Abstract
We present a finite element (FE) derived finite difference (FD) technique for solving cardiac activation problems over deforming geometries using a bidomain framework. The geometry of the solution domain is defined by a FE mesh and over these FEs a high resolution FD mesh is generated. The difference points are located at regular intervals in the normalized material space within each of the FEs. The bidomain equations are then transformed to the embedded FD mesh which provides a solution space that is both regular and orthogonal. The solution points move in physical space with any deformation of the solution domain, but the equations are set up in such a way that the solution is invariant as it is constructed in material space. The derivation of this new solution technique is presented along with a series of examples that demonstrate the accuracy of this bidomain framework. © 2003 Biomedical Engineering Society.
PAC2003: 8719Hh, 8710+e, 8719Rr
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References
Beeler, G. W., and H. Reuter. Reconstruction of the action potential of ventricular myocardial fibres. J. Physiol. (London)268:177–210, 1977.
Cherry, E. A space-time adaptive mesh refinement method for simulating complex cardiac electrical dynamics. Ph.D. thesis, Duke University, 2000.
Flugge, W. Tensor Analysis and Continuum Mechanics. New York: Springer, 1972.
Harrild, D. M., and C. S. Henriquez. A finite volume model of cardiac propagation. Adv. Biomed. Eng.25:315–334, 1997.
Harrild, D. M., C. Penland, and C. S. Henriquez. A flexible method for simulating cardiac conduction in three-dimensional complex geometries. J. Electrocardiol.33:241–251, 2000.
Henriquez, C. S.Simulating the electrical behaviour of cardiac tissue using the bidomain model. Crit. Rev. Biomed. Eng.21:1–77, 1993.
Huiskamp, G.Simulation of depolarization in a membrane-equations-based model of the anisotropic ventricle. IEEE Trans. Biomed. Eng.45:847–855, 1998.
Hunter, P. J., A. D. McCulloch, P. M. F. Nielsen, and B. H. Smaill. Computational methods in bioengineering. ASME, BED 9: 387–397, 1989.
Hunter, P. J., A. D. McCulloch, and H. E. D. J. ter Keurs. Modelling the mechanical properties of cardiac muscle. 69:289–331, 1998.
Hunter, P. J., P. A. McNaughton, and D. Noble. Analytical models of propagation in excitable cells. 30:99–144, 1975.
Hunter, P. J., and B. H. Smaill. The analysis of cardiac function: A continuum approach. 52:101–164, 1988.
Kreysig, E. Advanced Engineering Mathematics. New York: John Wiley, 1988.
Latimer, D. C., and B. J. Roth. Electrical stimulation of cardiac tissue by a bipolar electrode in a conductive bath. IEEE Trans. Biomed. Eng.45:1449–1458, 1998.
LeGrice, I. J., P. J. Hunter, and B. H. Smaill. Laminar structure of the heart: A mathematical model. () 272:H2466–H2476, 1997.
Miller, C. E., and C. S. Henriquez. Finite element analysis of bioelectric phenomena. Crit. Rev. Biomed. Eng.18:207–233, 1990.
Nash, M. P. Mechanics and material properties of an anatomically accurate mathematical model of the heart. Ph.D. thesis, The University of Auckland, New Zealand, 1998.
Nielsen, P. M. F., I. J. Le Grice, B. H. Smaill, and P. J. Hunter. Mathematical model of geometry and fibrous struc-ture of the heart. Am. J. Physiol.260:H1365–H1378, 1991.
Panfilov, S., and A. V. Holden Computational Biology of the Heart. New York: Wiley, 1996.
Quan, W., S. J. Evans, and H. M. Hastings. Efficient integration of a realistic two-dimensional cardiac tissue model by domain decomposition. IEEE Trans. Biomed. Eng.45:372–385, 1998.
Rogers, J. M., and A. D. McCulloch. A collocation-Galerkin finite element model of cardiac action potential propagation. IEEE Trans. Biomed. Eng.41:743–757, 1994.
Sepulveda, N. G., B. J. Roth, and J. P. Wikswo. Current injection into a two-dimensional anisotropic bidomain. Biophys. J.55:987–999, 1989.
Shampine, L. F., and M. K. Gordon. Computer Solution of Ordinary Differential Equations: The Initial Value Problem. San Francisco: W. H. Freeman, 1975.
Skouibine, K., N. Trayanova, and P. Moore. Anode/cathode make and break phenomena during defibrillation: Does electroporation make a difference?IEEE Trans. Biomed. Eng.46:769–777, 1999.
Tyldesley, J. R. An Introduction to Tensor Analysis. New York: Longman, 1975.
Wang, C. Y., J. B. Bassingthwaighte, and L. J. Weissman. Bifurcating distributive system using Monte Carlo method. 16:91–98, 1992.
Weixue, L., and X. Ling. Computer simulation of epicardial potentials using a heart-torso model with realistic geometry. IEEE Trans. Biomed. Eng.43:211–217, 1996.
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Buist, M., Sands, G., Hunter, P. et al. A Deformable Finite Element Derived Finite Difference Method for Cardiac Activation Problems. Annals of Biomedical Engineering 31, 577–588 (2003). https://doi.org/10.1114/1.1567283
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DOI: https://doi.org/10.1114/1.1567283