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Optimized Schwarz Methods for the Bidomain system in electrocardiology

Published online by Cambridge University Press:  18 January 2013

Luca Gerardo-Giorda  and
Mauro Perego
Luca Gerardo-Giorda
Affiliation:
BCAM, Basque Center for Applied Mathematics, Bilbao, Spain.. lgerardo@bcamath.org
Mauro Perego
Affiliation:
Dept. of Scientific Computing, The Florida State University, Thallahassee, FL, USA.; mperego@fsu.edu
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Abstract

The propagation of the action potential in the heart chambers is accurately described by the Bidomain model, which is commonly accepted and used in the specialistic literature. However, its mathematical structure of a degenerate parabolic system entails high computational costs in the numerical solution of the associated linear system. Domain decomposition methods are a natural way to reduce computational costs, and Optimized Schwarz Methods have proven in the recent years their effectiveness in accelerating the convergence of such algorithms. The latter are based on interface matching conditions more efficient than the classical Dirichlet or Neumann ones. In this paper we analyze an Optimized Schwarz approach for the numerical solution of the Bidomain problem. We assess the convergence of the iterative method by means of Fourier analysis, and we investigate the parameter optimization in the interface conditions. Numerical results in 2D and 3D are given to show the effectiveness of the method.

Keywords

Domain decompositionoptimized schwarz methodscomputational electrocardiology
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 , Issue 2 , March 2013 , pp. 583 - 608
Copyright
© EDP Sciences, SMAI, 2013

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References

Références

LifeV software. http://www.LifeV.org.
Trilinos software. http://trilinos.sandia.gov.
Alonso-Rodriguez, A. and Gerardo-Giorda, L., New non-overlapping domain decomposition methods for the time-harmonic Maxwell system. SIAM J. Sci. Comput. 28 (2006) 102–122. Google Scholar
Bochev, P. and Lehouc, R., On the finite element solution of the pure Neumann problem. SIAM Rev. 47 (2005) 50–66. Google Scholar
T.F. Chan and T.P. Mathew, Domain decomposition algorithms, in Acta Numerica 1994. Cambridge University Press (1994) 61–143.
Charton, P., Nataf, F. and Rogier, F., Méthode de décomposition de domaine pour l’équation d’advection-diffusion. C. R. Acad. Sci. 313 (1991) 623–626. Google Scholar
Chevalier, P. and Nataf, F., Symmetrized method with optimized second-order conditions for the Helmholtz equation, in Domain decomposition methods (Boulder, CO, 1997). Amer. Math. Soc. 10 (1998) 400–407. Google Scholar
Clayton, R.H., Bernus, O.M., 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. Bioph. Molec. Biol. 104 (2011) 22–48. Google ScholarPubMed
Clayton, R.H. and Panfilov, A.V., A guide to modelling cardiac electrical activity in anatomically detailed ventricles. Progr. Bioph. Molec. Biol. 96 (2008) 19–43. Google ScholarPubMed
Colli Franzone, P. and Pavarino, L.F., A parallel solver for reaction-diffusion systems in computational electrocardiology. Math. Models Methods Appl. Sci. 14 (2004) 883–911. Google Scholar
P. Colli Franzone, L. Pavarino and G. Savaré, Computational electrocardiology : mathematical and numerical modeling, in Complex Systems in Biomedicine – A. Quarteroni, edited by L. Formaggia and A. Veneziani. Springer, Milan (2006).
P. Colli Franzone and G. Savaré, Degenerate evolution systems modeling the cardiac electric field at micro and macroscopic level, in Evolution Equations, Semigroups and Functional Analysis, edited by A. Lorenzi and B. Ruf. Birkhauser (2002) 49–78.
Deng, Q., An analysis for a nonoverlapping domain decomposition iterative procedure. SIAM J. Sci. Comput. 18 (1997) 1517–1525. Google Scholar
V. Dolean and F. Nataf, An Optimized Schwarz Algorithm for the compressible Euler equations, in Domain Decomposition Methods in Science and Engineering XVI (Proceedings of the DD16 Conference). Springer-Verlag (2007) 173–180.
Dolean, V., Gander, M.J. and Gerardo-Giorda, L., Optimized Schwarz Methods for Maxwell’s equations. SIAM J. Sci. Comput. 31 (2009) 2193–2213. Google Scholar
O. Dubois, Optimized Schwarz Methods with Robin conditions for the Advection-Diffusion Equation, in Domain Decomposition Methods in Science and Engineering XVI (Proceedings of the DD16 Conference). Springer-Verlag (2007) 181–188.
Engquist, B. and Zhao, H.-K., Absorbing boundary conditions for domain decomposition. Appl. Numer. Math. 27 (1998) 341–365. Google Scholar
Faccioli, E., Maggio, F., Quarteroni, A. and Tagliani, A., Spectral domain decomposition methods for the solution of acoustic and elastic wave propagation. Geophys. 61 (1996) 1160–1174. Google Scholar
Faccioli, E., Maggio, F., Quarteroni, A. and Tagliani, A., 2d and 3d elastic wave propagation by pseudo-spectral domain decomposition method. J. Seismology 1 (1997) 237–251. Google Scholar
Gander, M.J., Optimized Schwarz methods. SIAM J. Numer. Anal. 44 (2006) 699–731. Google Scholar
Gander, M.J. and Halpern, L., Méthodes de relaxation d’ondes pour l’équation de la chaleur en dimension 1. C. R. Acad. Sci. Paris, Sér. I 336 (2003) 519–524. Google Scholar
Gander, M.J., Halpern, L. and Magoulès, F., An optimized Schwarz method with two-sided Robin transmission conditions for the Helmholtz equation. Int. J. Numer. Meth. Fluids 55 (2007) 163–175. Google Scholar
Gander, M.J., Halpern, L. and Nataf, F., Optimal Schwarz waveform relaxation for the one dimensional wave equation. SIAM J. Numer. Anal. 41 (2003) 1643–1681. Google Scholar
Gander, M.J., Magoulès, F. and Nataf, F., Optimized Schwarz methods without overlap for the Helmholtz equation. SIAM J. Sci. Comput. 24 (2002) 38–60. Google Scholar
L. Gerardo-Giorda, L. Mirabella, M. Perego and A. Veneziani, A model adaptive strategy for computational electrocardiology. Domain Decomposition Methods in Science and Engineering XXI (Proceedings of the DD21 Conference). Springer-Verlag. To appear (2013).
Gerardo-Giorda, L., 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., Nobile, F. and Vergara, C., Analysis and optimization of Robin–Robin partitioned procedures in Fluid-Structure Interaction problems. SIAM J. Numer. Anal. 48 (2010) 2091–2116. Google Scholar
Gerardo-Giorda, L., Perego, M. and Veneziani, A., Optimized Schwarz coupling of Bidomain and Monodomain models in electrocardiology. ESAIM : M2AN 45 (2011) 309–334. Google Scholar
Hagstrom, T., Tewarson, R.P. and Jazcilevich, A., Numerical experiments on a domain decomposition algorithm for nonlinear elliptic boundary value problems. Appl. Math. Lett. 1 (1988) 299–302. Google Scholar
Henriquez, C.S., Simulating the electrical behavior of cardiac tissue using the Bidomain model. Crit. Rev. Biomed. Eng. 21 (1993) 1–77. Google ScholarPubMed
Japhet, C., Nataf, F. and Rogier, F., The optimized order 2 method : Application to convection-diffusion problems. Future Gener. Comp. Syst. 18 (2001) 17–30. Google Scholar
Linge, S., Sundnes, J., Hanslien, M., Lines, G.T. and Tveito, A., Numerical solution of the bidomain equations. Phil. Trans. R. Soc. A. 367 (2009) 1931–1950. Google ScholarPubMed
Lines, G.T., Buist, M.L., Grottum, P., Pullan, A.J., Sundnes, J. and Tveito, A., Mathematical models and numerical methods for the forward problem in cardiac electrophysiology. Comput. Vis. Sci. 5 (2003) 215–239.Google Scholar
P.-L. Lions, On the Schwarz alternating method. III : a variant for nonoverlapping subdomains, in Third International Symposium on Domain Decomposition Methods for Partial Differential Equations, held in Houston, Texas, edited by T.F. Chan, R. Glowinski, J. Périaux and O. Widlund, SIAM Philadelphia, PA (1990).
Luo, L. and Rudy, Y., A model of the ventricular cardiac action potential : depolarization, repolarization and their interaction. Circ. Res. 68 (1991) 1501–1526. Google ScholarPubMed
Mirabella, L., Nobile, F. and Veneziani, A., An a posteriori error estimator for model adaptivity in electrocardiology. Comput. Methods Appl. Mech. Eng. 200 (2011) 2727–2737. 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. 3 (2009) 3861–3883. Google Scholar
Nataf, F. and Rogier, F., Factorization of the convection-diffusion operator and the Schwarz algorithm. M3AS 5 (1995) 67–93. 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. 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
M. Pennacchio and V. Simoncini, Non-symmetric Algebraic Multigrid Preconditioners for the Bidomain reaction-diffusion system, in Numerical Mathematics and Advanced Applications, ENUMATH 2009, Part 2 (2010) 729–736.
Perego, M. and Veneziani, A., An efficient generalization of the Rush-Larsen method for solving electro-physiology membrane equations. ETNA 35 (2009) 234–256. Google Scholar
Potse, M., Dubé, B., Richer, J. and Vinet, A., A comparison of Monodomain and Bidomain Reaction-Diffusion models for Action Potential Propagation in the Human Heart. IEEE Trans. Biomed. Eng. 53 (2006) 2425–2435,. Google ScholarPubMed
A.J. Pullan, M.L. Buist and L.K. Cheng, Mathematical Modelling the Electrical Activity of the Heart. World Scientific, Singapore (2005).
A. Quarteroni and A. Valli, Domain Decomposition Methods for Partial Differential Equations. Oxford Science Publications (1999).
Roth, B.J., Action potential propagation in a thick strand of cardiac muscle. Circ. Res. 68 (1991) 162–173. Google Scholar
F.B. Sachse, Computational Cardiology. Springer, Berlin (2004).
Scacchi, S., A hybrid multilevel Schwarz method for the Bidomain model. Comput. Methods Appl. Mech. Eng. 197 (2008) 4051–4061. Google Scholar
B.F. Smith, P.E. Bjørstad and W. Gropp. Domain Decomposition : Parallel Multilevel Methods for Elliptic Partial Differential Equations. Cambridge University Press (1996).
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
Toselli, A., Overlapping Schwarz methods for Maxwell’s equations in three dimensions. Numer. Math. 86 (2000) 733–752. Google Scholar
A. Toselli and O. Widlund, Domain Decomposition Methods – Algorithms and Theory. Springer Ser. Comput. Math. 34 (2004).
Veneroni, M., Reaction-diffusion systems for the macroscopic Bidomain model of the cardiac electric field. Nonlinear Anal. Real World Appl. 10 (2009) 849–868. 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., R. Weber dos Santos, A.J. Prassl, M. Deo and G. Plank, Solvers for the caridac Bidomain equations. Prog. Biophys. Mol. Biol. 96 (2008) 3–18. Google Scholar
Weber dos Santos, R., Planck, 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
Xu, J. and Zou, J., Some nonoverlapping domain decomposition methods. SIAM Review 40 (1998) 857–914. Google Scholar
Xu, J., Iterative methods by space decomposition and subspace correction. SIAM Review 34 (1992) 581–613. Google Scholar