A numerical guide to the solution of the bidomain equations of cardiac electrophysiology

Progress in Biophysics and Molecular Biology

Volumn 102, Issue 2-3, 2010, Pages 136-155

  Pathmanathan, Pras ^a   Bernabeu, Miguel O ^a   Bordas, Rafel ^a   Cooper, Jonathan ^a   Garny, Alan ^b   Pitt Francis, Joe M ^a   Whiteley, Jonathan P ^a   Gavaghan, David J ^a  

^a UNIVERSITY OF OXFORD   (United Kingdom)

^b UNIVERSITY OF OXFORD   (United Kingdom)

Author keywords

Bidomain; Cardiac electrophysiology; Computation

Indexed keywords

ALGORITHM; ANIMAL; BIOLOGICAL MODEL; COMPUTER SIMULATION; HEART; PHYSIOLOGY; REVIEW;

ALGORITHMS; ANIMALS; COMPUTER SIMULATION; HEART; MODELS, CARDIOVASCULAR;

EID: 77954215423     PISSN: 00796107     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.pbiomolbio.2010.05.006     Document Type: Review

References (40)

1. Austin T., Trew M., Pullan A. Solving the cardiac bidomain equations for discontinuous conductivities. IEEE Trans. Biomed. Eng. 2006, 53(7):1265-1272.
2. Bernabeu M., Bordas R., Pathmanathan P., Pitt-Francis J., Cooper J., Garny A., Gavaghan D., Rodriguez B., Southern J., Whiteley J. Chaste: incorporating a novel multi-scale spatial and temporal algorithm into a large-scale open source library. Phil. Trans. Roy. Soc. A 2009, 367(1895):1907-1930.
3. Bondarenko V., Szigeti G., Bett G., Kim S.-J., Rasmusson R. A computer model of the action potential of the mouse ventricular myocytes. Am. J. Physiol. Heart Circ. Physiol. 2004, 287:H1378-H1403.
4. Brown P., Walker H. GMRES on (nearly) singular systems. SIAM J. Matrix Analysis Appl. 1997, 18(1):37-51.
5. Cooper J., McKeever S., Garny A. On the application of partial evaluation to the optimisation of cardiac electrophysiological simulations. PEPM '06: Proceedings of the 2006 ACM SIGPLAN Symposium on Partial Evaluation and Semantics-based Program Manipulation 2006, 12-20. ACM Press, New York, NY, USA.
6. Cooper, J., 2009. Automatic validation and optimisation of biological models. Ph.D. thesis, University of Oxford.
7. Courtemanche M., Ramirez R., Nattel S. Ionic mechanisms underlying human atrial action potential properties: insights from a mathematical model. Am. J. Physiol. Heart Circ. Physiol. 1998, 275(1):H301-H321.
8. Dexter F., Saidel G., Levy M., Rudy Y. Mathematical model of dependence of heart rate on tissue concentration of acetylcholine. Am. J. Physiol. Heart Circ. Physiol. 1989, 256(2):H520-H526.
9. dos Santos R., Dickstein F. On the influence of a volume conductor on the orientation of currents in a thin cardiac tissue. Lecture Notes Comput. Sci. 2003, 111-121.
10. i overloaded cardiac myocytes: a simulation study. Biophys. J. 2000, 78(5):2392-2404.
11. Fox J., McHarg J., Gilmour R. Ionic mechanism of electrical alternans. Am. J. Physiol. Heart Circ. Physiol. 2002, 282:H516-H530.
12. Gamma E., Helm R., Johnson R., Vlissides J. Design Patterns: Elements of Reusable Object-Oriented Software 1995, Addison-Wesley.
13. Garny A., Kohl P., Hunter P., Boyett M., Noble D. One-dimensional rabbit sinoatrial node models: benefits and limitations. J. Cardiovasc. Electrophysiol. 2003, 14(s10):S121-S132.
14. Garny A., Nickerson D., Cooper J., dos Santos R., McKeever S., Nielsen P., Hunter P. CellML and associated tools and techniques. Phil. Trans. Roy. Soc. A 2008, 366(1878):3017-3043.
15. Hestenes M., Stiefel E. Methods of conjugate gradients for solving linear systems. J. Res. Natl. Bureau Standards 1952, 49(6):409-436.
16. Iserles A. A First Course in the Numerical Analysis of Differential Equations 1996, Cambridge University Press.
17. Jones N., Gomard C., Sestoft P. Partial Evaluation and Automatic Program Generation 1993, Prentice Hall.
18. Keener J., Sneyd J. Mathematical physiology. Interdisciplinary Applied Mathematics 1998, vol. 8. Springer.
19. Lloyd C., Halstead M., Nielsen P. CellML: its future, present and past. Prog. Biophys. Mol. Biol. 2004, 85:433-450.
20. Luo C., Rudy Y. A model of the ventricular cardiac action potential: depolarization, repolarization, and their interaction. Circ. Res. 1991, 68:1501-1526.
21. Noble D. A modification of the Hodgkin-Huxley equations applicable to Purkinje fibre action and pacemaker potentials. J. Physiol. 1962, 160:317-352.
22. Ks, length- and tension-dependent processes. Can. J. Cardiol. 1998, 14(1):123-134.
23. Paige C., Saunders M. Solution of sparse indefinite systems of linear equations. SIAM J. Numer. Anal. 1975, 617-629.
24. Pavarino L., Scacchi S. Multilevel additive Schwarz preconditioners for the Bidomain reaction-diffusion system. SIAM J. Sci. Comput. 2008, 31:420.
25. Pitt-Francis J., Bernabeu M., Cooper J., Garny A., Momtahan L., Osborne J., Pathmanathan P., Rodriguez B., Whiteley J., Gavaghan D. Chaste: using agile programming techniques to develop computational biology software. Phil. Trans. Roy. Soc. A 2008, 366(1878):3111-3136.
26. Pitt-Francis J., Pathmanathan P., Bernabeu M., Bordas R., Cooper J., Fletcher A., Mirams G., Murray P., Osborne J., Walter A., Chapman S., Garny A., van Leeuwen I., Maini K., Rodriguez B., Whiteley J., Byrne H., Gavaghan D. Chaste: a test-driven approach to software development for biological modelling. Comput. Phys. Commun. 2009, 180(12):2452-2471.
27. Plank G., Liebmann M., dos Santos R., Vigmond E., Haase G. Algebraic multigrid preconditioner for the cardiac bidomain model. IEEE Trans. Biomed. Eng. 2007, 54(4):585-596.
28. Reddy J. An Introduction to the Finite Element Method 1993, McGraw-Hill.
29. Rush S., Larsen H. A practical algorithm for solving dynamic membrane equations. IEEE Trans. Biomed. Eng. BME 1978, 25(4):389-392.
30. Saad Y., Schultz M.H. GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Comput. 1986, 7:856-869.
31. Southern J., Plank G., Vigmond E., Whiteley J. Solving the coupled system improves computational efficiency of the bidomain equations. IEEE Trans. Biomed. Eng. 2009, 56(10):2404-2412.
32. Sundnes J., Lines G., Tveito A. An operator splitting method for solving the bidomain equations coupled to a volume conductor model for the torso. Math. Biosci. 2005, 194(2):233-248.
33. Sundnes J., Nielsen B., Mardal K., Cai X., Lines G., Tveito A. On the computational complexity of the bidomain and the monodomain models of electrophysiology. Ann. Biomed. Eng. 2006, 34(7):1088-1097.
34. ten Tusscher K., Panfilov A. Alternans and spiral breakup in a human ventricular tissue model. Am. J. Physiol. Heart Circ. Physiol. 2006, 291(3):H1088-H1100.
35. Van der Vorst H. Iterative Krylov Methods for Large Linear Systems 2003, Cambridge Univ Pr.
36. Vigmond E., Aguel F., Trayanova N. Computational techniques for solving the bidomain equations in three dimensions. IEEE Trans. Biomed. Eng. 2002, 49(11):1260-1269.
37. Vigmond E., dos Santos R., Prassl A., Deo M., Plank G. Solvers for the cardiac bidomain equations. Prog. Biophys. Mol. Biol. 2008, 96(1-3):3-18.
38. Whiteley J. An efficient numerical technique for the solution of the monodomain and bidomain equations. IEEE Trans. Biomed. Eng. 2006, 53(11):2139-2147.
39. Whiteley J. Physiology driven adaptivity for the numerical solution of the bidomain equations. Ann. Biomed. Eng. 2007, 35(9):1510-1520.
40. Whiteley J. An efficient technique for the numerical solution of the bidomain equations. Ann. Biomed. Eng. 2008, 36(8):1398-1408.