Adaptive Macro Finite Elements for the Numerical Solution of Monodomain Equations in Cardiac Electrophysiology
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Many problems in Biology and Engineering are governed by anisotropic reaction–diffusion equations with a very rapidly varying reaction term. This usually implies the use of very fine meshes and small time steps in order to accurately capture the propagating wave while avoiding the appearance of spurious oscillations in the wave front. This work develops a family of macro finite elements amenable for solving anisotropic reaction–diffusion equations with stiff reactive terms. The developed elements are incorporated on a semi-implicit algorithm based on operator splitting that includes adaptive time stepping for handling the stiff reactive term. A linear system is solved on each time step to update the transmembrane potential, whereas the remaining ordinary differential equations are solved uncoupled. The method allows solving the linear system on a coarser mesh thanks to the static condensation of the internal degrees of freedom (DOF) of the macroelements while maintaining the accuracy of the finer mesh. The method and algorithm have been implemented in parallel. The accuracy of the method has been tested on two- and three-dimensional examples demonstrating excellent behavior when compared to standard linear elements. The better performance and scalability of different macro finite elements against standard finite elements have been demonstrated in the simulation of a human heart and a heterogeneous two-dimensional problem with reentrant activity. Results have shown a reduction of up to four times in computational cost for the macro finite elements with respect to equivalent (same number of DOF) standard linear finite elements as well as good scalability properties.
KeywordsCardiac modeling Efficient numerical schemes Pseudo-adaptive meshes Macro finite elements Monodomain equation Reaction diffusion equations
- 3.Bendahmane, M., R. Bürguer, and R. Ruiz-Baier. A multiresolution space-time adaptive scheme for the bidomain model in electrocardiology. Numer. Methods Partial Differ. Equ. 2010. doi: 10.1002/num.20495
- 4.Bernabeu, M. O., R. Bordas, P. Pathmanathan, J. Pitt-Francis, J. Cooper, A. Garny, D. J. Gavaghan, B. Rodriguez, J. A. Southern, and J. P. Whiteley. Chaste: incorporating a novel multi-scale spatial and temporal algorithm into a large-scale open source library. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 367(1895):1907–1930, 2009.CrossRefGoogle Scholar
- 5.Buttari, A., and S. Filippone. PSBLAS 2.3 User’s Guide. http://www.ce.uniroma2.it/psblas/. University of Rome and Tor Vergata, 2008.
- 11.Felippa, C. Introduction to Finite Element Methods. Boulder: Department of Aerospace Engineering Sciences, University of Colorado at Boulder, 2007.Google Scholar
- 17.Helm, P. A. A novel technique for quantifying variability of cardiac anatomy application to the dyssynchronous failing heart. PhD thesis, Johns Hopkins University, 2005.Google Scholar
- 18.Henriquez, C. S. Simulating the electrical behavior of cardiac tissue using the bidomain model. Crit. Rev. Bioeng. 21:1–77, 1993.Google Scholar
- 20.Hughes, T. J. R. The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Englewwog Cliffs, NJ: Prentice Hall Inc., 672 pp., 1987.Google Scholar
- 22.Katz, A. Physiology of the Heart. Philadelphia, USA: Lippincott Williams and Wilkins, 718 pp., 2001.Google Scholar
- 23.Karypis, G., and V. Kumar. METIS. A Software Package for Partitioning Unstructured Graphs, Partitioning Meshes, and Computing Fill-Reducing Orderings of Sparse Matrices. http://www.glaros.dtc.umn.edu/gkhome/metis/metis/overview. University of Minnesota, Department of Computer Science/Army HPC Research Center, Minneapolis, MN, version 4.0, 1998.
- 24.Keener, J., and J. Sneyd. Mathematical Physiology. New York: Springer-Verlag, 1148 pp., 2008.Google Scholar
- 42.Zienkiewicz, O. C., and R. L. Taylor. Finite Element Method, Vol. 1. Butterworth-Heinemann, Burlington, MA: Elsevier, 752 pp., 2005.Google Scholar