Abstract
Previous work [Whiteley, J. P. IEEE Trans. Biomed. Eng. 53:2139–2147, 2006] derived a stable, semi-implicit numerical scheme for solving the bidomain equations. This scheme allows the timestep used when solving the bidomain equations numerically to be chosen by accuracy considerations rather than stability considerations. In this study we modify this scheme to allow an adaptive numerical solution in both time and space. The spatial mesh size is determined by the gradient of the transmembrane and extracellular potentials while the timestep is determined by the values of: (i) the fast sodium current; and (ii) the calcium release from junctional sarcoplasmic reticulum to myoplasm current. For two-dimensional simulations presented here, combining the numerical algorithm in the paper cited above with the adaptive algorithm presented here leads to an increase in computational efficiency by a factor of around 250 over previous work, together with significantly less computational memory being required. The speedup for three-dimensional simulations is likely to be more impressive.
Similar content being viewed by others
References
Cherry E. M., Greenside H. S., Henriquez C. S. (2003) Efficient simulation of three-dimensional anisotropic cardiac tissue using an adaptive mesh refinement method. Chaos 13:853–865
Clancy C. E., Rudy Y. (2001) Cellular consequences of HERG mutations in the long QT syndrome: precursors to sudden cardiac death. Cardiovasc. Res. 50:301–313
Colli Franzone P., Pavarino L. F., Taccardi B. (2005) Simulating patterns of excitation, repolarization and action potential duration with cardiac bidomain and monodomain models. Math. Biosci. 197:35–66
Eriksson, K., D. Estep, P. Hansbo, and C. Johnson. Computational Differential Equations. Cambridge University Press, 1996.
Hooks D. A., Tomlinson K. A., Marsden S. G., LeGrice I. J., Smaill B. H., Pullan A. J., Hunter P. J. (2002) Cardiac microstructure: implications for electrical propagation and defibrillation in the heart. Circ. Res. 91:331–338
Iserles, A. A First Course in the Numerical Analysis of Differential Equations. Cambridge Texts in Applied Maths, 1996, Chapter 4.
Keener J. P., Bogar K. (1998) A numerical method for the solution of the bidomain equations in cardiac tissue. Chaos 8:234–241
Keener J. P., Sneyd J. (1998) Mathematical Physiology Chapter 11. Springer, New York
Latimer D. C., Roth B. J. (1998) Electrical stimulation of cardiac tissue by a bipolar electrode in a conductive bath. IEEE Trans. Biomed. Eng. 45:1449–1458
Luo C. H., Rudy Y. (1991) A model of the ventricular cardiac action potential. Depolarization, repolarization, and their interaction. Circ. Res. 68:1501–1526
Murillo M, Cai X.-C. (2004) A fully implicit parallel algorithm for simulating the non–linear electrical activity of the heart. Numer. Linear Algebra Appl. 11:261–277
Nash M. P., Panfilov A. V. (2004) Electromechanical model of excitable tissue to study reentrant cardiac arrhythmias. Prog. Biophys. Mol. Biol. 85:501–522
Noble D., Varghese A., Kohl P., Noble P. (1998) Improved guinea-pig ventricular cell model incorporating a diadic space, i Kr and i Ks , length- and tension-dependent processes. Can. J. Cardiol. 14:123–134
Pennacchio M. (2004) The mortar finite element method for the cardiac bidomain model of extracellular potential. J. Sci. Comput. 20:191–210
Qu Z., Garfinkel A. (1999) An advanced algorithm for solving partial differential equation in cardiac conduction. IEEE Trans. Biomed. Eng. 46:1166–1168
Qu Z., Weiss J. N., Garfinkel A. (1999) Cardiac electrical restitution properties and stability of reentrant spiral waves: a simulation study. Am. J. Physiol. 276:H269–H283
Quan W., Evans S. J., Hastings H. M. (1998) Efficient integration of a realistic two-dimensional cardiac tissue model by domain decomposition. IEEE Trans. Biomed. Eng. 45:372–385
Skouibine K., Trayanova N., Moore P. (2000) A numerically efficient model for simulation of defibrilation in an active bidomain sheet of myocardium. Math. Biosci. 166:85–100
Sundnes J., Lines G. T., Tveito A. (2001) Efficient solution of ordinary differential equations modeling electrical activity in cardiac cells. Math. Biosci. 172:55–72
Süli, E., and D. F. Mayers. An Introduction to Numerical Analysis. Cambridge University Press, 2003, Chapter 4.
ten Tusscher K. H. W. J., Noble D., Noble P. J., Panfilov A. V. (2004) A model for human ventricular tissue. Am. J. Physiol. 286:H1573–H1589
Vigmond E. J., Aguel F., Trayanova N. A. (2002) Computational techniques for solving the bidomain equations in three dimensions. IEEE Trans. Biomed. Eng. 49:1260–1269
Weber dos Santos R., Plank G., Bauer S., Vigmond E. J. (2004) Parallel multigrid preconditioner for the cardiac bidomain model. IEEE Trans. Biomed. Eng. 51:1960–1968
Whiteley J. P. (2006) An efficient numerical technique for the solution of the monodomain and bidomain equations. IEEE Trans. Biomed. Eng. 53:2139–2147
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Whiteley, J. Physiology Driven Adaptivity for the Numerical Solution of the Bidomain Equations. Ann Biomed Eng 35, 1510–1520 (2007). https://doi.org/10.1007/s10439-007-9337-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10439-007-9337-3