Skip to main content
SpringerLink
Log in
Book cover

Mathematical and Numerical Modeling of the Cardiovascular System and Applications pp 115–166Cite as

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

  • Chapter
  • First Online:

Part of the book series: SEMA SIMAI Springer 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.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Notes

  1. 1.

    More specifically, we would have g = p endo(t)n where n is the unit normal vector to the boundary, and p endo = p endo(t) is the external load applied by the fluid at the endocardium wall, which in this context is assumed to be prescribed.

References

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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

    Article  MathSciNet  Google Scholar 

  4. Ambrosi, D., Pezzuto, S.: Active stress vs. active strain in mechanobiology: constitutive issues. J. Elast. 107(2), 199–212 (2012)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  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. 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)

    Article  MathSciNet  Google Scholar 

  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. 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)

    Article  Google Scholar 

  13. Bonomi, D.: Reduced-order models for the parametrized cardiac electromechanical problem. Ph.D. thesis, Politecnico di Milano (2017)

    Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  16. Broyden, C.: A class of methods for solving nonlinear simultaneous equations. Math. Comput. 19(92), 577–593 (1965)

    Article  MathSciNet  Google Scholar 

  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

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  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

    Article  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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. 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)

    Article  MathSciNet  Google Scholar 

  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. Colli Franzone, P., Pavarino, L.F., Scacchi, S.: Parallel multilevel solvers for the cardiac electro-mechanical coupling. Appl. Numer. Math. 95, 140–153 (2015)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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. Gerbeau, J., Lombardi, D., Schenone, E.: Reduced order model in cardiac electrophysiology with approximated lax pairs. Adv. Comput. Math. 41(5), 1103–1130 (2015)

    Article  MathSciNet  Google Scholar 

  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 51/2017

    Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  41. Hesthaven, J., Rozza, G., Stamm, B.: Certified Reduced Basis Methods for Parametrized Partial Differential Equations. SpringerBriefs Mathematics. Springer, Cham (2016)

    Book  Google Scholar 

  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)

    Article  Google Scholar 

  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

    Article  MathSciNet  Google Scholar 

  44. Keldermann, R., Nash, M., Panfilov, A.: Modeling cardiac mechano-electrical feedback using reaction-diffusion-mechanics systems. Physica D 238(11), 1000–1007 (2009)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  46. Kuzmin, D.: A Guide to Numerical Methods for Transport Equations. University Erlangen-Nuremberg, Erlangen (2010)

    Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  51. Nardinocchi, P., Teresi, L.: On the active response of soft living tissues. J. Elast. 88(1), 27–39 (2007)

    Article  MathSciNet  Google Scholar 

  52. Nash, M., Hunter, P.: Computational mechanics of the heart. J. Elast. 61, 113–141 (2001). https://doi.org/10.1023/A:1011084330767

    Article  Google Scholar 

  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

    Article  Google Scholar 

  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. Negri, F., Manzoni, A., Amsallem, D.: Efficient model reduction of parametrized systems by matrix discrete empirical interpolation. J. Comput. Phys. 303, 431–454 (2015)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  58. Noble, D., Garny, A., Noble, P.: How the hodgkin-huxley equations inspired the cardiac physiome project. J. Physiol. 590(11), 2613–28 (2012)

    Article  Google Scholar 

  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. 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)

    Article  MathSciNet  Google Scholar 

  61. Pathmanathan, P., Whiteley, J.: A numerical method for cardiac mechanoelectric simulations. Ann. Biomed. Eng. 37(5), 860–873 (2009)

    Article  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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. Peherstorfer, B., Butnaru, D., Willcox, K., Bungartz, H.: Localized discrete empirical interpolation method. SIAM J. Sci. Comput. 36(1), A168–A192 (2014)

    Article  MathSciNet  Google Scholar 

  65. Pezzuto, S.: Mechanics of the heart – constitutive issues and numerical experiments. Ph.D. thesis, Politecnico di Milano (2013)

    Google Scholar 

  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. Quarteroni, A., Valli, A.: Numerical Approximation of Partial Differential Equations, vol. 23. Springer Science and Business Media, Berlin (2008)

    MATH  Google Scholar 

  68. Quarteroni, A., Manzoni, A., Negri, F.: Reduced Basis Methods for Partial Differential Equations: An Introduction. Unitext, vol. 92. Springer, Cham (2016)

    Book  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  70. Quarteroni, A., Manzoni, A., Vergara, C.: The cardiovascular system: mathematical modeling, numerical algorithms, clinical applications. Acta Numer. 26, 365–590 (2017)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  72. Rossi, S.: Anisotropic modeling of cardiac mechanical activation. Ph.D. thesis, Ecole Politechnique Federale de Lausanne (2014)

    Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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

    Article  MathSciNet  Google Scholar 

  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

    Article  MathSciNet  Google Scholar 

  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. 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)

    Article  Google Scholar 

  81. Taber, L., Perucchio, R.: Modeling heart development. J. Elast. 61(1–3), 165–197 (2000)

    Article  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  83. Tung, L.: A bi-domain model for describing ischemic myocardial DC potentials. Ph.D. thesis, Massachusetts Institute of Technology (1978)

    Google Scholar 

  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)

    Article  Google Scholar 

  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. 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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  88. Yang, H., Veneziani, A.: Efficient estimation of cardiac conductivities via pod-deim model order reduction. Appl. Numer. Math. 115, 180–199 (2017)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. MOX, Dipartimento di Matematica, Politecnico di Milano, Milano, Italy

    Andrea Manzoni, Diana Bonomi & Alfio Quarteroni

Authors
  1. Andrea Manzoni
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Diana Bonomi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Alfio Quarteroni
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Andrea Manzoni .

Editor information

Editors and Affiliations

  1. Department of Mathematics “F. Casorati”, University of Pavia, Pavia, Italy

    Daniele Boffi

  2. Department of Mathematics “F. Casorati”, University of Pavia, Pavia, Italy

    Luca F. Pavarino

  3. SISSA mathLab, Mathematics Area, International School for Advanced Studies, Trieste, Italy

    Gianluigi Rozza

  4. Department of Mathematics, University of Milan, Milano, Italy

    Simone Scacchi

  5. Department of Mathematics, Politecnico di Milano, Milano, Italy

    Christian Vergara

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Manzoni, A., Bonomi, D., Quarteroni, A. (2018). Reduced Order Modeling for Cardiac Electrophysiology and Mechanics: New Methodologies, Challenges and Perspectives. In: Boffi, D., Pavarino, L., Rozza, G., Scacchi, S., Vergara, C. (eds) Mathematical and Numerical Modeling of the Cardiovascular System and Applications. SEMA SIMAI Springer Series, vol 16. Springer, Cham. https://doi.org/10.1007/978-3-319-96649-6_6

Download citation

Publish with us

Policies and ethics

Search

Navigation