Advertisement

Reduced Order Modeling for Cardiac Electrophysiology and Mechanics: New Methodologies, Challenges and Perspectives

  • Andrea ManzoniEmail author
  • Diana Bonomi
  • Alfio Quarteroni
Chapter
Part of the SEMA SIMAI Springer Series book series (SEMA SIMAI, volume 16)

Abstract

Reduced-order modeling techniques enable a remarkable speed up in the solution of the parametrized electromechanical model for heart dynamics. Being able to rapidly approximate the solution of this problem allows to investigate the impact of significant model parameters querying the parameter-to-solution map in a very inexpensive way. The construction of reduced-order approximations for cardiac electromechanics faces several challenges from both modeling and computational viewpoints, because of the multiscale nature of the problem, the need of coupling different physics, and the nonlinearities involved. Our approach relies on the reduced basis method for parametrized PDEs. This technique performs a Galerkin projection onto low-dimensional spaces built from a set of snapshots of the high-fidelity problem by the Proper Orthogonal Decomposition technique. Snapshots are obtained for different values of the parameters and computed, e.g., by the finite element method. Then, suitable hyper-reduction techniques, in particular the Discrete Empirical Interpolation Method and its matrix version, are called into play to efficiently handle nonlinear and parameter-dependent terms. In this work we show how a fast and reliable approximation of both the electrical and the mechanical model can be achieved by developing two separate reduced order models where the interaction of the cardiac electrophysiology system with the contractile muscle tissue, as well as the sub-cellular activation-contraction mechanism, are included. Open challenges and possible perspectives are finally outlined.

References

  1. 1.
    Abdulle, A., Bai, Y.: Reduced basis finite element heterogeneous multiscale method for high-order discretizations of elliptic homogenization problems. J. Comput. Phys. 231(21), 7014–7036 (2012)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Abdulle, A., Budáč, O.: A reduced basis finite element heterogeneous multiscale method for stokes flow in porous media. Comput. Methods Appl. Mech. Eng. 307, 1–31 (2016)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Ambrosi, D., Arioli, G., Nobile, F., Quarteroni, A.: Electromechanical coupling in cardiac dynamics: the active strain approach. SIAM J. Appl. Math. 71(2), 605–621 (2011). https://doi.org/10.1137/100788379 MathSciNetCrossRefGoogle Scholar
  4. 4.
    Ambrosi, D., Pezzuto, S.: Active stress vs. active strain in mechanobiology: constitutive issues. J. Elast. 107(2), 199–212 (2012)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Amsallem, D., Zahr, M., Farhat, C.: Nonlinear model order reduction based on local reduced-order bases. Int. J. Numer. Meth. Eng. 92(10), 891–916 (2012)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Ashikaga, H., Coppola, B., Yamazaki, K., Villarreal, F., Omens, J., Covell, J.: Changes in regional myocardial volume during the cardiac cycle: implications for transmural blood flow and cardiac structure. Am. J. Physiol. Heart. Circ. Physiol. 295(2), H610–H618 (2008)CrossRefGoogle Scholar
  7. 7.
    Ballarin, F., Faggiano, E., Ippolito, S., Manzoni, A., Quarteroni, A., Rozza, G., Scrofani, R.: Fast simulations of patient-specific haemodynamics of coronary artery bypass grafts based on a POD–Galerkin method and a vascular shape parametrization. J. Comput. Phys. 315, 609–628 (2016)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Ballarin, F., Faggiano, E., Manzoni, A., Quarteroni, A., Rozza, G., Ippolito, S., Antona, C., Scrofani, R.: Numerical modeling of hemodynamics scenarios of patient-specific coronary artery bypass grafts. Biomech. Model. Mechanobiol. 16(4), 1373–1399 (2017)CrossRefGoogle Scholar
  9. 9.
    Balzani, D., Deparis, S., Fausten, S., Forti, D., Heinlein, A., Klawonn, A., Quarteroni, A., Rheinbach, O., Schroder, J.: Aspects of Arterial Wall Simulations: Nonlinear Anisotropic Material Models and Fluid Structure Interaction. Dekan der Fak. für Mathematik und Informatik (2014)Google Scholar
  10. 10.
    Barrault, M., Maday, Y., Nguyen, N., Patera, A.: An empirical interpolation method: application to efficient reduced-basis discretization of partial differential equations. C. R. Math. 339(9), 667–672 (2004)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Baumann, M.: Nonlinear model order reduction using pod/deim for optimal control of Burgers’ equation. Ph.D. thesis, TU Delft, Delft University of Technology (2013)Google Scholar
  12. 12.
    Biehler, J., Gee, M., Wall, W.: Towards efficient uncertainty quantification in complex and large scale biomechanical problems based on a Bayesian multi fidelity scheme. Biomech. Model. Mechanobiol. 14(3), 489–513 (2015)CrossRefGoogle Scholar
  13. 13.
    Bonomi, D.: Reduced-order models for the parametrized cardiac electromechanical problem. Ph.D. thesis, Politecnico di Milano (2017)Google Scholar
  14. 14.
    Bonomi, D., Manzoni, A., Quarteroni, A.: A matrix DEIM technique for model reduction of nonlinear parametrized problems in cardiac mechanics. Comput. Methods Appl. Mech. Eng. 324, 300–326 (2017)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Boulakia, M., Schenone, E., Gerbeau, J.: Reduced-order modeling for cardiac electrophysiology. Application to parameter identification. Int. J. Numer. Meth. Biomed. Eng. 28(6–7), 727–744 (2012)CrossRefGoogle Scholar
  16. 16.
    Broyden, C.: A class of methods for solving nonlinear simultaneous equations. Math. Comput. 19(92), 577–593 (1965)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Bueno-Orovio, A., Cherry, E., Fenton, F.: Minimal model for human ventricular action potentials in tissue. J. Theor. Biol. 253(3), 544–560 (2008). https://doi.org/10.1016/j.jtbi.2008.03.029 MathSciNetCrossRefGoogle Scholar
  18. 18.
    Carlberg, K., Farhat, C., Cortial, J., Amsallem, D.: The GNAT method for nonlinear model reduction: effective implementation and application to computational fluid dynamics and turbulent flows. J. Comput. Phys. 242, 623–647 (2013)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Carlberg, K., Tuminaro, R., Boggs, P.: Preserving Lagrangian structure in nonlinear model reduction with application to structural dynamics. SIAM J. Sci. Comput. 37(2), B153–B184 (2015)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Chapelle, D., Gariah, A., Sainte-Marie, J.: Galerkin approximation with proper orthogonal decomposition: new error estimates and illustrative examples. ESAIM: Math. Model. Numer. Anal. 46(4), 731–757 (2012)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Chaturantabut, S., Sorensen, D.: Nonlinear Model Reduction via Discrete Empirical Interpolation. SIAM J. Sci. Comput. 32(5), 2737–2764 (2010). https://doi.org/10.1137/090766498 MathSciNetCrossRefGoogle Scholar
  22. 22.
    Chaturantabut, S., Sorensen, D.: Application of POD and DEIM on dimension reduction of non-linear miscible viscous fingering in porous media. Math. Comp. Model. Dyn. 17(4), 337–353 (2011)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Cherubini, C., Filippi, S., Nardinocchi, P., Teresi, L.: An electromechanical model of cardiac tissue: constitutive issues and electrophysiological effects. Prog. Biophys. Mol. Biol. 97(2–3), 562–573 (2008)CrossRefGoogle Scholar
  24. 24.
    Clayton, R., Panfilov, A.: A guide to modelling cardiac electrical activity in anatomically detailed ventricles. Prog. Biophys. Mol. Biol. 96(1), 19–43 (2008). https://doi.org/10.1016/j.pbiomolbio.2007.07.004 CrossRefGoogle Scholar
  25. 25.
    Colciago, C., Deparis, S., Quarteroni, A.: Comparisons between reduced order models and full 3d models for fluid–structure interaction problems in haemodynamics. J. Comput. Appl. Math. 265, 120–138 (2014)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Colciago, C.M., Deparis, S., Forti, D.: Fluid-structure interaction for vascular flows: from supercomputers to laptops. In: Frei, S., Holm, B., Richter, T., Wick, T., Yang, H. (eds.) Fluid-Structure Interaction: Modeling, Adaptive Discretisations and Solvers. Radon Series on Computational and Applied Mathematics, vol. 20. De Gruyter, Berlin (2017)Google Scholar
  27. 27.
    Colli Franzone, P., Pavarino, L.F., Taccardi, B.: Simulating patterns of excitation, repolarization and action potential duration with cardiac bidomain and monodomain models. Math. Biosci. 197(1), 35–66 (2005)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Colli Franzone, P., Pavarino, L.F., Scacchi, S.: Mathematical Cardiac Electrophysiology. Modeling, Simulation and Applications Series, vol. 13. Springer, Milano (2014)Google Scholar
  29. 29.
    Colli Franzone, P., Pavarino, L.F., Scacchi, S.: Parallel multilevel solvers for the cardiac electro-mechanical coupling. Appl. Numer. Math. 95, 140–153 (2015)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Corrado, C., Lassoued, J., Mahjoub, M., Zemzemi, N.: Stability analysis of the POD reduced order method for solving the bidomain model in cardiac electrophysiology. Math. Biosci. 272, 81–91 (2016)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Dal, H., Goktepe, S., Kaliske, M., Kuhl, E.: A fully implicit finite element method for bidomain models of cardiac electromechanics. Comput. Methods Appl. Mech. Eng. 253, 323–336 (2013)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Deparis, S., Forti, D., Quarteroni, A.: A rescaled localized radial basis function interpolation on non-cartesian and nonconforming grids. SIAM J. Sci. Comput. 36(6), A2745–A2762 (2014)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Eriksson, T., Prassl, A., Plank, G., Holzapfel, G.: Influence of myocardial fiber/sheet orientations on left ventricular mechanical contraction. Math. Mech. Solids 18, 592–606 (2013)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Fedele, M., Faggiano, E., Barbarotta, L., Cremonesi, F., Formaggia, L., Perotto, S.: Semi-automatic three-dimensional vessel segmentation using a connected component localization of the region-scalable fitting energy. In: 2015 9th International Symposium on Image and Signal Processing and Analysis (ISPA), pp. 72–77. IEEE, Piscataway, NJ (2015)Google Scholar
  35. 35.
    Gerbeau, J., Lombardi, D., Schenone, E.: Reduced order model in cardiac electrophysiology with approximated lax pairs. Adv. Comput. Math. 41(5), 1103–1130 (2015)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Gerbi, A., Dede’, L., Quarteroni, A.: A monolithic algorithm for the simulation of cardiac electromechanics in the human left ventricle. Tech. rep., MOX - Politecnico di Milano (2017). Report 51/2017Google Scholar
  37. 37.
    Göktepe, S., Kuhl, E.: Electromechanics of the heart: a unified approach to the strongly coupled excitation–contraction problem. Comput. Mech. 45(2–3), 227–243 (2010)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Heidenreich, E., Ferrero, J., Doblare, M., Rodriguez, J.: Adaptive macro finite elements for the numerical solution of monodomain equations in cardiac electrophysiology. Ann. Biomed. Eng. 38(7), 2331–2345 (2010)CrossRefGoogle Scholar
  39. 39.
    Helfenstein, J., Jabareen, M., Mazza, E., Govindjee, S.: On non-physical response in models for fiber-reinforced hyperelastic materials. Int. J. Solids Struct. 47(16), 2056–2061 (2010)CrossRefGoogle Scholar
  40. 40.
    Hesthaven, J.S., Zhang, S., Zhu, X.: Reduced basis multiscale finite element methods for elliptic problems. Multiscale Model. Simul. 13(1), 316–337 (2015)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Hesthaven, J., Rozza, G., Stamm, B.: Certified Reduced Basis Methods for Parametrized Partial Differential Equations. SpringerBriefs Mathematics. Springer, Cham (2016)CrossRefGoogle Scholar
  42. 42.
    Hodgkin, A., Huxley, A.: A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 117(4), 500–544 (1952)CrossRefGoogle Scholar
  43. 43.
    Holzapfel, G., Ogden, R.: Constitutive modelling of passive myocardium: a structurally based framework for material characterization. Philos. Trans. A Math. Phys. Eng. Sci. 367(1902), 3445–3475 (2009).  https://doi.org/10.1098/rsta.2009.0091 MathSciNetCrossRefGoogle Scholar
  44. 44.
    Keldermann, R., Nash, M., Panfilov, A.: Modeling cardiac mechano-electrical feedback using reaction-diffusion-mechanics systems. Physica D 238(11), 1000–1007 (2009)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Krysl, P., Lall, S., Marsden, J.: Dimensional model reduction in non-linear finite element dynamics of solids and structures. Int. J. Numer. Meth. Eng. 51(4), 479–504 (2001)MathSciNetCrossRefGoogle Scholar
  46. 46.
    Kuzmin, D.: A Guide to Numerical Methods for Transport Equations. University Erlangen-Nuremberg, Erlangen (2010)Google Scholar
  47. 47.
    Lassila, T., Manzoni, A., Quarteroni, A., Rozza, G.: Boundary control and shape optimization for the robust design of bypass anastomoses under uncertainty. ESAIM-Math. Model. Numer. 47(4), 1107–1131 (2013)MathSciNetCrossRefGoogle Scholar
  48. 48.
    Lassila, T., Manzoni, A., Quarteroni, A., Rozza, G.: A reduced computational and geometrical framework for inverse problems in haemodynamics. Int. J. Numer. Methods Biomed. Eng. 29(7), 741–776 (2013)MathSciNetCrossRefGoogle Scholar
  49. 49.
    Maday, Y., Nguyen, N.C., Patera, A.T., Pau, G.S.H.: A general multipurpose interpolation procedure: the magic points. Commun. Pure Appl. Anal. 8(1), 383–404 (2009)MathSciNetCrossRefGoogle Scholar
  50. 50.
    Manzoni, A., Quarteroni, A., Rozza, G.: Shape optimization for viscous flows by reduced basis methods and free-form deformation. Int. J. Numer. Meth. Fluids 70(5), 646–670 (2012)MathSciNetCrossRefGoogle Scholar
  51. 51.
    Nardinocchi, P., Teresi, L.: On the active response of soft living tissues. J. Elast. 88(1), 27–39 (2007)MathSciNetCrossRefGoogle Scholar
  52. 52.
    Nash, M., Hunter, P.: Computational mechanics of the heart. J. Elast. 61, 113–141 (2001). https://doi.org/10.1023/A:1011084330767 CrossRefGoogle Scholar
  53. 53.
    Nash, M., Panfilov, A.: Electromechanical model of excitable tissue to study reentrant cardiac arrhythmias. Prog. Biophys. Mol. Biol. 85(2–3), 501–522 (2004). https://doi.org/10.1016/j.pbiomolbio.2004.01.016 CrossRefGoogle Scholar
  54. 54.
    Negri, F.: Efficient reduction techniques for the simulation and optimization of parametrized systems: Analysis and applications. Ph.D. thesis, Ecole Polytechnique Fédérale de Lausanne (2016)Google Scholar
  55. 55.
    Negri, F., Manzoni, A., Amsallem, D.: Efficient model reduction of parametrized systems by matrix discrete empirical interpolation. J. Comput. Phys. 303, 431–454 (2015)MathSciNetCrossRefGoogle Scholar
  56. 56.
    Neic, A., Campos, F., Prassl, A., Niederer, S., Bishop, M., Vigmond, E., Plank, G.: Efficient computation of electrograms and ECGs in human whole heart simulations using a reaction-eikonal model. J. Comput. Phys. 346, 191–211 (2017)MathSciNetCrossRefGoogle Scholar
  57. 57.
    Nguyen, N.: A multiscale reduced-basis method for parametrized elliptic partial differential equations with multiple scales. J. Comput. Phys. 227(23), 9807–9822 (2008)MathSciNetCrossRefGoogle Scholar
  58. 58.
    Noble, D., Garny, A., Noble, P.: How the hodgkin-huxley equations inspired the cardiac physiome project. J. Physiol. 590(11), 2613–28 (2012)CrossRefGoogle Scholar
  59. 59.
    Pagani, S.: Reduced-order models for inverse problems and uncertainty quantification in cardiac electrophysiology. Ph.D. thesis, Politecnico di Milano (2017)Google Scholar
  60. 60.
    Pagani, S., Manzoni, A., Quarteroni, A.: Numerical approximation of parametrized problems in cardiac electrophysiology by a local reduced basis method. Comput. Methods Appl. Mech. Eng. 340, 530–558 (2018)MathSciNetCrossRefGoogle Scholar
  61. 61.
    Pathmanathan, P., Whiteley, J.: A numerical method for cardiac mechanoelectric simulations. Ann. Biomed. Eng. 37(5), 860–873 (2009)CrossRefGoogle Scholar
  62. 62.
    Pathmanathan, P., Chapman, S., Gavaghan, D., Whiteley, J.: Cardiac electromechanics: the effect of contraction model on the mathematical problem and accuracy of the numerical scheme. J. Mech. Appl. Math. 63, 375–399 (2010)MathSciNetCrossRefGoogle Scholar
  63. 63.
    Pathmanathan, P., Mirams, G., Southern, J., Whiteley, J.: The significant effect of the choice of ionic current integration method in cardiac electro-physiological simulations. Int. J. Num. Meth. Biomed. Eng. 27(1), 1751–1770 (2011).  https://doi.org/10.1002/cnm. http://onlinelibrary.wiley.com/doi/10.1002/cnm.1494/full
  64. 64.
    Peherstorfer, B., Butnaru, D., Willcox, K., Bungartz, H.: Localized discrete empirical interpolation method. SIAM J. Sci. Comput. 36(1), A168–A192 (2014)MathSciNetCrossRefGoogle Scholar
  65. 65.
    Pezzuto, S.: Mechanics of the heart – constitutive issues and numerical experiments. Ph.D. thesis, Politecnico di Milano (2013)Google Scholar
  66. 66.
    Potse, M., Dubé, B., Vinet, A., Cardinal, R.: A comparison of monodomain and bidomain propagation models for the human heart. Conf. Proc. IEEE Eng. Med. Biol. Soc. 53(12), 3895–3898 (2006).  https://doi.org/10.1109/IEMBS.2006.259484 Google Scholar
  67. 67.
    Quarteroni, A., Valli, A.: Numerical Approximation of Partial Differential Equations, vol. 23. Springer Science and Business Media, Berlin (2008)zbMATHGoogle Scholar
  68. 68.
    Quarteroni, A., Manzoni, A., Negri, F.: Reduced Basis Methods for Partial Differential Equations: An Introduction. Unitext, vol. 92. Springer, Cham (2016)CrossRefGoogle Scholar
  69. 69.
    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
  70. 70.
    Quarteroni, A., Manzoni, A., Vergara, C.: The cardiovascular system: mathematical modeling, numerical algorithms, clinical applications. Acta Numer. 26, 365–590 (2017)MathSciNetCrossRefGoogle Scholar
  71. 71.
    Radermacher, A., Reese, S.: POD-based model reduction with empirical interpolation applied to nonlinear elasticity. Int. J. Numer. Meth. Eng. 107(6), 477–495 (2016)MathSciNetCrossRefGoogle Scholar
  72. 72.
    Rossi, S.: Anisotropic modeling of cardiac mechanical activation. Ph.D. thesis, Ecole Politechnique Federale de Lausanne (2014)Google Scholar
  73. 73.
    Rossi, S., Ruiz-Baier, R., Pavarino, L.F., Quarteroni, A.: Orthotropic active strain models for the numerical simulation of cardiac biomechanics. Int. J. Numer. Meth. Biomed. Eng. 28(6–7), 761–788 (2012)MathSciNetCrossRefGoogle Scholar
  74. 74.
    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/Sol. 48 (2013). https://doi.org/10.1016/j.euromechsol.2013.10.009 MathSciNetCrossRefGoogle Scholar
  75. 75.
    Ruiz-Baier, R., Gizzi, A., Rossi, S., Cherubini, C., Laadhari, A., Filippi, S., Quarteroni, A.: Mathematical modelling of active contraction in isolated cardiomyocytes. Math. Med. Biol. 31(3), 259–283 (2014)MathSciNetCrossRefGoogle Scholar
  76. 76.
    Sainte-Marie, J., Chapelle, D., Cimrman, R., Sorine, M.: Modeling and estimation of the cardiac electromechanical activity. Comput. Struct. 84(28), 1743–1759 (2006)MathSciNetCrossRefGoogle Scholar
  77. 77.
    Sansour, C.: On the physical assumptions underlying the volumetric-isochoric split and the case of anisotropy. Eur. J. Mech. A/Sol. 27(1), 28–39 (2008). https://doi.org/10.1016/j.euromechsol.2007.04.001 MathSciNetCrossRefGoogle Scholar
  78. 78.
    Smith, N., Nickerson, D., Crampin, E., Hunter, P.: Multiscale computational modelling of the heart. Acta Numer. 13, 371–431 (2004). https://doi.org/10.1017/S0962492904000200 MathSciNetCrossRefGoogle Scholar
  79. 79.
    Strobeck, J., Sonnenblick, E.: Myocardial contractile properties and ventricular performance. In: The Heart and Cardiovascular System: Scientific Foundations, pp. 31–49. Raven Press, New York (1986)Google Scholar
  80. 80.
    Sundnes, J., Wall, S., Osnes, H., Thorvaldsen, T., McCulloch, A.: Improved discretisation and linearisation of active tension in strongly coupled cardiac electro-mechanics simulations. Comput. Meth. Biomech. Biomed. Eng. 17(6), 604–615 (2014)CrossRefGoogle Scholar
  81. 81.
    Taber, L., Perucchio, R.: Modeling heart development. J. Elast. 61(1–3), 165–197 (2000)MathSciNetCrossRefGoogle Scholar
  82. 82.
    Trayanova, N., Eason, J., Aguel, F.: Computer simulations of cardiac defibrillation: a look inside the heart. Comput. Vis. Sci. 4(4), 259–270 (2002)CrossRefGoogle Scholar
  83. 83.
    Tung, L.: A bi-domain model for describing ischemic myocardial DC potentials. Ph.D. thesis, Massachusetts Institute of Technology (1978)Google Scholar
  84. 84.
    Wang, Y., Haynor, D., Kim, Y.: An investigation of the importance of myocardial anisotropy in finite-element modeling of the heart: methodology and application to the estimation of defibrillation efficacy. IEEE Trans. Biomed. Eng. 48(12), 1377–1389 (2001)CrossRefGoogle Scholar
  85. 85.
    Washabaugh, K., Amsallem, D., Zahr, M., Farhat, C.: Nonlinear model reduction for cfd problems using local reduced-order bases. In: 42nd AIAA Fluid Dynamics Conference and Exhibit, Fluid Dynamics and Co-located Conferences, AIAA Paper, vol. 2686, pp. 1–16 (2012)Google Scholar
  86. 86.
    Whiteley, J., Bishop, M., Gavaghan, D.: Soft tissue modelling of cardiac fibres for use in coupled mechano-electric simulations. Bull. Math. Biol. 69(7), 2199–2225 (2007)MathSciNetCrossRefGoogle Scholar
  87. 87.
    Wirtz, D., Sorensen, D., Haasdonk, B.: A posteriori error estimation for DEIM reduced nonlinear dynamical systems. SIAM J. Sci. Comput. 36(2), A311–A338 (2014)MathSciNetCrossRefGoogle Scholar
  88. 88.
    Yang, H., Veneziani, A.: Efficient estimation of cardiac conductivities via pod-deim model order reduction. Appl. Numer. Math. 115, 180–199 (2017)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Andrea Manzoni
    • 1
    Email author
  • Diana Bonomi
    • 1
  • Alfio Quarteroni
    • 1
  1. 1.MOX, Dipartimento di MatematicaPolitecnico di MilanoMilanoItaly

Personalised recommendations