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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 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
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)
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)
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
Ambrosi, D., Pezzuto, S.: Active stress vs. active strain in mechanobiology: constitutive issues. J. Elast. 107(2), 199–212 (2012)
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)
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)
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)
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)
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)
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)
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)
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)
Bonomi, D.: Reduced-order models for the parametrized cardiac electromechanical problem. Ph.D. thesis, Politecnico di Milano (2017)
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)
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)
Broyden, C.: A class of methods for solving nonlinear simultaneous equations. Math. Comput. 19(92), 577–593 (1965)
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
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)
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)
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)
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
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)
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)
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
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)
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)
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)
Colli Franzone, P., Pavarino, L.F., Scacchi, S.: Mathematical Cardiac Electrophysiology. Modeling, Simulation and Applications Series, vol. 13. Springer, Milano (2014)
Colli Franzone, P., Pavarino, L.F., Scacchi, S.: Parallel multilevel solvers for the cardiac electro-mechanical coupling. Appl. Numer. Math. 95, 140–153 (2015)
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)
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)
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)
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)
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)
Gerbeau, J., Lombardi, D., Schenone, E.: Reduced order model in cardiac electrophysiology with approximated lax pairs. Adv. Comput. Math. 41(5), 1103–1130 (2015)
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
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)
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)
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)
Hesthaven, J.S., Zhang, S., Zhu, X.: Reduced basis multiscale finite element methods for elliptic problems. Multiscale Model. Simul. 13(1), 316–337 (2015)
Hesthaven, J., Rozza, G., Stamm, B.: Certified Reduced Basis Methods for Parametrized Partial Differential Equations. SpringerBriefs Mathematics. Springer, Cham (2016)
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)
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
Keldermann, R., Nash, M., Panfilov, A.: Modeling cardiac mechano-electrical feedback using reaction-diffusion-mechanics systems. Physica D 238(11), 1000–1007 (2009)
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)
Kuzmin, D.: A Guide to Numerical Methods for Transport Equations. University Erlangen-Nuremberg, Erlangen (2010)
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)
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)
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)
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)
Nardinocchi, P., Teresi, L.: On the active response of soft living tissues. J. Elast. 88(1), 27–39 (2007)
Nash, M., Hunter, P.: Computational mechanics of the heart. J. Elast. 61, 113–141 (2001). https://doi.org/10.1023/A:1011084330767
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
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)
Negri, F., Manzoni, A., Amsallem, D.: Efficient model reduction of parametrized systems by matrix discrete empirical interpolation. J. Comput. Phys. 303, 431–454 (2015)
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)
Nguyen, N.: A multiscale reduced-basis method for parametrized elliptic partial differential equations with multiple scales. J. Comput. Phys. 227(23), 9807–9822 (2008)
Noble, D., Garny, A., Noble, P.: How the hodgkin-huxley equations inspired the cardiac physiome project. J. Physiol. 590(11), 2613–28 (2012)
Pagani, S.: Reduced-order models for inverse problems and uncertainty quantification in cardiac electrophysiology. Ph.D. thesis, Politecnico di Milano (2017)
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)
Pathmanathan, P., Whiteley, J.: A numerical method for cardiac mechanoelectric simulations. Ann. Biomed. Eng. 37(5), 860–873 (2009)
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)
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
Peherstorfer, B., Butnaru, D., Willcox, K., Bungartz, H.: Localized discrete empirical interpolation method. SIAM J. Sci. Comput. 36(1), A168–A192 (2014)
Pezzuto, S.: Mechanics of the heart – constitutive issues and numerical experiments. Ph.D. thesis, Politecnico di Milano (2013)
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
Quarteroni, A., Valli, A.: Numerical Approximation of Partial Differential Equations, vol. 23. Springer Science and Business Media, Berlin (2008)
Quarteroni, A., Manzoni, A., Negri, F.: Reduced Basis Methods for Partial Differential Equations: An Introduction. Unitext, vol. 92. Springer, Cham (2016)
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)
Quarteroni, A., Manzoni, A., Vergara, C.: The cardiovascular system: mathematical modeling, numerical algorithms, clinical applications. Acta Numer. 26, 365–590 (2017)
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)
Rossi, S.: Anisotropic modeling of cardiac mechanical activation. Ph.D. thesis, Ecole Politechnique Federale de Lausanne (2014)
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)
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
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)
Sainte-Marie, J., Chapelle, D., Cimrman, R., Sorine, M.: Modeling and estimation of the cardiac electromechanical activity. Comput. Struct. 84(28), 1743–1759 (2006)
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
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
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)
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)
Taber, L., Perucchio, R.: Modeling heart development. J. Elast. 61(1–3), 165–197 (2000)
Trayanova, N., Eason, J., Aguel, F.: Computer simulations of cardiac defibrillation: a look inside the heart. Comput. Vis. Sci. 4(4), 259–270 (2002)
Tung, L.: A bi-domain model for describing ischemic myocardial DC potentials. Ph.D. thesis, Massachusetts Institute of Technology (1978)
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)
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)
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)
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)
Yang, H., Veneziani, A.: Efficient estimation of cardiac conductivities via pod-deim model order reduction. Appl. Numer. Math. 115, 180–199 (2017)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer Nature Switzerland AG
About this chapter
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
DOI: https://doi.org/10.1007/978-3-319-96649-6_6
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-96648-9
Online ISBN: 978-3-319-96649-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)