Mathematical and numerical methods for reaction-diffusion models in electrocardiology
This paper presents a review of current mathematical and numerical models of the bioelectrical activity in the ventricular myocardium, describing cardiac cells excitability and the action-potential propagation in cardiac tissue. The degenerate reaction-di
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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 models of the bioelectrical activity in the ventricular myocardium, describing cardiac cells excitability and the action-potential propagation in cardiac tissue. The degenerate reaction-diffusion system called the Bidomain model is introduced and interpreted 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 considered, such as Monodomain and eikonal-curvature models. The main numerical methods for the Bidomain and Monodomain models are then reviewed. In particular, 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 excitation processes generated by virtual electrode response in 3D orthotropic blocks of myocardial tissue.
5.1 Introduction Electrocardiology deals with the investigation of intracardiac bioelectric phenomena and the evolution of cardiac potential fields at the body surface is one of the main purposes of Electrocardiology. Clinic Electrocardiography deals with the dePiero Colli-Franzone University of Pavia, Department of Mathematics, via Ferrata 1, 27100 Pavia, Italy e-mail: [email protected] Luca F. Pavarino University of Milano, Department of Mathematics, via Saldini 50, 20133 Milano, Italy e-mail: [email protected] Simone Scacchi University of Milano, Department of Mathematics, via Saldini 50, 20133 Milano, Italy e-mail: [email protected]
Ambrosi D., Quarteroni A., Rozza G. (Eds.): Modeling of Physiological Flows. DOI 10.1007/978-88-470-1935-5 5, © Springer-Verlag Italia 2012
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P. Colli-Franzone, L.F. Pavarino, and S. Scacchi
tection and interpretation of the morphology of the usual electrocardiograms (ECG) at a few points on the body surface or from the evolution of body surface maps (see [135, 136, 160] for a survey). The information content of ECGs and body maps is limited, due to the strong signal attenuation and smoothing associated with current conduction from heart to thorax; thus it is a difficult task to detect from these signals detailed information on pathological heart states. The scientific ground of Electrocardiology is the so called Forward Problem of Electrocardiology, that studies the bioelectric cardiac sources and conducting media in order to derive the potential field. Of considerable interest are also the so called Inverse Problems of Electrocardiography in terms of potentials or cardiac sources (see e.g. the reviews [54, 96, 112]). In this paper, we will focus on the modelling and simulation techniques for describing the bioelectrical activity in the myocardium at the macroscopic leve
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