Well-Posedness for a Modified Bidomain Model Describing Bioelectric Activity in Damaged Heart Tissues

We prove the existence and the uniqueness of a solution for a modified bidomain model, describing the electrical behaviour of the cardiac tissue in pathological situations. The leading idea is to reduce the problem to an abstract parabolic setting, which requires to introduce several auxiliary differential systems and a non-standard bilinear form. The main difficulties are due to the degeneracy of the bidomain system and to its non-standard coupling with a diffusion equation, accounting for the presence of the pathological zone in the heart tissue.

[1]  S. Yoshizawa,et al.  An Active Pulse Transmission Line Simulating Nerve Axon , 1962, Proceedings of the IRE.

[2]  E. Zeidler.  Nonlinear functional analysis and its applications , 1988 .

[3]  Jean-Frédéric Gerbeau,et al.  Etude mathématique et numérique de modèles issus du domaine biomédical , 2018 .

[4]  Marco Veneroni,et al.  Reaction–diffusion systems for the macroscopic bidomain model of the cardiac electric field , 2009 .

[5]  Clair Poignard,et al.  “Classical” Electropermeabilization Modeling at the Cell Scale , 2014, Journal of mathematical biology.

[6]  乔花玲,et al.  关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解 , 2003 .

[7]  Miguel A. Fernández,et al.  Mathematical Modeling of Electrocardiograms: A Numerical Study , 2010, Annals of Biomedical Engineering.

[8]  Piero Colli Franzone,et al.  Multiscale Modeling for the Bioelectric Activity of the Heart , 2005, SIAM J. Math. Anal..

[9]  C. Poignard,et al.  Modified bidomain model with passive periodic heterogeneities , 2020, Discrete & Continuous Dynamical Systems - S.

[10]  Kenneth H. Karlsen,et al.  Analysis of a class of degenerate reaction-diffusion systems and the bidomain model of cardiac tissue , 2006, Networks Heterog. Media.

[11]  Y. Bourgault,et al.  Existence and uniqueness of the solution for the bidomain model used in cardiac electrophysiology , 2009 .

[12]  W. Krassowska,et al.  Homogenization of syncytial tissues. , 1993, Critical reviews in biomedical engineering.

[13]  Kenneth H. Karlsen,et al.  The cardiac bidomain model and homogenization , 2018, Networks Heterog. Media.

[14]  Nejib Zemzemi,et al.  Theoretical and Numerical study of the electric activity of the heart. Modeling and Numerical simulation of electrocardiograms. , 2009 .

[15]  C. Jerez-Hanckes,et al.  Derivation of cable equation by multiscale analysis for a model of myelinated axons , 2018, Discrete & Continuous Dynamical Systems - B.

[16]  Miguel A. Fernández,et al.  Towards the Numerical Simulation of Electrocardiograms , 2007, FIMH.

[17]  Daniele Andreucci,et al.  Homogenization of a modified bidomain model involving imperfect transmission , 2021, Communications on Pure & Applied Analysis.

[18]  Miguel A. Fernández,et al.  A Coupled System of PDEs and ODEs Arising in Electrocardiograms Modeling , 2010 .

[19]  A. Collin,et al.  Mathematical analysis and 2-scale convergence of a heterogeneous microscopic bidomain model , 2018 .

[20]  A. Haraux,et al.  An Introduction to Semilinear Evolution Equations , 1999 .

[21]  Existence and uniqueness for an elliptic problem with evolution arising in electrodynamics , 2003 .

[22]  Bernd Eggers,et al.  Nonlinear Functional Analysis And Its Applications , 2016 .

[23]  Yves Coudière,et al.  The modified bidomain model with periodic diffusive inclusions , 2014, Computing in Cardiology 2014.

[24]  Daniele Andreucci,et al.  Existence and uniqueness for an elliptic problem with evolution arising in electrodynamics , 2005 .

[25]  Andjela Davidovic.  Multiscale mathematical modelling of structural heterogeneities in cardiac electrophysiology , 2016 .

[26]  H Zhang,et al.  Models of cardiac tissue electrophysiology: progress, challenges and open questions. , 2011, Progress in biophysics and molecular biology.

[27]  James P. Keener,et al.  Mathematical physiology , 1998 .