Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-10T15:35:57.824Z Has data issue: false hasContentIssue false
  • English
  • Français

A comparison of coupled and uncoupled solversfor the cardiac Bidomain model∗∗∗

Published online by Cambridge University Press:  07 June 2013

P. Colli Franzone ,
L. F. Pavarino  and
S. Scacchi
P. Colli Franzone
Affiliation:
Dipartimento di Matematica, Università di Pavia, Via Ferrata, 27100 Pavia, Italy.. colli@imati.cnr.it
L. F. Pavarino
Affiliation:
Dipartimento di Matematica, Università di Milano, Via Saldini 50, 20133 Milano, Italy.; luca.pavarino@unimi.it; simone.scacchi@unimi.it
S. Scacchi
Affiliation:
Dipartimento di Matematica, Università di Milano, Via Saldini 50, 20133 Milano, Italy.; luca.pavarino@unimi.it; simone.scacchi@unimi.it
Get access

Abstract

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.

Keywords

Operator splittingmultilevel preconditionersparallel computing
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 , Issue 4: Direct and inverse modeling of the cardiovascular and respiratory systems , July 2013 , pp. 1017 - 1035
Copyright
© EDP Sciences, SMAI, 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Austin, T.M., Trew, M.L. and Pullan, A.J., Solving the cardiac Bidomain equations for discontinuous conductivities. IEEE Trans. Biomed. Eng. 53 (2006) 1265–1272. Google ScholarPubMed
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).
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).
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) 1071–1097. Google ScholarPubMed
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) 22–48. Google ScholarPubMed
Colli Franzone, P. and Pavarino, L.F., A parallel solver for reaction-diffusion systems in computational electrocardiology. Math. Mod. Meth. Appl. Sci. 14 (2004) 883–911. Google Scholar
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.
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) 33–66. Google ScholarPubMed
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) 942–962. Google Scholar
Deuflhard, P., Erdmann, B., Roitzsch, R. and Lines, G.T., Adaptive finite element simulation of ventricular fibrillation dynamics. Comput. Visual. Sci. 12 (2009) 201–205. Google Scholar
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) 313–348. Google Scholar
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) 58–69. Google Scholar
Dryja, M. and Widlund, O.B., Domain decomposition algorithms with small overlap. SIAM J. Sci. Comput. 15 (1994) 604–620. Google Scholar
Ethier, M. and Bourgault, Y., Semi-implicit time-discretization schemes for the Bidomain model. SIAM J. Numer. Anal. 46 (2008) 2443–2468. Google Scholar
Fernandez, M.A. and Zemzemi, N., Decoupled time–marching schemes in computational cardiac electrophysiology and ECG numerical simulation. Math. Biosci. 226 (2010) 58–75. Google ScholarPubMed
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) 2–21. Google ScholarPubMed
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) 3625–3639. Google Scholar
Gerardo Giorda, L., Perego, M. and Veneziani, A., Optimized Schwarz coupling of Bidomain and Monodomain models in electrocardiology. Math. Model. Numer. Anal. 45 (2011) 309–334. Google Scholar
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) H571–H582. Google ScholarPubMed
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) 1931–1950. Google ScholarPubMed
Luo, C. and Rudy, Y., A model of the ventricular cardiac action potential: depolarization, repolarization, and their interaction. Circ. Res. 68 (1991) 1501–1526. Google ScholarPubMed
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) 83–98. Google Scholar
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).
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) 1065–1097. Google Scholar
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) 3861–3883. Google Scholar
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) 261–277. Google Scholar
Neu, J.S. and Krassowska, W., Homogenization of syncytial tissues. Crit. Rev. Biomed. Eng. 21 (1993) 137–199. Google ScholarPubMed
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) 136–155. Google Scholar
Pavarino, L.F. and Scacchi, S., Multilevel additive Schwarz preconditioners for the Bidomain reaction-diffusion system. SIAM J. Sci. Comput. 31 (2008) 420–443. Google Scholar
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) 1897–1919. Google Scholar
Pennacchio, M., Savaré, G. and Franzone, P.C.. Multiscale modeling for the bioelectric activity of the heart. SIAM J. Math. Anal. 37 (2006) 1333–1370. Google Scholar
Pennacchio, M. and Simoncini, V., Efficient algebraic solution of reaction-diffusion systems for the cardiac excitation process. J. Comput. Appl. Math. 145 (2002) 49–70. Google Scholar
Pennacchio, M. and Simoncini, V., Algebraic multigrid preconditioners for the bidomain reaction-diffusion system. Appl. Numer. Math. 59 (2009) 3033–3050. Google Scholar
Pennacchio, M. and Simoncini, V., Fast structured AMG preconditioning for the bidomain model in electrocardiology. SIAM J. Sci. Comput. 33 (2011) 721–745. Google Scholar
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) 585–596. Google ScholarPubMed
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) 2425–2434. Google ScholarPubMed
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.
Qu, Z. and Garfinkel, A., An advanced algorithm for solving partial differential equation in cardiac conduction. IEEE Trans. Biomed. Eng. 46 (1999) 1166–1168. Google ScholarPubMed
A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations. Springer (1997).
Scacchi, S., A hybrid multilevel Schwarz method for the bidomain model. Comput. Methods Appl. Mech. Eng. 197 (2008) 4051–4061. Google Scholar
Scacchi, S., A multilevel hybrid Newton-Krylov-Schwarz method for the Bidomain model of electrocardiology. Comput. Methods Appl. Mech. Eng. 200 (2011) 717–725. Google Scholar
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) 1089–1127. Google Scholar
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) 85–100. Google Scholar
B.F. Smith, P. Bjørstad and W.D. Gropp, Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations, Cambridge University Press (1996).
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) 2404–2412. Google ScholarPubMed
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) 397–409. Google ScholarPubMed
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) 233–248. Google ScholarPubMed
H. Si, http://tetgen.berlios.de/. Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany.
V. Thomée, Galerkin Finite Element Methods for Parabolic Problems. Springer (1997).
A. Toselli and O.B. Widlund, Domain Decomposition Methods: Algorithms and Theory. Comput. Math. Springer-Verlag, Berlin 34 (2004).
Trangenstein, J.A. and Kim, C., Operator splitting and adaptive mesh refinement for the Luo-Rudy I model. J. Comput. Phys. 196 (2004) 645–679. Google Scholar
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) 1260–1269. Google ScholarPubMed
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) 3–18. Google ScholarPubMed
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) 1960–1968. Google ScholarPubMed
Whiteley, J.P., An efficient numerical technique for the solution of the monodomain and bidomain equations. IEEE Trans. Biomed. Eng. 53 (2006) 2139–2147. Google ScholarPubMed
Zaniboni, M., 3D current-voltage-time surfaces unveil critical repolarization differences underlying similar cardiac action potentials: A model study. Math. Biosci. 233 (2011) 98–110. Google ScholarPubMed
Zhang, X., Multilevel Schwarz methods. Numer. Math. 63 (1992) 521–539. Google Scholar