Role and modelling of some heterogeneities for cardiac electrophysiology

International audienceIntroduction: The most used model in the elctrophysiology of theheart, known as the bidomain model, is the system of degenerate parabolicPDEs coupled with the non-linear ODE. Even though these equations pro-vide quite accurate results, they are based on the fact that active cardiomy-ocytes are present everywhere in the heart, while it is known that non-smallregions exist where fibroblasts and other non-excitable cells or additionalextracellular media take place. These regions, which play an importantrole in diseased hearts, are often taken into account through ad-hoc roughtuning of the tissue conductivities. In this work, we introduce a rigorousway to derive these conductivities from a microscopic description of theheterogeneities in the tissue.Method: We assume a periodic alternation of the healthy tissue (bido-main model) and the fibrotic tissue (diffusive part). In order to reducethe computational cost, we derive a homogenized model at the macroscopicscale, following a two-scale convergence method. There are two problemsrising here. First one has to deal with the degeneracy of parabolic equa-tions and second one comes from the non-linearity of the ionic model of thecardiac cells. In order to study the model and illustrate its relevance, wecomputed numerical simulations of both the microscopic and homogenizedmodels based on a non-physical linear model, and then on the Mitchell-Schaeffer ionic model.Results: Interestingly, we recover a bidomain type model, but withmodified conductivities, that depend on the volume fraction of the diffusiveinclusions but also on their geometries. The numerical results confirm theconvergence of the microscopic model to the homogenized equations in thelinear case. We are currently working on the numerical simulations for thenon-linear case, where we expect to observe the influence of the diffusiveinclusions on the propagation of action potentials.Conclusion: With the final non-linear model, we shall provide cheapmodeling tools to account for tissue heterogeneities at intermediate scales,as can be observed, e.g., in the fibrotic tissue