Scalable Cardiac Electro-Mechanical Solvers and Reentry Dynamics

  • P. Colli Franzone
  • L. F. Pavarino
  • S. Scacchi
  • Stefano Zampini
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 125)


We present a scalable solver for the three-dimensional cardiac electro-mechanical coupling (EMC) model, which represents, currently, the most complete mathematical description of the interplay between the electrical and mechanical phenomena occurring during a heartbeat. The most computational demanding parts of the EMC model are: the electrical current flow model of the cardiac tissue, called Bidomain model, consisting of two non-linear partial differential equations of reaction-diffusion type; the quasi-static finite elasticity model for the deformation of the cardiac tissue. Our finite element parallel solver is based on: Block Jacobi and Multilevel Additive Schwarz preconditioners for the solution of the linear systems deriving from the discretization of the Bidomain equations; Newton-Krylov-Algebraic-Multigrid or Newton-Krylov-BDDC algorithms for the solution of the non-linear algebraic system deriving from the discretization of the finite elasticity equations. Three-dimensional numerical test on two linux clusters show the effectiveness and scalability of the EMC solver in simulating both physiological and pathological cardiac dynamics.


  1. 1.
    S. Balay et al., PETSc users manual. Tech. Rep. ANL-95/11 - Revision 3.3, Argonne National Laboratory, 2012.Google Scholar
  2. 2.
    P. Colli Franzone, L.F. Pavarino, S. Scacchi, Mathematical Cardiac Electrophysiology, MSA, vol. 13 (Springer, New York, 2014)zbMATHGoogle Scholar
  3. 3.
    P. Colli Franzone, L.F. Pavarino, S. Scacchi, Bioelectrical effects of mechanical feedbacks in a strongly coupled cardiac electro-mechanical model. Math. Models Methods Appl. Sci. 26, 27–57 (2016)MathSciNetCrossRefGoogle Scholar
  4. 4.
    P. Colli Franzone, L.F. Pavarino, S. Scacchi, Effects of mechanical feedback on the stability of cardiac scroll waves: a bidomain electro-mechanical simulation study. Chaos 27, 093905 (2017)MathSciNetCrossRefGoogle Scholar
  5. 5.
    C.R. Dohrmann, A preconditioner for substructuring based on constrained energy minimization. SIAM J. Sci. Comput. 25, 246–258 (2003)MathSciNetCrossRefGoogle Scholar
  6. 6.
    T.S.E. Eriksson et al., Influence of myocardial fiber/sheet orientations on left ventricular mechanical contraction. Math. Mech. Solids 18, 592–606 (2013)MathSciNetCrossRefGoogle Scholar
  7. 7.
    V. Gurev et al., Models of cardiac electromechanics based on individual hearts imaging data: image-based electromechanical models of the heart. Biomech. Model Mechanobiol. 10, 295–306 (2011)CrossRefGoogle Scholar
  8. 8.
    A. Klawonn, O. Rheinbach, Highly scalable parallel domain decomposition methods with an application to biomechanics. ZAMM-Z. Angew. Math. Mech. 90, 5–32 (2010)MathSciNetCrossRefGoogle Scholar
  9. 9.
    A. Klawonn, O.B. Widlund, Dual-primal FETI methods for linear elasticity. Commun. Pure Appl. Math., 59, 1523–1572 (2006)MathSciNetCrossRefGoogle Scholar
  10. 10.
    S. Land et al., An analysis of deformation-dependent electromechanical coupling in the mouse heart. J. Physiol. 590, 4553–4569 (2012)CrossRefGoogle Scholar
  11. 11.
    J. Mandel, C.R. Dohrmann, Convergence of a balancing domain decomposition by constraints and energy minimization. Linear Algebra Appl. 10, 639–659 (2003)MathSciNetCrossRefGoogle Scholar
  12. 12.
    S.A. Niederer, N.P. Smith, A mathematical model of the slow force response to stretch in rat ventricular myocites. Biophys. J. 92, 4030–4044 (2007)CrossRefGoogle Scholar
  13. 13.
    L.F. Pavarino, S. Scacchi, Multilevel additive Schwarz preconditioners for the Bidomain reaction-diffusion system. SIAM J. Sci. Comput. 31, 420–443 (2008)MathSciNetCrossRefGoogle Scholar
  14. 14.
    L.F. Pavarino, S. Zampini, O.B. Widlund, BDDC preconditioners for spectral element discretizations of almost incompressible elasticity in three dimensions. SIAM J. Sci. Comput. 32(6), 3604–3626 (2010)MathSciNetCrossRefGoogle Scholar
  15. 15.
    L.F. Pavarino, S. Scacchi, S. Zampini, Newton-krylov-BDDC solvers for non-linear cardiac mechanics. Comput. Methods Appl. Mech. Eng. 295, 562–580 (2015)CrossRefGoogle Scholar
  16. 16.
    S. Rossi et al., Orthotropic active strain models for the numerical simulation of cardiac biomechanics. Int. J. Numer. Methods Biomed. Eng. 28, 761–788 (2012)MathSciNetCrossRefGoogle Scholar
  17. 17.
    J. Sundnes et al., Improved discretisation and linearisation of active tension in strongly coupled cardiac electro-mechanics simulations. Comput. Methods Biomech. Biomed. Eng. 17, 604–615 (2014)CrossRefGoogle Scholar
  18. 18.
    K.H.W.J. ten Tusscher et al., A model for human ventricular tissue. Am. J. Phys. Heart. Circ. Physiol. 286, H1573–H1589 (2004)CrossRefGoogle Scholar
  19. 19.
    K.H.W.J. ten Tusscher, A.V. Panfilov, Alternans and spiral breakup in a human ventricular tissue model. Am. J. Physiol. Heart Circ. Physiol. 291, H1088–H1100 (2006)CrossRefGoogle Scholar
  20. 20.
    A. Toselli, O.B. Widlund, Domain Decomposition Methods: Algorithms and Theory (Springer, Berlin, 2004)zbMATHGoogle Scholar
  21. 21.
    F.J. Vetter, A.D. McCulloch, Three-dimensional stress and strain in passive rabbit left ventricle: a model study. Ann. Biomed. Eng. 28, 781–792 (2000)CrossRefGoogle Scholar
  22. 22.
    S. Zampini, Dual-primal methods for the cardiac bidomain model. Math. Models Methods Appl. Sci. 24, 667–696 (2014)MathSciNetCrossRefGoogle Scholar
  23. 23.
    S. Zampini, PCBDDC: a class of robust dual-primal preconditioners in PETSc. SIAM J. Sci. Comput. 38(5), S282–S306 (2016)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.University of PaviaDept. of MathematicsPaviaItaly
  2. 2.University of MilanoDept. of MathematicsMilanoItaly
  3. 3.Extreme Computing Research Center, Computer Electrical and Mathematical Sciences & Engineering DepartmentKing Abdullah University of Science and TechnologyThuwalSaudi Arabia

Personalised recommendations