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An oscillation-free fully staggered algorithm for velocity-dependent active models of cardiac mechanics. (English) Zbl 1506.74213

Summary: In this paper we address an unresolved problem in the numerical modeling of cardiac electromechanics, that is the onset of numerical oscillations due to the dependence of force generation models on the fibers shortening velocity. A way to avoid numerical oscillations is to use monolithic schemes for the solution of the coupled problem of active-passive mechanics. However, staggered strategies, which foresee the sequential solution of the models of force generation and of tissue mechanics, are preferable, due to their reduced computational cost and low implementation effort. In this paper we propose a cure for this issue, by introducing, with respect to the standard staggered scheme, a numerically consistent stabilization term. This term is derived in virtue of the identification of the cause of instability in the mismatch between macroscopic and microscopic strains, inconsistently expressed in Lagrangian and Eulerian coordinates, respectively. By considering a model problem of active mechanics we prove that the proposed scheme is unconditionally absolutely stable (i.e. it is stable for any time step size), yet within a fully staggered framework. As such, the new scheme removes the non-physical oscillations, as we prove by applying it to three force generation models, namely the Niederer-Hunter-Smith model, the model by Land and coworkers, and the mean-field force generation model that we have recently proposed.

MSC:

74L15 Biomechanical solid mechanics
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
92C50 Medical applications (general)
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[1] Jenkins, G. W.; Kemnitz, C. P.; Tortora, G. J., Anatomy and Physiology: From Science to Life (2007), Wiley Hoboken
[2] Tortora, G. J.; Derrickson, B. H., Principles of Anatomy and Physiology (2008), John Wiley & Sons
[3] Katz, A. M., Physiology of the Heart (2010), Lippincott Williams & Wilkins
[4] Smith, N.; Nickerson, D.; Crampin, E.; Hunter, P., Multiscale computational modelling of the heart, Acta Numer., 13, 371-431 (2004) · Zbl 1112.92021
[5] Nordsletten, D.; Niederer, S.; Nash, M.; Hunter, P.; Smith, N., Coupling multi-physics models to cardiac mechanics, Prog. Biophys. Mol. Biol., 104, 1-3, 77-88 (2011)
[6] Fink, M.; Niederer, S.; Cherry, E.; Fenton, F.; Koivumäki, J.; Seemann, G.; Thul, R.; Zhang, H.; Sachse, F.; Beard, D.; Crampin, E.; Smith, N., Cardiac cell modelling: observations from the heart of the cardiac physiome project, Prog. Biophys. Mol. Biol., 104, 1, 2-21 (2011)
[7] Chabiniok, R.; Wang, V.; Hadjicharalambous, M.; Asner, L.; Lee, J.; Sermesant, M.; Kuhl, E.; Young, A.; Moireau, P.; Nash, M.; Chapelle, D.; Nordsletten, D., Multiphysics and multiscale modelling, data-model fusion and integration of organ physiology in the clinic: ventricular cardiac mechanics, Interface Focus, 6, 2, Article 20150083 pp. (2016)
[8] Quarteroni, A.; Dede’, L.; Manzoni, A.; Vergara, C., (Mathematical Modelling of the Human Cardiovascular System: Data, Numerical Approximation, Clinical Applications. Mathematical Modelling of the Human Cardiovascular System: Data, Numerical Approximation, Clinical Applications, Cambridge Monographs on Applied and Computational Mathematics (2019), Cambridge University Press) · Zbl 1411.92003
[9] 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. Engrg., 314, 345-407 (2017) · Zbl 1439.74208
[10] Niederer, S. A.; Smith, N. P., An improved numerical method for strong coupling of excitation and contraction models in the heart, Prog. Biophys. Mol. Biol., 96, 1-3, 90-111 (2008)
[11] 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, Quart. J. Mech. Appl. Math., 63, 3, 375-399 (2010) · Zbl 1250.74019
[12] Gerbi, A., Numerical Approximation of Cardiac Electro-fluid-mechanical Models: Coupling Strategies for Large-scale Simulation (2018), Ecole Polytechnique Fédérale de Lausanne, (Ph.D. thesis)
[13] Dedè, L.; Gerbi, A.; Quarteroni, A., Segregated algorithms for the numerical simulation of cardiac electromechanics in the left human ventricle, (Ambrosi, D.; Ciarletta, P., The Mathematics of Mechanobiology (2020), Springer) · Zbl 1446.74210
[14] 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, 7, 2199-2225 (2007) · Zbl 1296.92120
[15] Pathmanathan, P.; Whiteley, J. P., A numerical method for cardiac mechanoelectric simulations, Ann. Biomed. Eng., 37, 5, 860-873 (2009)
[16] Regazzoni, F., Mathematical Modeling and Machine Learning for the Numerical Simulation of Cardiac Electromechanics (2020), Politecnico di Milano, (Ph.D. thesis)
[17] Levrero-Florencio, F.; Margara, F.; Zacur, E.; Bueno-Orovio, A.; Wang, Z.; Santiago, A.; Aguado-Sierra, J.; Houzeaux, G.; Grau, V.; Kay, D., Sensitivity analysis of a strongly-coupled human-based electromechanical cardiac model: Effect of mechanical parameters on physiologically relevant biomarkers, Comput. Methods Appl. Mech. Engrg., 361, Article 112762 pp. (2020) · Zbl 1442.74132
[18] Hill, A., The heat of shortening and the dynamic constants of muscle, Proc. R. Soc. Lond. Ser. B: Biol. Sci., 126, 843, 136-195 (1938)
[19] Caremani, M.; Pinzauti, F.; Reconditi, M.; Piazzesi, G.; Stienen, G. J.; Lombardi, V.; Linari, M., Size and speed of the working stroke of cardiac myosin in situ, Proc. Natl. Acad. Sci., 113, 13, 3675-3680 (2016)
[20] Colli Franzone, P.; Pavarino, L.; Scacchi, S., Mathematical Cardiac Electrophysiology, Vol. 13 (2014), Springer · Zbl 1318.92002
[21] Zhang, W.; Capilnasiu, A.; Sommer, G.; Holzapfel, G.; Nordsletten, D., An efficient and accurate method for modeling nonlinear fractional viscoelastic biomaterials (2019), arXiv preprint arXiv:1910.02141
[22] Huxley, A. F., Muscle structure and theories of contraction, Prog. Biophys. Biophys. Chem., 7, 255-318 (1957)
[23] Zahalak, G. I., A distribution-moment approximation for kinetic theories of muscular contraction, Math. Biosci., 55, 1-2, 89-114 (1981) · Zbl 0475.92010
[24] Bestel, J.; Clément, F.; Sorine, M., A biomechanical model of muscle contraction, (International Conference on Medical Image Computing and Computer-Assisted Intervention (2001), Springer), 1159-1161 · Zbl 1041.68560
[25] Chapelle, D.; Le Tallec, P.; Moireau, P.; Sorine, M., Energy-preserving muscle tissue model: formulation and compatible discretizations, Int. J. Multiscale Comput. Eng., 10, 2 (2012)
[26] Antman, S. S., Nonlinear Problems of Elasticity (1995), Springer · Zbl 0820.73002
[27] Quarteroni, A.; Valli, A., Numerical Approximation of Partial Differential Equations, Vol. 23 (2008), Springer Science & Business Media · Zbl 1151.65339
[28] Quarteroni, A.; Sacco, R.; Saleri, F., Numerical Mathematics, Vol. 37 (2010), Springer Science & Business Media
[29] Niederer, S.; Hunter, P.; Smith, N., A quantitative analysis of cardiac myocyte relaxation: a simulation study, Biophys. J., 90, 5, 1697-1722 (2006)
[30] Land, S.; Park-Holohan, S.; Smith, N.; dos Remedios, C.; Kentish, J.; Niederer, S., A model of cardiac contraction based on novel measurements of tension development in human cardiomyocytes, J. Mol. Cell. Cardiol., 106, 68-83 (2017)
[31] Regazzoni, F.; Dedè, L.; Quarteroni, A., Biophysically detailed mathematical models of multiscale cardiac active mechanics, PLoS Comput. Biol., 16, 10, Article e1008294 pp. (2020)
[32] Washio, T.; Okada, J.; Sugiura, S.; Hisada, T., Approximation for cooperative interactions of a spatially-detailed cardiac sarcomere model, Cell. Mol. Bioeng., 5, 1, 113-126 (2012)
[33] Zygote 3D models, https://www.zygote.com/.
[34] Bayer, J. D.; Blake, R. C.; Plank, G.; Trayanova, N. A., A novel rule-based algorithm for assigning myocardial fiber orientation to computational heart models, Ann. Biomed. Eng., 40, 10, 2243-2254 (2012)
[35] Colli Franzone, P.; Pavarino, L. F.; Savaré, G., Computational electrocardiology: mathematical and numerical modeling, (Complex Systems in Biomedicine (2006), Springer), 187-241 · Zbl 1387.92056
[36] Colli Franzone, P.; Pavarino, L. F.; Scacchi, S., Mathematical Cardiac Electrophysiology, Vol. 13 (2014), Springer · Zbl 1318.92002
[37] Ten Tusscher, K. H.; Panfilov, A. V., Alternans and spiral breakup in a human ventricular tissue model, Amer. J. Physiol.-Heart Circ. Physiol., 291, 3, H1088-H1100 (2006)
[38] Usyk, T. P.; LeGrice, I. J.; McCulloch, A. D., Computational model of three-dimensional cardiac electromechanics, Comput. Vis. Sci., 4, 4, 249-257 (2002) · Zbl 1001.92005
[39] Pfaller, M. R.; Hörmann, J. M.; Weigl, M.; Nagler, A.; Chabiniok, R.; Bertoglio, C.; Wall, W. A., The importance of the pericardium for cardiac biomechanics: From physiology to computational modeling, Biomech. Model. Mechanobiol., 18, 2, 503-529 (2019)
[40] Gerbi, A.; Dedè, L.; Quarteroni, A., A monolithic algorithm for the simulation of cardiac electromechanics in the human left ventricle, Math. Eng., 1, 1, 1-37 (2018) · Zbl 1442.92006
[41] Regazzoni, F.; Dedè, L.; Quarteroni, A., Machine learning of multiscale active force generation models for the efficient simulation of cardiac electromechanics, Comput. Methods Appl. Mech. Engrg., 370, Article 113268 pp. (2020) · Zbl 1506.74212
[42] Westerhof, N.; Lankhaar, J.-W.; Westerhof, B. E., The arterial windkessel, Med. Biol. Eng. Comput., 47, 2, 131-141 (2009)
[43] Regazzoni, F.; Dedè, L.; Quarteroni, A., Active force generation in cardiac muscle cells: mathematical modeling and numerical simulation of the actin-myosin interaction, Vietnam J. Math., 1-32 (2020)
[44] Bergel, D.; Hunter, P., The mechanics of the heart, Quant. Cardiovasc. Stud., 151-213 (1979)
[45] Hunter, P., Myocardial constitutive laws for continuum models of the heart, (Sideman, S.; Beyar, R., Molecular and Subcellular Cardiology (1995), Plenum Press: Plenum Press New York)
[46] Hunter, P.; McCulloch, A.; Ter Keurs, H., Modelling the mechanical properties of cardiac muscle, Prog. Biophys. Mol. Biol., 69, 2, 289-331 (1998)
[47] Land, S.; Niederer, S. A.; Aronsen, J. M.; Espe, E. K.; Zhang, 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, 18, 4553-4569 (2012)
[48] Razumova, M.; Bukatina, A.; Campbell, K., Stiffness-distortion sarcomere model for muscle simulation, J. Appl. Physiol., 87, 5, 1861-1876 (1999)
[49] Rice, J.; Wang, F.; Bers, D.; de Tombe, P., Approximate model of cooperative activation and crossbridge cycling in cardiac muscle using ordinary differential equations, Biophys. J., 95, 5, 2368-2390 (2008)
[50] Ford, S. J.; Chandra, M.; Mamidi, R.; Dong, W.; Campbell, K. B., Model representation of the nonlinear step response in cardiac muscle, J. Gen. Physiol., 136, 2, 159-177 (2010)
[51] Regazzoni, F.; Dedè, L.; Quarteroni, A., Active contraction of cardiac cells: a reduced model for sarcomere dynamics with cooperative interactions, Biomech. Model. Mechanobiol., 17, 1663-1686 (2018)
[52] Washio, T.; Okada, J.; Takahashi, A.; Yoneda, K.; Kadooka, Y.; Sugiura, S.; Hisada, T., Multiscale heart simulation with cooperative stochastic cross-bridge dynamics and cellular structures, Multiscale Model. Simul., 11, 4, 965-999 (2013) · Zbl 1284.92022
[53] Washio, T.; Yoneda, K.; Okada, J.; Kariya, T.; Sugiura, S.; Hisada, T., Ventricular fiber optimization utilizing the branching structure, Int. J. Numer. Methods Biomed. Eng. (2015)
[54] Hussan, J.; de Tombe, P.; Rice, J., A spatially detailed myofilament model as a basis for large-scale biological simulations, IBM J. Res. Dev., 50, 6, 583-600 (2006)
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