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Multi-Level Parallelism for the Cardiac Bidomain Equations

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Abstract

Cardiovascular diseases are associated with high mortality rates in the globe. The development of new drugs, new medical equipment and non-invasive techniques for the heart demand multidisciplinary efforts towards the characterization of cardiac anatomy and function from the molecular to the organ level. Computational modeling has demonstrated to be a useful tool for the investigation and comprehension of the complex biophysical processes that underlie cardiac function. The set of Bidomain equations is currently one of the most complete mathematical models for simulating the electrical activity in cardiac tissue. Unfortunately, large scale simulations, such as those resulting from the discretization of an entire heart, remain a computational challenge. In order to reduce simulation execution times, parallel implementations have traditionally exploited data parallelism via numerical schemes based on domain-decomposition. However, it has been verified that the parallel efficiency of these implementations severely degrades as the number of processors increases. In this work we propose and implement a new parallel algorithm for the solution of cardiac models. By relaxing the coherence of the execution, a new level of parallelism could be identified and exploited: pipelining. A synchronous parallel algorithm that uses both pipelining and data decomposition techniques was implemented and used the MPI library for communication. Numerical tests were performed in two different cluster configurations. Our preliminary results indicated that the proposed algorithm is able to increase the parallel efficiency up to 20% on an 8-core cluster. On a 32-core cluster the multi-level algorithm was 1.7 times faster than the traditional domain decomposition algorithm. In addition, the numerical precision was kept under control (relative errors under 6%) when the relaxed coherence execution was adopted.

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References

  1. Hodgkin A. L., Huxley A. F.: A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 117, 500–544 (1952)

    Google Scholar 

  2. Plonsey, R.: Bioelectric sources arising in excitable fibers (ALZA lecture). Ann. Biomed. Eng. 16(6), 519–546 (1988). [Online]. Available: http://eutils.ncbi.nlm.nih.gov/entrez/eutils/elink.fcgi?cmd=prlinks&dbfrom=pubmed&retmode=ref&id=3067629

  3. Gima K., Rudy Y.: Ionic current basis of electrocardiographic waveforms: A model study. Circ. Res. 90, 889–896 (2002)

    Article  Google Scholar 

  4. Santos R., Olaf K., Steinhoff U., Bauer S., Trahms L., Koch H.: Mcg to ecg source differences: measurements and a 2d computer model study. J. Electrocardiol. 37(4), 123–127 (2004)

    Google Scholar 

  5. Vigmond E., Aguel F., Trayanova N.: Computational techniques for solving the bidomain equations in three dimensions. IEEE Trans. Biomed. Eng. 49(11), 1260–1269 (2002)

    Article  Google Scholar 

  6. Sundnes, J., Lines, G. T., Tveito, A.: An operator splitting method for solving the bidomain equations coupled to a volume conductor model for the torso. Math. Biosci. 194(2), 233–248 (2005). [Online]. Available: http://eutils.ncbi.nlm.nih.gov/entrez/eutils/elink.fcgi?cmd=prlinks&dbfrom=pubmed&retmode=ref&id=15854678

  7. Weber dos Santos, R., Plank, G., Bauer, S., Vigmond, E.: Parallel multigrid preconditioner for the cardiac bidomain model. IEEE Trans. Biomed. Eng. 51(11), 1960–1968 (2004). [Online]. Available: http://eutils.ncbi.nlm.nih.gov/entrez/eutils/elink.fcgi?cmd=prlinks&dbfrom=pubmed&retmode=ref&id=15536898

  8. Plank G., Liebmann M., Weber dos Santos R., Vigmond E., Haase G.: Algebraic Multigrid Preconditioner for the Cardiac Bidomain Model. IEEE Trans. Biomed. Eng. 54, 585–596 (2007)

    Article  Google Scholar 

  9. Felippa C. A., Park K. C., Farhat C.: Partitioned analysis of coupled mechanical systems. Comput. Methods Appl. Mech. Eng. 190, 3247–3270 (2001)

    Article  MATH  Google Scholar 

  10. Henriquez C.: Simulating the electrical behavior of cardiac tissue using the bidomain model. Crit. Rev. Biomed. Eng. 21(1), 1–77 (1993)

    MathSciNet  Google Scholar 

  11. Strikwerda, J.: Finite difference schemes and partial differential equations. Soc. Ind. Math. (2004)

  12. Message Passing Interface Forum: MPI, a message-passing interface standard. Int. J. Supercomp. 8, 159–416 (1994)

    Google Scholar 

  13. Balay, S., Buschelman, K., Gropp, W.D., Kaushik, D., Knepley, M., McInnes, L.C., Smith, B.F., Zhang, H.: PETSc users manual, Argonne National Laboratory, Tech. Rep. ANL-95/11 - Revision 2.1.5 (2002)

  14. Santos R., Campos F., Neumann L., Nygren A., Giles W.: ATX-II effects on the apparent location of m cells in a computational human left ventricular wedge. J. Cardiovasc. Electrophysiol. 17, S86–S95 (2006)

    Article  Google Scholar 

  15. Bauer, S., dos Santos, R., Schmal, T., Nagel, E., Baer, M., Koch, H.: QRS width and QT time alteration due to geometry change in modelled human cardiac magnetograms. In: Proceedings of IEEE Computers in Cardiology, pp. 639–642 (2005)

  16. Muzikant A., Henriquez C.: Validation of three-dimensional conduction models using experimental mapping: are we getting closer?. Prog. Biophys. Mol. Biol. 69(2–3), 205–223 (1998)

    Article  Google Scholar 

  17. Ten Tusscher K., Noble D., Noble P. J., Panfilov A. V.: A model for human ventricular tissue. Am. J. Physiol. Heart Circ. Physiol. 286(4), 1573–1589 (2004)

    Article  Google Scholar 

  18. Skouibine, K., Trayanova, N., Moore, P.: A numerically efficient model for simulation of defibrillation in an active bidomain sheet of myocardium. Math. Biosci. 166(1), 85–100 (2000) [Online]. Available: http://eutils.ncbi.nlm.nih.gov/entrez/eutils/elink.fcgi?cmd=prlinks&dbfrom=pubmed&retmode=ref&id=10882801

  19. Roth B.: Meandering of spiral waves in anisotropic cardiac tissue. Physica D 150, 127–136 (2001)

    Article  MATH  Google Scholar 

  20. Jacquemet, V., Henriquez, C.: Finite volume stiffness matrix for solving anisotropic cardiac propagation in 2-D and 3-D unstructured meshes. IEEE Trans. Biomed. Eng., 52(8), 1490–1492 (2005). [Online]. Available: http://eutils.ncbi.nlm.nih.gov/entrez/eutils/elink.fcgi?cmd=prlinks&dbfrom=pubmed&retmode=ref&id=16119246

  21. Plank, G., Leon, L. J., Kimber, S., Vigmond, E. J.: Defibrillation depends on conductivity fluctuations and the degree of disorganization in reentry patterns. J. Cardiovasc. Electrophysiol. 16(2), 205–216 (2005). [Online]. Available: http://eutils.ncbi.nlm.nih.gov/entrez/eutils/elink.fcgi?cmd=prlinks&dbfrom=pubmed&retmode=ref&id=15720461

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Authors and Affiliations

  1. Department of Computer Science, Universidade Federal de Juiz de Fora, Juiz de Fora, MG, Brazil

    Carolina Ribeiro Xavier, Rafael Sachetto Oliveira, Vinicius da Fonseca Vieira & Rodrigo Weber dos Santos

  2. Department of Computer Science, Universidade Federal de Minas Gerais, Belo Horizonte, MG, Brazil

    Wagner Meira Jr.

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  1. Carolina Ribeiro Xavier
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  2. Rafael Sachetto Oliveira
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  3. Vinicius da Fonseca Vieira
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  4. Rodrigo Weber dos Santos
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  5. Wagner Meira Jr.
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Corresponding author

Correspondence to Rodrigo Weber dos Santos.

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Xavier, C.R., Oliveira, R.S., da Fonseca Vieira, V. et al. Multi-Level Parallelism for the Cardiac Bidomain Equations. Int J Parallel Prog 37, 572–592 (2009). https://doi.org/10.1007/s10766-009-0110-0

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  • DOI: https://doi.org/10.1007/s10766-009-0110-0

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