow contour plots show the nature of the flow through the artery at different cross-sections. Results were then taken from the CFD for each time-step and plots show WSS occurring at inner and outer locations at bends in the geometry were created.

the application of the finite element method in the investigation of physiologically realistic artery wall stresses and deformations. Finite element solvers have a wider variety of element types within their libraries, from simple first order 2D quadrilateral elements to higher order 3D tetrahedral elements. The choice of element is dependant on the geometry of the model to be investigated and also on the required accuracy and the nature of the deformations. As with CFD, FEA consist of three main stages.87 The first, pre-processing, involves creating a geometrical representation of a body using a 2D or 3D CAD system. This model may then be imported into a mesh generation software package, where the geometrical model is discretized into a finite number of individual elements. Concentrations of elements are placed at regions of interest in the structure and where stresses are likely to be most complex. During the pre-processing stage, names tags are attached to edges or surfaces for constraining or loading, and to planes or volumes for defining continuums. Also, edges and surfaces of interest may be pre-defined during pre-processing to aid post-processing of results. The second stage involves exporting the element mesh to a solver where the necessary material properties, constraints and loads are defined. During this stage, the solution method and convergence criteria are also set for the numerical procedure. The model is run until the solution converges to a required limit, otherwise, the model must be redefined or remeshed.

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The final stage in FEA is to take the solution and to analyse the results using a post-processor. Here, geometrical definitions enable specific surfaces, edges and volumes to be selected and various quantities, such as Von Mises stress and directional strain, may be recorded and plotted.

4.2.1. Application Solid models of two bypass graft anastomoses were generated using ProEngineer Wildfire (Parametric Technology Corp.). One model consisted of a 6 mm diameter artery with a 1 mm thick wall attached to a 6 mm diameter graft with a 1 mm thick wall. The second model was that of a 6 mm diameter graft attached to a 6 mm diameter artery at an anastomotic angle of 45◦ . The models were exported to ProMechanica (Parametric Technology Corp.) were automeshing was performed using 9 node quadrilateral elements. A multipass solution process was performed until the wall stresses converged to within 2% of their limiting value. The Von Mises Stress along the suture lines (Fig. 12) shows how the wall stress characteristics of the end-to-end anastomosis are superior to those of the end-to-side anastomosis. This observation supports previously published work.86

5. Experimental Investigations In vitro modeling of arterial systems using experimental techniques enables researchers to validate numerical results and also to investigate phenomena that are beyond the capability of numerical solvers. While numerical simulation enables quick and efficient investigation, complex phenomena such as particulate flows frequently need to be studied using experimental methods. In addition, numerical simulations

Fig. 12. A comparison between the stress distributions between a Dacron graft and a compliant artery for an end-to-end anastomosis and an end-to-side anastomosis in femoral bypass procedures. A mean pressure of 10 kPa was applied to the internal walls of the models. The peak stress measured in the ESA was 56 kPa while in the EEA it was 28 kPa. For a normal healthy artery under similar loads it is 18 kPa. It is evident from this study that the ESA produces wall stresses that are significantly higher than those in the EEA and over three times those present in a healthy artery. From the stress distributions, it is also evident that significant gradients in the wall exist.

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need to be validated using experimental and theoretical models in order to establish the accuracy and reliability of a modeling technique.88–90

5.1. Physiological flow modeling The basic element required for enabling experimental investigations of the arterial system is the flow circuit. A wide range of different flow circuits are possible each one being tailored to fulfill a particular requirement.91–94 A basic flow circuit requires a pressure head to drive fluid through the system under investigation, usually a pump or reservoir. The fluid then passes through the test section, which may be a geometrical model of an artery lumen or experimental medical device. In general, steady flow conditions are modeled using a constant pressure head.93,95 Pressure within the system may be set by controlling the head height or pressure within the reservoir while a valve controls the flow rate. Steady flow models have been used to investigate flow fields, wall stresses and device performances in the arterial system.27 However, steady flow modeling is limited in that it fails to capture the main attribute of the cardiovascular system, that of its pulsatile nature. Pulsating flow circuits are required to accurately replicate the in vivo flow conditions. Numerous studies have used pulsatile flow circuits in order to create flow-rate waveforms similar to those found in the body.20,91,92,94,96 A basic method to generate pulsatile flowrates involves using a piston pump with one-way inlet and outlet valves.29 A motor is used to drive the piston and the resulting fluid displacement through the test section is proportional to the displacement of the piston. Complex waveforms may be modeled by using computer controlled stepper or servo-motors to drive the piston. Fluid is drawn into the piston from a reservoir and is pumped through a valve to the test section. The motion of the piston may be related to the motion of the fluid by means of the continuity equation. Therefore, the temporal integral of the desired mean velocity in the test section is proportional to the displacement of the piston. While a piston pump is capable of producing flow-rates similar to those found in the arterial system, the modeling of physiologically realistic pressure waveforms is more complex. While addition of a valve downstream from the test section can serve to increase pressures acting within the test section to realistic levels, accurate reproducibility of both pressure and velocity is particularly difficult due to the fact that the arterial system is complex, and that pressure waveforms are composed of numerous harmonics. These harmonics may result from the pumping nature of the heart, from the elasticity of the arterial walls or from the resistance of the peripheral system. In addition, wave reflections from branches and bifurcations also play a major role in influencing the nature of the pressure waveform. In a flow circuit which is predominately rigid, and the test section is compliant, it may be necessary to incorporate a Windkessel, or pressure reservoir, to mimic the contraction of the arteries located elsewhere in the arterial system. The compliance inherent in the Windkessel serves to increase the steady system

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Fig. 13. A computer controlled piston pump drives fluid through a test section at physiologically realistic flow-rates. Pressures within the test section are controlled by valves located downstream from the test section. Transient flow-rates and pressures are recorded and a laser Doppler anemometer is used to determine the flow velocity at specific locations in the test section.

pressure to more physiologically realistic levels after the transient pressure pulse has passed through. A schematic diagram of a pulsing flow circuit is shown below (Fig. 13). In this example, pulsatile pressure and flow waveforms were passed through compliant models of an end-to-side bypass graft. The pressure transducer was used to record the pressure within the model while laser Doppler anemometry was used to investigate the nature of the flow through the junction.

5.2. Laser Doppler anemometry Laser Doppler anemometry (LDA) is a non-contact optical measurement system which offers excellent directional response and high temporal and spatial resolution. It determines the average velocity of particles through a small finite volume of fluid and, depending on the system, is capable of acquiring measurements in all three Cartesian co-ordinates. The principle behind LDA is that of the Doppler shift; light reflecting from a body experiences a shift in frequency. In operation, two laser beams intersect at a point, with one of the beams at a slightly different frequency than the other. As micro-particles pass through the intersection (measurement volume), light fluctuating in intensity is reflected off them and into a photo-detector. The frequency of this light is equivalent to the Doppler shift between the incident and scattered light and is proportional to the component of particle velocity which lies in the plane of the

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two beams and perpendicular to their bisector. All 3 components of velocity can be measured by using additional lasers set in the plane of each of the 3 Cartesian axes. LDA is often termed as being a ‘single point measurement’ system. However, LDA relies on particles passing through a small finite volume. The velocity is sampled each time a particle passes through the finite volume. The main limitations of LDA are experienced due to the nature of the small finite volume. If the measurement volume is large with respect to the flow volume being studied, a significant velocity gradient may exist across the measurement volume. Also, if the velocity increases, so too does the number of particles passing through the finite volume, thereby affecting the accuracy of the system. The operation of an LDA system requires that particles be suspended in the fluid. Therefore, the medium through which the laser beams pass must be transparent. While LDA offers fast and accurate single point measurement, the set-up and alignment of the laser beams can be difficult. The beams must be aligned around the axis of rotation of the optics set and after passing through a convex lens, must intersect at the focal point of the lens. Each beam must have the same cross-sectional area at intersection. LDA has been used in a variety of different applications in arterial biomechanics.4,67,97 Studies have been performed investigating the nature of pulsatile flow94 and on the specifics of flow through arterial models such as the carotid artery,67 the aorta,20 aneurysms98 and femoral bypasses.27 Rigid and compliant walled artery models have been investigated, and Newtonian and non-Newtonian flows have also been studied. Complex three dimensional pulsatile flow analyses have been investigated in addition to simpler one-dimensional steady flow investigations.11,27 With the advent of CFD, complex and expensive LDA systems are presently primarily used for validation and for performing investigations on models unsuitable for CFD. Recent studies have investigated arterial biomechanics using both LDA and CFD as a means of validation and also to investigate the effectiveness of experimental and numerical modeling techniques.88,99 The following example illustrates the use of LDA in CFD validation. A computer controlled piston pump was used to drive a Newtonian 42%/58% water/glycerine fluid (by weight) through an idealised model of a femoral bypass graft.88 A 55x series, one-dimensional LDA system (Dantec Dynamics Ltd.) using a 20 mW HeNe laser operating in forward scatter configuration was used to record the flow at various points through the aneurysm. The focal length of the lens was 160 mm with a beam separation of 76 mm. The refractive index of the fluid was 1.41 producing a laser wavelength of 448.8 nm in the fluid. A 1.95 beam expander was used and the beam waist was calculated after expansion to be 2.15 mm with a diameter after focussing to be 0.042 mm. The resulting beam intersection volume dimensions were calculated to be 0.043 mm × 0.043 mm × 0.255 mm giving a volume of 0.23 pm.3 45 fringes with a spacing of 2.72 µm were determined through the volume. Polystyrene latex particles of 0.5 µm diameter were used to seed the fluid.

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Fig. 14. Comparison between various investigative techniques applied to bypass junctions. MRI and finite volume CFD are compared (top) while 1D LDA is used to validate finite element and finite volume flow profiles (bottom).

The resulting profiles at this time are shown (Fig. 14) together with those acquired from corresponding CFD studies. It is evident from the results that there is excellent agreement between the experimental and numerical results. Results are also shown for steady flow through an idealised model of an abdominal aortic aneurysm (Fig. 15).

5.3. Particle image velocimetry While LDA is a point measurement technique, particle image velocimetry (PIV) is a whole field method providing instantaneous velocity measurement in a cross-section of flow.100,101 Two velocity components are measured while a stereo arrangement enables measurement of all three components. The use of high-speed charge-coupled device (CCD) cameras and digital processing enables computation of real-time velocity maps. As with LDA, lasers are used in PIV. A pulsed laser light sheet is targeted at the cross section of interest while velocity motion of particles suspended in the flow is tracked using CCD cameras. Once a sequence of two light pulses is recorded, the images are divided into subsections called interrogation areas. The interrogation areas for each image are then cross-correlated with each other, pixel-by-pixel. The correlation produces a signal peak, identifying the common particle displacement, and dividing by time yields the speed of the particle. The direction is determined from the initial location of the

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Fig. 15. Validation of numerical results through an idealised model of an abdominal aortic aneurysm using 1D LDA. Excellent agreement between the LDA and the CFD is evident. Steady flow through a rigid model of the abdominal aorta was investigated.

Fig. 16. A comparison between a flowfield determined using PIV and one using CFD for a bypass graft junction model. A pulsatile flow circuit was used and the velocities are shown for the peak time in the inlet pulse.

particle to the final location by means of Cartesian co-ordinate system applied to the field. A velocity vector map is then created over the whole area by cross-correlation of all interrogation areas. PIV has been used to validate global flow fields in arterial biomechanics. Figure 16 illustrates a comparison between flow through and end-to-side anastomosis visualised using PIV and calculated using CFD.

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5.4. Video extensometry CCD cameras have been used in the past for a variety of different biomedical investigations from monitoring cellular fluorescence to recording anatomical changes.102,103 Video extensometry is a technique, which uses CCD cameras to record the deformation of materials over time. It has been used in biomechanics to record the deformations of compliant walled arterial models.20,104 A major advantage of the technique is its portability, enabling researchers to investigate model wall deformations as well as arterial deformations during surgery. A video extensometer consists of a camera, which captures real-time images of the deformation, and software, which processes and records the images. The system must be calibrated using known dimensions of the material under investigation and then it may determine the strain in the material over the course of the deformation. The strains in compliant models of femoral bypass grafts may be determined in such a manner. As an example, a video extensometer (Messphysik Laborgeraete GmbH.) was used to record the deformation of a rigid graft/compliant artery model. The system was pressurized to physiologically realistic pressures and the resulting deformations were recoded using the video extensometer. Figure 17 illustrates the geometrical nature of the model together with the deformation that occurs due to physiologically realistic pressures.

Fig. 17. Video extensometry. The bypass junction model is pressurised from the unloaded state to the loaded state and deformations are calculated using associated software. Bed rise and side-wall bulging is evident from the images.

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5.5. Pressure and flow-rate measurement The measurement of pressure and flow-rates in in vitro flow studies is possible through the use of medical grade products such as pressure catheters and ultrasonic flowmeters. While traditional measurement systems such as elastic pressure transducers, hot-wire anemometry, pitot tubes and turbine meters can play a role in arterial biomechanics, piezoelectric, electromagnetic and ultrasonic transducers offer minimal flow disturbance and high frequency operation. Medical grade pressure catheter and ultrasonic systems enable researchers to perform both in vivo and in vitro experiments. A metal diaphragm with directly deposited resistive strain gauges (Gaeltec Ltd.) was used to measure the pressure in the aortic aneurysm model below (Fig. 19). An ultrasonic flowmeter (Transonic Systems Inc.) incorporating a transittime flowsensor was used to measure corresponding flowrates. 5.6. Photoelasticity A variety of different non-contact experimental strain measurement techniques have been used to investigate compliant arterial models. Video extensometry,21,104,105 photonic sensor,106 photocell combined with light-emitting diode or scanning laser107,108 have previously been used to quantify arterial wall strains. These methods enable measurement of surface stresses, however, they do not provide data on the whole stress field unless numerous measurements are taken. However, the reflection photoelastic method enables viewing of the whole stress field on a loaded part. Photoelastic fringe analysis is a widely used method for experimental visualizing of stresses during static and dynamic loading. The photoelastic method has been applied to the stress analysis of biomechanical applications. Photoelasticity works on the principle of birefringence, which is exhibited by certain transparent materials. A ray of light passing through a birefringent material experiences two refractive indices. The light is resolved along two principal stress lines and each component experiences two different refractive indices. The difference between the refractive indices produces phase retardation between the two component waves proportional to the two principal stresses. As the level of birefringence is proportional to the stress, differently stressed areas will produce different colors. In photoelasticity, the birefringent property is expressed only when a stress is applied to the material. Models are made from a birefringent material such as PL-3 epoxy resin. The photoelastic technique may be used to investigate the stress distributions in an aortic aneurysm.109 In this example, a photoelastic model of an idealized abdominal aortic aneurysm is manufactured using an injection moulding technique (Fig. 18). The model is manufactured using PL-3 epoxy resin and its interior surface is coated in PC-11 reflective adhesive. A computer controlled piston pump is used to provide a physiologically realistic pressure waveform to the interior of the model. Internal pressure is monitored using a pressure catheter. The model was illuminated by plane polarized light using a

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Fig. 18. Photographs of an aneurysm model used for photoelastic experiments. (a) An injection moulded model manufactured from PL-3 epoxy resin. The model is removed from the mould, the male cast is removed by cutting the model open which is then coated in PC-11 reflective adhesive and glued back to its original shape using an elastic glue. The result is a photoelastic model shown in (b) which may be inserted into a flow loop.

Fig. 19. A photograph of the photoelastic model under load. The photo is taken through the optical head of the polariscope. The fringes corresponding to different stress levels are clearly evident with maximum stress evident at the proximal and distal aneurysm ends.

030 series reflection polariscope (Measurement Group Inc.). A 12 V 100W tungstenhalogen lamp was used. A model 232 null-balance compensator was used to measure the fringe orders with a resolution of 0.01 fringe. In the unloaded state, there was no fringe pattern, while under load, a fringe pattern was visible (Fig. 19). As the

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fringe orders observed in the material were proportional to the difference between the principal strains in the material, it is possible to determine the principal stresses from the elastic modulus and Poisson’s ratio of the epoxy resin. This method found that hypertensive patients may experience wall stresses up to 35% greater than normal. These findings also correlated with those from previous studies which show that most aneurysm ruptures occur at the postero-lateral wall.110,111

6. Medical Device Research and Development The techniques discussed in previous sections may be used for the study of arterial biomechanics with a view to better understanding the mechanical environment and to aid in the development of medical devices to treat various medical conditions. This section is concerned with illustrating how a medical device may be developed to address a certain condition. 6.1. Vascular grafts for abdominal aortic aneurysms The problem under examination is that of the increase in cardiac load which results following the implantation of an abdominal aortic aneurysm graft. In order to study this problem, a sample of a conventional aortic aneurysm graft for open surgery was sourced from a commercial manufacturer. The dimensions of the graft were measured and a solid model of the graft was generated using a 3D solid modeler. The resulting conventional graft geometry is shown in Fig. 20. The graft is a woven Dacron graft consisting of an aortic section of uniform diameter, two iliac legs of uniform diameter, and a bifurcation region where the three components are woven together. Comparison of the cross-sectional area of the aortic graft section to the sum of the cross-sectional areas of the iliac graft sections shows that the area ratio between the aorta to the iliac arteries is approximately 2.

Fig. 20. An illustration of the abdominal aortic aneurysm, conventional graft and tapered graft computer solid models. Two views of the tapered graft is shown clearly illustrating the gradual change from the aorta inlet to the iliac outlets.

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An MRI scan of a healthy aorta was then used to determine the physiologically realistic area ratio (Fig. 2). The scans showed that the aorta and iliac arteries are not of constant cross sectional area, rather, the cross-sectional area reduces along the axial distance from the heart. Therefore, the area ratio was determined averaging the diameter of each vessel a short distance (∼ 10 mm from the apex of the bifurcation. The resulting area ratio between the aorta and the iliac arteries in the healthy case was found to be 1.3. Previous work has cited an area ratio of approximately 1.27.112 It was then hypothesized that the uniform cross-sectional areas and nonphysiologically realistic area ratio associated with the aortic graft may have a role to play in augmenting the aortic pressure. There was a suggestion that pressure wave reflections from the iliac aortic bifurcation may result in increases in aortic pressure.113 Therefore, a study was performed in order to redesign the vascular graft to a more physiologically realistic specification, and then to assess this new design using in vitro experimental techniques. The initial step involved redesigning the graft. It was specified that the new graft would have to have the same aortic inlet and iliac outlet diameters as those of the original graft. In addition, the graft would have the same axial length and be manufactured from the same materials. The only variable would be that of the cross-sectional area, which would no longer be uniform in each vessel, rather, it would be tapered gradually along the length of the graft. In addition, the area ratio between the aortic and iliac segments would be less than that determined from the physiologically realistic MRI scan, approximately 1.1. A novel tapered cross-section was defined which involved sweeping a variable cross-sectional area aorta into two variable cross-sectioned iliacs (Fig. 20). The graft was then subjected to in vitro tests in order to assess the effects of the geometrical design on the system fluid dynamics. The first test involved using CFD on the new graft. Two models were investigated, namely, the original commercial graft and the new tapered graft. The models were meshed using a similar meshing technique in a CFD pre-processor and then exported for solution in a finite volume processor. Here, a transient, pulsatile, Newtonian analysis was performed and the fluid velocity profiles at various locations through the geometries were compared. It was found that the tapered design produced less skewing of the flow in the iliac legs and that the fluid acceleration from the aorta into the iliac arteries was less pronounced than in the commercial graft. The second test involved creating experimental models of the commercial and new graft and subjecting them to pressure tests using an in vitro experimental flow circuit. Aluminium moulds were manufactured corresponding to the graft lumen dimensions using a CNC machining center. The resulting moulds were then used to produce wax casts of the lumen. These were then placed in a box and surrounded by a two-part liquid polyurethane that was allowed to cure. Once cured, the wax lumens were melted out and the resulting lumen models were inserted into the flow circuit.

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A computer controlled piston pump was used to produce physiologically realistic flow and pressure waveforms acquired from the literature.114,115 Water was pumped at pressure through the models in turn. Each model was mounted into the flow circuit and a series of pressure measurements were taken using a 1 mm diameter pressure catheter located 20 mm upstream from the aortic bifurcation. These sets of results were then compared to each other and it was determined that the tapered design resulting in a reduction in aortic pressure of approximately 9 mmHg. The second test proved that the in vitro aortic pressure could be reduced by tapering the graft along its length and designing the bifurcation so that a sudden change in area ratio did not occur. The results of these in vitro investigations have since been used in the development of a new medical device incorporating the design attributes described here. The final stage in the process, performing in vivo tests and establishing the efficacy of the proposed device, involves significant clinical input and lies beyond the scope of this chapter. It should be noted, however, that though the in vitro results appear promising, full in vivo studies are required to capture phenomena assumed negligible in the in vitro flow circuit.

6.2. Vascular grafts for femoral bypass procedures Atherosclerosis is a disease characterized by the deposition of low-density lipoproteins onto the intima of an artery. These depositions gradually increase in size until they lead to the development of plaques, stenoses or occlusions. When an artery becomes stenosed or occluded, blood supply to a location is reduced or stopped, and this may result in tissue necrosis. In the case of the femoral artery, stenosis leads to a reduction in blood supply to the leg, while complete occlusion of the artery can lead ultimately to amputation of the affected limb. A variety of different treatments are available to treat atherosclerosis in the femoral artery, with the endovascular methods of angioplasty and stenting and the open surgical technique of femoro-popliteal bypass being the most commonly used. The open surgery technique involves bypassing the diseased portion of an artery with a new conduit, typically made from autologous vein or from a synthetic material such as ePTFE or Dacron. However, the long-term patency rates of prosthetic grafts remain low. There have been two fundamental disease formation mechanisms suggested which contribute to the poor long-term patency of femoral artery bypasses. Disease has been attributed to the abnormal hemodynamic patterns associated with the geometrical configuration of the end-to-side anastomosis.116,117 In addition, evidence is available which suggests disease forms due to wall stress gradients resulting from the material mismatch between the stiff graft and the compliant host artery.1 Two different configurations may be used to create the distal anastomosis in a femoropopliteal bypass, end-to-end (ETE) anastomosis and end-to-side (ETS) anastomosis. The latter is more commonly used and has a number of different geometrical configurations, including the Miller cuff and Taylor patch, which may be used when anastomosing a synthetic graft to arteries below the knee.79 Cuffs and patches have

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been developed in an attempt to buffer the material mismatch between the synthetic graft and the artery by incorporating a section of vein between them. Also, increasing the volume of the graft/artery junction can lead to a reduction in the fluid forces impinging on the host artery wall. A recent review of these approaches concluded that an optimum geometry for the ETS anastomosis may not exist.82 Previous work has shown that the wall stresses and hemodynamic patterns associated with the ETE anastomosis are less physiologically abnormal than those associated with the ETS anastomosis. Therefore, a potential treatment was hypothesized which seeks to improve the wall stress and hemodynamic characteristics of the ETS anastomosis by means of the bifurcated graft design shown in Fig. 21. The bifurcated graft means that two ETE anastomoses are used to attach the graft to the host artery rather than the traditional single ETS anastomosis. It is hypothesized that this device design may increase the long-term patency of femoral bypass procedures, however, to date, only in vitro evidence supporting this claim has been produced.27 Due to its bifurcated nature, flow through the graft into the host arteries is more streamlined than in the traditional ETS anastomosis. A previous in vitro study used CFD to illustrate how this streamlined flow serves to reduce abnormal wall shear stress magnitudes and gradients near the graft/artery junction. A comparative CFD study of idealized models of a traditional ETS bypass and a bifurcated ETE bypass graft was performed and velocity vector plots and WSS contour plots showed significant advantage of streamlined flow over the disturbed flow associated with the traditional ETS anastomosis. In addition to this study, FEA has been used to study the wall stresses at the ETE anastomosis and the ETS anastomosis (Fig. 13). Experimental investigations on the two anastomoses have also been carried out using steady flow idealized models. LDA has been used to investigate the flow profiles associated with the ETE and the ETS anastomses while video extensometry was used concurrently to quantify the wall deformations. Evidence from these studies also suggest that improved flow dynamics and wall strain characteristics result from using two ETE anastomoses rather than one ETS anastomosis.

Fig. 21.

A hypothetical femoral artery bypass graft design.

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7. Conclusions and Future Directions Numerical and experimental techniques offer researchers powerful tools for the study of arterial biomechanics, disease and potential treatment. Incorporating a variety of multi-disciplinary techniques and methodologies is key to successful implementation of a biomechanics research program. It should be noted, however, that arterial biomechanics is just one aspect of the greater study of the cardiovascular system. While biomechanics concerns itself with flow and structural phenomena, ever increasing effort is being placed on arterial biochemistry, pharmacology and genetics. As arterial biomechanics progresses into the future, it is expected that the technologies and techniques described here will be used to further the development of machines and systems capable of investigating, diagnosing and treating disease at the microscopic level. Indeed, it is this fusion of newly developed biomechanical investigative techniques and biochemical and genetic advances which offer the greatest potential in furthering future understanding of the artery, its complications and its treatment.

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