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. 2020 Mathematics Subject Classification. 65N55 and 35A01 and 35B27 and 35K57 and 65M..

Microscopic tridomain model of electrical activity in the heart with dynamical gap junctions. Part 2 – Derivation of the macroscopic tridomain model by unfolding homogenization method

“…Remark 10. Proceeding exactly as in [4], we prove that the limits v k and s coincide respectively with u k i −u e for k = 1, 2 and u 1 i −u 2 i . Furthermore, since we have assumed that the initial data v k 0,ε , w k 0,ε for k = 1, 2 and s 0,ε introduced in (7), are also uniformly bounded in the adequate norm (see assumption (11)).…”

“…The first method [3] is a formal and intuitive method based on a new three-scale asymptotic expansion method applied to the meso-and microscopic model. The second one [4] based on unfolding operators which not only derive the homogenized equation but also prove the convergence and rigorously justify the mathematical writing of the preceding asymptotic expansion method.…”

Microscopic Tridomain Model of Electrical Activity in the Heart with Dynamical Gap Junctions. Part 1 -- Modeling and Well-Posedness

“…Remark 8. The authors in [5,2,3] treated the microscopic bidomain problem where the gap junction is ignored. They considered that there are only intra-and extracellular media separated by the membrane (sarcolemma).…”

Homogenization of a nonlinear reaction‐diffusion problem in a perforated domain with different size obstacles

“…For instance, a cardiac bidomain model was studied in ref. [6] which relies on three different (micro-meso-macro) scales, one can also read the refs. [7,8] for more multi-scale related models.…”

“…Refs. [6,[39][40][41][42] studied the homogenization problems in a porous domain with two different periodic scales 𝜖 and 𝜖𝛿(𝜖) but only containing 𝜖𝛿(𝜖)-size obstacles with period 𝜖𝛿(𝜖). Difference from the above mentioned references, just like for Gahn et al [26], in the present paper we consider a nonlinear parabolic problem with low diffusion scale of order (𝜖𝛿) 2 in Ω 𝜖𝛿 and a nonlinear Neumann condition is prescribed on the boundary of the smaller obstacles, and the final homogenized macroscopic model depends on three scales.…”