Three Scale Unfolding Homogenization Method Applied to Cardiac Bidomain Model

Abstract

In this paper, we are dealing with a rigorous homogenization result at two different levels for the bidomain model of cardiac electro-physiology. The first level associated with the mesoscopic structure such that the cardiac tissue consists of extracellular and intracellular domains separated by an interface (the sarcolemma). The second one related to the microscopic structure in such a way that the intracellular medium can only be viewed as a periodical layout of unit cells (mitochondria). At the interface between intra- and extracellular media, the fluxes are given by nonlinear functions of ionic and applied currents. A rigorous homogenization process based on unfolding operators is applied to derive the macroscopic (homogenized) model from our meso-microscopic bidomain model. We apply a three-scale unfolding method in the intracellular problem to obtain its homogenized equation at two levels. The first level upscaling of the intracellular structure yields the mesoscopic equation. The second step of the homogenization leads to obtain the intracellular homogenized equation. To prove the convergence of the nonlinear terms, especially those defined on the microscopic interface, we use the boundary unfolding method and a Kolmogorov-Riesz compactness’s result. Next, we use the standard unfolding method to homogenize the extracellular problem. Finally, we obtain, at the limit, a reaction-diffusion system on a single domain (the superposition of the intracellular and extracellular media) which contains the homogenized equations depending on three scales. Such a model is widely used for describing the macroscopic behavior of the cardiac tissue, which is recognized to be an important messengers between the cytoplasm (intracellular) and the other extracellular inside the biological cells.

Appendix:  Compactness Result for the Space \(L^{p}(\varOmega ,B)\)

In this part, we give a characterization of relatively compact sets $F$ in $L^{p}(\varOmega , B)$ for $p\in [1;+\infty )$, $\varOmega \subset \mathbb{R}^{d}$ open and bounded set and $B$ a Banach space.

Proposition 26

Kolmogorov-Riesz type compactness result

Let $\varOmega \subset \mathbb{R}^{d}$ be an open and bounded set. Let $F \subset L^{p}(\varOmega ,B)$ for a Banach space B and $p\in [1;+\infty )$. For $f\in F$ and $h\in \mathbb{R}^{d}$, we define $\tau _{h}f(x):=f(x+h)$. Then $F$ is relatively compact in $L^{p}(\varOmega ,B)$ if and only if

• (i) for every measurable set $C\subset \varOmega$ the set $\lbrace \int _{C} f dx \ : \ f \in F \rbrace$ is relatively compact in $B$,

• (ii) for all $\lambda >0$, $h\in \mathbb{R}^{d}$ and $h_{i}\geq 0$, $i=1,\dots ,d$, there holds

  $$ \underset{f\in F}{\sup } \left \| \tau _{h}f-f\right \| _{L^{p} \left (\varOmega _{\lambda }^{h},B\right )}\rightarrow 0, \textit{ for } h \rightarrow 0, $$

  where $\varOmega _{\lambda }^{h}:=\lbrace x\in \varOmega _{\lambda }: x+h \in \varOmega _{\lambda }\rbrace$ and $\varOmega _{\lambda }:=\lbrace x\in \varOmega : dist(x,\partial \varOmega )> \lambda \rbrace$,

• (iii) for $\lambda >0$, there holds $\underset{f\in F}{\sup } \int _{\varOmega \setminus \varOmega _{\lambda }} \left |f(x)\right |^{p} dx\rightarrow 0$ for $\lambda \rightarrow 0$.

Proof

The proof of the proposition can be found as Corollary 2.5 in [25]. □