Role and modelling of some heterogeneities for cardiac electrophysiology
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.
Domains
Mathematics [math]
Origin : Files produced by the author(s)