Equations for estimating muscle fiber stress in the left ventricular wall

Left ventricular muscle fiber stress is an important parameter in cardiac energetics. Hence, we developed equations for estimating regional fiber stresses in rotationally symmetric chambers, and equatorial and apical fiber stresses in prolate spheroidal chambers. The myocardium was modeled as a soft incompressible material embedding muscle fibers that support forces only in their longitudinal direction. A thin layer of muscle fibers then contributes with a pressure increment determined by the fiber stress and curvature. The fiber curvature depends on the orientation of the fibers, which varies continuously across the wall. However, by assuming rotational symmetry about the long axis of the ventricle and including a longitudinal force balance, we obtained equations where fiber stress is completely determined by the principal curvatures of the middle wall surface, wall thickness, and cavity pressure. The equations were validated against idealized prolate spheroidal chambers, whose wall thicknesses are such that the fiber stress is uniform from the equator to the apex. Because the apex is free to rotate, the resultant moment about the long axis of the LV must be zero. By using this constraint together with our fiber-stress equations, we were able to estimate a muscle fiber orientation distribution across the wall that was in qualitative agreement with published measurements.