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Mathematical and numerical methods for reaction-diffusion models in electrocardiology

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Part of the book series: MS&A — Modeling, Simulation and Applications ((MS&A,volume 5))

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.

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Acknowledgements

The authors would like to thank Bruno Taccardi for introducing them to the field of Mathematical Physiology and for many stimulating discussions.

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  1. Department of Mathematics, University of Pavia, via Ferrata 1, 27100, Pavia, Italy

    Piero Colli-Franzone

  2. Department of Mathematics, University of Milano, via Saldini 50, 20133, Milano, Italy

    Luca F. Pavarino & Simone Scacchi

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  1. MOX - Department of Mathematics, Politecnico di Milano, Italy

    Davide Ambrosi  & Alfio Quarteroni  & 

  2. Mathematics Institute of Computational Science and Engineering, École Polytechnique Fédérale de Laussane, Switzerland

    Alfio Quarteroni  & Gianluigi Rozza  & 

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Colli-Franzone, P., Pavarino, L.F., Scacchi, S. (2012). Mathematical and numerical methods for reaction-diffusion models in electrocardiology. In: Ambrosi, D., Quarteroni, A., Rozza, G. (eds) Modeling of Physiological Flows. MS&A — Modeling, Simulation and Applications, vol 5. Springer, Milano. https://doi.org/10.1007/978-88-470-1935-5_5

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