Mathematical aspects of the mechanics of left ventricular contraction

Abstract

Large elastic deformation is used in order to model the mechanics of left ventricular contraction. The active force generated by the myocardium is modeled as force/unit myocardial volume in the mathematical formal- ism describing the local equilibrium of forces in the myocardium. Expressions for the stress components are derived by assuming a cylindrical geometry for the left ventricle, the total stress is expressed as the sum of a component due to the deformation of the passive medium of the myocardium and an active component induced by the tension in the muscular fibers. It is shown that knowledge of the tension generated by the muscular fiber in the myocardium can lead to useful information for the estimation of the pseudo-strain energy function used to express the stress-strain constitutive relations in a non-linear model. Keywords: active force of the myocardium, cardiac mechanics, mathematical modeling of ventricular contraction, pressure‑volume relation in the left ventricle. Many articles have been devoted to the study of stress-strain relations in the myocardium and how to formulate a pseudo-strain energy function W that is used to derive the constitutive relations between stress and strain (1-4). Previous studies by the authors (5-9) have shown how the active force developed by the myocardium can be modeled as force generated by unit volume of the myo- cardium in the mathematical formalism used to describe the local equilibrium of forces. This mathematical approach was successfully developed by using large elastic deformation (5, 8) as well as linear elasticity (6). For this purpose the total stress in the myocardium is expressed as the sum of a contribution due to the stress generated by the deformation of the passive isotropic medium of the myocardium, and a contribution coming from the stress induced by the active muscular fibers. A mathematical description of how this splitting is done can be found in the work of Spencer (10). The approach used in this study is similar to the approach used in some studies in which the pseudo- strain energy function W is split into the sum of an isotropic component and an anisotropic component in the form W = W iso + W aniso (11), the link between the two approaches is evident from what follows. We also show in this study how the calculation of the stress in the myocardium can be based on the knowledge of the muscular fiber tension T without an explicit knowledge of the pseudo-strain energy function W, which presents a possible way to estimate W as a result of the method of calculation shown in this study.