IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. BME-30, NO. 1, JANUARY 1983 Analysis of Left Ventricular Mechanics During Filling, Isovolumic Contraction, and Ejection DENNIS J. ARENA AND WILLIAM J. OHLEY, MEMBER, IEEE Abstract-The left ventricle (LV) was modeled by two confocal ellip- soids truncated in a plane corresponding to the base of the LV. The ellipsoids were approximated by a series of cylindrical shells. During passive filling, the pressure within the ventricular chamber was deter- mined from chamber volume and a stress-strain relationship for myo- cardium in the relaxed state. The rapid filling phase of diastole was not analyzed. During isovolumic contraction, the cylindrical shells assumed the properties of myocardium in active contraction. Contraction was sequential, beginning at the ventricular apex, and progressing toward the ventricular base. Geometric changes occurred in the LV model as a function of wall stress, material properties, and timing of myocardial ac- tivation. During ejection, viscous and inertial forces were determined as were model output pressure and flow waveforms. The pressure-volume (P-V) relationship, obtained by using strain cal- culated at 1 LV wall thickness, is similar to P-V relationships obtained experimentally. The pressure and flow waveforms obtained from the model are also similar to experimental results. The model reveals that pressure distribution throughout the ventricular chamber during iso- volumic contraction and ejection is such that blood flow tends to be in the direction of the aortic valve. INTRODUCTION A MAJOR problem in utilizing cellular mechanics to model the left ventricle (LV) arises due to the asynchrony of LV excitation [1]. LV fibers do not contract in a uniform fashion. During isovolumic contraction, some fibers are short- ening while others are being stretched [2] . Pressure generated within the ventricular chamber is a func- tion of LV geometry and stress developed within the ventricu- lar wall. For a given myocardial stress, intraventricular pressure is determined by chamber radius [3] and wall thickness [4]. Stress in the LV wall is not only a function of fiber tension, but also of the spatial orientation of the fibers [5]. Another major problem in modeling the LV arises due to the fact that the load against which the LV contracts is a complex function of time [1]. This load contains inertial, frictional, and elastic components [6]. Hence, an understanding of the relationship between forces of contraction in muscle fibers to the pressures and flows gen- erated by the LV requires detailed information such as fiber orientation, wall curvature, sequence of contraction, and wall thickness [7] , plus detailed information on LV afterload [ 1]. Various geometries have been used to model the LV in the past. For instance, Hanna [8] and Mirsky [9] utilized spheres. Manuscript received Septemiber 28,1981; revised July 13, 1982. D. J. Arena was with the University of Rhode Island, Kingston, RI 02881. He is now with the Department of Internal Medicine, Miriam Hospital, Providence, RI 02906. W. J. Ohley is with Datascope Corporation, Paramus, NJ 07652, on leave from the University of Rhode Island, Kingston, RI 02881. Recently, Arts et al. [10] proposed a cylindrical representa- tion of the LV. Streeter and Hanna [11] and Dieudonne [12] used confocal ellipsoids in their analyses of the LV. The asyn- chrony of contraction of the LV was modeled by an electrical analog which displayed contraction progressing from apex to base [13] . However, an analysis of the LV has not previously been described which takes into account the spread of excita- tion during isovolumic contraction from LV apex to base and the simultaneous geometric changes. PRINCIPLES OF THE MATHEMATICAL MODEL The LV was modeled by representing the endocardial and epicardial surfaces at end-diastole as confocal ellipsoids. The ellipsoids were truncated perpendicular to the long axis in a plane corresponding to the base of the LV. This representa- tion conforms to the cutoff eggshell shape of the LV and ac- counts for the spatial variation in wall thickness. The end- diastolic dimensions were chosen from measurements on ca- nine LV's [ 1 11], [ 12]. The ellipsoids were approximated by a series of ten cylindrical shells of equal height. The resulting end-diastolic configuration of the model is displayed in Fig. 1. Other model constraints include end-diastolic pressure, cardiac output, heart rate, duration of isovolumic contraction, and duration of systole. The assumptions used in the analysis were 1) blood is an incompressible Newtonian fluid, 2) stress is circumferential and uniform throughout the wall of each cylindrical shell, 3) the volume of cardiac muscle is conserved during contrac- tion, and 4) flow through each cylindrical shell is laminar and unidi- rectional. Stress in the wall of each cylindrical shell is related to the pressure within the shell by Pr h (1) where a = stress in the wall of the shell P = pressure within the shell r = inside radius h = wall thickness. This expression is ordinarily applied to thin wall structures, but with the assumption of uniform and circumferential stress, it can be used for thick wall cylinders. 0018-9294/83/0100-0035$01.00 © 1983 IEEE 35