The Modified Bidomain Model with Diffusive Inclusions

Bidomain equations are the standard way to model the electric potential in cardiac tissue. They are based on the fact that active cardiomyocytes are present everywhere in the heart, while it is known that non-small regions exist where additional extracellular media take place. These regions, which play an important role in diseased hearts, are often taken into account through ad-hoc rough tuning of the tissue conductivities. In this work, we introduce a rigorous way to derive these conductivities from a microscopic description of the heterogeneities in the tissue. We assume a periodic alternation of the healthy tissue and the fibrotic tissue. Such a microscopic model can be simulated directly, at the price of a very high computational cost. Instead we derive a homogenized model at the macroscopic scale, following a standard multiscale technique.We recover a bidomain type model, but with modified conductivities, that depend on the volume fraction of the diffusive inclusions but also on their geometries. The numerical results confirm the convergence of the microscopic model to the homogenized equations. We observe the influence of the diffusive inclusions on the propagation of action potentials. With the final model we shall provide cheap modeling tools to account for tissue heterogeneities at intermediate scales. The diffusive volume ratio, that enters the model, might be available through functional imaging, which enlightens the practical interest of the method