Role and modelling of some heterogeneities for cardiac electrophysiology

Anđela Davidović 1, 2, 3
1 CARMEN - Modélisation et calculs pour l'électrophysiologie cardiaque
IMB - Institut de Mathématiques de Bordeaux, Inria Bordeaux - Sud-Ouest, IHU-LIRYC
Abstract : Introduction: The most used model in the elctrophysiology of the heart, known as the bidomain model, is the system of degenerate parabolic PDEs 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-small regions exist where fibroblasts and other non-excitable cells or 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. Method: We assume a periodic alternation of the healthy tissue (bido- main model) and the fibrotic tissue (diffusive part). In order to reduce the computational cost, we derive a homogenized model at the macroscopic scale, following a two-scale convergence method. There are two problems rising 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 the cardiac cells. In order to study the model and illustrate its relevance, we computed numerical simulations of both the microscopic and homogenized models based on a non-physical linear model, and then on the Mitchell- Schaeffer ionic model. Results: Interestingly, 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 in the linear case. We are currently working on the numerical simulations for the non-linear case, where we expect to observe the influence of the diffusive inclusions on the propagation of action potentials. Conclusion: With the final non-linear model, we shall provide cheap modeling tools to account for tissue heterogeneities at intermediate scales, as can be observed, e.g., in the fibrotic tissue.
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Submitted on : Tuesday, February 17, 2015 - 3:25:15 PM
Last modification on : Thursday, January 11, 2018 - 6:23:41 AM

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Anđela Davidović. Role and modelling of some heterogeneities for cardiac electrophysiology. MATHEMATICAL MODELLING WORKSHOP: MATHMODS & EMA ALUMNI MEETING, Jul 2014, L'Aquila, Italy. 2014. 〈hal-01117279〉

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