Volume 47, Number 4, July-August 2013
Direct and inverse modeling of the cardiovascular and respiratory systems
Page(s) 1017 - 1035
Published online 07 June 2013
  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. [CrossRef] [PubMed] [Google Scholar]
  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). [Google Scholar]
  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). [Google Scholar]
  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. [Google Scholar]
  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. [CrossRef] [Google Scholar]
  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. [CrossRef] [Google Scholar]
  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. [Google Scholar]
  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. [CrossRef] [Google Scholar]
  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. [CrossRef] [MathSciNet] [Google Scholar]
  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. [CrossRef] [Google Scholar]
  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. [CrossRef] [MathSciNet] [Google Scholar]
  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. [Google Scholar]
  13. M. Dryja and O.B. Widlund, Domain decomposition algorithms with small overlap. SIAM J. Sci. Comput. 15 (1994) 604–620. [CrossRef] [MathSciNet] [Google Scholar]
  14. M. Ethier and Y. Bourgault, Semi-implicit time-discretization schemes for the Bidomain model. SIAM J. Numer. Anal. 46 (2008) 2443–2468. [Google Scholar]
  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. [Google Scholar]
  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. [CrossRef] [PubMed] [Google Scholar]
  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. [CrossRef] [Google Scholar]
  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. [CrossRef] [EDP Sciences] [Google Scholar]
  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. [Google Scholar]
  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. [CrossRef] [Google Scholar]
  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. [Google Scholar]
  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. [CrossRef] [Google Scholar]
  23. G. Karypis and V. Kumar, MeTis: Unstructured Graph Partitioning and Sparse Matrix Ordering System, Version 4.0. University of Minnesota, Minneapolis, MN (2009). [Google Scholar]
  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. [CrossRef] [Google Scholar]
  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. [CrossRef] [Google Scholar]
  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. [CrossRef] [Google Scholar]
  27. J.S. Neu and W. Krassowska, Homogenization of syncytial tissues. Crit. Rev. Biomed. Eng. 21 (1993) 137–199. [PubMed] [Google Scholar]
  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. [CrossRef] [Google Scholar]
  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. [CrossRef] [Google Scholar]
  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. [CrossRef] [Google Scholar]
  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. [CrossRef] [Google Scholar]
  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. [CrossRef] [MathSciNet] [Google Scholar]
  33. M. Pennacchio and V. Simoncini, Algebraic multigrid preconditioners for the bidomain reaction-diffusion system. Appl. Numer. Math. 59 (2009) 3033–3050. [CrossRef] [Google Scholar]
  34. M. Pennacchio and V. Simoncini, Fast structured AMG preconditioning for the bidomain model in electrocardiology. SIAM J. Sci. Comput. 33 (2011) 721–745. [CrossRef] [Google Scholar]
  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. [CrossRef] [PubMed] [Google Scholar]
  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. [CrossRef] [PubMed] [Google Scholar]
  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. [Google Scholar]
  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. [CrossRef] [PubMed] [Google Scholar]
  39. A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations. Springer (1997). [Google Scholar]
  40. S. Scacchi, A hybrid multilevel Schwarz method for the bidomain model. Comput. Methods Appl. Mech. Eng. 197 (2008) 4051–4061. [CrossRef] [Google Scholar]
  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. [CrossRef] [Google Scholar]
  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. [CrossRef] [Google Scholar]
  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. [CrossRef] [PubMed] [Google Scholar]
  44. B.F. Smith, P. Bjørstad and W.D. Gropp, Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations, Cambridge University Press (1996). [Google Scholar]
  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. [CrossRef] [PubMed] [Google Scholar]
  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. [CrossRef] [Google Scholar]
  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. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  48. H. Si, Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany. [Google Scholar]
  49. V. Thomée, Galerkin Finite Element Methods for Parabolic Problems. Springer (1997). [Google Scholar]
  50. A. Toselli and O.B. Widlund, Domain Decomposition Methods: Algorithms and Theory. Comput. Math. Springer-Verlag, Berlin 34 (2004). [Google Scholar]
  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. [CrossRef] [Google Scholar]
  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. [CrossRef] [PubMed] [Google Scholar]
  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. [CrossRef] [PubMed] [Google Scholar]
  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. [CrossRef] [PubMed] [Google Scholar]
  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. [CrossRef] [PubMed] [Google Scholar]
  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. [CrossRef] [PubMed] [Google Scholar]
  57. X. Zhang, Multilevel Schwarz methods. Numer. Math. 63 (1992) 521–539. [CrossRef] [Google Scholar]

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