Computational electrocardiology : mathematical and numerical modeling

This paper deals with mathematical models of cardiac bioelectric activity at both the cell and tissue levels, their integration in coupled models and their numerical simulation. The macroscopic bidomain model of the cardiac tissue is derived by the two-scale homogenization method. Existence and uniqueness results for the cellular and bidomain models are reviewed. A rigorous derivation of the bidomain model is presented in the framework of Cyrillic capital letter GHE-convergence theory, and approximation results concerning its time and space discretization are given. The bidomain model of the myocardium is coupled with the extracardiac medium and extracardiac potentials, computed from given cardiac sources by means of differential or integral representations in order to obtain body surface maps and electrograms. Various approximate models of the bidomain model are examined and discussed such as the monodomain model, the eikonal equations and a relaxed monodomain model. These continuous cardiac models are then numerically approximated by isoparametric finite elements in space and adaptive finitie difference methods in time. Numerical simulations of the monodomain and bidomain models are discussed and examples of large-scale parallel computations are reported; these simulate excitation and repolarization processes in three-dimensional anisotropic domains.