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A Deformable Finite Element Derived Finite Difference Method for Cardiac Activation Problems

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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

  1. Beeler, G. W., and H. Reuter. Reconstruction of the action potential of ventricular myocardial fibres. J. Physiol. (London)268:177–210, 1977.

    Google Scholar 

  2. Cherry, E. A space-time adaptive mesh refinement method for simulating complex cardiac electrical dynamics. Ph.D. thesis, Duke University, 2000.

  3. Flugge, W. Tensor Analysis and Continuum Mechanics. New York: Springer, 1972.

    Google Scholar 

  4. Harrild, D. M., and C. S. Henriquez. A finite volume model of cardiac propagation. Adv. Biomed. Eng.25:315–334, 1997.

    Google Scholar 

  5. 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.

    Google Scholar 

  6. Henriquez, C. S.Simulating the electrical behaviour of cardiac tissue using the bidomain model. Crit. Rev. Biomed. Eng.21:1–77, 1993.

    Google Scholar 

  7. Huiskamp, G.Simulation of depolarization in a membrane-equations-based model of the anisotropic ventricle. IEEE Trans. Biomed. Eng.45:847–855, 1998.

    Google Scholar 

  8. Hunter, P. J., A. D. McCulloch, P. M. F. Nielsen, and B. H. Smaill. Computational methods in bioengineering. ASME, BED 9: 387–397, 1989.

    Google Scholar 

  9. Hunter, P. J., A. D. McCulloch, and H. E. D. J. ter Keurs. Modelling the mechanical properties of cardiac muscle. 69:289–331, 1998.

    Google Scholar 

  10. Hunter, P. J., P. A. McNaughton, and D. Noble. Analytical models of propagation in excitable cells. 30:99–144, 1975.

    Google Scholar 

  11. Hunter, P. J., and B. H. Smaill. The analysis of cardiac function: A continuum approach. 52:101–164, 1988.

    Google Scholar 

  12. Kreysig, E. Advanced Engineering Mathematics. New York: John Wiley, 1988.

    Google Scholar 

  13. 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.

    Google Scholar 

  14. LeGrice, I. J., P. J. Hunter, and B. H. Smaill. Laminar structure of the heart: A mathematical model. () 272:H2466–H2476, 1997.

    Google Scholar 

  15. Miller, C. E., and C. S. Henriquez. Finite element analysis of bioelectric phenomena. Crit. Rev. Biomed. Eng.18:207–233, 1990.

    Google Scholar 

  16. 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.

    Google Scholar 

  17. 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.

    Google Scholar 

  18. Panfilov, S., and A. V. Holden Computational Biology of the Heart. New York: Wiley, 1996.

    Google Scholar 

  19. 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.

    Google Scholar 

  20. 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.

    Google Scholar 

  21. Sepulveda, N. G., B. J. Roth, and J. P. Wikswo. Current injection into a two-dimensional anisotropic bidomain. Biophys. J.55:987–999, 1989.

    Google Scholar 

  22. Shampine, L. F., and M. K. Gordon. Computer Solution of Ordinary Differential Equations: The Initial Value Problem. San Francisco: W. H. Freeman, 1975.

    Google Scholar 

  23. 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.

    Google Scholar 

  24. Tyldesley, J. R. An Introduction to Tensor Analysis. New York: Longman, 1975.

    Google Scholar 

  25. Wang, C. Y., J. B. Bassingthwaighte, and L. J. Weissman. Bifurcating distributive system using Monte Carlo method. 16:91–98, 1992.

    Google Scholar 

  26. 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.

    Google Scholar 

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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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