Skip to main content Accessibility help

A comparison of coupled and uncoupled solvers for the cardiac Bidomain model∗∗

  • P. Colli Franzone (a1), L. F. Pavarino (a2) and S. Scacchi (a2)


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.



Hide All
[1] Austin, T.M., Trew, M.L. and Pullan, A.J., Solving the cardiac Bidomain equations for discontinuous conductivities. IEEE Trans. Biomed. Eng. 53 (2006) 12651272.
[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. (2001).
[4] Boulakia, M., Cazeau, S., Fernandez, M.A., Gerbeau, J.-F. and Zemzemi, N., Mathematical modeling of electrocardiograms: a numerical study. Ann. Biomed. Eng. 38 (2010) 10711097.
[5] Clayton, R.H., Bernus, O., Cherry, E.M., Dierckx, H., Fenton, F.H., Mirabella, L., Panfilov, A.V., Sachse, F.B., Seemann, G. and Zhang, H., Models of cardiac tissue electrophysiology: Progress, challenges and open questions. Progr. Biophys. Molec. Biol. 104 (2011) 2248.
[6] Colli Franzone, P. and Pavarino, L.F., A parallel solver for reaction-diffusion systems in computational electrocardiology. Math. Mod. Meth. Appl. Sci. 14 (2004) 883911.
[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] Colli Franzone, P., Pavarino, L.F. and Taccardi, B., Simulating patterns of excitation, repolarization and action potential duration with cardiac bidomain and monodomain models. Math. Biosci. 197 (2005) 3366.
[9] Colli Franzone, P., Deuflhard, P., Erdmann, B., Lang, J. and Pavarino, L.F., Adaptivity in space and time for reaction-diffusion systems in Electrocardiology. SIAM J. Sci. Comput. 28 (2006) 942962.
[10] Deuflhard, P., Erdmann, B., Roitzsch, R. and Lines, G.T., Adaptive finite element simulation of ventricular fibrillation dynamics. Comput. Visual. Sci. 12 (2009) 201205.
[11] Dryja, M., Sarkis, M.V. and Widlund, O.B., Multilevel Schwarz methods for elliptic problems with discontinuous coefficients in three dimensions. Numer. Math. 72 (1996) 313348.
[12] Dryja, M. and Widlund, O.B., Multilevel additive methods for elliptic finite element problems. Parallel algorithms for partial differential equations (Kiel 1990) Notes Numer. Fluid Mech. 31 (1991) 5869.
[13] Dryja, M. and Widlund, O.B., Domain decomposition algorithms with small overlap. SIAM J. Sci. Comput. 15 (1994) 604620.
[14] Ethier, M. and Bourgault, Y., Semi-implicit time-discretization schemes for the Bidomain model. SIAM J. Numer. Anal. 46 (2008) 24432468.
[15] Fernandez, M.A. and Zemzemi, N., Decoupled time–marching schemes in computational cardiac electrophysiology and ECG numerical simulation. Math. Biosci. 226 (2010) 5875.
[16] Fink, M., Niederer, S.A., Cherry, E.M., Fenton, F.H., Koivumaki, J.T., Seemann, G., Rudiger, T., Zhang, H., Sachse, F.B., Beard, D., Crampin, E.J. and Smith, N.P., Cardiac cell modelling: observations from the heart of the cardiac physiome project. Prog. Biophys. Mol. Biol. 104 (2011) 221.
[17] Giorda, L.G., Mirabella, L., Nobile, F., Perego, M. and Veneziani, A., A model-based block-triangular preconditioner for the Bidomain system in electrocardiology. J. Comput. Phys. 228 (2009) 36253639.
[18] Gerardo Giorda, L., Perego, M. and Veneziani, A., Optimized Schwarz coupling of Bidomain and Monodomain models in electrocardiology. Math. Model. Numer. Anal. 45 (2011) 309334.
[19] LeGrice, I.J., Smaill, B.H., Chai, L.Z., Edgar, S.G., Gavin, J.B. and Hunter, P.J., Laminar structure of the heart: ventricular myocyte arrangement and connective tissue architecture in the dog. Amer. J. Physiol. Heart Circ. Physiol. 269 (1995) H571H582.
[20] Linge, S., Sundnes, J., Hanslien, M., Lines, G.T. and Tveito, A., Numerical solution of the bidomain equations. Philos. Trans. R. Soc. A 367 (2009) 19311950.
[21] Luo, C. and Rudy, Y., A model of the ventricular cardiac action potential: depolarization, repolarization, and their interaction. Circ. Res. 68 (1991) 15011526.
[22] Mardal, K.-A., Nielsen, B.F., Cai, X. and Tveito, A., An order optimal solver for the discretized bidomain equations. Numer. Linear Algebra Appl. 14 (2007) 8398.
[23] G. Karypis and V. Kumar, MeTis: Unstructured Graph Partitioning and Sparse Matrix Ordering System, Version 4.0. University of Minnesota, Minneapolis, MN (2009).
[24] Munteanu, M. and Pavarino, L.F., Decoupled Schwarz algorithms for implicit discretization of nonlinear Monodomain and Bidomain systems. Math. Mod. Meth. Appl. Sci. 19 (2009) 10651097.
[25] Munteanu, M., Pavarino, L.F. and Scacchi, S.. A scalable Newton-Krylov-Schwarz method for the Bidomain reaction-diffusion system. SIAM J. Sci. Comput. 31 (2009) 38613883.
[26] Murillo, M. and Cai, X.-C., A fully implicit parallel algorithm for simulating the non-linear electrical activity of the heart. Numer. Linear Algebra Appl. 11 (2004) 261277.
[27] Neu, J.S. and Krassowska, W., Homogenization of syncytial tissues. Crit. Rev. Biomed. Eng. 21 (1993) 137199.
[28] Pathmanathan, P., Bernabeu, M.O., Bordas, R., Cooper, J., Garny, A., Pitt-Francis, J.M., Whiteley, J.P. and Gavaghan, D.J., A numerical guide to the solution of the bidomain equations of cardiac electrophysiology. Progr. Biophys. Molec. Biol. 102 (2010) 136155.
[29] Pavarino, L.F. and Scacchi, S., Multilevel additive Schwarz preconditioners for the Bidomain reaction-diffusion system. SIAM J. Sci. Comput. 31 (2008) 420443.
[30] Pavarino, L.F. and Scacchi, S., Parallel Multilevel Schwarz and Block Preconditioners for the Bidomain Parabolic-Parabolic and Parabolic-Elliptic Formulations. SIAM J. Sci. Comput. 33 (2011) 18971919.
[31] Pennacchio, M., Savaré, G. and Franzone, P.C.. Multiscale modeling for the bioelectric activity of the heart. SIAM J. Math. Anal. 37 (2006) 13331370.
[32] Pennacchio, M. and Simoncini, V., Efficient algebraic solution of reaction-diffusion systems for the cardiac excitation process. J. Comput. Appl. Math. 145 (2002) 4970.
[33] Pennacchio, M. and Simoncini, V., Algebraic multigrid preconditioners for the bidomain reaction-diffusion system. Appl. Numer. Math. 59 (2009) 30333050.
[34] Pennacchio, M. and Simoncini, V., Fast structured AMG preconditioning for the bidomain model in electrocardiology. SIAM J. Sci. Comput. 33 (2011) 721745.
[35] Plank, G., Liebmann, M., Weber dos Santos, R., Vigmond, E.J. and Haase, G., Algebraic Multigrid Preconditioner for the Cardiac Bidomain Model. IEEE Trans. Biomed. Eng. 54 (2007) 585596.
[36] Potse, M., Dubè, B., Richer, J., Vinet, A. and Gulrajani, R., A comparison of Monodomain and Bidomain reaction–diffusion models for action potential propagation in the human heart. IEEE Trans. Biomed. Eng. 53 (2006) 24252434.
[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.
[38] Qu, Z. and Garfinkel, A., An advanced algorithm for solving partial differential equation in cardiac conduction. IEEE Trans. Biomed. Eng. 46 (1999) 11661168.
[39] A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations. Springer (1997).
[40] Scacchi, S., A hybrid multilevel Schwarz method for the bidomain model. Comput. Methods Appl. Mech. Eng. 197 (2008) 40514061.
[41] Scacchi, S., A multilevel hybrid Newton-Krylov-Schwarz method for the Bidomain model of electrocardiology. Comput. Methods Appl. Mech. Eng. 200 (2011) 717725.
[42] Scacchi, S., Colli Franzone, P., Pavarino, L.F. and Taccardi, B., Computing cardiac recovery maps from electrograms and monophasic action potentials under heterogeneous and ischemic conditions. Math. Mod. Methods Appl. Sci. 20 (2010) 10891127.
[43] Skouibine, K.B., Trayanova, N. and Moore, P., A numerically efficient model for the simulation of defibrillation in an active bidomain sheet of myocardium. Math. Biosci. 166 (2000) 85100.
[44] B.F. Smith, P. Bjørstad and W.D. Gropp, Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations, Cambridge University Press (1996).
[45] Southern, J.A., Plank, G., Vigmond, E.J. and Whiteley, J.P., Solving the coupled system improves computational efficiency of the Bidomain equations. IEEE Trans. Biomed. Eng. 56 (2009) 24042412.
[46] Sundnes, J., Lines, G.T., Mardal, K.A. and Tveito, A., 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) 397409.
[47] Sundnes, J., Lines, G.T. and Tveito, A., An operator splitting method for solving the bidomain equations coupled to a volume conductor model for the torso. Math. Biosci. 194 (2005) 233248.
[48] H. Si, Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany.
[49] V. Thomée, Galerkin Finite Element Methods for Parabolic Problems. Springer (1997).
[50] A. Toselli and O.B. Widlund, Domain Decomposition Methods: Algorithms and Theory. Comput. Math. Springer-Verlag, Berlin 34 (2004).
[51] Trangenstein, J.A. and Kim, C., Operator splitting and adaptive mesh refinement for the Luo-Rudy I model. J. Comput. Phys. 196 (2004) 645679.
[52] Vigmond, E.J., Aguel, F. and Trayanova, N.A., Computational techniques for solving the bidomain equations in three dimensions. IEEE Trans. Biomed. Eng. 49 (2002) 12601269.
[53] Vigmond, E.J., Weber dos Santos, R., Prassl, A.J., Deo, M. and Plank, G., Solvers for the cardiac bidomain equations. Progr. Biophys. Molec. Biol. 96 (2008) 318.
[54] Weber dos Santos, R., Plank, G., Bauer, S. and Vigmond, E.J., Parallel multigrid preconditioner for the cardiac bidomain model. IEEE Trans. Biomed. Eng. 51 (2004) 19601968.
[55] Whiteley, J.P., An efficient numerical technique for the solution of the monodomain and bidomain equations. IEEE Trans. Biomed. Eng. 53 (2006) 21392147.
[56] Zaniboni, M., 3D current-voltage-time surfaces unveil critical repolarization differences underlying similar cardiac action potentials: A model study. Math. Biosci. 233 (2011) 98110.
[57] Zhang, X., Multilevel Schwarz methods. Numer. Math. 63 (1992) 521539.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

ESAIM: Mathematical Modelling and Numerical Analysis
  • ISSN: 0764-583X
  • EISSN: 1290-3841
  • URL: /core/journals/esaim-mathematical-modelling-and-numerical-analysis
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed