Section: New Results

Mathematical models

• Mathematical derivation of a bilayer surface model of the atria using asymptotic analysis methods [28] , [16]

  We derived rigorously, by using asymptotic analysis tools, a bilayer model of atrial electrophysiology. Starting with a 3D model of atrial tissue that includes two layers with distinct electrophysiological characteristics and with an aspect ratio of $ϵ$, we obtained an asymptotic equivalent model when $ϵ⟶0$ made up of two surface models coupled by a coupling term. The bilayer model discribes the evolution of the mean in the thickness of the 3D potential in each layer. This approach is an improvement of the classical surface model of cardiac electrophysiology, because it guaranties a higher convergence speed, and allows to take into account transmural heterogeneities. We numerically implemented the 3D and bilayer models and compared it to the classical surface model. We observed a second order accuracy of the bilayer model and drastically reduced computational times respectively to the 3D model.

• Formal derivation of a macroscopic model of propagation that includes the non linear behavior of gap junctions [16]

  A macroscopic model of electrical propagation that take into account the non linear conductivity of the gap junction is obtained by a formal homogenization method. We derived a one dimensional macroscopic model which diffusive tensor varies in time. We compared this macroscopic model with a cell-to-cell propagation. This is an very important improvement of existing models that only consider a linear cell-to-cell coupling. The introduction of this non linear phenomenon in homogenized models gives a simulation tool to investigate the impact of the microscopic nonlinear mechanisms on the macroscopic propagation.

• Influence of periodic diffusive inclusions on the bidomain model [29]

  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 homogenisation 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.