Advertisement

Preconditioning Techniques for the Bidomain Equations

  • Rodrigo Weber Dos Santos
  • G. Plank
  • S. Bauer
  • E.J. Vigmond
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 40)

Summary

In this work we discuss parallel preconditioning techniques for the bidomain equations, a non-linear system of partial differential equations which is widely used for describing electrical activity in cardiac tissue. We focus on the solution of the linear system associated with the elliptic part of the bidomain model, since it dominates computation, with the preconditioned conjugate gradient method. We compare different parallel preconditioning techniques, such as block incomplete LU, additive Schwarz and multigrid. The implementation is based on the PETSc library and we report results for a 16-node HP cluster. The results suggest the multigrid preconditioner is the best option for the bidomain equations.

Keywords

Cardiac Tissue Preconditioned Conjugate Gradient Method Parallel Speedup Conjugate Gradient Iteration Incomplete Factorization 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. S. Balay, K. Buschelman, W. Gropp, D. Kaushik, M. Knepley, L. McInnes, B. Smith, and H. Zhang. PETSc users manual. Technical Report ANL-95/11 — Revision 2.1.5, Argonne National Laboratory, 2002.Google Scholar
  2. X.-C. Cai and M. Sarkis. A restricted additive Schwarz preconditioner for general sparse linear systems. SIAM Journal on Scientific Computing, 21: 239–247, 1999.MathSciNetCrossRefGoogle Scholar
  3. J. Eason and R. Malkin. A simulation study evaluating the performance of high-density electrode arrays on myocardial tissue. IEEE Trans Biomed Eng, 47(7):893–901, 2000.CrossRefGoogle Scholar
  4. A. Hodgkin and A. Huxley. A quantitative description of membrane current and its application to conduction and excitation in nerve. Journal of Physiology, 117:500–544, 1952.Google Scholar
  5. N. Hooke, C. Henriquez, P. Lanzkron, and D. Rose. Linear algebraic transformations of the bidomain equations: implications for numerical methods. Math Biosci, 120(2):127–45, 1994.MathSciNetCrossRefGoogle Scholar
  6. J. Keener and K. Bogar. A numerical method for the solution of the bidomain equations in cardiac tissue. Chaos, 8(1):234–241, 1998.CrossRefGoogle Scholar
  7. J. Keener and J. Sneyd. Mathematical physiology. Springer, 1998.Google Scholar
  8. W. Krassowska and J. Neu. Effective boundary conditions for syncytial tissues. IEEE Trans. on Biomed. Eng., 41:143–150, 1994.CrossRefGoogle Scholar
  9. D. Latimer and B. Roth. Electrical stimulation of cardiac tissue by a bipolar electrode in a conductive bath. IEEE Trans. on Biomed. Eng., 45(12): 1449–1458, 1998.CrossRefGoogle Scholar
  10. D. Lindblad, C. Murphey, J. Clark, and W. Giles. A model of the action potential and the underlying membrane currents in a rabbit atrial cell. The American Physiological Society, (0363-6125):H1666–H1696, 1996.Google Scholar
  11. Message Passing Interface library. MPI, a message-passing interface standard. Int. J. Supercomp., 8:159–416, 1994.Google Scholar
  12. L. Pavarino and P. Franzone. Parallel solution of cardiac reaction-diffusion models. In R. Kornhuber, R. Hoppe, D. Keyes, J. Periaux, O. Pironneau, and J. Xu, editors, Procedings of the 15th International Conference on Domain Decomposition Methods, Lecture Notes in Computational Science and Engineering. Springer, 2004.Google Scholar
  13. M. Pennacchio and V. Simoncini. Efficient algebraic solution of reaction-diffusion systems for the cardiac excitation process. Journal of Computational and Applied Mathematics, 145(1):49–70, 2002. ISSN 0377-0427.MathSciNetCrossRefGoogle Scholar
  14. J. Pormann. Computer simulations of cardiac electrophysiology. In Proceedings of SC2000, 2000.Google Scholar
  15. H. Saleheen and Kwong. A new three-dimensional finite-difference bidomain formulation for inhomogeneous anisotropic cardiac tissues. IEEE Trans. on Biomed. Eng., 45(1):15–25, 1998.CrossRefGoogle Scholar
  16. K. Skouibine and W. Krassowska. Increasing the computational efficiency of a bidomain model of defibrillation using a time-dependent activating function. Annals of Biomedical Engineering, 28:772–780, 2000.CrossRefGoogle Scholar
  17. G. Strang. On the construction and comparison of difference scheme. SIAM Journal on Numerical Analysis, 5:506–517, 1968.MATHMathSciNetCrossRefGoogle Scholar
  18. A. Street and R. Plonsey. Propagation in cardiac tissue adjacent to connective tissue: Two-dimensional modeling studies. IEEE Transactions on Biomedical Engineering, 46:19–25, 1999.CrossRefGoogle Scholar
  19. J. Sundnes, G. Lines, and A. Tveito. Efficient solution of ordinary differential equations modeling electrical activity in cardiac cells. Math Biosci, 172(2): 55–72, 2001.MathSciNetCrossRefGoogle Scholar
  20. E. Vigmond, F. Aguel, and N. Trayanova. Computational techniques for solving the bidomain equations in three dimensions. IEEE Trans Biomed Eng, 49(11):1260–9, 2002.CrossRefGoogle Scholar
  21. R. Weber dos Santos. Modelling cardiac electrophysiology. PhD thesis, Federal University of Rio de Janeiro, Mathematics dept., Rio de Janeiro, Brazil, 2002.Google Scholar
  22. R. Weber dos Santos and F. Dickstein. On the influence of a volume conductor on the orientation of currents in a thin cardiac tissue. In I. Magnin, J. Montagnat, P. Clarysse, J. Nenonen, and T. Katila, editors, Lecture Notes in Computer Science, pages 111–121. Springer, Berlin, 2003.Google Scholar
  23. R. Weber dos Santos, U. Steinhoff, E. Hofer, D. Sanchez-Quintana, and H. Koch. Modelling the electrical propagation in cardiac tissue using detailed histological data. Biomedizinische Technik, 2003.Google Scholar
  24. C. Yung. Application of a stiff, operator-splitting scheme to the computational modeling of electrical propagation of cardiac ventricles. Engineering dept., Johns Hopkins University, Maryland, 2000.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Rodrigo Weber Dos Santos
    • 1
  • G. Plank
    • 2
  • S. Bauer
    • 1
  • E.J. Vigmond
    • 3
  1. 1.Dept. of BiosignalsPhysikalisch-Technische BundesanstaltBerlinGermany
  2. 2.Inst. für Medizinische Physik und BiophysikUniversität GrazAustria
  3. 3.Dept. of Electrical and Computer EngineeringUniversity of CalgaryCanada

Personalised recommendations