Abstract
There is a large number of ordinary differential equations (ODEs) characterize the electrical behavior generated by ionic movements in human myocardial cell. In this paper, several approaches were investigated in order to improve the efficiency of solving the ODE systems for ten Tusscher et al.’s ionic model of human ventricular tissue. By using non-standard finite difference (NSFD) scheme, the stiffness of the ODEs system will be successfully reduced, so a larger step-size can be used. A popular multi-step method called backward differentiation formulation (BDF) was also incorporated into the computational model for testing the largest possible time steps. The results show that NSFD can be as much as 10 times more efficient than standard forward Euler in single cell model simulation while maintaining an acceptable level of accuracy. The investigation of BDF method shows that a large step size is not recommended for the single cell simulation application. All solvers were coupled to the partial differential equations for the complete simulation of heart tissue, and such computation scheme may be a good calculation technique in heart modeling and simulation.
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References
Lines, G.T., Buist, M.L., Buist, P.G., Pullan, A.J., Sundnes, J., Tveito, A.: Mathematical Models and Numerical Methods for The Forward Problem in Cardiac Electrophysiology. Computing and Visualization in Science 5, 215–239 (2003)
Luo, C.H., Rudy, Y.: A Model of The Ventricular Cardiac Action Potential Depolarization, Repolarization, and Their Interaction. Circ. Res. 68, 1501–1526 (1991)
Winslow, R.L., Rice, J., Jafri, S., Marban, E., O’Rourke, B.: Mechanisms of Altered Excitation-Contraction Coupling in Canine Tachycardia-Induced Heart Failure. II: Model Studies. Circ. Res. 84, 571–586 (1999)
Lambert, J.D.: Numerical Methods for Ordinary Differential Systems. John Wiley and Sons, Chichester (1991)
Mickens, R.E.: Nonstandard Finite Difference Models of Differential Equations. World Scientific, Singapore (1994)
Mickens, R.E.: Applications of Nonstandard Finite Difference Schemes. World Scientific, Singapore (2000)
Tentusscher, K.H., Noble, D., Noble, P.J., Panfilov, A.V.: A Model for Human Ventricular Tissue. Am Physiol Heart Circ Physiol, vol. 286, H1573–89 (2004)
Hille, B.: Ionic Channels of Excitable Membranes. 2 edn. (1992)
Hodgkin, A.L., Huxley, A.F.: A Quantitative Description of Membrane Current and Its Application to Conduction and Excitation in Nerve. J. Physiol 117, 500–544 (1952)
Sundnes, J., Nielsen, B.F., Mardal, K.A., Cai, X., Lines, G.T., Tveito, A.: On The Computational Complexity of The Bidomain and The Monodomain Models of Electrophysiology. Ann. Biomed. Eng. 34, 1088–1097 (2006)
Zwillinger, D.: Handbook of Differential Equations. Academic Press, Boston (1989)
Mickens, R.E.: Finite-Difference Schemes Having the Correct Linear Stability Properties for All Finite Step-Sizes II. Dynamic Systems and Applications 1, 329–340 (1992)
Mickens, R.E., Smith, A.: Finite Difference Models of Ordinary Differential Equations: Influence of Denominator Functions. Journal of the Franklin Institute 327, 143–145 (1990)
Brown, P.N., Byrne, G.D., Hindmarsh, A.C.: VODE: A Variable-Coefficient ODE Solver. SIAM J. Sci. Comput. 10, 1038–1051 (1989)
Whea, P.: Numerical Recipes in FORTRAN:The Art of Scientific Computing. Cambridge University Press, NY, USA (1992)
Zhang, Y., Xia, L., Hou, G.: Efficient Solution of Bidomain Equations in Simulation of Cardiac Excitation Anisotropic Propagation. In: Huang, D.-S., Li, K., Irwin, G.W. (eds.) ICIC 2006. LNCS (LNBI), vol. 4115, pp. 571–581. Springer, Heidelberg (2006)
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Zhang, Y., Xia, L., Gong, Y. (2007). Application of Efficient Numerical Methods in Solution of Ordinary Differential Equations for Modeling Electrical Activity in Cardiac Cells. In: Huang, DS., Heutte, L., Loog, M. (eds) Advanced Intelligent Computing Theories and Applications. With Aspects of Contemporary Intelligent Computing Techniques. ICIC 2007. Communications in Computer and Information Science, vol 2. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74282-1_49
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DOI: https://doi.org/10.1007/978-3-540-74282-1_49
Publisher Name: Springer, Berlin, Heidelberg
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