Numerical approximation of heart electromechanics
In this talk we present an overview of the numerical simulation of the interaction between cardiac electrophysiology, sub-cellular activation mechanisms, and macroscopic tissue contraction; that together comprise the essential elements of the electromechanical function of the heart. We will discuss the development of some mathematical models tailored for the simulation of the cardiac excitation-contraction mechanisms, which are primarily based on nonlinear elasticity theory and phenomenological descriptions of the mechano-electrical feedback. Here the link between contraction and the biochemical reactions at microscales is described by an active strain decomposition model. Then we turn to the mathematical analysis of a simplified version of the model problem consisting in a reaction-diffusion system governing the dynamics of ionic quantities, intra and extra-cellular potentials, and the electrodynamics equations describing the motion of an incompressible material. Under the assumption of linearized elastic behavior and a truncation of the updated nonlinear diffusivities, we are able to prove existence of weak solutions to the underlying coupled system and uniqueness of regular solutions. A finite element formulation is also introduced, for which we establish existence of discrete solutions and show convergence to a weak solution of the original problem. We next introduce a novel method based on mixed-primal formulations of the coupled system, and we close with some examples illustrating properties of the numerical schemes in recovering salient features of the model.