Solvers for the cardiac bidomain equations

Progress in Biophysics and Molecular Biology

Volumn 96, Issue 1-3, 2008, Pages 3-18

  Vigmond, E J ^a   Weber dos Santos, R ^b   Prassl, A J ^c   Deo, M ^a   Plank, G ^c  

^a UNIVERSITY OF CALGARY   (Canada)

^b FEDERAL UNIVERSITY OF JUIZ DE FORA   (Brazil)

^c MEDICAL UNIVERSITY OF GRAZ   (Austria)

Author keywords

Bidomain model; Cardiac electrical modeling; Multigrid; Preconditioning

Indexed keywords

ALGORITHM; ANIMAL; BIOLOGICAL MODEL; BIOLOGY; HEART; HEART MUSCLE CONDUCTION SYSTEM; HUMAN; PHYSIOLOGY; REVIEW;

ALGORITHMS; ANIMALS; COMPUTATIONAL BIOLOGY; HEART; HEART CONDUCTION SYSTEM; HUMANS; MODELS, CARDIOVASCULAR;

EID: 43049141917     PISSN: 00796107     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.pbiomolbio.2007.07.012     Document Type: Review

References (70)

1. Anwander A., Kuhn M., Reitzinger S., and Wolters C. A parallel algebraic multigrid solver for finite element method based source localization in the human brain. Comput. Visualization Sci. 5 3 (2002) 167-177
2. Austin T., Trew M., and Pullan A. Solving the cardiac biodomain equations for discontinuous conductivities. IEEE Trans. Biomed. Eng. 53 7 (2006) 1265-1272
3. Brezina M., Falgout R., MacLachlan S., Manteuffel T., McCormick S., and Ruge J. Adaptive smoothed aggregation αSA multigrid. SIAM Rev. 47 2 (2005) 317-346
4. Burton A., Plank G., Schneider J., Grau V., Ahammer H., Keeling S., Lee J., Smith N., Trayanova N., and Kohl P. 3-Dimensional models of individual cardiac histo-anatomy: tools and challenges. Ann. N Y Acad. Sci. 1080 (2006) 301-319
5. Chartier T., Falgout R.D., Henson V.E., Jones J., Manteuffel T., Mc-Cormick S., Ruge J., and Vassilevski P.S. Spectral AMGe (ρAMGe). SIAM J. Sci. Comput. 25 1 (2003) 1-26
6. Colli-Franzone P., and Pavarino L. A parallel solver for reaction-diffusion systems in computational electrophysiology. Math. Models Methods Appl. Sci. 14 6 (2004) 883-911
7. Courant R., Friedrichs K., and Lewy H. Über die partiellen Differenzen-gleichungen der mathematischen Physik. Math. Ann. 100 1 (1928) 32-74
8. Cuthill E., and McKee J. Reducing the bandwidth of sparse symmetric matrices. Proceedings of the 24th National Conference of the ACM (1969) 157-172
9. Eason J., and Malkin R. A simulation study evaluating the performance of high-density electrode arrays on myocardial tissue. IEEE Trans. Biomed. Eng. 47 7 (2000) 893-901
10. Efimov I.R., Nikolski V.P., and Salama G. Optical imaging of the heart. Circ. Res. 95 1 (2004) 21-33
11. Fischer G., Tilg B., Modre R., Huiskamp G., Fetzer J., Rucker W., and Wach P. A bidomain model based BEM-FEM coupling formulation for anisotropic cardiac tissue. Ann. Biomed. Eng. 28 10 (2000) 1229-1243
12. Franzone P., and Guerri L. Spreading of excitation in 3-D models of the anisotropic cardiac tissue. I: validation of the eikonal model. Math. Biosci. 113 (1993) 145-209
13. George A. Nested dissection of a regular finite element mesh. SIAM J. Numer. Anal. 10 (1973) 345-363
14. Haase G., and Reitzinger S. Cache issues of algebraic multigrid methods for linear systems with multiple right-hand sides. SIAM J. Sci. Comput. 27 1 (2005) 1-18
15. Haase G., Langer U., Reitzinger S., and Schöberl J. Algebraic multigrid methods based on element preconditioning. Int. J. Comput. Math. 78 4 (2001) 575-598
16. Haase G., Kuhn M., and Reitzinger S. Parallel AMG on distributed memory computers. SIAM J. Sci. Comput. 24 2 (2002) 410-427
17. Hodgkin A.L., and Huxley A.F. A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 117 (1952) 500-544
18. Holst M., and Saied F. Multigrid and domain decomposition methods for electrostatics problems. In: Keyes D., and Xu J. (Eds). Domain Decomposition Methods in Science and Engineering (1995), American Mathematical Society, Providence, RI
19. Hooke N., Henriquez C.S., Lanzkron P., and Rose D. Linear algebraic transformations of the bidomain equations: implications for numerical methods. Math. Biosci. 120 2 (1994) 127-145
20. Hund T.J., and Rudy Y. Rate dependence and regulation of action potential and calcium transient in a canine cardiac ventricular cell model. Circulation 110 20 (2004) 3168-3174
21. Iyer V., Mazhari R., and Winslow R.L. A computational model of the human left-ventricular epicardial myocyte. Biophys. J. 87 3 (2004) 1507-1525
22. Jacquemet V., and Henriquez C. Finite volume stiffness matrix for solving anisotropic cardiac propagation in 2-D and 3-D unstructured meshes. IEEE Trans. Biomed. Eng. 52 8 (2005) 1490-1492
23. Johnson C., Mohr M., Rude U., Samsonov A., and Zyp K. Multilevel methods for inverse bioelectric field problems. In: Barth T., Chan T., and Haimes R. (Eds). Multiscale and Multiresolution Methods in Computational Science and Engineering vol. 20 (2003), Springer, Berlin
24. Kaltenbacher M., Reitzinger S., and Schoberl J. Algebraic multigrid method for solving 3D nonlinear electrostatic and magnetostatic field problems. IEEE Trans. Magn. 36 4 (2000) 1561-1564
25. Keener J., and Bogar K. A numerical method for the solution of the bidomain equations in cardiac tissue. Chaos 8 1 (1998) 234-241
26. Keener J., and Sneyd J. Mathematical Physiology (1998), Springer, New York
27. Krassowska W., and Neu J.C. Effective boundary conditions for syncytial tissues. IEEE Trans. Biomed. Eng. 41 2 (1994) 143-150
28. Latimer D., and Roth B. Electrical stimulation of cardiac tissue by a bipolar electrode in a conductive bath. IEEE Trans. Biomed. Eng. 45 12 (1998) 1449-1458
29. Li X., and Demmel J. SuperLU DIST: a scalable distributed-memory sparse direct solver for unsymmetric linear systems. ACM Trans. Math. Software 29 2 (2003) 110-140
30. Lin S., and Wikswo J. New perspectives in electrophysiology from the cardiac bidomain. In: Rosenbaum D., and Jalife J. (Eds). Optical Mapping of Cardiac Excitation and Arrhythmias (2001), Futura, New York 335-359
31. Miller W.T., and Geselowitz D.B. Simulation studies of the electrocardiogram. I. The normal heart. Circ. Res. 43 2 (1978) 301-315
32. Muzikant A., and Henriquez C. Validation of three-dimensional conduction models using experimental mapping: are we getting closer?. Prog. Biophys. Mol. Biol. 69 2-3 (1998) 205-223
33. Pennacchio M. The mortar finite element method for the cardiac "bidomain" model of extracellular potential. J. Sci. Comput. 20 2 (2004) 191-210
34. 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
35. Plank G., Leon L.J., Kimber S., and Vigmond E.J. Defibrillation depends on conductivity fluctuations and the degree of disorganization in reentry patterns. J. Cardiovasc. Electrophysiol. 16 2 (2005) 205-216
36. Plank, G., Liebmann, M., Weber dos Santos, R., Vigmond, E., Haase, G., 2007. Algebraic multigrid preconditioner for the cardiac bidomain model. IEEE Trans. Biomed. Eng. 54 (4), 585-596.
37. Plonsey R. Bioelectric sources arising in excitable fibers (ALZA lecture). Ann. Biomed. Eng. 16 6 (1988) 519-546
38. Pollard A.E., Hooke N., and Henriquez C.S. Cardiac propagation simulation. Crit. Rev. Biomed. Eng. 20 3-4 (1992) 171-210
39. Potse, M., Dubé, B., Richer, J., Vinet, A., Gulrajani, R., 2006. A comparison of monodomain and bidomain reaction-diffusion models for action potential propagation in the human heart. IEEE Trans. Biomed. Eng. 53 (12) 1, 2425-2435.
40. Press W.H., Teukolsky S.A., Vetterling W.T., and Flannery B.P. Numerical Recipes in C: The Art of Scientific Computing. second ed (1994), Cambridge University Press, New York
41. Priebe L., and Beuckelmann D.J. Simulation study of cellular electric properties in heart failure. Circ. Res. 82 11 (1998) 1206-1223
42. Roth B. Meandering of spiral waves in anisotropic cardiac tissue. Physica D 150 (2001) 127-136
43. Ruge J.W., and Stüben K. Algebraic multigrid AMG. In: McCormick S. (Ed). Multigrid Methods. Frontiers in Applied Mathematics vol. 5 (1986), SIAM, Philadelphia 73-130
44. Saleheen H., and Ng K. A new three-dimensional finite-difference bidomain formulation for inhomogeneous anisotropic cardiac tissues. IEEE Trans. Biomed. Eng. 45 1 (1998) 15-25
45. Saleheen H.I., and Ng K.T. A new three-dimensional finite-difference bidomain formulation for inhomogeneous anisotropic cardiac tissues. IEEE Trans. Biomed. Eng. 45 1 (1998) 15-25
46. Sepulveda N.G., Roth B.J., and Wikswo Jr. J.P. Current injection into a two-dimensional anisotropic bidomain. Biophys. J. 55 (1989) 987-999
47. Shewchuk, J., 1994. An introduction to Conjugate Gradient Method Without the Agonizing Pain, first ed.
48. Skouibine K., and Krassowska W. Increasing the computational efficiency of a bidomain model of defibrillation using a time-dependent activating function. Ann. Biomed. Eng. 28 7 (2000) 772-780
49. Skouibine K., Trayanova N.A., and Moore P.K. Anode/cathode make and break phenomena in a model of defibrillation. IEEE Trans. Biomed. Eng. 46 7 (1999) 769-777
50. Skouibine K., Trayanova N., and Moore P. A numerically efficient model for simulation of defibrillation in an active bidomain sheet of myocardium. Math. Biosci. 166 1 (2000) 85-100
51. Strikwerda J. Finite Difference Schemes and Partial Differential Equations (1989), Chapman & Hall, Pacific Grove
52. Sundnes J., Lines G., and Tveito A. Efficient solution of ordinary differential equations modeling electrical activity in cardiac cells. Math. Biosci. 172 2 (2001) 55-72
53. 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 6 (2002) 397-409
54. Tatebe O., and Oyanagi Y. Efficient implementation of the multigrid preconditioned conjugate gradient method on distributed memory machines. Proceedings of Super Computing 94 (1994), IEEE Computing Society 194-203
55. Trayanova N.A. Defibrillation of the heart insights into mechanisms from modelling studies. Exp. Physiol. 91 2 (2006) 323-337
56. Trayanova N.A., Skouibine K., and Aguel F. The role of cardiac tissue structure in defibrillation. Chaos 8 (1998) 221-233
57. Trew M., LeGrice I., Smaill B., and Pullan A. A finite volume method for modeling discontinuous electrical activation in cardiac tissue. Ann. Biomed. Eng. 33 5 (2005) 590-602
58. Trew M., Smaill B., Bullivant D., Hunter P., and Pullan A. A generalized finite difference method for modeling cardiac electrical activation on arbitrary, irregular computational meshes. Math. Biosci. 198 2 (2005) 169-189
59. Tung, L., 1978. A bi-domain model for describing ischemic myocardial D-C potentials. Ph.D. Thesis, MIT, Cambridge, MA.
60. van der Vorst H. Bi-CGSTAB: a fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 13 (1992) 631-644
61. Vigmond E., Aguel F., and Trayanova N. Computational techniques for solving the bidomain equations in three dimensions. IEEE Trans. Biomed. Eng. 49 11 (2002) 1260-1269
62. Vigmond, E., Hughes, M., Plank, G., Leon, L., 2003. Computational tools for modeling electrical activity in cardiac tissue. J. Electrocardiol. 3 (suppl.), 69-74.
63. Weber dos Santos, R., 2002. Modelling cardiac electrophysiology. Ph.D. Thesis, Mathematics Department, Federal University of Rio de Janeiro, Rio de Janeiro, Brazil.
64. Weber dos Santos R., and Dickstein F. On the influence of a volume conductor on the orientation of currents in a thin cardiac tissue. In: Magnin I., Montagnat J., Clarysse P., Nenonen J., and Katila T. (Eds). Lecture Notes in Computer Science (2003), Springer, Berlin 111-121
65. Weber dos Santos R., Dickstein F., and Marchesin D. Transversal versus longitudinal current propagation on a cardiac tissue and its relation to MCG. Biomed. Tech. 47 1 (2003) 249-252
66. Weber dos Santos R., Plank G., Bauer S., and Vigmond E. Parallel multigrid preconditioner for the cardiac bidomain model. IEEE Trans. Biomed. Eng. 51 11 (2004) 1960-1968
67. Weber dos Santos, R., Plank, G., Bauer, S., Vigmond, E., 2004b. Preconditioning Techniques for the Bidomain Equations. Lecture Notes in Computational Science and Engineering, vol. 40, Springer, Berlin, pp. 571-580.
68. Wikswo Jr. J.P. Tissue anisotropy, the cardiac bidomain, and the virtual cathode effect. In: Zipes D.P., and Jalife J. (Eds). Cardiac Electrophysiology: From Cell to Bedside. second ed (1994), WB Saunders Co. 348-361
69. Wikswo Jr. J.P., Lin S.-F., and Abbas R.A. Virtual electrodes in cardiac tissue: a common mechanism for anodal and cathodal stimulation. Biophys. J. 69 (1995) 2195-2210
70. Zozor S., Blanc O., Jacquemet V., Virag N., Vesin J., Pruvot E., Kappen-berger L., and Henriquez C. A numerical scheme for modeling wavefront propagation on a monolayer of arbitrary geometry. IEEE Trans. Biomed. Eng. 50 4 (2003) 412-420