Proceedings of the Sixth New England Bioengineering Conference: March 23-24, 1978, University of Rhode Island, Kingston, Rhode Island

Proceedings of the Sixth New England Bioengineering Conference

• Language: English
• Publisher: Elsevier Science
• Release date: Jun 28, 2014
• ISBN: 9781483182049

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KEYNOTE ADDRESS: WHAT IS NEW IN VENTRICULAR DEFIBRILLATION?

Leslie A. Geddes, Ph.D.,     Showalter Distinguished Professor Director, Biomedical Engineering Center Purdue University, West Lafayette, Indiana 47907

Publisher Summary

This chapter presents details of a keynote address regarding new developments in the field of ventricular defibrillation. Ventricular fibrillation is a condition in which the muscle cells in the ventricles, the main pumping chambers in the heart, contract and relax randomly, and no blood is pumped from the heart. Irreversible brain damage starts to occur in about three minutes unless the circulation is restored. Ventricular fibrillation can result from a myocardial infarct, electric shock, drugs, and a variety of other conditions. At present, three types of defibrillators are used: (1) the damped sine wave; (2) the square wave; and (3) the trapezoidal wave. In addition, two techniques of defibrillation are used: (1) single shock; and (2) multiple shocks. The strength of the shock is controlled by the energy setting on the defibrillator. Animal studies with these three waveforms indicate that the minimum energy required is nonlinearly selected to body weight.

Ventricular fibrillation is a condition in which the muscle cells in the ventricles, the main pumping chambers in the heart, contract and relax randomly and no blood is pumped from the heart. Irreversible brain damage starts to occur in about three minutes unless the circulation is restored. Ventricular fibrillation can result from a myocardial infarct, electric shock, drugs, and a variety of other conditions. The only safe and effective method of arresting fibrillation is to pass a pulse of current through the heart. If the current pulse is adequately strong, all of the heart muscle cells are stimulated at once and the random excitation that is fibrillation is extinguished.

Despite the fact that animal hearts have been defibrillated electrically since the dawn of the 20th century and despite the fact that human hearts have been defibrillated successfully since 1947, there is still considerable controversy over what electrical dose is required to achieve ventricular fibrillation with the various current waveforms available from defibrillators. At present three types of defibrillators are used: the damped sine wave, the square wave and the trapezoidal wave. In addition, two techniques of defibrillation are used: single shock and multiple shock. The strength of the shock is controlled by the energy setting on the defibrillator. The commercially available damped sine wave and high-tilt (90%) trapezoidal units store 400 joules of energy. The square wave defibrillators store about 250 joules of energy.

Defibrillation of adult subjects is successful with these three types of defibrillator. However, the energy requirements differ when single and multiple shocks are used. Nonetheless, there appears to be some difficulty in defibrillating very heavy subjects. Among such subjects, the success rate is not well established.

Animal studies with these three waveforms indicate that the minimum energy required is non-linearly selected to body weight. The same studies indicate that the minimum current required for defibrillation is linearly related to body weight. In addition, animal and human studies indicate that the trans-thoracic impedance decreases slightly with successive shocks. Thus, for the same energy setting on a defibrillator, slightly more current will be delivered. At present there is very little information on the current used in human ventricular defibrillation. It may be that the slight increase in current with successive shocks is, in part, responsible for the increased success with multiple shocks.

The burning question among clinicians at present relates to choice of the appropriate electrical dose, i.e., the energy and current required for small (children), large and very large subjects. The other area of controversy relates to what constitutes an overdose, i.e., the amount of energy above the minimum required for defibrillation. At present these two issues are only partly resolved. However, there is no doubt that, for each current waveform used for defibrillation, a dose concept must be adopted.

Session 1

CARDIOVASCULAR SYSTEM

Outline

Chapter 2: PERFORMANCE OF THE RHONE-POULENC NON-OCCLUSIVE ROLLER BLOOD PUMP

Chapter 3: DESIGN OF A SYSTEM TO SIMULATE THE FLUID MECHANICS OF THE HUMAN LEFT VENTRICLE

Chapter 4: ANALYSIS OF THE DIASTOLIC PRESSURE-VOLUME RELATIONSHIP USING AN ELLIPSOIDAL REPRESENTATION OF THE LEFT VENTRICLE

Chapter 5: NON-INVASIVE CARDIAC OUTPUT ESTIMATION BASED UPON AN ANALOG MODEL OF THE AORTA; COMPARISON WITH THERMO-DILUTION METHOD IN 13 PATIENTS

Chapter 6: AN ANALYTICAL STUDY OF THE SELF CLEANING HEART VALVE

Chapter 7: RELATIONSHIP BETWEEN CORONARY ARTERY STENOSIS AND LEFT VENTRICULAR ASYNERGY: A COMPUTERIZED STUDY TO EVALUATE LV WALL MOTION

Chapter 8: A FOUR-CHANNEL CARDIAC DIMENSION GAUGE USING INDUCTIVELY COUPLED COILS

Chapter 9: KOROTKOFF SOUNDS - A PHENOMENON ASSOCIATED WITH PARAMETRIC INSTABILITY OF FLUID FILLED ELASTIC TUBE

PERFORMANCE OF THE RHONE-POULENC NON-OCCLUSIVE ROLLER BLOOD PUMP

P.D. Richardson, P.M. Galletti and L.A. Trudell,     Brown University, Providence, R. I. 02912

Publisher Summary

This chapter describes the performance of Rhone-Poulenc nonocclusive roller blood pump. The Rhone-Poulenc blood pump consists of a rotor, motor, support frame, and two pumping tubes, one for venous blood and the other arterial. The variation of volume pumped per revolution of the pump is imitative of Starling’s Law of the heart. According to this law, the volume pumped by the ventricle depends upon the filling pressure available to distend the pump before systole. The outlet pressure has an effect on the performance of the pump. The reason for this is the nonocclusive feature of the rollersystem. The pump outflow can be brought to zero by an outflow pressure that is sufficiently high but that is modest in terms of the risk of rupture of an extracorporeal circuit.

INTRODUCTION

The Rhone-Poulenc blood pump consists of a rotor, motor, support frame, and two pumping tubes, one for venous blood and the other arterial. The motor is totally enclosed, together with reduction gear, so that the rotor is the only exposed mechanically-driven component. The rotor consists of three equi-sized coaxial solid disks with spacers between them, the spacers providing sufficient width for the pump tubes to be stretched flat across the rollers mounted near the periphery of the disks. For each pump tube three 10mm diameter rollers are provided on a pitch circle of 95mm diameter with 120° mutual separation; rollers for the two tubes are staggered at 60° to each other. The disposable pump tubes have an unstretched length of 610mm between the clamping faces of integral collars which fit against the corresponding faces of the yoke that is part of the frame, these yoke faces being 200mm below the rotor axis and holding the tube axes 140mm apart. The pump tubes are produced to have different natural cross-sections when exposed to zero transmural pressure, neither being exactly circular and the venous pump tube more non-circular (i.e., flattened) than the arterial. There are no valves. Both tubes are lined with silicon-free silicone rubber.

PUMP DISPLACEMENT

When the pump tubes have been assembled onto the pump they are somewhat stretched, and also bent where they pass over the rollers. This tends to flatten both tubes further than when they are lying free from the pump. When the rotor rotates the rollers catch a bolus of liquid three times each revolution and carry it round. Even if each roller in rolling against the tube serves as a perfect occluder, the volume displaced in each bolus is limited by the extent to which the tube has filled. This extent depends upon the transmural pressure for the tube, and because the external pressure is atmospheric the size of the bolus depends on the internal (filling) pressure. With the venous pump tube in particular the bolus is small when the filling pressure is sub-atmospheric i.e., below 0 mmHg. As the inlet pressure is raised the effect is to distend the pump tube somewhat so that the volume of the bolus is increased. However, there is a finite range over which the increase of bolus size with increase of inlet pressure is pronounced, because it requires only a finite pressure to distend the pump tube to a circular cross-section. This is achieved first in the region of the pump tube mid-way between the pump rollers. Further gain in bolus volume is achieved by distending the cross-section closer and closer to the rollers which tend to keep the tube flat, and the rate of gain with increase in pressure is relatively small.

The variation of volume pumped per revolution of the pump is imitative of Starling’s Law of the heart. According to this law, the volume pumped by the ventricle depends upon the filling pressure available to distend the pump before systole. This pump characteristic is maintained over the range of pump speeds provided. The manufacturer has limited the maximum rotor speed to approximately 50 r.p.m., and a graph of volume pumped per revolution as a function of filling pressure is relatively invariant with pump speed provided the outlet pressure is low enough, Fig. 1.

Fig. 1 Venous pump performance at different speeds as a function of inlet pressure with outlet pressure of 100 mmHg.

EFFECT OF OUTLET PRESSURE

The outlet pressure has an effect on the performance of the pump. The reason for this is the non-occlusive feature of the roller system. The pump tubes simply pass over the rollers with some tension applied because of the tube fixture system. When the outlet pressure is raised sufficiently the tube is held open slightly as the roller passes underneath it. Normally there is a pressure difference between the liquid in the pump tube on one side of a roller and on the other side, with that on the distal side being higher, so that when the tube is held open in this way there is a backflow. The backflow rate depends primarily on the size of the opening and thereby on the outlet pressure. The impact of this on pump performance is more pronounced at lower pump speeds because there is then more time per pump revolution for the backflow to occur. Thus, if the volume pumped per revolution with a constant inlet pressure is plotted as a function of outlet pressure, there is a progressive effect of pump speed, Fig. 2.

Fig. 2 Venous pump performance at different speeds as a function of outlet pressure with inlet pressure of 50 mmHg.

The pump outflow can be brought to zero by an outflow pressure which is sufficiently high–– but which is modest in terms of the risk of rupture of an extracorporeal circuit. This outflow pressure rises with the pump speed, but is very much lower than the pressure which can be developed by an occlusive roller pump. This characteristic provides a safety feature in use, but it also limits the maximum pressure against which the pump can deliver a useful blood flow rate.

The pumping characteristics typical of the venous blood pump are repeated by the arterial blood pump in general form, but with some quantitative displacement–– the inlet pressure required to achieve a given degree of filling of the tube is lower for the arterial pump than for the venous pump. This is due to the different cross-sectional shape of the arterial pump tube when there is zero transmural pressure: it is distinctly rounder.

PUMP PERFORMANCE IN AN EXTRACORPOREAL CIRCUIT

In normal use the output of the venous pump is connected to the input of the arterial pump via a blood oxygenator. The inlet pressure of the arterial pump is lower than the output pressure of the venous pump because of the flow resistance of the blood oxygenator. The flow resistance of a blood oxygenator varies depending on its type and on the operating conditions (1,2).

With a bubble oxygenator the input pressure to the arterial pump is usually governed by the level of blood in the arterial reservoir, and this does not vary much. With a membrane oxygenator there is no similar control of the pressure, but it is usually desired to keep the pressure in the blood phase above the gas pressure in the gas phase to avoid the risk of bubbling of gas through pinholes in the membrane. This risk is much greater with a microporous membrane, of course. In steady state operation the flow rates through the venous and arterial pumps must equal each other, and with the capacity of the arterial pump to pump a greater stroke volume for a given inlet pressure one might suspect that the pump-oxygenator-pump system would reach a steady state where the blood pressure at the oxygenator outlet is very low i.e., limiting the overall flow rate by limiting the inlet pressure of the arterial pump. With some membrane oxygenators one finds considerable compliance effects (3), whereby the flow resistance is increased as the oxygenator outlet pressure falls, an effect which would help sustain low inlet pressure to the arterial pump. In fact, the flow limitation arises more from the outflow pressure required from the arterial pump. The pressure which the pump must sustain is not just the subject’s arterial pressure (when on veno-arterial bypass) but also the flow resistance of the blood return line and the cannula. Depending on the cannulation technique and blood flow rate, this may involve an additional 50-300mmHg.

In raising the backpressure on the arterial pump, a pump-jump phenomenon can be produced, Fig. 3. For small increases in outlet pressure the blood flow rate decreases slowly, but the backflow leakage increases in the arterial pump and a point can be reached where the arterial pump unloads and actually becomes a resistor. The whole pumping load then falls on the venous pump and the blood flow rate drops. The oxygenator is subjected to the maximum pressure in the system. The condition can be cured by increasing the tension on the arterial pump tube at the price of a shorter tube life.

Fig. 3 Pump-oxygenator performance with a Travenol TefloR oxygenator. Blood flow and the pressure rise in venous and arterial pumps as a function of outlet pressure. Pump at 40 rpm. The pump-jump occurs here around 350 mmHg outlet pressure.

DISCUSSION

The Rhone-Poulenc blood pump has two distinguishing features in comparison with ordinary roller pumps: the pump rollers are non-occlusive, and the tubes are manufactured so that they are naturally out-of-round (i.e., in a partially collapsed state). One consequence of these features is that the inflow or outflow lines can be clamped or blocked while the pump is running without causing cavitation in the blood drainage line or rupture of the parts of the circuit exposed to higher pressure. This is a distinct safety feature obtained without mechanical complication, and is particularly attractive in long-term perfusions with awake subjects whose movements might occlude the flow. Another consequence of the design features is that the pump adjusts its output to changes in the inflow pressure. This response is more physiological than the usual roller pump, and it eliminates the need for a blood reservoir on the drainage line often used with ordinary roller pumps. The mutual adjustability of the flow rates of the two pumps in series eliminates the need for the by-pass line often incorporated in two-pump extracorporeal circuits. The simplicity of the circuit can then lead to a smaller priming volume, another advantage in bypass.

One disadvantage of this blood pump is that acceptable output blood pressures are limited, which may eliminate use with some methods of cannulation that have been applied for long-term bypass, or put the user close to the pump-jump condition. Fortunately the latter is readily visible when it occurs so that remedies can be applied. Pump tubes tend to fail by cracking along the main flex lines of the out-of-round shapes: our experience is that an average life of several days can be expected. With attention paid to tube crack development, we have found the pump very convenient for multi-day partial bypass, with the RP-03 model giving typically 3 L/min blood flow. This study has been supported in part by Grant HL-11945.

REFERENCES

1. Murphy, W.R.C., Galletti, P.M., Richardson, P.D. Determinants of performance in spiral coil membrane oxygenators. Proc. 3rd Annual New England Bioeng. Conf., 1975; 279.

2. Richardson, P. D., Galletti, P. M. Correlation of effects of blood flow rates, viscosity and design features on artificial lung performance. In: Dawids S.G., Engell H.C., eds. Physiological and Clinical Aspects of Oxygenator Design. Elsevier/North Holland; 1976:29.

3. Laska, E., Richardson, P.D., Galletti, P.M. Dynamical characteristics of a shefet-flow membrane oxygenator. Proc. 5th Annual New England Bioeng. Conf.: Pergamon, 1977:58

DESIGN OF A SYSTEM TO SIMULATE THE FLUID MECHANICS OF THE HUMAN LEFT VENTRICLE

Edward A. Boucheron, David A. Kodl and J.Richard Shanebrook,     Union College, Department of Mechanical Engineering, Schenectady, New York 12308

Publisher Summary

This chapter focuses on the design of a system to simulate the fluid mechanics of the human left ventricle. Simulation of left ventricular fluid mechanics is particularly difficult because of the complexities of left ventricular geometry and flow conditions. The left ventricular chamber consists of stationary and moving parts that are similar to those of the Davila et al. test chamber. However, instead of a postmortem heart, the left atrium and ventricle are constructed from rectangular blocks of plexiglas that are machined internally according to specifications reported by Wieting. These chambers are assembled in a vertical plane with the atrium directly above the ventricle. Pumping is accomplished by displacing a diaphragm at the base of the ventricle. Flow exits from the left ventricle through the aortic branch that is located adjacent to the mitral valve at the top of the ventricle and offset from the center of the chamber as in the normal human heart. This provides for a pattern of flow within the left ventricular chamber that is similar to that of a natural heart in that the flow must turn nearly 180 degrees before leaving the ventricle through the aortic valve. The system is also provided with capacitance, resistance and an elevated main reservoir to enable adjustment of pressure readings to physiological values. Other features of the system include a weir in the main reservoir, a siphon tube leading to the left atrial chamber and a diffuser located directly below the diaphragm. The function of the siphon tube is to help maintain the mean left atrial pressure at a reasonable level since if the atrial reservoir were placed above the left atrial chamber the mean atrial pressure would be too high.

INTRODUCTION

Simulation of left ventricular fluid mechanics is particularly difficult due to the complexities of left ventricular geometry and flow conditions. Davila et al [1] presented an in vitro test system for the simulation of left ventricular fluid mechanics consisting of a postmortem heart with the left ventricle connected to a piston pump through an opening in the left ventricle. Motion of the piston produced pulsating flow through a mechanical circuit. This system was used to record the motion of human heart valve leaflets. No attempt was made to measure the performance of the natural valves tested.

DESIGN OF THE SYSTEM

Figures 1 and 2 illustrate the system presented here for simulating the fluid mechanics of the human left ventricle. The left ventricular chamber consists of stationary and moving parts that are similar to those of the Davila et al [1] test chamber. However, instead of a postmortem heart, the left atrium and ventricle are constructed from rectangular blocks of plexiglas that are machined internally according to specifications reported by Wieting [2]. These chambers are assembled in a vertical plane with the atrium directly above the ventricle. Pumping is accomplished by displacing a diaphragm (latex dental dam-medium gauge) at the base of the ventricle. The diaphragm, which operates as a remote pumping head, is hydraulically coupled to a Harvard Apparatus Model 1423 pulsatile blood pump. The Harvard pump allows variation of stroke rate, stroke volume and systole to diastole time ratio. Flow exits from the left ventricle through the aortic branch which is located adjacent to the mitral valve at the top of the ventricle and offset from the center of the chamber as in the normal human heart. This provides for a pattern of flow within the left ventricular chamber that is similar to that of a natural heart in that the flow must turn nearly 180 degrees before leaving the ventricle through the aortic valve. The system is also provided with capacitance, resistance and an elevated main reservoir to enable adjustment of pressure readings to physiological values.

Fig. 1 Photograph of the Test System (Harvard pump not visible)

Fig. 2 Schematic Diagram of the Test System

Other features of the system include a weir in the main reservoir, a siphon tube leading to the left atrial chamber and a diffuser located directly below the diaphragm. The weir acts as a flow stabilizer in the return line to the atrial reservoir thus enabling flow rate measurements to be made with a Fischer & Porter Co. flowmeter (model 10A 3565). The function of the siphon tube is to help maintain the mean left atrial pressure at a reasonable level since if the atrial reservoir were placed above the left atrial chamber the mean atrial pressure would be too high. The diffuser aids in reducing turbulence in the line from the Harvard pump to the diaphragm. Previously, without the diffuser, turbulence in the line was responsible for low frequency disturbances in the left ventricular pressure traces.

SAMPLE RESULTS

Left atrial pressure, left ventricular pressure and differential pressure across the mitral valve were measured with three Validyne pressure transducers (model DP15TL) and recorded on a Sanborn four channel recorder (model 150). The pressure transducers were provided with damping in order to achieve a low pass filtering effect that eliminated high frequency noise in the pressure signals and a dynamic calibrator was used to determine the effect of damping on frequency response. Static calibration was accomplished by applying a known pressure (measured by manometer) and adjusting each of the transducer’s amplifier gain to give conveniently scaled output.

Figure 3 is a sample recording of left ventricular pressure (LVP) taken at a pulse rate of 70 cycles per minute, a chart speed of 25 mm per second and a Starr-Edwards ball valve (Model 6320, size 3M) in the mitral position. The recorded waveform resembles closely the left ventricular pressure traces reported by Zimmerman [3, page 303, Fig. 14] for a catheterized patient.

Fig. 3 Sample Recording of Left Ventricular Pressure (LVP).

(1) Harvard Pump

(2) Diaphragm

(3) Pressure taps

(4) Atrial reservoir

(5) Siphon tube

(6) Aortic valve chamber

(7) Left atrial chamber

(8) Mitral valve

(9) Left ventricular chamber

(10) Diffuser

(11) Systemic capacitance

(12) Systemic resistance

(13) Main reservoir

(14) Flowmeter

The system has been designed to compare the performance of prosthetic mitral valves by measuring the mean diastolic pressure drop (ΔP) across each valve and the amount of backflow on a relative basis. Table 1 lists results found for three commercially available prosthetic mitral valves under identical test conditions of left ventricular pressure (140/0 mm Hg), mean left atrial pressure (10 mm Hg) and a pulse rate of 72 cycles per minute. All tests were conducted with a blood analog fluid (glycerine and distilled water) as described by Wieting [2]. The percent backflow results reported in Table 1 were obtained by comparing the measured flow rates of the prosthetic valves with that of the spring-loaded, non-regurgitating valve pictured in Fig. 4. That is, it was assumed that this valve is perfect in that its spring return and positive seating allows no backflow when operating in the mitral position.

TABLE 1

RESULTS FOR THREE PROSTHETIC MITRAL VALVES

Fig. 4 Photograph of Non-Regurgitating Valve.

ACKNOWLEDGEMENTS

The authors express their thanks to Henry T. Jacobson and to the Health Fund of United Way of Schenectady County, Inc. for their support of this work.

REFERENCES

Davila, J. C., Trout, R. G., Sunner, J. E., Glover, R. P. A Simple Mechanical Pulse Duplicator for Cinematography of Cardiac Valves in Action. Annals of Surgery. 1956; Vol. 143:544–551.

Wieting, D. W. Dynamic Flow Characteristics of Heart Valves. In: Ph.D. dissertation, The University of Texas at Austin. Available from University Microfilms, Inc.; 1969. [Ann Arbor, MI (O.N. 69-21, 904).].

Zimmerman, H.A. Intravascular Catheterization. C.C. Thomas, Springfiled, I11., 1966.

ANALYSIS OF THE DIASTOLIC PRESSURE-VOLUME RELATIONSHIP USING AN ELLIPSOIDAL REPRESENTATION OF THE LEFT VENTRICLE

Dennis J. Arena, William J. Ohley and Dov Jaron,     Biomedical Engineering Program, Department of Electrical Engineering, University of Rhode Island, Kingston, Rhode Island 02881

Publisher Summary

This chapter presents an analysis of the diastolic pressure–volume relationship using an ellipsoidal representation of the left ventricle. It presents a time domain representation of the left ventricle. Previously, this model was used to analyze the isovolumic contraction phase and the ejection phase of the cardiac cycle. Results obtained from the simulation agreed with results from animal experiments. The chapter explores the properties of the model under conditions where the distributed strain in the ventricular wall was replaced by strain concentrated along a single surface within the myocardium. In the study described in the chapter, the left ventricle was modeled by two ellipses of revolution that represent the endocardial and epicardial surfaces of the chamber. The ellipsoids were truncated perpendicular to their long axis by a plane corresponding to the base of the left ventricle. The ellipsoids were approximated by a series of cylindrical shells. The model was tested utilizing 10 cylindrical segments. The stress in the walls of the cylindrical shells at end diastole was found from the end-diastolic pressure and end-diastolic configuration. The corresponding strain in the walls of the cylindrical shells was calculated from the stress–strain relationship of the model. As the assumed location of the strain is allowed to shift toward the epicardium, the results obtained from the model indicate that left ventricular internal pressure sensitivity to changes in chamber volume diminishes.

Introduction

We have developed a time domain representation of the left ventricle. Previously, this model was used to analyze the iso-volumic contraction phase and the ejection phase of the cardiac cycle (1). Results obtained from the simulation agreed with results from animal experiments. In the present work, the model was used to analyze the passive diastolic compliance of the healthy left ventricle. In particular, we wanted to explore the properties of the model under conditions where the distributed strain in the ventricular wall was replaced by strain concentrated along a single surface within the myocardium.

The Model

The left ventricle was modeled by two ellipses of revolution which represent the endocardial and epicardial surfaces of the chamber. The ellipsoids were truncated perpendicular to their long axis by a plane corresponding to the base of the left ventricle. The ellipsoids were approximated by a series of cylindrical shells.

Left ventricular end diastole was chosen as the initial time state for the model. At that instant, the two ellipsoids are confocal. End diastolic pressure, end diastolic volume and equatorial wall thickness were specified for the model using values obtained from the literature for a canine left ventricle (2,3).

Throughout the analysis, stress was assumed to be circumferential and uniform in the wall of each cylindrical shell. Stress in the wall of each shell was related to the pressure within the shell by a pressure-stress relationship derived from the law of Laplace (4). Blood was assumed to be incompressible.

Strain of the myocardium in the shell wall was related to stress in the wall by a stress-strain relationship for myocardium in the passive state. The relationship used was one derived for soft biological tissues in general (5) with constants evaluated for passive myocardium (6). During the simulation, as the radius of a cylindrical shell changed, the wall thickness was adjusted to conserve the volume of the shell wall.

The stress in the walls of the cylindrical shells at end diastole was found from the end-diastolic pressure and end-diastolic configuration. The corresponding strain in the walls of the cylindrical shells was calculated from the stress-strain relationship of the model. From this information, left ventricular volume at zero stress and zero strain was obtained. This unstrained configuration was then used in subsequent analysis to calculate wall strain at any ventricular