Modified bidomain model with passive periodic heterogeneities

Yves Coudière; Anđela Davidović; Clair Poignard

doi:10.3934/dcdss.2020126

Article Contents

2020, Volume 13, Issue 8: 2231-2258. Doi: 10.3934/dcdss.2020126

This issue Previous Article Global-in-time Gevrey regularity solutions for the functionalized Cahn-Hilliard equation Next Article Fractional Cauchy problems and applications

Modified bidomain model with passive periodic heterogeneities

1.
Institut de Mathématiques de Bordeaux, CNRS UMR 5251, L'Institut de rythmologie et modélisation cardiaque LIRYC, Université de Bordeaux, Carmen, Inria Bordeaux–Sud-Ouest, Bordeaux, France
2.
Carmen, Inria Bordeaux–Sud-Ouest, Institut de Mathématiques de Bordeaux, CNRS UMR 5251, L'Institut de rythmologie et modélisation cardiaque LIRYC, Université de Bordeaux, Département de biologie computationnelle, Institut Pasteur, USR 3756 CNRS, Paris, France
3.
Monc, Inria Bordeaux–Sud-Ouest, Institut de Mathématiques de Bordeaux, CNRS UMR 5251, Université de Bordeaux, Bordeaux, France

^* Corresponding author: Anđela Davidović
^* Corresponding author: Anđela Davidović

Received: March 31, 2018

Revised: December 31, 2018

Early access: October 2019

Published: May 2020

Y.C. and A.D. have been partially funded by ANR projects, references: ANR-13-MONU-0004 and ANR-10-IAHU-04. C.P. has been partially funded by the Plan Cancer projects DYNAMO (PC201515) and NUMEP (PC201615), and Inria associate team NUM4SEP

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Abstract

In this paper we study how mesoscopic heterogeneities affect electrical signal propagation in cardiac tissue. The standard model used in cardiac electrophysiology is a bidomain model - a system of degenerate parabolic PDEs, coupled with a set of ODEs, representing the ionic behviour of the cardiac cells. We assume that the heterogeneities in the tissue are periodically distributed diffusive regions, that are significantly larger than a cardiac cell. These regions represent the fibrotic tissue, collagen or fat, that is electrically passive. We give a mathematical setting of the model. Using semigroup theory we prove that such model has a uniformly bounded solution. Finally, we use two–scale convergence to find the limit problem that represents the average behviour of the electrical signal in this setting.

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Mathematics Subject Classification: Primary: 35B27, 35Q92, 35K57, 47H20.

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Figure 1. On the left: the idealised full 2D domain, $ \Omega $. On the right: the periodic cell, $ Y $

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Figure 2. The convergence study for $ L^2 $ errors of $ v_{\varepsilon} $ and $ h_{\varepsilon} $ in log-log scale. Observed convergence rates are $ 1.39 $ for $ v_{\varepsilon} $, and $ 0.63 $ for $ h_{\varepsilon} $

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