Three Scale Unfolding Homogenization Method Applied to Cardiac Bidomain Model
BENDAHMANE, Mostafa
Institut de Mathématiques de Bordeaux [IMB]
Modélisation et calculs pour l'électrophysiologie cardiaque [CARMEN]
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Institut de Mathématiques de Bordeaux [IMB]
Modélisation et calculs pour l'électrophysiologie cardiaque [CARMEN]
BENDAHMANE, Mostafa
Institut de Mathématiques de Bordeaux [IMB]
Modélisation et calculs pour l'électrophysiologie cardiaque [CARMEN]
< Réduire
Institut de Mathématiques de Bordeaux [IMB]
Modélisation et calculs pour l'électrophysiologie cardiaque [CARMEN]
Langue
en
Article de revue
Ce document a été publié dans
Acta Applicandae Mathematicae. 2021-12, vol. 176, n° 1, p. 14
Springer Verlag
Résumé en anglais
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 ...Lire la suite >
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. Contents 2020 Mathematics Subject Classification. 65N55 and 35A01 and 35B27 and 35K57 and 65M..< Réduire
Mots clés en anglais
double periodic media
reaction-diffusion system
Bidomain model
homogenization theory
periodic unfolding method
convergence
Project ANR
Centre de Mathématiques Henri Lebesgue : fondements, interactions, applications et Formation - ANR-11-LABX-0020
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