The cardiac bidomain model and homogenization

Erik Grandelius; Kenneth H. Karlsen

doi:10.3934/nhm.2019009

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2019, Volume 14, Issue 1: 173-204. Doi: 10.3934/nhm.2019009

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The cardiac bidomain model and homogenization

Erik Grandelius^, and
Kenneth H. Karlsen

Department of Mathematics, University of Oslo, P.O. Box 1053, Blindern, N–0316 Oslo, Norway

^* Corresponding author: Erik Grandelius
^* Corresponding author: Erik Grandelius

Received: August 2018

Revised: November 2018

Published: January 2019

This work was supported by the Research Council of Norway (project 250674/F20)

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Abstract

We provide a rather simple proof of a homogenization result for the bidomain model of cardiac electrophysiology. Departing from a microscopic cellular model, we apply the theory of two-scale convergence to derive the bidomain model. To allow for some relevant nonlinear membrane models, we make essential use of the boundary unfolding operator. There are several complications preventing the application of standard homogenization results, including the degenerate temporal structure of the bidomain equations and a nonlinear dynamic boundary condition on an oscillating surface.

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Mathematics Subject Classification: Primary: 35K57, 35B27; Secondary: 35K65, 92C30.

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Figure 1. The rescaled sets $ \Omega_i^{\varepsilon} $, $ \Omega_e^{\varepsilon} $, $ \Gamma^{\varepsilon} $ (left) and the unit cell $ Y $ (right)

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