A porous-medium approach for modeling heart mechanics. I. theory

A method is developed for evaluating the distribution of the blood pressure, stresses and strains of the muscle fibers, and motion of the cardiac wall due to the cyclic contractions of the heart. The cardiac system is subdivided into two media: the chambers and the wall; the latter is enclosed by two impermeable surfaces (with one interface separating the two media and the other confining the wall). The momentum balance equation for the blood in the (two) cardiac ventricles is averaged, yielding a modification of Forchheimer's law, namely inclusion of the time derivative of the flux. The contracting muscle provides the driving force for the blood flow, and the endocardial velocity is thus taken as identical to that of the blood in the cavity next to the wall. Translation of the endocardium is governed by the blood pressure gradients in the ventricles. The blood pressure and stress-strain pattern in the myocardium are analyzed by applying concepts of the theory of mixtures to the blood and to the saturated solid matrix. With the blood pressure simulated by a modified Darcy law with relative fluid-solid velocity, fiber stresses and strains can be assessed with the aid of appropriate constitutive and compatibility laws.