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) no. 4, pp. 987-1016.

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.

DOI : https://doi.org/10.1051/m2an/2012054
Classification : 34C15,  35B40,  78A70,  92C50
Mots clés : asymptotic analysis, cardiac electrophysiology, Mitchell−Schaeffer model
     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},
     pages = {987--1016},
     publisher = {EDP-Sciences},
     volume = {47},
     number = {4},
     year = {2013},
     doi = {10.1051/m2an/2012054},
     mrnumber = {3082286},
     language = {en},
     url = {www.numdam.org/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) no. 4, pp. 987-1016. doi : 10.1051/m2an/2012054. http://www.numdam.org/item/M2AN_2013__47_4_987_0/

[1] R.R. Aliev and A.V. Panfilov, A simple two-variable model of cardiac excitation. Chaos Soliton. Fract. 7 (1996) 293-301.

[2] M. Beck, C.K.R.T. Jones, D. Schaeffer and M. Wechselberger, Electrical Waves in a One-Dimensional Model of Cardiac Tissue. SIAM J. Appl. Dynam. Syst. 7 (2008) 1558-1581. | MR 2470977 | Zbl 1167.34364

[3] G. W. Beeler and H. Reuter, Reconstruction of the action potential of ventricular myocardial fibres. J. Physiol. 268 (1977) 177-210.

[4] M. Boulakia, M. Fernàndez, J.-F. Gerbeau and N. Zemzemi, Towards the numerical simulation of electrocardiograms, in Functional Imaging and Modeling of the Heart, vol. 4466 of Lect. Notes Comput. Sci., edited by F. Sachse and G. Seemann. Springer, Berlin/Heidelberg (2007) 240-249. | MR 2404069

[5] N. F. Britton, Essential Mathematical Biology. Springer Undergrad. Math. Series (2005). | Zbl 1037.92001

[6] J.W. Cain, Taking math to the heart: Mathematical challenges in cardiac electrophysiology. Notices of the AMS 58 (2011) 542-549. | MR 2807520 | Zbl 1233.37057

[7] R.H. Clayton and A.V. Panfilov, A guide to modelling cardiac electrical activity in anatomically detailed ventricles. Prog. Biophys. Mol. Bio. 96 (2008) 19-43.

[8] P. Colli Franzone, L. Guerri and S. Rovida, Wavefront propagation in an activation model of the anisotropic cardiac tissue: asymptotic analysis and numerical simulations. J. Math. Biol. 28 (1990) 121-176. DOI: 10.1007/BF00163143. | MR 1042483 | Zbl 0733.92006

[9] B. Deng, The existence of infinitely many traveling front and back waves in the Fitzhugh - Nagumo equations. SIAM J. Math. Anal. 22 (1991) 1631-1650. | MR 1129402 | Zbl 0752.35025

[10] K. Djabella, M. Landau and M. Sorine, A two-variable model of cardiac action potential with controlled pacemaker activity and ionic current interpretation. 46th IEEE Conf. Decis. Control (2007) 5186-5191.

[11] E.G. Tolkacheva, D.G. Schaeffer, D.J. Gauthier and C.C. Mitchell, Analysis of the Fenton-Karma model through an approximation by a one-dimensional map. Chaos 12 (2002) 1034-1042 . | MR 1946777 | Zbl 1080.92509

[12] F. Fenton and A. Karma, Vortex dynamics in three-dimensional continuous myocardium with fiber rotation: Filament instability and fibrillation. Chaos 8 (1998) 20-47. | Zbl 1069.92503

[13] R.A. Fitzhugh, Impulses and physiological states in theoretical models of nerve membrane. Biophys. J. 1 (1961) 445-466.

[14] S. Hastings, Single and multiple pulse waves for the Fitzhugh-Nagumo equations. SIAM J. Appl. Math. 42 (1982) 247-260. | MR 650220 | Zbl 0503.92009

[15] A.L. Hodgkin and A.F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 117 (1952) 500-544.

[16] J. Keener and J. Sneyd, Mathematical Physiology. Springer (2004). | MR 1673204 | Zbl 1273.92017

[17] J.P. Keener, Modeling electrical activity of cardiac cells, Two variable models, Mitchell-Schaeffer revised. Available at www.math.utah.edu/˜keener/lectures/ionic_models/Two_variable_models.

[18] J.P. Keener, An eikonal-curvature equation for action potential propagation in myocardium. J. Math. Biol. 29 (1991) 629-651. DOI: 10.1007/BF00163916. | MR 1119208 | Zbl 0744.92015

[19] K.H. Ten Tusscher, D. Noble, P.J. Noble and A.V. Panfilov, A model for human ventricular tissue. Am. J. Physiol. Heart Circ. Physiol. 286 (2004) H1973-H1589.

[20] R. Killmann, P. Wach and F. Dienstl, Three-dimensional computer model of the entire human heart for simulation of reentry and tachycardia: gap phenomenon and Wolff-Parkinson-White syndrome. Basic Res. Cardiol. 86 (1991) 485-501.

[21] C.H. Luo and Y. Rudy, A dynamic model of the cardiac ventricular action potential: I. simulations of ionic currents and concentration changes. Circ. Res. 74 (1994) 1071-1096.

[22] C. Mitchell and D. Schaeffer, A two-current model for the dynamics of cardiac membrane. Bull. Math. Bio. 65 (2003) 767-793.

[23] B.R. Munson, D.F. Young and T.H. Okiishi, Fundamentals of Fluid Mechanics. Wiley and Sons (2001). | Zbl 0747.76001

[24] J. Nagumo, S. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axon. Proc. IRE. 50 (1962) 2061-2070.

[25] D. Noble, A modification of the Hodgkin-Huxley equations applicable to purkinje fibre action and pacemaker potentials. J. Physiol. 160 (1962) 317-352.

[26] C. Pierre, Modélisation et simulation de l'activité électrique du coeur dans le thorax, analyse numérique et méthodes de volumes finis. PhD thesis, University of Nantes (2005).

[27] J. Relan, M. Sermesant, H. Delingette, M. Pop, G.A. Wright and N. Ayache, Quantitative comparison of two cardiac electrophysiology models using personalisation to optical and mr data, in Proc. Sixth IEEE Int. Symp. Biomed. Imaging 2009 (ISBI'09).

[28] J. Relan, M. Sermesant, M. Pop, H. Delingette, M. Sorine, G.A. Wright and N. Ayache, Parameter estimation of a 3d cardiac electrophysiology model including the restitution curve using optical and MR data, in World Congr. on Med. Phys. and Biomed. Eng., WC 2009, München (2009).

[29] D. Schaeffer, J. Cain, D. Gauthier, S. Kalb, R. Oliver, E. Tolkacheva, W. Ying and W. Krassowska, An ionically based mapping model with memory for cardiac restitution. Bull. Math. Bio. 69 (2007) 459-482. DOI: 10.1007/s11538-006-9116-6. | MR 2320707 | Zbl 1139.92309

[30] D. Schaeffer, W. Ying and X. Zhao, Asymptotic approximation of an ionic model for cardiac restitution. Nonlinear Dyn. 51 (2008) 189-198. DOI: 10.1007/s11071-007-9202-9. | Zbl 1169.92014

[31] M. Sermesant, Y. Coudière, V. Moreau Villéger, K.S. Rhode, D.L.G. Hill and R. Ravazi, A fast-marching approach to cardiac electrophysiology simulation for XMR interventional imaging, in Proc. of MICCAI'05, vol. 3750 of Lect. Notes Comput. Sci., Palm Springs, California. Springer Verlag (2005) 607-615.

[32] J. Sundnes, G.T. Lines, X. Cai, B.F. Nielsen, K.-A. Mardal and A. Tveito, Computing the Electrical Activity in the Heart. Springer, Monogr. Comput. Sci. Eng. 1 (2006). | MR 2258456 | Zbl 1182.92020