A comparison of coupled and uncoupled solvers for the cardiac Bidomain model
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 47 (2013) no. 4, p. 1017-1035

The aim of this work is to compare a new uncoupled solver for the cardiac Bidomain model with a usual coupled solver. The Bidomain model describes the bioelectric activity of the cardiac tissue and consists of a system of a non-linear parabolic reaction-diffusion partial differential equation (PDE) and an elliptic linear PDE. This system models at macroscopic level the evolution of the transmembrane and extracellular electric potentials of the anisotropic cardiac tissue. The evolution equation is coupled through the non-linear reaction term with a stiff system of ordinary differential equations (ODEs), the so-called membrane model, describing the ionic currents through the cellular membrane. A novel uncoupled solver for the Bidomain system is here introduced, based on solving twice the parabolic PDE and once the elliptic PDE at each time step, and it is compared with a usual coupled solver. Three-dimensional numerical tests have been performed in order to show that the proposed uncoupled method has the same accuracy of the coupled strategy. Parallel numerical tests on structured meshes have also shown that the uncoupled technique is as scalable as the coupled one. Moreover, the conjugate gradient method preconditioned by Multilevel Hybrid Schwarz preconditioners converges faster for the linear systems deriving from the uncoupled method than from the coupled one. Finally, in all parallel numerical tests considered, the uncoupled technique proposed is always about two or three times faster than the coupled approach.

DOI : https://doi.org/10.1051/m2an/2012055
Classification:  65N55,  65M55,  65F10,  92C30
Keywords: operator splitting, multilevel preconditioners, parallel computing
@article{M2AN_2013__47_4_1017_0,
author = {Colli Franzone, P. and Pavarino, L. F. and Scacchi, S.},
title = {A comparison of coupled and uncoupled solvers for the cardiac Bidomain model},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {47},
number = {4},
year = {2013},
pages = {1017-1035},
doi = {10.1051/m2an/2012055},
mrnumber = {3082287},
language = {en},
url = {http://www.numdam.org/item/M2AN_2013__47_4_1017_0}
}

Colli Franzone, P.; Pavarino, L. F.; Scacchi, S. A comparison of coupled and uncoupled solvers for the cardiac Bidomain model. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 47 (2013) no. 4, pp. 1017-1035. doi : 10.1051/m2an/2012055. http://www.numdam.org/item/M2AN_2013__47_4_1017_0/

[1] T.M. Austin, M.L. Trew and A.J. Pullan, Solving the cardiac Bidomain equations for discontinuous conductivities. IEEE Trans. Biomed. Eng. 53 (2006) 1265-1272.

[2] S. Balay, K. Buschelman, W.D. Gropp, D. Kaushik, M. Knepley, L. Curfman Mcinnes, B.F. Smith and H. Zhang, PETSc Users Manual.Tech. Rep. ANL-95/11 - Revision 2.1.5, Argonne National Laboratory (2002).

[3] S. Balay, K. Buschelman, W.D. Gropp, D. Kaushik, M. Knepley, L. Curfman Mcinnes, B.F. Smith and H. Zhang, PETSc home page. http://www.mcs.anl.gov/petsc (2001).

[4] M. Boulakia, S. Cazeau, M.A. Fernandez, J.-F. Gerbeau and N. Zemzemi, Mathematical modeling of electrocardiograms: a numerical study. Ann. Biomed. Eng. 38 (2010) 1071-1097.

[5] R.H. Clayton, O. Bernus, E.M. Cherry, H. Dierckx, F.H. Fenton, L. Mirabella, A.V. Panfilov, F.B. Sachse, G. Seemann and H. Zhang, Models of cardiac tissue electrophysiology: Progress, challenges and open questions. Progr. Biophys. Molec. Biol. 104 (2011) 22-48.

[6] P. Colli Franzone and L.F. Pavarino, A parallel solver for reaction-diffusion systems in computational electrocardiology. Math. Mod. Meth. Appl. Sci. 14 (2004) 883-911. | MR 2069498 | Zbl 1068.92024

[7] P. Colli Franzone, L.F. Pavarino and S. Scacchi, Mathematical and numerical methods for reaction-diffusion models in electrocardiology, in Modeling of Physiological flows, edited by D. Ambrosi, A. Quarteroni and G. Rozza. Springer (2011) 107-142.

[8] P. Colli Franzone, L.F. Pavarino and B. Taccardi, Simulating patterns of excitation, repolarization and action potential duration with cardiac bidomain and monodomain models. Math. Biosci. 197 (2005) 33-66. | MR 2167484 | Zbl 1074.92004

[9] P. Colli Franzone, P. Deuflhard, B. Erdmann, J. Lang and L.F. Pavarino, Adaptivity in space and time for reaction-diffusion systems in Electrocardiology. SIAM J. Sci. Comput. 28 (2006) 942-962. | MR 2240798 | Zbl 1114.65110

[10] P. Deuflhard, B. Erdmann, R. Roitzsch and G.T. Lines, Adaptive finite element simulation of ventricular fibrillation dynamics. Comput. Visual. Sci. 12 (2009) 201-205. | MR 2507225

[11] M. Dryja, M.V. Sarkis and O.B. Widlund, Multilevel Schwarz methods for elliptic problems with discontinuous coefficients in three dimensions. Numer. Math. 72 (1996) 313-348. | MR 1367653 | Zbl 0857.65131

[12] M. Dryja and O.B. Widlund, Multilevel additive methods for elliptic finite element problems. Parallel algorithms for partial differential equations (Kiel 1990) Notes Numer. Fluid Mech. 31 (1991) 58-69. | MR 1167868 | Zbl 0783.65057

[13] M. Dryja and O.B. Widlund, Domain decomposition algorithms with small overlap. SIAM J. Sci. Comput. 15 (1994) 604-620. | MR 1273155 | Zbl 0802.65119

[14] M. Ethier and Y. Bourgault, Semi-implicit time-discretization schemes for the Bidomain model. SIAM J. Numer. Anal. 46 (2008) 2443-2468. | MR 2421042 | Zbl 1182.92009

[15] M.A. Fernandez and N. Zemzemi, Decoupled time-marching schemes in computational cardiac electrophysiology and ECG numerical simulation. Math. Biosci. 226 (2010) 58-75. | MR 2675617 | Zbl 1193.92024

[16] M. Fink, S.A. Niederer, E.M. Cherry, F.H. Fenton, J.T. Koivumaki, G. Seemann, T. Rudiger, H. Zhang, F.B. Sachse, D. Beard, E.J. Crampin and N.P. Smith, Cardiac cell modelling: observations from the heart of the cardiac physiome project. Prog. Biophys. Mol. Biol. 104 (2011) 2-21.

[17] L.G. Giorda, L. Mirabella, F. Nobile, M. Perego and A. Veneziani, A model-based block-triangular preconditioner for the Bidomain system in electrocardiology. J. Comput. Phys. 228 (2009) 3625-3639. | MR 2511070 | Zbl 1187.92053

[18] L. Gerardo Giorda, M. Perego and A. Veneziani, Optimized Schwarz coupling of Bidomain and Monodomain models in electrocardiology. Math. Model. Numer. Anal. 45 (2011) 309-334. | Numdam | MR 2804641 | Zbl 1274.92022

[19] I.J. Legrice, B.H. Smaill, L.Z. Chai, S.G. Edgar, J.B. Gavin and P.J. Hunter, Laminar structure of the heart: ventricular myocyte arrangement and connective tissue architecture in the dog. Amer. J. Physiol. Heart Circ. Physiol. 269 (1995) H571-H582.

[20] S. Linge, J. Sundnes, M. Hanslien, G.T. Lines and A. Tveito, Numerical solution of the bidomain equations. Philos. Trans. R. Soc. A 367 (2009) 1931-1950. | MR 2512073 | Zbl 1185.65169

[21] C. Luo and Y. Rudy, A model of the ventricular cardiac action potential: depolarization, repolarization, and their interaction. Circ. Res. 68 (1991) 1501-1526.

[22] K.-A. Mardal, B.F. Nielsen, X. Cai and A. Tveito, An order optimal solver for the discretized bidomain equations. Numer. Linear Algebra Appl. 14 (2007) 83-98. | MR 2292297 | Zbl 1199.65111

[23] G. Karypis and V. Kumar, MeTis: Unstructured Graph Partitioning and Sparse Matrix Ordering System, Version 4.0. http://www.cs.umn.edu/~metis/. University of Minnesota, Minneapolis, MN (2009).

[24] M. Munteanu and L.F. Pavarino, Decoupled Schwarz algorithms for implicit discretization of nonlinear Monodomain and Bidomain systems. Math. Mod. Meth. Appl. Sci. 19 (2009) 1065-1097. | MR 2553178 | Zbl 1178.65116

[25] M. Munteanu, L.F. Pavarino and S. Scacchi. A scalable Newton-Krylov-Schwarz method for the Bidomain reaction-diffusion system. SIAM J. Sci. Comput. 31 (2009) 3861-3883. | MR 2556566 | Zbl 1205.65261

[26] M. Murillo and X.-C. Cai, A fully implicit parallel algorithm for simulating the non-linear electrical activity of the heart. Numer. Linear Algebra Appl. 11 (2004) 261-277. | MR 2065816 | Zbl 1114.65112

[27] J.S. Neu and W. Krassowska, Homogenization of syncytial tissues. Crit. Rev. Biomed. Eng. 21 (1993) 137-199.

[28] P. Pathmanathan, M.O. Bernabeu, R. Bordas, J. Cooper, A. Garny, J.M. Pitt-Francis, J.P. Whiteley and D.J. Gavaghan, A numerical guide to the solution of the bidomain equations of cardiac electrophysiology. Progr. Biophys. Molec. Biol. 102 (2010) 136-155.

[29] L.F. Pavarino and S. Scacchi, Multilevel additive Schwarz preconditioners for the Bidomain reaction-diffusion system. SIAM J. Sci. Comput. 31 (2008) 420-443. | MR 2460784 | Zbl 1185.65179

[30] L.F. Pavarino and S. Scacchi, Parallel Multilevel Schwarz and Block Preconditioners for the Bidomain Parabolic-Parabolic and Parabolic-Elliptic Formulations. SIAM J. Sci. Comput. 33 (2011) 1897-1919. | MR 2831039 | Zbl 1233.65070

[31] M. Pennacchio, G. Savaré and P.C. Franzone. Multiscale modeling for the bioelectric activity of the heart. SIAM J. Math. Anal. 37 (2006) 1333-1370. | MR 2192297 | Zbl 1113.35019

[32] M. Pennacchio and V. Simoncini, Efficient algebraic solution of reaction-diffusion systems for the cardiac excitation process. J. Comput. Appl. Math. 145 (2002) 49-70. | MR 1914350 | Zbl 1006.65102

[33] M. Pennacchio and V. Simoncini, Algebraic multigrid preconditioners for the bidomain reaction-diffusion system. Appl. Numer. Math. 59 (2009) 3033-3050. | MR 2560833 | Zbl 1171.92017

[34] M. Pennacchio and V. Simoncini, Fast structured AMG preconditioning for the bidomain model in electrocardiology. SIAM J. Sci. Comput. 33 (2011) 721-745. | MR 2785969 | Zbl 1227.92034

[35] G. Plank, M. Liebmann, R. Weber dos Santos, E.J. Vigmond and G. Haase, Algebraic Multigrid Preconditioner for the Cardiac Bidomain Model. IEEE Trans. Biomed. Eng. 54 (2007) 585-596.

[36] M. Potse, B. Dubè, J. Richer, A. Vinet and R. Gulrajani, A comparison of Monodomain and Bidomain reaction-diffusion models for action potential propagation in the human heart. IEEE Trans. Biomed. Eng. 53 (2006) 2425-2434.

[37] P.-A. Raviart, The use of numerical integration in finite element methods for solving parabolic equations. In Topics in Numerical Analysis, edited by J.J.H. Miller. Academic Press (1973) 233-264. | MR 345428 | Zbl 0293.65086

[38] Z. Qu and A. Garfinkel, An advanced algorithm for solving partial differential equation in cardiac conduction. IEEE Trans. Biomed. Eng. 46 (1999) 1166-1168.

[39] A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations. Springer (1997). | MR 1299729 | Zbl 0803.65088

[40] S. Scacchi, A hybrid multilevel Schwarz method for the bidomain model. Comput. Methods Appl. Mech. Eng. 197 (2008) 4051-4061. | MR 2458128 | Zbl 1194.78048

[41] S. Scacchi, A multilevel hybrid Newton-Krylov-Schwarz method for the Bidomain model of electrocardiology. Comput. Methods Appl. Mech. Eng. 200 (2011) 717-725. | MR 2749030 | Zbl 1225.92011

[42] S. Scacchi, P. Colli Franzone, L.F. Pavarino and B. Taccardi, Computing cardiac recovery maps from electrograms and monophasic action potentials under heterogeneous and ischemic conditions. Math. Mod. Methods Appl. Sci. 20 (2010) 1089-1127. | MR 2673412 | Zbl 1194.92050

[43] K.B. Skouibine, N. Trayanova and P. Moore, A numerically efficient model for the simulation of defibrillation in an active bidomain sheet of myocardium. Math. Biosci. 166 (2000) 85-100. | Zbl 0963.92019

[44] B.F. Smith, P. Bjørstad and W.D. Gropp, Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations, Cambridge University Press (1996). | MR 1410757 | Zbl 0857.65126

[45] J.A. Southern, G. Plank, E.J. Vigmond and J.P. Whiteley, Solving the coupled system improves computational efficiency of the Bidomain equations. IEEE Trans. Biomed. Eng. 56 (2009) 2404-2412.

[46] J. Sundnes, G.T. Lines, K.A. Mardal and A. Tveito, Multigrid block preconditioning for a coupled system of partial differential equations modeling the electrical activity in the heart. Comput. Methods Biomech. Biomed. Eng. 5 (2002) 397-409.

[47] J. Sundnes, G.T. Lines and A. Tveito, An operator splitting method for solving the bidomain equations coupled to a volume conductor model for the torso. Math. Biosci. 194 (2005) 233-248. | MR 2142490 | Zbl 1063.92018

[48] H. Si, http://tetgen.berlios.de/. Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany.

[49] V. Thomée, Galerkin Finite Element Methods for Parabolic Problems. Springer (1997). | MR 1479170 | Zbl 1105.65102

[50] A. Toselli and O.B. Widlund, Domain Decomposition Methods: Algorithms and Theory. Comput. Math. Springer-Verlag, Berlin 34 (2004). | Zbl 1069.65138

[51] J.A. Trangenstein and C. Kim, Operator splitting and adaptive mesh refinement for the Luo-Rudy I model. J. Comput. Phys. 196 (2004) 645-679. | MR 2054352 | Zbl 1056.92014

[52] E.J. Vigmond, F. Aguel and N.A. Trayanova, Computational techniques for solving the bidomain equations in three dimensions. IEEE Trans. Biomed. Eng. 49 (2002) 1260-1269.

[53] E.J. Vigmond, R. Weber dos Santos, A.J. Prassl, M. Deo and G. Plank, Solvers for the cardiac bidomain equations. Progr. Biophys. Molec. Biol. 96 (2008) 3-18.

[54] R. Weber dos Santos, G. Plank, S. Bauer and E.J. Vigmond, Parallel multigrid preconditioner for the cardiac bidomain model. IEEE Trans. Biomed. Eng. 51 (2004) 1960-1968.

[55] J.P. Whiteley, An efficient numerical technique for the solution of the monodomain and bidomain equations. IEEE Trans. Biomed. Eng. 53 (2006) 2139-2147.

[56] M. Zaniboni, 3D current-voltage-time surfaces unveil critical repolarization differences underlying similar cardiac action potentials: A model study. Math. Biosci. 233 (2011) 98-110. | MR 2856847 | Zbl 1226.92013

[57] X. Zhang, Multilevel Schwarz methods. Numer. Math. 63 (1992) 521-539. | MR 1189535 | Zbl 0796.65129