Role and modelling of some heterogeneities for cardiac electrophysiology

Introduction: The most used model in the elctrophysiology of the heart,known as the bidomain model, is the system of degenerate parabolic PDEs cou-pled with the non-linear ODE. Even though these equations provide quite ac-curate results, they are based on the fact that active cardiomyocytes are presenteverywhere in the heart, while it is known that non-small regions exist wherefibroblasts and other non-excitable cells or additional extracellular media takeplace. These regions, which play an important role in diseased hearts, are oftentaken into account through ad-hoc rough tuning of the tissue conductivities. Inthis work, we introduce a rigorous way to derive these conductivities from amicroscopic description of the heterogeneities in the tissue.Method: We assume a periodic alternation of the healthy tissue (bidomainmodel) and the fibrotic tissue (diffusive part). Such a microscopic model canbe simulated directly, at the price of a very fine discretization and a high com-putational cost. Instead we derive a homogenized model at the macroscopicscale, following a two-scale method technique. There are two problems risinghere. First one has to deal with the degeneracy of parabolic equations and sec-ond one comes from the non-linearity of the ionic model of the cardiac cells.In order to study the model and illustrate its relevance, we computed numeri-cal simulations of both the microscopic and homogenized models based on anon-physical linear model, and then on the Mitchell-Schaeffer ionic model.Results: Interestingly, we recover a bidomain type model, but with modifiedconductivities, that depend on the volume fraction of the diffusive inclusionsbut also on their geometries. The numerical results confirm the convergenceof the microscopic model to the homogenized equations in the linear case. Weare currently working on the numerical simulations for the non-linear case,where we expect to observe the influence of the diffusive inclusions on thepropagation of action potentials.Conclusion: With the final non-linear model, we shall provide cheap mod-eling tools to account for tissue heterogeneities at intermediate scales, as canbe observed, e.g., in the fibrotic tissue