Biomedical Engineering Reference
In-Depth Information
5
Mathematical and numerical methods
for reaction-diffusion models
in electrocardiology
Piero Colli-Franzone, Luca F. Pavarino, and Simone Scacchi
Abstract.
This paper presents a review of current mathematical and numerical mod-
els of the bioelectrical activity in the ventricular myocardium, describing cardiac
cells excitability and the action-potential propagation in cardiac tissue. The degen-
erate reaction-diffusion system called the Bidomain model is introduced and inter-
preted as macroscopic averaging of a cellular model on a periodic assembling of
myocytes. The main theoretical results for the cellular and Bidomain models are
given. Various approximate models based on some relaxed approaches are also con-
sidered, such as Monodomain and eikonal-curvature models. The main numerical
methods for the Bidomain and Monodomain models are then reviewed. In particu-
lar, we focus on isoparametric finite elements, semi-implicit time discretizations and
a parallel iterative solver based on a multilevel Schwarz preconditioned conjugate
gradient method. The Bidomain solver is finally applied to the study of the excita-
tion processes generated by virtual electrode response in 3D orthotropic blocks of
myocardial tissue.
5.1 Introduction
Electrocardiology deals with the investigation of intracardiac bioelectric phenom-
ena and the evolution of cardiac potential fields at the body surface is one of the
main purposes of Electrocardiology. Clinic Electrocardiography deals with the de-
Piero Colli-Franzone
University of Pavia, Department of Mathematics, via Ferrata 1, 27100 Pavia, Italy
e-mail: colli@imati.cnr.it
Luca F. Pavarino
University of Milano, Department of Mathematics, via Saldini 50, 20133 Milano, Italy
e-mail: luca.pavarino@unimi.it
Simone Scacchi
University of Milano, Department of Mathematics, via Saldini 50, 20133 Milano, Italy
e-mail: simone.scacchi@unimi.it
