A coupled system of PDEs and ODEs arising in electrocardiograms modelling

نویسندگان

  • Muriel Boulakia
  • Jean-Frédéric Gerbeau
  • Nejib Zemzemi
  • Miguel A. Fernández
چکیده

We study the well-posedness of a coupled system of PDEs and ODEs arising in the numerical simulation of electrocardiograms. It consists of a system of degenerate reaction-diffusion equations, the so-called bidomain equations, governing the electrical activity of the heart, and a diffusion equation governing the potential in the surrounding tissues. Global existence of weak solutions is proved for an abstract class of ionic models including MitchellSchaeffer, FitzHugh-Nagumo, Aliev-Panfilov and MacCulloch. Uniqueness is proved in the case of the FitzHugh-Nagumo ionic model. The proof is based on a regularisation argument with a Faedo-Galerkin/compactness procedure. Key-words: Bidomain model, reaction-diffusion system, electrocardiograms modelling ∗ Université Pierre et Marie Curie Paris 6, Laboratoire Jacques-Louis Lions, REO team; e-mail: [email protected] † INRIA, REO team; e-mail: [email protected] ‡ INRIA, REO team; e-mail: [email protected] § INRIA, REO team & Université Paris 11, Laboratoire de mathématiques d’Orsay; e-mail: [email protected] Etude d’un système couplé d’EDO et EDP intervenant dans la modélisation d’éléctrocardiogrammes Résumé : Nous étudions le caractère bien posé d’un système d’équations intervenant dans la simulation numérique d’électrocardiogrammes. Ce système, qui couple des équations aux dérivées ordinaires et des équations aux dérivées partielles, est constitué d’équations de réaction-diffusion dégénérées, les équations bidomaine qui modélisent l’activité électrique du coeur, et d’une équation de diffusion qui modélise l’activité électrique du tissu environnant (le thorax). L’existence globale de solutions faibles est obtenue pour un ensemble de modèles ioniques, dont les modèles de Mitchell-Schaeffer, FitzHugh-Nagumo, Aliev-Panfilov et MacCulloch. L’unicité est prouvée pour le modèle de FitzHugh-Nagumo. La preuve s’appuie sur un argument de régularisation et la méthode d’approximation de Faedo-Galerkin, combinés avec des résultats de compacité. Mots-clés : Modèle bidomaine, système de reaction-diffusion, modélisation de l’électrocardiogramme A coupled system of PDEs and ODEs arising in ECGs modelling 3

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تاریخ انتشار 2007