Electro-Mechanical Modeling and Simulation of Reentry Phenomena in the Presence of Myocardial Infarction

  • Piero Colli Franzone
  • Luca F. Pavarino
  • Simone ScacchiEmail author
Part of the SEMA SIMAI Springer Series book series (SEMA SIMAI, volume 16)


In this work we present a parallel solver for the numerical simulation of the cardiac electro-mechanical activity. We first review the most complete mathematical model of cardiac electro-mechanics, the so-called electro-mechanical coupling (EMC) model, which consists of the following four sub-models, strongly coupled together: the Bidomain model for the electrical activity at tissue scale, constituted by a parabolic system of two reaction-diffusion partial differential equations (PDEs); the finite elasticity system for the mechanical behavior at tissue scale; the membrane model for the bioelectrical activity at cellular scale, consisting of a stiff system of ordinary differential equations (ODEs); the active tension model for the mechanical activity at cellular scale, consisting of a system of ODEs. The discretization of the EMC model is performed by finite elements in space and an operator splitting strategy in time, based on semi-implicit finite differences. As a result of the discretization techniques adopted, the most computational demanding part at each time step is the solution of the non-linear algebraic system, deriving from the discretization of the finite elasticity equations, and of the linear system deriving from the discretization of the Bidomain equations. The former is solved by a Newton-GMRES-BDDC solver, i.e. the Jacobian system at each Newton iteration is solved by GMRES accelerated by the Balancing Domain Decomposition by Constraints (BDDC) preconditioner. The latter is solved by the Conjugate Gradient method, preconditioned by the Multilevel Additive Schwarz preconditioner. The performance of the resulting parallel solver is studied on the simulation of the induction of ventricular tachycardia in an idealized left ventricle affected by an infarct scar. The simulations are run on the Marconi-KNL cluster of the Cineca laboratory.


  1. 1.
    Adeniran, I., Hancox, J.C., Zhang, H.: Effect of cardiac ventricular mechanical contraction on the characteristics of the ECG: a simulation study. J. Biomed. Sci. Eng. 6, 47–60 (2013)CrossRefGoogle Scholar
  2. 2.
    Ambrosi, D., Pezzuto, S.: Active stress vs. active strain in mechanobiology: constitutive issues. J. Elast. 107, 199–212 (2012)Google Scholar
  3. 3.
    Arevalo, H.J., Vadakkumpadan, F., Guallar, E., Jebb, A., Malamas, P., Wu, K.C., Trayanova, N.A.: Arrhythmia risk stratification of patients after myocardial infarction using personalized heart models. Nat. Commun. 7, 11437 (2016)CrossRefGoogle Scholar
  4. 4.
    Augustin, C.M., Neic, A., Liebmann, M., Prassl, A.J., Niederer, S.A., Haase, G., Plank, G.: Anatomically accurate high resolution modeling of human whole heart electromechanics: A strongly scalable algebraic multigrid solver method for nonlinear deformation. J. Comput. Phys. 305, 622–646 (2016)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Austin, T.M., Trew, M.L., Pullan, A.J.: Solving the cardiac Bidomain equations for discontinuous conductivities. IEEE Trans. Biomed. Eng. 53(7), 1265–1272 (2006)CrossRefGoogle Scholar
  6. 6.
    Balay, S., Abhyankar, S., Adams, M.F., Brown, J., Brune, P., Buschelman, K., Dalcin, L., Eijkhout, V., Gropp, W.D., Kaushik, D., Knepley, M.G., Curfman McInnes, L., Rupp, K., Smith, B.F., Zampini, S., Zhang, H.: PETSc Web page,, 2015
  7. 7.
    Bers, D.M.: Excitation-Contraction Coupling and Cardiac Contractile Force, 2nd edn. Kluwer Academic Publisher (2001)Google Scholar
  8. 8.
    Brands, D., Klawonn, A., Rheinbach, O., Schroeder, J.: Modelling and convergence in arterial wall simulations using a parallel FETI solution strategy. Comput. Methods Biomech. Biomed. Eng. 11(5), 569–583 (2008)CrossRefGoogle Scholar
  9. 9.
    Cabo, C., Boyden, P.: Electrical remodeling of the epicardial border zone in the canine infarcted heart: a computational analysis. Am. J. Physiol. Heart Circ. Physiol. 284, H372–H384 (2003)CrossRefGoogle Scholar
  10. 10.
    Chabiniok, R., Wang, V.Y., Hadjicharalambous, M., Asner, L., Lee, J., Sermesant, M., Kuhl, E., Young, A.A., Moireau, P., Nash, M.P., Chapelle, D., Nordsletten, D.A.: Multiphysics and multiscale modelling, data-model fusion and integration of organ physiology in the clinic: ventricular cardiac mechanics. Interface Focus 6, 20150083 (2016)CrossRefGoogle Scholar
  11. 11.
    Cherubini, C., Filippi, S., Gizzi, A., Ruiz-Baier, R.: A note on stress-driven anisotropic diffusion and its role in active deformable media. J. Theor. Biol. 430, 221–228 (2017)CrossRefGoogle Scholar
  12. 12.
    Cherubini, C., Filippi, S., Nardinocchi, P., Teresi, L.: An electromechanical model of cardiac tissue: constitutive issues and electrophysiological effects. Prog. Biophys. Mol. Biol. 97, 562–573 (2008)CrossRefGoogle Scholar
  13. 13.
    Colli Franzone, P., Guerri, L., Pennacchio, M., Taccardi, B.: Spread of excitation in 3-D models of he anisotropic cardiac tissue. II. Effects of fiber architecture and ventricular geometry. Math. Biosci. 147, 131–171 (1998)zbMATHGoogle Scholar
  14. 14.
    Colli Franzone, P., Pavarino, L.F.: A parallel solver for reaction-diffusion systems in computational electrocardiology. Math. Models Methods. Appl. Sci. 14(6), 883–911 (2004)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Colli Franzone, P., Pavarino, L.F., Scacchi, S.: A comparison of coupled and uncoupled solvers for the cardiac Bidomain model. ESAIM: Math. Mod. Numer. Anal. 47(4), 1017–1035 (2013)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Colli Franzone, P., Pavarino, L.F., Scacchi, S.: Mathematical Cardiac Electrophysiology, MSA Vol. 13. Springer, New York (2014)Google Scholar
  17. 17.
    Colli Franzone, P., Pavarino, L.F., Scacchi, S.: Bioelectrical effects of mechanical feedbacks in a strongly coupled cardiac electro-mechanical model. Math. Mod. Meth. Appl. Sci. 26(1), 27–57 (2016)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Colli Franzone, P., Pavarino, L.F., Scacchi, S.: Joint influence of transmural heterogeneities and wall deformation on cardiac bioelectrical activity: A simulation study. Math. Biosci. 280, 71–86 (2016)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Colli Franzone, P., Pavarino, L.F., Scacchi, S.: Effects of mechanical feedback on the stability of cardiac scroll waves: A bidomain electro-mechanical simulation study. Chaos 27, 093905 (2017)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Coppola1, B.A., Omens, J.H.: Role of tissue structure on ventricular wall mechanics. Mol. Cell. Biomech. 5(3), 183–196 (2008)Google Scholar
  21. 21.
    Costa, K.D., Holmes, J.W., McCulloch, A.D.: Modelling cardiac mechanical properties in three dimensions. Philos. Trans. R. Soc. Lond. A 359, 1233–1250 (2001)CrossRefGoogle Scholar
  22. 22.
    Dohrmann, C.R.: A Preconditioner for substructuring based on constrained energy minimization. SIAM J. Sci. Comput. 25, 246–258 (2003)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Doll, S., Schweizerhof, K.: On the development of volumetric strain energy functions. J. Appl. Mech. 67, 17–21 (2000)CrossRefGoogle Scholar
  24. 24.
    Eriksson, T.S.E., Prassl, A.J., Plank, G., Holzapfel, G.A.: Influence of myocardial fiber/sheet orientations on left ventricular mechanical contraction. Math. Mech. Solids 18, 592–606 (2013)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Farhat, C., Lesoinne, M., LeTallec, P., Pierson, K., Rixen, D.: FETI-DP: a dual-primal unified FETI method – part I. A faster alternative to the two-level FETI method. Int. J. Numer. Meth. Eng. 50, 1523–1544 (2001)zbMATHGoogle Scholar
  26. 26.
    Gerardo Giorda, L., Mirabella, L., Nobile, F., Perego, M., Veneziani, A.: A model-based block-triangular preconditioner for the Bidomain system in electrocardiology. J. Comput. Phys. 228(10), 3625–3639 (2009)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Geselowitz, D.B., Miller, W.T.: A bidomain model for anisotropic cardiac muscle. Ann. Biomed. Eng. 11, 191–206 (1983)CrossRefGoogle Scholar
  28. 28.
    G\(\ddot {\mbox{o}}\)ktepe, S., Kuhl, E.: Electromechanics of the heart - A unified approach to the strongly coupled excitation-contraction problem. Comput. Mech. 80, 227–243 (2010)Google Scholar
  29. 29.
    Guccione, J.M., McCulloch, A.D., Waldman, L.K.: Passive material properties of intact ventricular myocardium determined from a cylindrical model. J. Biomech. Eng. 113(1), 42–55 (1991)CrossRefGoogle Scholar
  30. 30.
    Guccione, J.M., Costa, K.D., McCulloch, A.D.: Finite element stress analysis of left ventricular mechanics in the beating dog heart. J. Biomech. 28, 1167–1177 (1995)CrossRefGoogle Scholar
  31. 31.
    Gurev, V., Constantino, J., Rice, J.J., Trayanova, N.A.: Distribution of electromechanical delay in the heart: insights from a three-dimensional electromechanical model. Biophys. J. 99, 745–754 (2010)CrossRefGoogle Scholar
  32. 32.
    Hodgkin, A., Huxley, A.: A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 117, 500–544 (1952)CrossRefGoogle Scholar
  33. 33.
    Holmes, J.W., Borg, T.K., Covell, J.W.: Structure and mechanics of healing myocardial infarcts. Annu. Rev. Biomed. Eng. 7, 223–253 (2005)CrossRefGoogle Scholar
  34. 34.
    Holmes, J.W., Laksman, Z., Gepstein, L.: Making better scar: Emerging approaches for modifying mechanical and electrical properties following infarction and ablation. Prog. Biophys. Mol. Biol. 120(1–3), 134–148 (2016)CrossRefGoogle Scholar
  35. 35.
    Holzapfel, G.A., Ogden, R.W.: Constitutive modelling of passive myocardium. A structurally-based framework for material characterization. Philos. Trans. R. Soc. Lond. A 367, 3445–3475 (2009)CrossRefGoogle Scholar
  36. 36.
    Humphrey, J.D.: Cardiovascular Solid Mechanics, Cells, Tissues and Organs. Springer, New York (2001)Google Scholar
  37. 37.
    Hunter, P.J., McCulloch, A.D., ter Keurs, H.E.D.J.: Modelling the mechanical properties of cardiac muscle. Prog. Biophys. Mol. Biol. 69, 289–331 (1998)CrossRefGoogle Scholar
  38. 38.
    Hunter, P.J., Nash, M.P., Sands, G.B.: Computational electromechanics of the heart. In: Panfilov, A.V., Holden, A.V. (eds.) Computational Biology of the Heart. Wiley (1997)Google Scholar
  39. 39.
    Jie, X., Gurev, V., Trayanova, N.A.: Mechanisms of mechanically induced spontaneous arrhythmias in acute regional ischemia. Circ. Res. 106, 185–192 (2010)CrossRefGoogle Scholar
  40. 40.
    Keldermann, R.H., Nash, M.P., Gelderblom, H., Wang, V.Y., Panfilov, A.V.: Electromechanical wavebreak in a model of the human left ventricle. Am. J. Physiol. Heart Circ. Physiol. 299, H134–H143 (2010)CrossRefGoogle Scholar
  41. 41.
    Kerckhoffs, R.C.P., Bovendeerd, P.H.M., Kotte, J.C.S., Prinzen, F.W., Smits, K., Arts, T.: Homogeneity of cardiac contraction despite physiological asyncrony of depolarization: a model study. Ann. Biomed. Eng. 31, 536–547 (2003)CrossRefGoogle Scholar
  42. 42.
    Kerckhoffs, R.C.P., Neal, M.L., Gu, Q., Bassingthwaighte, J.B., Omens, J.H., McCulloch, A.D.: Coupling of a 3D finite element model of cardiac ventricular mechanics to lumped systems models of the systemic and pulmonic circulation. Ann. Biomed. Eng. 35(1), 1–18 (2007)CrossRefGoogle Scholar
  43. 43.
    Klawonn, A., Rheinbach, O.: Highly scalable parallel domain decomposition methods with an application to biomechanics. ZAMM - Z. Angew. Math. Mech. 90(1), 5–32 (2010)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Klawonn, A., Widlund, O.B.: Dual-primal FETI methods for linear elasticity. Comm. Pure Appl. Math. 59, 1523–1572 (2006)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Kohl, P., Sachs, F., Franz, M.R.: Cardiac Mechano-Electric Feedback and Arrhythmias: From Pipette to Patient. Elsevier Sauders (2011)Google Scholar
  46. 46.
    Krishnamurthy, A., Villongco, C.T., Chuang, J., Frank, L.R., Nigam, V., Belezzuoli, E., Stark, P., Krummen, D.E., Narayan, S., Omens, J.H., McCulloch, A.D., Kerckhoffs, R.C.P.: Patient-specific models of cardiac biomechanics. J. Comput. Phys. 244, 4–21 (2013)CrossRefGoogle Scholar
  47. 47.
    Land, S., Niederer, S.A., Aronsen, J.M., Espe, E.K.S., Zhang, L.L., Louch, W.E., Sjaastad, I., Sejersted, O.M., Smith, N.P.: An analysis of deformation-dependent electromechanical coupling in the mouse heart. J. Physiol. 590, 4553–4569 (2012)CrossRefGoogle Scholar
  48. 48.
    LeGrice, I.J., Smaill, B.H., Chai, L.Z., Edgar, S.G., Gavin, J.B., Hunter, P.J.: Laminar structure of the heart: ventricular myocyte arrangement and connective tissue architecture in the dog. Am. J. Physiol. Heart Circ. Physiol. 269, H571–H582 (1995)CrossRefGoogle Scholar
  49. 49.
    Li, X.T., Dyachenko, V., Zuzarte, M., Putzke, C., Preisig-Muller, R., Isenberg, G., Daut, J.: The stretch-activated potassium channel TREK-1 in rat cardiac ventricular muscle. Cardiovasc. Res. 69, 86–97 (2006)CrossRefGoogle Scholar
  50. 50.
    Li, J., Widlund, O.B.: FETI-DP, BDDC, and block Cholesky methods. Int. J. Numer. Meth. Eng. 66(2), 250–271 (2006)MathSciNetCrossRefGoogle Scholar
  51. 51.
    Mandel, J., Dohrmann, C.R.: Convergence of a balancing domain decomposition by constraints and energy minimization. Numer. Lin. Alg. Appl. 10(7), 639–659 (2003)MathSciNetCrossRefGoogle Scholar
  52. 52.
    Mandel, J., Dohrmann, C.R., Tezaur, R.: An algebraic theory for primal and dual substructuring methods by constraints. Appl. Numer. Math. 54(2), 167–193 (2005)MathSciNetCrossRefGoogle Scholar
  53. 53.
    Mardal, K.-A., Nielsen, B.F., Cai, X., Tveito, A.: An order optimal solver for the discretized bidomain equations. Numer. Linear Algebr. Appl. 14(2), 83–98 (2007)MathSciNetCrossRefGoogle Scholar
  54. 54.
    Munteanu, M., Pavarino, L.F.: Decoupled Schwarz algorithms for implicit discretization of nonlinear Monodomain and Bidomain systems. Math. Mod. Meth. Appl. Sci. 19(7), 1065–1097 (2009)MathSciNetCrossRefGoogle Scholar
  55. 55.
    Munteanu, M., Pavarino, L.F., Scacchi, S.: A scalable Newton-Krylov-Schwarz method for the Bidomain reaction-diffusion system. SIAM J. Sci. Comput. 31(5), 3861–3883 (2009)MathSciNetCrossRefGoogle Scholar
  56. 56.
    Murillo, M., Cai, X.: A fully implicit parallel algorithm for simulating the nonlinear electrical activity of the heart. Numer. Linear Algebr. Appl. 11, 261–277 (2004)CrossRefGoogle Scholar
  57. 57.
    Nardinocchi, P., Teresi, L.: Electromechanical modeling of anisotropic cardiac tissues. Math. Mech. Sol. 18(6), 576–591 (2013)MathSciNetCrossRefGoogle Scholar
  58. 58.
    Nash, M.P., Hunter, P.J.: Computational mechanics of the heart. From tissue structure to ventricular function. J. Elast. 61, 113–141 (2000)zbMATHGoogle Scholar
  59. 59.
    Nash, M.P., Panfilov, A.V.: Electromechanical model of excitable tissue to study reentrant cardiac arrhythmias. Prog. Biophys. Mol. Biol. 85, 501–522 (2004)CrossRefGoogle Scholar
  60. 60.
    Nayak, A.R., Panfilov, A.V., Pandit, R.: Spiral-wave dynamics in a mathematical model of human ventricular tissue with myocytes and Purkinje fibers. Phys. Rev. E 95(2), 022405 (2017)MathSciNetCrossRefGoogle Scholar
  61. 61.
    Niederer, S.A., Hunter, P.J., Smith, N.P.: A quantitative analysis of cardiac myocyte relaxation: a simulation study. Biophys. J. 90, 1697–1722 (2006)CrossRefGoogle Scholar
  62. 62.
    Niederer, S.A., Smith, N.P.: A mathematical model of the slow force response to stretch in rat ventricular myocites, Biophys. J. 92, 4030–4044 (2007)CrossRefGoogle Scholar
  63. 63.
    Nobile, F., Quarteroni, A., Ruiz-Baier, R.: An active strain electromechanical model of cardiac tissue. Int. J. Num. Methods Biomed. Eng. 28, 52–71 (2012)MathSciNetCrossRefGoogle Scholar
  64. 64.
    Noble, D., Rudy, Y.: Models of cardiac ventricular action potentials: iterative interaction between experiment and simulation. Philos. Trans. R. Soc. Lond. A 359, 1127–1142 (2001)CrossRefGoogle Scholar
  65. 65.
    Palamara, S., Vergara, C., Faggiano, E., Nobile, F.: An effective algorithm for the generation of patient-specific Purkinje networks in computational electrocardiology. J. Comput. Phys. 283, 495–517 (2015)MathSciNetCrossRefGoogle Scholar
  66. 66.
    Pathmanathan, P.J., Chapman, S.J., Gavaghan, D.J., Whiteley, J.P.: Cardiac electromechanics: the effect of contraction model on the mathematical problem and accuracy of the numerical scheme. Quart. J. Mech. Appl. Math. 63(3), 375–399 (2010)MathSciNetCrossRefGoogle Scholar
  67. 67.
    Pathmanathan, P.J., Whiteley, J.P.: A numerical method for cardiac mechanoelectric simulations. Ann. Biomed. Eng. 37, 860–873 (2009)CrossRefGoogle Scholar
  68. 68.
    Pavarino, L.F., Scacchi, S.: Multilevel additive Schwarz preconditioners for the Bidomain reaction-diffusion system. SIAM J. Sci. Comput. 31(1), 420–443 (2008)MathSciNetCrossRefGoogle Scholar
  69. 69.
    Pavarino, L.F., Scacchi, S.: Parallel multilevel Schwarz and block preconditioners for the Bidomain parabolic-parabolic and parabolic-elliptic formulations. SIAM J. Sci. Comp. 33(4), 1897–1919 (2011)MathSciNetCrossRefGoogle Scholar
  70. 70.
    Pavarino, L.F., Scacchi, S., Zampini, S.: Newton–Krylov–BDDC solvers for non-linear cardiac mechanics. Comput. Meth. Appl. Mech. Eng. 295, 562–580 (2015)CrossRefGoogle Scholar
  71. 71.
    Pavarino, L.F., Widlund, O.B., Zampini, S.: BDDC Preconditioners for spectral element discretizations of almost incompressible elasticity in three dimensions. SIAM J. Sci. Comp. 32(6), 3604–3626 (2010)MathSciNetCrossRefGoogle Scholar
  72. 72.
    Pennacchio, M., Savaré, G., Colli Franzone, P.: Multiscale modeling for the bioelectric activity of the heart. SIAM J. Math. Anal. 37, 1333–1370 (2006)MathSciNetCrossRefGoogle Scholar
  73. 73.
    Pennacchio, M., Simoncini, V.: Algebraic multigrid preconditioners for the bidomain reaction–diffusion system. Appl. Numer. Math. 59, 3033–3050 (2009)MathSciNetCrossRefGoogle Scholar
  74. 74.
    Pennacchio, M., Simoncini, V.: Fast structured AMG preconditioning for the bidomain model in electrocadiology. SIAM J. Sci. Comput. 33, 721–745 (2011)MathSciNetCrossRefGoogle Scholar
  75. 75.
    Pezzuto, S., Ambrosi, D., Quarteroni, A.: An orthotropic active-strain model for the myocardium mechanics and its numerical approximation. Eur. J. Mech. A Solids 48(1), 83–96 (2014)MathSciNetCrossRefGoogle Scholar
  76. 76.
    Plank, G., Liebmann, M., Weber dos Santos, R., Vigmond, E.J., Haase, G.: Algebraic multigrid preconditioner for the cardiac bidomain model. IEEE Trans. Biomed. Eng. 54(4), 585–596 (2007)CrossRefGoogle Scholar
  77. 77.
    Pullan, A.J., Buist, M.L., Cheng, L.K.: Mathematically Modelling the Electrical Activity of the Heart. World Scientific, Singapore (2005)CrossRefGoogle Scholar
  78. 78.
    Quarteroni, A., Lassila, T., Rossi, S., Ruiz-Baier, R.: Integrated Heart–Coupling multiscale and multiphysics models for the simulation of the cardiac function. Comput. Methods Appl. Mech. Eng. 314, 345–407 (2017)MathSciNetCrossRefGoogle Scholar
  79. 79.
    Quarteroni, A., Manzoni, A., Vergara, C.: The cardiovascular system: mathematical modelling, numerical algorithms and clinical applications. Acta Numerica 26, 365–590 (2017)MathSciNetCrossRefGoogle Scholar
  80. 80.
    Richardson, W.J., Clarke, S.A., Quinn, T.A., Holmes, J.W.: Physiological implications of myocardial scar structure. Compr. Physiol. 5(4), 1877–1909 (2015)CrossRefGoogle Scholar
  81. 81.
    Rice, J.J., Wang, F., Bers, D.M., de Tombe, P.P.: Approximate model of cooperative activation and crossbridge cycling in cardiac muscle using ordinary differential equations. Biophys. J. 95, 2368–2390 (2008)CrossRefGoogle Scholar
  82. 82.
    Rossi, S., Ruiz-Baier, R., Pavarino, L.F., Quarteroni, A.: Orthotropic active strain models for the numerical simulation of cardiac biomechanics. Int. J. Num. Methods Biomed. Eng. 28, 761–788 (2012)MathSciNetCrossRefGoogle Scholar
  83. 83.
    Rossi, S., Lassila, T., Ruiz-Baier, R., Sequeira, A., Quarteroni, A.: Thermodynamically consistent orthotropic activation model capturing ventricular systolic wall thickening in cardiac electromechanics. Eur. J. Mech. A Solids 48, 129–142 (2014)MathSciNetCrossRefGoogle Scholar
  84. 84.
    Rudy, Y., Silva, J.R.: Computational biology in the study of cardiac ion channels and cell electrophysiology. Quart. Rev. Biophys. 39(1), 57–116 (2006)CrossRefGoogle Scholar
  85. 85.
    Sahli Costabal, F., Concha, F.A., Hurtado, D.E., Kuhl, E.: The importance of mechano-electrical feedback and inertia in cardiac electromechanics. Comput. Methods Appl. Mech. Eng. 320, 352–368 (2017)MathSciNetCrossRefGoogle Scholar
  86. 86.
    Sainte-Marie, J., Chapelle, D., Cimrman, R., Sorine, M.: Modeling and estimation of cardiac electromechanical activity. Comput. Struct. 84, 1743–1759 (2006)MathSciNetCrossRefGoogle Scholar
  87. 87.
    Scacchi, S.: A multilevel hybrid Newton-Krylov-Schwarz method for the Bidomain model of electrocardiology. Comput. Methods Appl. Mech. Eng. 200(5–8), 717–725 (2011)MathSciNetCrossRefGoogle Scholar
  88. 88.
    Smith, N.P., Nickerson, D.P., Crampin, E.J., Hunter, P.J.: Multiscale computational modelling of the heart. Acta Numerica, 371–431 (2004)Google Scholar
  89. 89.
    Sundnes, J., Lines, G.T., Mardal, K., Tveito, A.: Multigrid block preconditioning for a coupled system of partial differential equations modeling the electrical activity in the heart. Comput. Methods Biomech. Biomed. Eng. 5, 397–409 (2002)CrossRefGoogle Scholar
  90. 90.
    Tagliabue, A., Dedè, L., Quarteroni, A.: Fluid dynamics of an idealized left ventricle: the extended Nitsche’s method for the treatment of heart valves as mixed time varying boundary conditions. Int. J. Numer. Methods Fluids 85(3), 135–164 (2017)MathSciNetCrossRefGoogle Scholar
  91. 91.
    Tagliabue, A., Dedè, L., Quarteroni, A.: Complex blood flow patterns in an idealized left ventricle: A numerical study. Chaos 27(9), 093939 (2017)MathSciNetCrossRefGoogle Scholar
  92. 92.
    K. H. W. J. ten Tusscher, D. Noble, P. J. Noble and A. Pan, V.: A model for human ventricular tissue. Am. J. Phys. Heart Circ. Physiol. 286, H1573–H1589 (2004)CrossRefGoogle Scholar
  93. 93.
    ten Tusscher, K.H.W.J., Panfilov, A.V.: Alternans and spiral breakup in a human ventricular tissue model. Am. J. Phys. Heart Circ. Physiol. 291, H1088–H1100 (2006)CrossRefGoogle Scholar
  94. 94.
    Toselli, A., Widlund, O.B.: Domain Decomposition Methods: Algorithms and Theory. Computational Mathematics, Vol. 34. Springer, Berlin (2004)Google Scholar
  95. 95.
    Tung, L.: A bidomain model for describing ischemic myocardiacl D.C. potentials. PhD dissertation, MIT, Cambridge, MA (1978)Google Scholar
  96. 96.
    Usyk, T.P., LeGrice, I.J., McCulloch, A.D.: Computational model of three-dimensional cardiac electromechanics. Comput. Visual. Sci. 4, 249–257 (2002)CrossRefGoogle Scholar
  97. 97.
    Usyk, T.P., Mazharia, R., McCulloch, A.D.: Effect of laminar orthotropic myofiber architecture on regional stress and strain in the canine left ventricle. J. Elast. 61(1–3), 143–164 (2000)CrossRefGoogle Scholar
  98. 98.
    Veneroni, M.: Reaction-diffusion systems for the macroscopic Bidomain model of the cardiac electric field. Nonlin. Anal. Real World Appl. 10, 849–868 (2009)MathSciNetCrossRefGoogle Scholar
  99. 99.
    Vergara, C., Lange, M., Palamara, S., Lassila, T., Frangi, A., Quarteroni, A.: A coupled 3D–1D numerical monodomain solver for cardiac electrical activation in the myocardium with detailed Purkinje network. J. Comput. Phys. 308, 218–238 (2016)MathSciNetCrossRefGoogle Scholar
  100. 100.
    Vetter, F.J., McCulloch, A.D.: Three-dimensional stress and strain in passive rabbit left ventricle: a model study. Ann. Biomed. Eng. 28, 781–792 (2000)CrossRefGoogle Scholar
  101. 101.
    Vigmond, E.J., Aguel, F., Trayanova, N.A.: Computational techniques for solving the bidomain equations in three dimensions. IEEE Trans. Biomed. Eng. 49(11), 1260–1269 (2002)CrossRefGoogle Scholar
  102. 102.
    Wall, S.T., Guccione, J.M., Ratcliffe, M.B., Sundnes, J.S.: Electromechanical feedback with reduced cellular connectivity alters electrical activity in an infarct injures left ventricle: a finite element model study. Am. J. Physiol. Heart Circ. Physiol. 302, H206–H214 (2012)CrossRefGoogle Scholar
  103. 103.
    Whiteley, J.P., Bishop, M.J., Gavaghan, D.J.: Soft tissue modelling of cardiac fibres for use in coupled mechano-electric simulations. Bull. Math. Biol. 69, 2199–2225 (2007)MathSciNetCrossRefGoogle Scholar
  104. 104.
    Zampini, S.: Balancing Neumann-Neumann methods for the cardiac Bidomain model. Numer. Math. 123(2), 363–393 (2013)MathSciNetCrossRefGoogle Scholar
  105. 105.
    Zampini, S.: Dual-primal methods for the cardiac bidomain model. Math. Models Methods Appl. Sci. 24(4), 667–696 (2014)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Piero Colli Franzone
    • 1
  • Luca F. Pavarino
    • 1
  • Simone Scacchi
    • 2
    Email author
  1. 1.Department of MathematicsUniversity of PavioPaviaItaly
  2. 2.Department of MathematicsUniversity of MilanoMilanoItaly

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