Mathematical and numerical methods for reaction-diffusion models in electrocardiology

Abstract

This paper presents a review of current mathematical and numerical models of the bioelectrical activity in the ventricular myocardium, describing cardiac cells excitability and the action-potential propagation in cardiac tissue. The degenerate reaction-diffusion system called the Bidomain model is introduced and interpreted as macroscopic averaging of a cellular model on a periodic assembling of myocytes. The main theoretical results for the cellular and Bidomain models are given. Various approximate models based on some relaxed approaches are also considered, such as Monodomain and eikonal-curvature models. The main numerical methods for the Bidomain and Monodomain models are then reviewed. In particular, we focus on isoparametric finite elements, semi-implicit time discretizations and a parallel iterative solver based on a multilevel Schwarz preconditioned conjugate gradient method. The Bidomain solver is finally applied to the study of the excitation processes generated by virtual electrode response in 3D orthotropic blocks of myocardial tissue.