The relations between end-diastolic (D) and end-systolic (S) fiber angles (a) and sarcomere lengths (s) have been previously studied at different sites in canine left ventricular myocardium. However, no postulates have been advanced for predicting a and s in successive states of the ejection cycle (D or S) or at different sites in one state when the semimajor (Z) and semiminor (R) axes of the wall surfaces for successive states and the fiber orientations and sarcomere lengths at one site in one state are known. In this study, the myocardial fibers were regarded as the matrix of a myocardial continuum: they are prisoners of the heart wall and must comply with the movements of the wall. Using the same values as in the preceding article, the wall was treated as a nested set of truncated ellipsoidal shells of revolution with shell volumes preserved from D to S. Both confocal and nonconfocal configurations were analyzed. In each shell, the fibers were assumed to follow a "helical" path with a constant advance in each turn about the Z axis (the simplest possible path). The results of this assumption were compatible with previously reported values of a and s measured at various sites in the left ventricular free wall in states D and S. These results suggest a postulate for the heart wall: in the beating heart, each muscle fiber changes direction and length uniquely as the wall changes shape. KEY WORDSdog heart fiber orientation helical path ellipsoid of revolution confocal ellipses nested set of shells left ventricular wall shape truncated ellipsoid grid spacing• A precise knowledge of the myocardial fiber structure of the left ventricular wall (1) is important to the physiologist, because it leads to the establishment of laws relating possible fiber motion (2) to the overall motion of the wall. This paper shows that changes in individual left ventricular fiber angle (3-5) and sarcomere length (5) can be related to the change in the shape of the left ventricular wall throughout the cardiac cycle. There are no statistically significant data supporting the change in fiber angle (4, 5 ) , although there are data supporting the change in sarcomere length (5). The modeling of left ventricular wall shape rests on the assumptions previously given for a nested set of truncated ellipsoids of revolution (1). The fibers in left ventricular myocardium can be

The relations between end-diastolic (D) and end-systolic (S) cavitary volumes (V c), wall volumes (V w), and cavitary dimensions have been studied in the canine and human left ventricle. However, the models selected for left ventricular myocardium do not represent the real heart adequately for a fiber-by-flber analysis of fiber orientation and sarcomere length during successive states of the ejection cycle. In this study, the endocardial and epicardial surfaces were postulated to be a nested set of truncated ellipsoidal shells of revolution where wall volumes were preserved from D to S. Shell dimensions on the semiminor and semimajor axes, R and Z, respectively, were related to V c and V w by two representations: confocal and nonconfocal. If the focal length C = (Z 2-R 2)t and C is the same for each shell, then the shells are confocal, otherwise they are nonconfocal. From measured V c , V w , and epicardial Z in D, shell dimensions were calculated for states D and S, using both confocal and nonconfocal representations , and compared with the measured dimensions, When no empirical corrections were made, the calculated endocardial R in S underestimated the measured R in S by 12%; moreover, the calculated epicardial R in S overestimated the measured R in S by 4%. Endocardial and epicardial C measured 3.73 ± 0.33 (SE) cm and 3.79 ± 0.34 cm, respectively, in D and 3.77-0.11 cm and 3.71 ±0.10 cm, respectively, in S. KEY WORDS cavitary volumes wall volumes confocal ellipses papillary muscle volumes ellipsoid of revolution grid dimensions left ventricular wall thickness eccentricity of ellipse focal length of ellipse • The shape of the left ventricular myocardium can be likened to that of an eggshell with its top cut off: the aortic and mitral valve openings represent the rim (the base) and the wall is thickest at the greatest circumference (the equator) and thinnest at the bottom (the apex) (Fig. 1). The left ventricular wall is the site at which local fiber orientation, sarcomere length, and fiber stress should be measured; a mathematical description is needed. Eighty years ago, the left ventricular wall was viewed as a curved membrane with two principal radii of curvature and a finite thickness (1). There were no systematic intramural data and fiber stresses were thought to be uniformly distributed across the thickness of the wall. Since that time, great strides have been made in mapping the interior of this structure at both the gross (2-6) and the ultrastructural level (7-10). Yet the simple membrane concept still prevails (11), although some investigators depict the left ventricle as a sphere (12) or as a full ellipsoid (13-15) with uniform wall thickness. A sophisticated representation, devised by Dieudonne (16), assumes that the endo-cardial and epicardial surfaces are confocal ellip-soids of revolution, thus realistically accounting for the longitudinal variation in wall thickness. Because of the lagging development of realistic models, clinical techniques for measuring the wall do not adequately describe the m...

A simulation of the function of the human heart and heart muscle has been developed in the form of a digital computer code. For a given set of values for the input variables, realistic values of the cardiac output variables are predicted. A detailed discussion of the simulation and some results obtained from its application are presented. This simulation represents a unique combination of what was known in muscle mechanics, muscle thermodynamics, and of the structure, size, and shape of the heart, into an engineering model to improve the understanding of human heart muscle function. The left ventricle (LV) is treated as a thick-walled sphere whose wall is composed entirely of muscle fibers. Force-length velocity relationships are used to determine the tension in each fiber. The pressure in the LV is computed from fiber tension and fiber structure in the LV. A lumped-parameter simulation of the arterial tree provides a load impedance for the LV. Results are presented for simulation of normal human LV performance.