Influence of periodic diffusive inclusions on the bidomain model

We present a new mathematical model of the electric activity of the heart. In the standard bidomain model we can distinguish the intra- and the extracellular space with different conductivities for excitable cells and the fibrotic tissue around them. The main drawback is that it assumes the existence of excitable cells everywhere in the heart, while it is known that there exist non small regions where fibroblasts take place. The fibroblasts are equally distributed and since they are non excitable cells, they can be considered as a diffusive part. Hence we extend the standard bidomain model as follows: we assume that we have periodic alternation of the healthy tissue (linear bidomain model) and fibrotic extracellular space (diffusive part). We use homogenization techniques to derive our macroscopic partial differential equations. Interestingly, we obtain again a bidomain type model with modified conductivities that involve the volume fraction of the diffusive domain. Preliminary numerical experiments will conclude on the influence of these diffusive inclusions.