Research article

Projection-based reduced order models for parameterized nonlinear time-dependent problems arising in cardiac mechanics

  • Received: 18 November 2021 Revised: 08 March 2022 Accepted: 11 March 2022 Published: 21 April 2022
  • The numerical simulation of several virtual scenarios arising in cardiac mechanics poses a computational challenge that can be alleviated if traditional full-order models (FOMs) are replaced by reduced order models (ROMs). For example, in the case of problems involving a vector of input parameters related, e.g., to material coefficients, projection-based ROMs provide mathematically rigorous physics-driven surrogate ROMs. In this work we demonstrate how, once trained, ROMs yield extremely accurate predictions (according to a prescribed tolerance) – yet cheaper than the ones provided by FOMs – of the structural deformation of the left ventricular tissue over an entire heartbeat, and of related output quantities of interest, such as the pressure-volume loop, for any desired input parameter values within a prescribed parameter range. However, the construction of ROM approximations for time-dependent cardiac mechanics is not straightforward, because of the highly nonlinear and multiscale nature of the problem, and almost never addressed. Our approach relies on the reduced basis method for parameterized partial differential equations. This technique performs a Galerkin projection onto a low-dimensional space for the displacement variable; the reduced space is built from a set of solution snapshots – obtained for different input parameter values and time instances – of the high-fidelity FOM, through the proper orthogonal decomposition technique. Then, suitable hyper-reduction techniques, such as the Discrete Empirical Interpolation Method, are exploited to efficiently handle nonlinear and parameter-dependent terms. In this work we show how a fast and reliable approximation of the time-dependent cardiac mechanical model can be achieved by a projection-based ROM, taking into account both passive and active mechanics for the left ventricle providing all the building blocks of the methodology, and highlighting those challenging aspects that are still open.

    Citation: Ludovica Cicci, Stefania Fresca, Stefano Pagani, Andrea Manzoni, Alfio Quarteroni. Projection-based reduced order models for parameterized nonlinear time-dependent problems arising in cardiac mechanics[J]. Mathematics in Engineering, 2023, 5(2): 1-38. doi: 10.3934/mine.2023026

    Related Papers:

  • The numerical simulation of several virtual scenarios arising in cardiac mechanics poses a computational challenge that can be alleviated if traditional full-order models (FOMs) are replaced by reduced order models (ROMs). For example, in the case of problems involving a vector of input parameters related, e.g., to material coefficients, projection-based ROMs provide mathematically rigorous physics-driven surrogate ROMs. In this work we demonstrate how, once trained, ROMs yield extremely accurate predictions (according to a prescribed tolerance) – yet cheaper than the ones provided by FOMs – of the structural deformation of the left ventricular tissue over an entire heartbeat, and of related output quantities of interest, such as the pressure-volume loop, for any desired input parameter values within a prescribed parameter range. However, the construction of ROM approximations for time-dependent cardiac mechanics is not straightforward, because of the highly nonlinear and multiscale nature of the problem, and almost never addressed. Our approach relies on the reduced basis method for parameterized partial differential equations. This technique performs a Galerkin projection onto a low-dimensional space for the displacement variable; the reduced space is built from a set of solution snapshots – obtained for different input parameter values and time instances – of the high-fidelity FOM, through the proper orthogonal decomposition technique. Then, suitable hyper-reduction techniques, such as the Discrete Empirical Interpolation Method, are exploited to efficiently handle nonlinear and parameter-dependent terms. In this work we show how a fast and reliable approximation of the time-dependent cardiac mechanical model can be achieved by a projection-based ROM, taking into account both passive and active mechanics for the left ventricle providing all the building blocks of the methodology, and highlighting those challenging aspects that are still open.



    加载中


    [1] A. Quarteroni, A. Manzoni, C. Vergara, The cardiovascular system: mathematical modelling, numerical algorithms and clinical applications, Acta Numer., 26 (2017), 365–590. http://doi.org/10.1017/S0962492917000046 doi: 10.1017/S0962492917000046
    [2] A. Quarteroni, L. Dede', A. Manzoni, C. Vergara, Mathematical modelling of the human cardiovascular system. data, numerical approximation, clinical applications, Cambridge: Cambridge University Press, 2019. https://doi.org/10.1017/9781108616096
    [3] P. Hauseux, J. Hale, S. Cotin, S. P. A. Bordas, Quantifying the uncertainty in a hyperelastic soft tissue model with stochastic parameters, Appl. Math. Model., 62 (2018), 86–102. https://doi.org/10.1016/j.apm.2018.04.021 doi: 10.1016/j.apm.2018.04.021
    [4] J. Campos, J. Sundnes, R. Dos Santos, B. M. Rocha, Uncertainty quantification and sensitivity analysis of left ventricular function during the full cardiac cycle, Phil. Trans. R. Soc. A, 378 (2020), 20190381. https://doi.org/10.1098/rsta.2019.0381 doi: 10.1098/rsta.2019.0381
    [5] L. Marx, M. A. F. Gsell, A. Rund, F. Caforio, A. J. Prassl, G. Toth-Gayor, et al., Personalization of electro-mechanical models of the pressure-overloaded left ventricle: fitting of windkessel-type afterload models, Phil. Trans. R. Soc. A, 378 (2020), 20190342. https://doi.org/10.1098/rsta.2019.0342 doi: 10.1098/rsta.2019.0342
    [6] R. Rodríguez-Cantano, J. Sundnes, M. Rognes, Uncertainty in cardiac myofiber orientation and stiffnesses dominate the variability of left ventricle deformation response, Int. J. Numer. Meth. Biomed. Eng., 35 (2020), e3178. https://doi.org/10.1002/cnm.3178 doi: 10.1002/cnm.3178
    [7] S. Pagani, A. Manzoni, Enabling forward uncertainty quantification and sensitivity analysis in cardiac electrophysiology by reduced order modeling and machine learning, Int. J. Numer. Meth. Biomed. Eng., 37 (2021), e3450. https://doi.org/10.1002/cnm.3450 doi: 10.1002/cnm.3450
    [8] A. Quarteroni, A. Manzoni, F. Negri, Reduced basis methods for partial differential equations: an Introduction, Cham: Springer, 2016. https://doi.org/10.1007/978-3-319-15431-2
    [9] P. Benner, S. Gugercin, K. Willcox, A survey of projection-based model reduction methods for parametric dynamical systems, SIAM Rev., 57 (2015), 483–531. https://doi.org/10.1137/130932715 doi: 10.1137/130932715
    [10] M. Morales, M. van den Boomen, C. Nguyen, J. Kalpathy-Cramer, B. R. Rosen, C. M. Stultz, et al., Deepstrain: A deep learning workflow for the automated characterization of cardiac mechanics, Front. Cardiovasc. Med., 8 (2021), 1041. https://doi.org/10.3389/fcvm.2021.730316 doi: 10.3389/fcvm.2021.730316
    [11] P. Di Achille, A. Harouni, S. Khamzin, O. Solovyova, J. J. Rice, V. Gurev, Gaussian process regressions for inverse problems and parameter searches in models of ventricular mechanics, Front. Physiol., 9 (2018), 1002. https://doi.org/10.3389/fphys.2018.01002 doi: 10.3389/fphys.2018.01002
    [12] U. Noè, A. Lazarus, H. Gao, V. Davies, B. Macdonald, K. Mangion, et al., Gaussian process emulation to accelerate parameter estimation in a mechanical model of the left ventricle: a critical step towards clinical end-user relevance, J. R. Soc. Interface, 16 (2019), 20190114. https://doi.org/10.1098/rsif.2019.0114 doi: 10.1098/rsif.2019.0114
    [13] S. Longobardi, A. Lewalle, S. Coveney, I. Sjaastad, E. K. S. Espe, W. E. Louch, et al., Predicting left ventricular contractile function via gaussian process emulation in aortic-banded rats, Phil. Trans. R. Soc. A, 378 (2019), 20190334. https://doi.org/10.1098/rsta.2019.0334 doi: 10.1098/rsta.2019.0334
    [14] S. Buoso, T. Joyce, S. Kozerke, Personalising left-ventricular biophysical models of the heart using parametric physics-informed neural networks, Med. Image Anal., 71 (2021), 102066. https://doi.org/10.1016/j.media.2021.102066 doi: 10.1016/j.media.2021.102066
    [15] D. Bonomi, A. Manzoni, A. Quarteroni, A matrix DEIM technique for model reduction of nonlinear parametrized problems in cardiac mechanics, Comput. Meth. Appl. Mech. Eng., 324 (2017), 300–326. https://doi.org/10.1016/j.cma.2017.06.011 doi: 10.1016/j.cma.2017.06.011
    [16] A. Manzoni, D. Bonomi, A. Quarteroni, Reduced order modeling for cardiac electrophysiology and mechanics: New methodologies, challenges and perspectives, In: Mathematical and numerical modeling of the cardiovascular system and applications, Cham: Springer, 2018,115–166. https://doi.org/10.1007/978-3-319-96649-6_6
    [17] D. Chapelle, A. Gariah, P. Moireau, J. Sainte-Marie, A Galerkin strategy with proper orthogonal decomposition for parameter-dependent problems – analysis, assessments and applications to parameter estimation, ESAIM: M2AN, 47 (2013), 1821–1843. https://doi.org/10.1051/m2an/2013090 doi: 10.1051/m2an/2013090
    [18] M. Pfaller, M. Cruz Varona, J. Lang, C. Bertoglio, W. A. Wall, Using parametric model order reduction for inverse analysis of large nonlinear cardiac simulations, Int. J. Numer. Meth. Biomed. Eng., 36 (2020), e3320. https://doi.org/10.1002/cnm.3320 doi: 10.1002/cnm.3320
    [19] J. M. Guccione, A. D. McCulloch, Finite element modeling of ventricular mechanics, In: Theory of heart, New York, NY: Springer, 1991,121–144. https://doi.org/10.1007/978-1-4612-3118-9_6
    [20] J. Guccione, A. McCulloch, Mechanics of active contraction in cardiac muscle: Part Ⅰ–Constitutive relations for fiber stress that describe deactivation, J. Biomech. Eng., 115 (1993), 72–81. http://doi.org/10.1115/1.2895473 doi: 10.1115/1.2895473
    [21] K. Costa, J. Holmes, A. McCulloch, Modelling cardiac mechanical properties in three dimensions, Phil. Trans. R. Soc. A, 359 (2001), 1233–1250. https://doi.org/10.1098/rsta.2001.0828 doi: 10.1098/rsta.2001.0828
    [22] G. Holzapfel, R. Ogden, Constitutive modelling of passive myocardium: a structurally based framework for material characterization, Phil. Trans. R. Soc. A, 367 (2009), 3445–3475. https://doi.org/10.1098/rsta.2009.0091 doi: 10.1098/rsta.2009.0091
    [23] D. Nordsletten, S. Niederer, M. Nash, P. J. Hunter, N. P. Smith, Coupling multi-physics models to cardiac mechanics, Prog. Biophys. Mol. Biol., 104 (2011), 77–88. https://doi.org/10.1016/j.pbiomolbio.2009.11.001 doi: 10.1016/j.pbiomolbio.2009.11.001
    [24] H. M. Wang, H. Gao, X. Y. Luo, C. Berry, B. E. Griffith, R. W. Ogden, et al., Structure-based finite strain modelling of the human left ventricle in diastole, Int. J. Numer. Meth. Biomed. Eng., 29 (2013), 83–103. https://doi.org/10.1002/cnm.2497 doi: 10.1002/cnm.2497
    [25] L. Barbarotta, S. Rossi, L. Dede', A. Quarteroni, A transmurally heterogeneous orthotropic activation model for ventricular contraction and its numerical validation, Int. J. Numer. Meth. Biomed. Eng., 34 (2018), e3137. https://doi.org/10.1002/cnm.3137 doi: 10.1002/cnm.3137
    [26] L. Dede', A. Gerbi, A. Quarteroni, Segregated algorithms for the numerical simulation of cardiac electromechanics in the left human ventricle, In: The mathematics of mechanobiology, Cham: Springer, 2020, 81–116. https://doi.org/10.1007/978-3-030-45197-4_3
    [27] D. Lin, F. Yin, A multiaxial constitutive law for mammalian left ventricular myocardium in steady-state barium contracture or tetanus, J. Biomech. Eng., 120 (1998), 504–517. https://doi.org/10.1115/1.2798021 doi: 10.1115/1.2798021
    [28] D. Ambrosi, S. Pezzuto, Active stress vs. active strain in mechanobiology: constitutive issues, J. Elast., 107 (2012), 199–212. https://doi.org/10.1007/s10659-011-9351-4 doi: 10.1007/s10659-011-9351-4
    [29] G. A. Holzapfel, Nonlinear solid mechanics. A continuum approach for engineering, Chichester: Wiley, 2001.
    [30] C. G. Broyden, A class of methods for solving nonlinear simultaneous equations, Math. Comp., 19 (1965), 577–593. https://doi.org/10.1090/S0025-5718-1965-0198670-6 doi: 10.1090/S0025-5718-1965-0198670-6
    [31] A. Radermacher, S. Reese, POD-based model reduction with empirical interpolation applied to nonlinear elasticity, Int. J. Numer. Meth. Eng., 107 (2016), 477–495. https://doi.org/10.1002/nme.5177 doi: 10.1002/nme.5177
    [32] J. Edmonds, Matroids and the greedy algorithm, Mathematical Programming, 1 (1971), 127–136. https://doi.org/10.1007/BF01584082 doi: 10.1007/BF01584082
    [33] C. Farhat, S. Grimberg, A. Manzoni, A. Quarteroni, Computational bottlenecks for proms: precomputation and hyperreduction, In: Model order reduction, Berlin: De Gruyter, 2020,181–244. https://doi.org/10.1515/9783110671490-005
    [34] M. Barrault, Y. Maday, N. Nguyen, A. T. Patera, An 'empirical interpolation' method: application to efficient reduced-basis discretization of partial differential equations, C. R. Math., 339 (2004), 667–672. https://doi.org/10.1016/j.crma.2004.08.006 doi: 10.1016/j.crma.2004.08.006
    [35] M. Grepl, Y. Maday, N. Nguyen, A. T. Patera, Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations, ESAIM: M2AN, 41 (2007), 575–605. https://doi.org/10.1051/m2an:2007031 doi: 10.1051/m2an:2007031
    [36] Y. Maday, N. Nguyen, A. Patera, S. H. Pau, A general multipurpose interpolation procedure: The magic points, Commun. Pure Appl. Anal., 8 (2008), 383–404. https://doi.org/10.3934/cpaa.2009.8.383 doi: 10.3934/cpaa.2009.8.383
    [37] S. Chaturantabut, D. Sorensen, Nonlinear model reduction via discrete empirical interpolation, SIAM J. Sci. Comput., 32 (2010), 2737–2764. https://doi.org/10.1137/090766498 doi: 10.1137/090766498
    [38] D. Wirtz, D. Sorensen, B. Haasdonk. A posteriori error estimation for DEIM reduced nonlinear dynamical systems, SIAM J. Sci. Comput., 36 (2014), A311–A338. https://doi.org/10.1137/120899042 doi: 10.1137/120899042
    [39] K. Carlberg, R. Tuminaro, P. Boggs, Preserving lagrangian structure in nonlinear model reduction with application to structural dynamics, SIAM J. Sci. Comput., 37 (2015), B153–B184. https://doi.org/10.1137/140959602 doi: 10.1137/140959602
    [40] F. Negri, A. Manzoni, D. Amsallem, Efficient model reduction of parametrized systems by matrix discrete empirical interpolation, J. Comput. Phys., 303 (2015), 431–454. https://doi.org/10.1016/j.jcp.2015.09.046 doi: 10.1016/j.jcp.2015.09.046
    [41] P. Astrid, S. Weiland, K. Willcox, T. Backx, Missing point estimation in models described by proper orthogonal decomposition, IEEE Trans. Automat. Contr., 53 (2008), 2237–2251. https://doi.org/10.1109/TAC.2008.2006102 doi: 10.1109/TAC.2008.2006102
    [42] K. Carlberg, C. Bou-Mosleh, C. Farhat, Efficient non-linear model reduction via a least-squares petrov–galerkin projection and compressive tensor approximations, Int. J. Numer. Meth. Eng., 86 (2011), 155–181. https://doi.org/10.1002/nme.3050 doi: 10.1002/nme.3050
    [43] M. Drohmann, B. Haasdonk, M. Ohlberger, Reduced basis approximation for nonlinear parametrized evolution equations based on empirical operator interpolation, SIAM J. Sci. Comput., 34 (2012), A937–A969. https://doi.org/10.1137/10081157X doi: 10.1137/10081157X
    [44] D. Amsallem, M. Zahr, C. Farhat, Nonlinear model order reduction based on local reduced-order bases, Int. J. Numer. Meth. Eng., 92 (2012), 891–916. https://doi.org/10.1002/nme.4371 doi: 10.1002/nme.4371
    [45] P. Tiso, D. J. Rixen, Discrete empirical interpolation method for finite element structural dynamics, In: Topics in nonlinear dynamics, volume 1, New York, NY: Springer, 2013,203–212. https://doi.org/10.1007/978-1-4614-6570-6_18
    [46] F. Ghavamian, P. Tiso, A. Simone, POD-DEIM model order reduction for strain-softening viscoplasticity, Comput. Meth. Appl. Mech. Eng., 317 (2017), 458–479. https://doi.org/10.1016/j.cma.2016.11.025 doi: 10.1016/j.cma.2016.11.025
    [47] S. Land, V. Gurev, S. Arens, C. M. Augustin, L. Baron, R. Blake, et al., Verification of cardiac mechanics software: benchmark problems and solutions for testing active and passive material behaviour, Proc. R. Soc. A, 471 (2015), 20150641. https://doi.org/10.1098/rspa.2015.0641 doi: 10.1098/rspa.2015.0641
    [48] D. Arndt, W. Bangerth, B. Blais, T. C. Clevenger, M. Fehling, A. V. Grayver, et al., The deal.Ⅱ library, version 9.2, J. Numer. Math., 28 (2020), 131–146. https://doi.org/10.1515/jnma-2020-0043 doi: 10.1515/jnma-2020-0043
    [49] J. Bayer, R. Blake, G. Plank, N. A. Trayanova, A novel rule-based algorithm for assigning myocardial fiber orientation to computational heart models, Ann. Biomed. Eng., 40 (2012), 2243–2254. https://doi.org/10.1007/s10439-012-0593-5 doi: 10.1007/s10439-012-0593-5
    [50] J. Wong, E. Kuhl, Generating fibre orientation maps in human heart models using poisson interpolation, Comput. Meth. Biomech. Biomed. Eng., 17 (2014), 1217–1226. https://doi.org/10.1080/10255842.2012.739167 doi: 10.1080/10255842.2012.739167
    [51] R. Doste, D. Soto-Iglesias, G. Bernardino, A. Alcaine, R. Sebastian, S. Giffard-Roisin, et al., A rule-based method to model myocardial fiber orientation in cardiac biventricular geometries with outflow tracts, Int. J. Numer. Meth. Biomed. Eng., 35 (2019), e3185. http://doi.org/10.1002/cnm.3185 doi: 10.1002/cnm.3185
    [52] R. Piersanti, P. Africa, M. Fedele, C. Vergara, L. Dedè, A. F. Corno, et al., Modeling cardiac muscle fibers in ventricular and atrial electrophysiology simulations, Comput. Meth. Appl. Mech. Eng., 373 (2021), 113468. https://doi.org/10.1016/j.cma.2020.113468 doi: 10.1016/j.cma.2020.113468
    [53] S. Rossi, T. Lassila, R. Ruiz-Baier, A. Sequeira, A. Quarteroni, Thermodynamically consistent orthotropic activation model capturing ventricular systolic wall thickening in cardiac electromechanics, Eur. J. Mech-A/Solids, 48 (2014), 129–142. https://doi.org/10.1016/j.euromechsol.2013.10.009 doi: 10.1016/j.euromechsol.2013.10.009
    [54] A. Saltelli, M. Ratto, T. Andres, F. Campolongo, J. Cariboni, D. Gatelli, et al., Global sensitivity analysis: the primer, John Wiley & Sons, 2008. https://doi.org/10.1002/9780470725184
    [55] F. Regazzoni, L. Dede', A. Quarteroni, Machine learning of multiscale active force generation models for the efficient simulation of cardiac electromechanics, Comput. Meth. Appl. Mech. Eng., 370 (2020), 113268. https://doi.org/10.1016/j.cma.2020.113268 doi: 10.1016/j.cma.2020.113268
    [56] Zygote solid 3d heart generation ii development report, Zygote Media Group Inc., 2014.
    [57] F. Regazzoni, M. Salvador, P. C. Africa, M. Fedele, L. Dedè, A. Quarteroni, A cardiac electromechanical model coupled with a lumped-parameter model for closed-loop blood circulation, J. Comput. Phys., 457 (2022), 111083. https://doi.org/10.1016/j.jcp.2022.111083 doi: 10.1016/j.jcp.2022.111083
    [58] F. Regazzoni, L. Dede', A. Quarteroni, Biophysically detailed mathematical models of multiscale cardiac active mechanics, PLoS Comput. Biol., 16 (2020), e1008294. https://doi.org/10.1371/journal.pcbi.1008294 doi: 10.1371/journal.pcbi.1008294
    [59] N. Halko, P. Martinsson, J. Tropp, Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions, SIAM Rev., 53 (2011), 217–288. https://doi.org/10.1137/090771806 doi: 10.1137/090771806
  • Reader Comments
  • © 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(2346) PDF downloads(246) Cited by(3)

Article outline

Figures and Tables

Figures(20)  /  Tables(4)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog