Role and modelling of some heterogeneities for cardiac electrophysiology
Résumé
Introduction: The most used model in the elctrophysiology of the heart,
known as the bidomain model, is the system of degenerate parabolic PDEs cou-
pled with the non-linear ODE. Even though these equations provide quite ac-
curate results, 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
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 (bidomain
model) and the fibrotic tissue (diffusive part). Such a microscopic model can
be simulated directly, at the price of a very fine discretization and a high com-
putational cost. Instead we derive a homogenized model at the macroscopic
scale, following a two-scale method technique. There are two problems rising
here. First one has to deal with the degeneracy of parabolic equations and sec-
ond 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 numeri-
cal 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 mod-
eling tools to account for tissue heterogeneities at intermediate scales, as can
be observed, e.g., in the fibrotic tissue.
Domaines
Mathématiques [math]Origine | Fichiers produits par l'(les) auteur(s) |
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