Advertisement

Gating-enhanced IMEX splitting methods for cardiac monodomain simulation

  • Kevin R. GreenEmail author
  • Raymond J. Spiteri
Original Paper

Abstract

The electrical activity in excitable cardiac tissue can be simulated using the so-called monodomain model. The monodomain model is a continuum-based multi-scale model that consists of non-linear ordinary differential equations describing the electrical activity at the cellular scale along with a semi-linear parabolic partial differential equation describing electrical propagation at the tissue scale. The standard “scale-based” splitting method for simulating the monodomain model is to split the tissue and cell models, applying different integrators to each. Typically, the tissue model is simulated with an implicit time-integration method, and the cell model is simulated with an explicit or explicit-exponential one. We demonstrate that the application of implicit-explicit (IMEX) linear multistep or Runge–Kutta methods to this splitting can have poor stability properties when the cell model is stiff. We propose a novel “gating-enhanced” IMEX splitting that treats the tissue variable and the (typically stiff) cell model gating variables together implicitly. The performance of 14 different IMEX methods using both splittings is measured in a variety of one- and two-dimensional experiments. The low incremental overhead combined with the substantially improved stability of the gating-enhanced splitting is shown to result in a performance increase of approximately a factor of four for simulations of the monodomain model with the stiff ten Tusscher–Panfilov model of human endocardial cells.

Keywords

Monodomain model Cardiac electrophysiology Implicit-explicit time-integration methods 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

References

  1. 1.
    Ascher, U.M., Ruuth, S.J., Spiteri, R.J.: Implicit-explicit Runge–Kutta methods for time-dependent partial differential equations. App. Num. Math 25(2-3), 151–167 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ascher, U.M., Ruuth, S.J., Wetton, B.T.R.: Implicit-explicit methods for time-dependent partial differential equations. SIAM J. Num. Analy 32(3), 797–823 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Auckland Bioengineering Institute: The CellML project., http://www.cellml.org/ (2011)
  4. 4.
    Burrage, K., Butcher, J.: Non-linear stability of a general class of differential equation methods. BIT 20, 185–203 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cervi, J., Spiteri, R.J.: High-order operator splitting for the bidomain and monodomain models. SIAM J. Sci. Comput. 40(2), A769–A786 (2018).  https://doi.org/10.1137/17M1137061 MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cooper, J., Spiteri, R.J., Mirams, G.R.: Cellular cardiac electrophysiology modeling with chaste and cellml. Front. Physiol. 5, 511 (2015).  https://doi.org/10.3389/fphys.2014.00511. https://www.frontiersin.org/article/10.3389/fphys.2014.00511 CrossRefGoogle Scholar
  7. 7.
    Ethier, M., Bourgault, Y.: Semi-implicit time-discretization schemes for the bidomain model. SIAM J. Numer. Anal. 46(5), 2443–2468 (2008).  https://doi.org/10.1137/070680503 MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    FitzHugh, R.: Impulses and physiological states in theoretical models of nerve membrane. Biophys. J 1(6), 445–466 (1961)CrossRefGoogle Scholar
  9. 9.
    Hooke, N., Henriquez, C., Lanzkron, P., Rose, D.: Linear algebraic transformations of the bidomain equations: Implications for numerical methods. Math. Biosci. 120(2), 127–145 (1994).  https://doi.org/10.1016/0025-5564(94)90049-3. http://www.sciencedirect.com/science/article/pii/0025556494900493 MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Karniadakis, G., Sherwin, S.: Spectral/hp element methods for computational fluid dynamics, 2nd edn. Oxford University Press, Oxford (2005)CrossRefzbMATHGoogle Scholar
  11. 11.
    Keener, J.P., Bogar, K.: A numerical method for the solution of the bidomain equations in cardiac tissue. Chaos: An Interdisciplinary Journal of Nonlinear Science 8(1), 234–241 (1998).  https://doi.org/10.1063/1.166300 CrossRefzbMATHGoogle Scholar
  12. 12.
    Marsh, M.E., Torabi Ziaratgahi, S., Spiteri, R.J.: The secrets to the success of the rush-larsen method and its generalizations. IEEE Trans. Biomed. Eng. 59(9), 2506–2515 (2012).  https://doi.org/10.1109/TBME.2012.2205575 CrossRefGoogle Scholar
  13. 13.
    Mirin, A.A., Richards, D.F., Glosli, J.N., Draeger, E.W., Chan, B., Fattebert, J.L., Krauss, W.D., Oppelstrup, T., Rice, J.J., Gunnels, J.A., Gurev, V., Kim, C., Magerlein, J., Reumann, M., Wen, H.F.: Toward real-time modeling of human heart ventricles at cellular resolution: Simulation of drug-induced arrhythmias, pp 2:1–2:11. IEEE Computer Society Press, Los Alamitos (2012). http://dl.acm.org/citation.cfm?id=2388996.2388999 Google Scholar
  14. 14.
    Nektar++: Spetral/hp Element Framework. Users Guide - Version 4.4.1: http://doc.nektar.info/userguide/4.4.1 (2017). [Online; accessed 24-Jan-2019]
  15. 15.
    Niederer, S.A., Kerfoot, E., Benson, A.P., Bernabeu, M.O., Bernus, O., Bradley, C., Cherry, E.M., Clayton, R., Fenton, F.H., Garny, A., Heidenreich, E., Land, S., Maleckar, M., Pathmanathan, P., Plank, G., Rodriguez, J.F., Roy, I., Sachse, F.B., Seemann, G., Skavhaug, O., Smith, N.P.: Verification of cardiac tissue electrophysiology simulators using an N-version benchmark. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 369(1954), 4331–4351 (2011).  https://doi.org/10.1098/rsta.2011.0139 MathSciNetCrossRefGoogle Scholar
  16. 16.
    Pandit, S.V., Clark, R.B., Giles, W.R., Demir, S.S.: A mathematical model of action potential heterogeneity in adult rat left ventricular myocytes. Biophys. J. 81(6), 3029–3051 (2001).  https://doi.org/10.1016/S0006-3495(01)75943-7. http://www.sciencedirect.com/science/article/pii/S0006349501759437 CrossRefGoogle Scholar
  17. 17.
    Richardson, L.F.: On the approximate arithmetical solution by finite differences of physical problems involving differential equations, with an application to the stresses in a masonry dam. Proc R Soc London A: Math Phys Eng Sci 83(563), 335–336 (1910).  https://doi.org/10.1098/rspa.1910.0020. http://rspa.royalsocietypublishing.org/content/83/563/335 CrossRefzbMATHGoogle Scholar
  18. 18.
    Spiteri, R.J., Dean, R.C.: Stiffness analysis of cardiac electrophysiological models. Ann. Biomed. Eng. 38(12), 3592–3604 (2010).  https://doi.org/10.1007/s10439-010-0100-9 CrossRefGoogle Scholar
  19. 19.
    Spiteri, R.J., Torabi Ziaratgahi, S.: Operator splitting for the bidomain model revisited. J. Comput. Appl. Math. 296, 550–563 (2016).  https://doi.org/10.1016/j.cam.2015.09.015. http://linkinghub.elsevier.com/retrieve/pii/S0377042715004677 MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Sundnes*, J., Artebrant, R., Skavhaug, O., Tveito, A.: A second-order algorithm for solving dynamic cell membrane equations. IEEE Trans. Biomed. Eng. 56(10), 2546–2548 (2009).  https://doi.org/10.1109/TBME.2009.2014739 CrossRefGoogle Scholar
  21. 21.
    Sundnes, J., Lines, G.T., Cai, X., Nielsen, B.F., Mardal, K.A., Tveito, A.: Computing the electrical activity in the heart. Springer-Verlag, Berlin (2006)zbMATHGoogle Scholar
  22. 22.
    Tung, L.: A bi-domain model for describing ischemic myocardial D-C potentials. Ph.D. thesis, MIT (978). Department of Electrical Engineering and Computer ScienceGoogle Scholar
  23. 23.
    ten Tusscher, K., Noble, D., Noble, P.J., Panfilov, A.V.: A model for human ventricular tissue. AJP - Heart and Circulatory Physiology 286(4), 1573–1589 (2004). http://ajpheart.physiology.org/cgi/content/abstract/286/4/H1573 CrossRefGoogle Scholar
  24. 24.
    ten Tusscher, K.H.W.J., Panfilov, A.V.: Alternans and spiral breakup in a human ventricular tissue model. Am. J. Physiol. Heart Circ. Physiol. 291(3), 1088–1100 (2006).  https://doi.org/10.1152/ajpheart.00109.2006 CrossRefGoogle Scholar
  25. 25.
    Vos, P.E.J., Eskilsson, C., Bolis, A., Chun, S., Kirby, R.M., Sherwin, S.J.: A generic framework for time-stepping partial differential equations (pdes): general linear methods, object-oriented implementation and application to fluid problems. Int. J. Compt. Fluid. Dyn. 25(3), 107–125 (2011).  https://doi.org/10.1080/10618562.2011.575368 MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Vos, P.E.J., Sherwin, S.J., Kirby, M.R.: From h to p efficiently: Implementing finite and spectral/hp element discretisations to achieve optimal performance at low and high order approximations. J. Compt. Phys. 229(13), 5161–5181 (2010).  https://doi.org/10.1016/j.jcp.2010.03.031 CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of SaskatchewanSaskatoonCanada
  2. 2.Department of Mathematics and StatisticsUniversity of SaskatchewanSaskatoonCanada

Personalised recommendations