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Optimized Schwarz coupling of Bidomain and Monodomain models in electrocardiology

Published online by Cambridge University Press:  20 August 2010

Luca Gerardo-Giorda ,
Mauro Perego  and
Alessandro Veneziani
Luca Gerardo-Giorda
Affiliation:
Dept. of Mathematics and Computer Science, Emory University, Atlanta, USA. [email protected]; [email protected]; [email protected]
Mauro Perego
Affiliation:
Dept. of Mathematics and Computer Science, Emory University, Atlanta, USA. [email protected]; [email protected]; [email protected]
Alessandro Veneziani
Affiliation:
Dept. of Mathematics and Computer Science, Emory University, Atlanta, USA. [email protected]; [email protected]; [email protected]
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Abstract

The Bidomain model is nowadays one of the most accurate mathematical descriptions of the action potential propagation in the heart. However, its numerical approximation is in general fairly expensive as a consequence of the mathematical features of this system. For this reason, a simplification of this model, called Monodomain problem is quite often adopted in order to reduce computational costs. Reliability of this model is however questionable, in particular in the presence of applied currents and in the regions where the upstroke or the late recovery of the action potential is occurring. In this paper we investigate a domain decomposition approach for this problem, where the entire computational domain is suitably split and the two models are solved in different subdomains. Since the mathematical features of the two problems are rather different, the heterogeneous coupling is non trivial. Here we investigate appropriate interface matching conditions for the coupling on non overlapping domains. Moreover, we pursue an Optimized Schwarz approach for the numerical solution of the heterogeneous problem. Convergence of the iterative method is analyzed by means of a Fourier analysis. We investigate the parameters to be selected in the matching radiation-type conditions to accelerate the convergence. Numerical results both in two and three dimensions illustrate the effectiveness of the coupling strategy.

Keywords

Computational electrocardiologyOptimized Schwarz Methodsheterogeneous models
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 45 , Issue 2 , March 2011 , pp. 309 - 334
Copyright
© EDP Sciences, SMAI, 2010

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References

Alonso-Rodriguez, A. and Gerardo-Giorda, L., New non-overlapping domain decomposition methods for the time-harmonic Maxwell system. SIAM J. Sci. Comp. 28 (2006) 102–122. CrossRef
Bendahmane, M. and Karlsen, K.H., Analysis of a class of degenerate reaction-diffusion systems and the bidomain model of cardiac tissue. Netw. Heterog. Media 1 (2006) 185–218. CrossRef
Bourgault, Y., Coudière, Y. and Pierre, C., Existence and uniqueness of the solution for the bidomain model used in cardiac electrophysiology. Nonlinear Anal.: Real World Appl. 10 (2009) 458–482. CrossRef
Clayton, R.H. and Panfilov, A.V., A guide to modelling cardiac electrical activity in anatomically detailed ventricles. Prog. Biophys. Mol. Biol. 96 (2008) 19–43. CrossRef
R.H. Clayton, O.M. 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. Prog. Biophys. Mol. Biol. (2010) DOI: 10.1016/j.pbiomolbio.2010.05.008. CrossRef
Clements, J.C., Nenonen, J., Li, P.K.J. and Horacek, M., Activation dynamics in anisotropic cardiac tissue via decoupling. Ann. Biomed. Eng. 32 (2004) 984–990. CrossRef
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. CrossRef
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, A. Lorenzi and B. Ruf Eds., Birkhauser (2002) 49–78.
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. Biosc. 197 (2005) 35–66. CrossRef
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. CrossRef
V. Dolean and F. Nataf, An Optimized Schwarz Algorithm for the compressible Euler equations, in Domain Decomposition Methods in Science and Engineering, 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. CrossRef
Fox, J.J., McHarg, J.L. and Gilmour, R.F., Ionic mechanism of electrical alternans. Am. J. Physiol. (Heart Circ. Physiol.) 282 (2002) H516–H530. CrossRef
Gander, M.J., Optimized Schwarz methods. SIAM J. Num. Anal. 44 (2006) 699–731. CrossRef
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. CrossRef
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. Comp. Phys. 228 (2009) 3625–3639. CrossRef
Keener, J.P., Direct activation and defibrillation of cardiac tissue. J. Theor. Biol. 178 (1996) 313–324. CrossRef
J.P. Keener and J. Sneyd, Mathematical Physiology. Springer-Verlag, New York (1998).
Latimer, D.C. and Roth, B.J., Electrical stimulation of cardiac tissue by a bipolar electrode in a conductive bath. IEEE Trans. Biomed. Eng. 45 (1998) 1449–1458. CrossRef
J. Le Grice, 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. Am. J. Physiol. (Heart Circ. Physiol.) 269 (1995) H571–H582.
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, March 20–22, 1989, Philadelphia, R. Glowinski, J. Périaux, T.F. Chan and O. Widlund Eds., SIAM (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. CrossRef
L. Mirabella, F. Nobile and A. Veneziani, An a posteriori error estimator for model adaptivity in electrocardiology. Technical Report TR-2009-025, Dept. MathCS, Emory University (2009).
Nielsen, B.F., Ruud, T.S., Lines, G.T. and Tveito, A., Optimal monodomain approximation of the bidomain equations. Appl. Math. Comp. 184 (2007) 276–290. CrossRef
Nygren, A., Fiset, C., Firek, L., Clark, J.W., Lindblad, D.S., Clark, R.B. and Giles, W.R., Mathematical model of an adult human atrial cell: the role of K + currents in repolarization. Circ. Res. 82 (1998) 63–81. CrossRef
Pavarino, L.F. and Scacchi, S., Multilevel additive Schwarz preconditioners for the Bidomain reaction-diffusion system. SIAM J. Sci. Comp. 31 (2008) 420–443. CrossRef
Pennacchio, M. and Simoncini, V., Efficient algebraic solution of rection-diffusion systems for the cardiac excitation process. J. Comput. Appl. Math. 145 (2002) 49–70. CrossRef
Perego, M. and Veneziani, A., An efficient generalization of the Rush-Larsen method for solving electro-physiology membrane equations. Electronic Transaction on Numerical Analysis 35 (2009) 234–256.
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. CrossRef
A. Quarteroni and A. Valli, Domain Decompostion Methods for Partial Differential Equations. Oxford University Press, Oxford (1999).
A. Quarteroni, L. Formaggia and A. Veneziani, Complex Systems in Biomedicine, in Computational electrocardiology: mathematical and numerical modeling, P. Colli Franzone, L. Pavarino and G. Savaré Eds., Springer, Milan (2006).
Roth, B., A comparison of two boundary conditions used with the bidomain model of cardiac tissue. Ann. Biomed. Eng. 19 (1991) 669–678. CrossRef
Scacchi, S., A hybrid multilevel Schwarz method for the bidomain model. Comp. Meth. Appl. Mech. Eng. 197 (2008) 4051–4061. CrossRef
B.F. Smith, P.E. Bjørstad and W. Gropp, Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations. Cambridge University Press, Cambridge (1996).
D. Streeter, Gross morphology and fiber geometry in the heart, in Handbook of Physiology 1 (Sect. 2), R.M. Berne Ed., Williams and Wilnkins (1979) 61–112.
A. Toselli and O. Widlund, Domain Decomposition Methods. 1st edition, Springer (2004).
Trayanova, N., Defibrillation of the heart: insights into mechanisms from modelling studies. Exp. Physiol. 91 (2006) 323–337. CrossRef
Veneroni, M., Reaction-diffusion systems for the macroscopic bidomain model of the cardiac electric field. Nonlinear Anal.: Real World Appl. 10 (2009) 849–868. CrossRef
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. CrossRef
Vigmond, E.J., Weber dos, R. Santos, A.J. Prassl, M. Deo and G. Plank, Solvers for the caridac bidomain equations. Prog. Biophys. Mol. Biol. 96 (2008) 3–18. CrossRef
Weber dos, R. Santos, G. Planck, S. Bauer and E.J. Vigmond, Parallel multigrid preconditioner for the cardiac bidomain model. IEEE Trans. Biomed. Eng. 51 (2004) 1960–1968. CrossRef