One of the current debate about simulating the electrical activity in the heart is the following: Using a realistic anatomical setting, i.e. realistic geometries, fibres orientations, etc., is it enough to use a simplified 2-variable phenomenological model to reproduce the main characteristics of the cardiac action potential propagation, and in what sense is it sufficient? Using a combination of dimensional and asymptotic analysis, together with the well-known Mitchell - Schaeffer model, it is shown that it is possible to accurately control (at least locally) the solution while spatial propagation is involved. In particular, we reduce the set of parameters by writing the bidomain model in a new nondimensional form. The parameters of the bidomain model with Mitchell - Schaeffer ion kinetics are then set and shown to be in one-to-one relation with the main characteristics of the four phases of a propagated action potential. Explicit relations are derived using a combination of asymptotic methods and ansatz. These relations are tested against numerical results. We illustrate how these relations can be used to recover the time/space scales and speed of the action potential in various regions of the heart.
@article{M2AN_2013__47_4_987_0,
author = {Rioux, M. and Bourgault, Y.},
title = {A predictive method allowing the use of a single ionic model in numerical cardiac electrophysiology},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
volume = {47},
year = {2013},
pages = {987-1016},
doi = {10.1051/m2an/2012054},
mrnumber = {3082286},
language = {en},
url = {http://dml.mathdoc.fr/item/M2AN_2013__47_4_987_0}
}
Rioux, M.; Bourgault, Y. A predictive method allowing the use of a single ionic model in numerical cardiac electrophysiology. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 47 (2013) pp. 987-1016. doi : 10.1051/m2an/2012054. http://gdmltest.u-ga.fr/item/M2AN_2013__47_4_987_0/
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