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A mathematical model of the human heart suitable to address clinical problems

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Abstract

In this paper, we present a mathematical model capable of simulating the human cardiac function. We review the basic equations of the model, their coupling, the numerical approach for the computer solution of this mathematical model, and a few examples of application to specific problems of clinical interest.

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Notes

  1. Notice that here \({\textbf{u}}_\Gamma\) is not intended as the absolute valve velocity, rather as the relative valve velocity with respect to the myocardial motion.

References

  1. Abraham, W.T., Hayes, D.L.: Cardiac resynchronization therapy for heart failure. Circulation 108(21), 2596–2603 (2003)

    Google Scholar 

  2. Arevalo, H.J., Vadakkumpadan, F., Guallar, E., Jebb, A., Malamas, P., Wu, K.C., Trayanova, N.A.: Arrhythmia risk stratification of patients after myocardial infarction using personalized heart models. Nat. Commun. 7(1), 1–8 (2016)

    Google Scholar 

  3. Augustin, C.M., Neic, A., Liebmann, M., Prassl, A.J., Niederer, S.A., Haase, G., Plank, G.: Anatomically accurate high resolution modeling of human whole heart electromechanics: A strongly scalable algebraic multigrid solver method for nonlinear deformation. J. Comput. Phys. 305, 622–646 (2016). https://doi.org/10.1016/j.jcp.2015.10.045

    Article  MathSciNet  MATH  Google Scholar 

  4. Basting, S., Quaini, A., Čanić, S., Glowinski, R.: Extended ale method for fluid-structure interaction problems with large structural displacements. J. Comput. Phys. 331, 312–336 (2017)

    MathSciNet  MATH  Google Scholar 

  5. Bayer, J., Blake, R., Plank, G., Trayanova, N.: A novel rule-based algorithm for assigning myocardial fiber orientation to computational heart models. Ann. Biomed. Eng. 40(10), 2243–2254 (2012)

    Google Scholar 

  6. Bazilevs, Y., Calo, V., Cottrell, J., Hughes, T., Reali, A., Scovazzi, G.: Variational multiscale residual-based turbulence modeling for large eddy simulation of incompressible flows. Comput. Methods Appl. Mech. Eng. 197(1–4), 173–201 (2007)

    MathSciNet  MATH  Google Scholar 

  7. Blanco, P.J., Feijóo, R.A.: A 3D–1D-0D computational model for the entire cardiovascular system. Comput. Mech. 24, 5887–5911 (2010)

    Google Scholar 

  8. Bucelli, M., Zingaro, A., Africa, P.C., Fumagalli, I., Dede, L., Quarteroni, A.: A mathematical model that integrates cardiac electrophysiology, mechanics and fluid dynamics: application to the human left heart. arXiv preprint arXiv:2208.05551 (2022)

  9. 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), 20150083 (2016)

    Google Scholar 

  10. Colli Franzone, P., Pavarino, L.F., Scacchi, S.: Mathematical Cardiac Electrophysiology. Springer (2014)

    MATH  Google Scholar 

  11. Cookson, A., Lee, J., Michler, C., Chabiniok, R., Hyde, E., Nordsletten, D., Sinclair, M., Siebes, M., Smith, N.: A novel porous mechanical framework for modelling the interaction between coronary perfusion and myocardial mechanics. J. Biomech. 45(5), 850–855 (2012)

    Google Scholar 

  12. Crampin, E.J., Halstead, M., Hunter, P., Nielsen, P., Noble, D., Smith, N., Tawhai, M.: Computational physiology and the physiome project. Exp. Physiol. 89(1), 1–26 (2004)

    Google Scholar 

  13. Del Corso, G., Verzicco, R., Viola, F.: A fast computational model for the electrophysiology of the whole human heart. J. Comput. Phys. 457, 111084 (2022). https://doi.org/10.1016/j.jcp.2022.111084

    Article  MathSciNet  MATH  Google Scholar 

  14. Di Gregorio, S., Fedele, M., Pontone, G., Corno, A.F., Zunino, P., Vergara, C., Quarteroni, A.: A computational model applied to myocardial perfusion in the human heart: from large coronaries to microvasculature. J. Comput. Phys. 424, 109836 (2021)

    MathSciNet  MATH  Google Scholar 

  15. Di Gregorio, S., Vergara, C., Pelagi, G.M., Baggiano, A., Zunino, P., Guglielmo, M., Fusini, L., Muscogiuri, G., Rossi, A., Rabbat, M.G., et al.: Prediction of myocardial blood flow under stress conditions by means of a computational model. Eur. J. Nucl. Med. Mol. Imaging 49(6), 1894–1905 (2022)

    Google Scholar 

  16. Fedele, M., Faggiano, E., Dedè, L., Quarteroni, A.: A patient-specific aortic valve model based on moving resistive immersed implicit surfaces. Biomech. Model. Mechanobiol. 16(5), 1779–1803 (2017)

    Google Scholar 

  17. Fedele, M., Piersanti, R., Regazzoni, F., Salvador, M., Africa, P.C., Bucelli, M., Zingaro, A., Dede, L., Quarteroni, A.: A comprehensive and biophysically detailed computational model of the whole human heart electromechanics. arXiv preprint arXiv:2207.12460 (2022)

  18. Fedele, M., Quarteroni, A.: Polygonal surface processing and mesh generation tools for the numerical simulation of the cardiac function. Int. J. Numer. Methods Biomed. Eng. 37(4), e3435 (2021). https://doi.org/10.1002/cnm.3435

    Article  MathSciNet  Google Scholar 

  19. 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)

    Google Scholar 

  20. Forti, D., Dedè, L.: Semi-implicit bdf time discretization of the navier-stokes equations with vms-les modeling in a high performance computing framework. Comput. Fluids 117, 168–182 (2015)

    MathSciNet  MATH  Google Scholar 

  21. Frontera, A., Limite, L., Pagani, S., Cireddu, M., Vlachos, K., et al.: Electrogram fractionation during sinus rhythm occurs in normal voltage atrial tissue in patients with atrial fibrillation. Pacing Clin. Electrophysiol. 45(2), 219–228 (2022)

    Google Scholar 

  22. Frontera, A., Pagani, S., Limite, L., Hadjis, A., Manzoni, A., Dede’, L., Quarteroni, A., Della Bella, P.: Outer loop and isthmus in ventricular tachycardia circuits: characteristics and implications. Heart Rhythm 17(10), 1719–1728 (2020)

    Google Scholar 

  23. Frontera, A., Pagani, S., Limite, L.R., Peirone, A., Fioravanti, F., Enache, B., Cuellar Silva, J., Vlachos, K., Meyer, C., Montesano, G., et al.: Slow conduction corridors and pivot sites characterize the electrical remodeling in atrial fibrillation. JACC Clin. Electrophysiol. 8(5), 561–577 (2022). https://doi.org/10.1016/j.jacep.2022.01.019

    Article  Google Scholar 

  24. Fumagalli, I., Fedele, M., Vergara, C., Ippolito, S., Nicolò, F., Antona, C., Scrofani, R., Quarteroni, A., et al.: An image-based computational hemodynamics study of the systolic anterior motion of the mitral valve. Comput. Biol. Med. 123, 103922 (2020)

    Google Scholar 

  25. Fumagalli, I., Polidori, R., Renzi, F., Fusini, L., Quarteroni, A., Pontone, G., Vergara, C.: Fluid-structure interaction analysis of transcatheter aortic valve implantation. MOX Report n.29 (2022)

  26. Fumagalli, I., Vitullo, P., Vergara, C., Fedele, M., Corno, A.F., Ippolito, S., Scrofani, R., Quarteroni, A.: Image-based computational hemodynamics analysis of systolic obstruction in hypertrophic cardiomyopathy. Front. Physiol. 12, 787082 (2022)

  27. Genet, M., Lee, L.C., Baillargeon, B., Guccione, J.M., Kuhl, E.: Modeling pathologies of diastolic and systolic heart failure. Ann. Biomed. Eng. 44(1), 112–127 (2016)

    Google Scholar 

  28. Gerach, T., Schuler, S., Fröhlich, J., Lindner, L., Kovacheva, E., Moss, R., Wülfers, E.M., Seemann, G., Wieners, C., Loewe, A.: Electro-mechanical whole-heart digital twins: a fully coupled multi-physics approach. Mathematics 9(11), 1247 (2021). https://doi.org/10.3390/math9111247

    Article  Google Scholar 

  29. Glass, L., Hunter, P., McCulloch, A.: Theory of Heart: Biomechanics, Biophysics, and Nonlinear Dynamics of Cardiac Function. Springer Science & Business Media (2012)

    Google Scholar 

  30. Göktepe, S., Kuhl, E.: Electromechanics of the heart: a unified approach to the strongly coupled excitation-contraction problem. Comput. Mech. 45, 227–243 (2010)

    MathSciNet  MATH  Google Scholar 

  31. Gray, R.A., Pathmanathan, P.: Patient-specific cardiovascular computational modeling: diversity of personalization and challenges. J. Cardiovasc. Transl. Res. 11(2), 80–88 (2018)

    Google Scholar 

  32. Griffith, B.E., Peskin, C.S.: Electrophysiology. Commun. Pure Appl. Math. 66(12), 1837–1913 (2013). https://doi.org/10.1002/cpa.21484

    Article  MathSciNet  MATH  Google Scholar 

  33. Guccione, J.M., McCulloch, A.D.: Finite element modeling of ventricular mechanics. In: Glass, L., Hunter, P., McCulloch, A. (eds) Theory of Heart, pp. 121–144. Springer (1991). https://doi.org/10.1007/978-1-4612-3118-9_6

  34. Guccione, J.M., McCulloch, A.D., Waldman, L.K.: Passive material properties of intact ventricular myocardium determined from a cylindrical model. J. Biomech. Eng. 113, 42–55 (1991)

    Google Scholar 

  35. Hirschvogel, M., Bassilious, M., Jagschies, L., et al.: A monolithic 3D–0D coupled closed-loop model of the heart and the vascular system: Experiment-based parameter estimation for patient-specific cardiac mechanics. Int. J. Numer. Methods Biomed. Eng. 33(8), e2842 (2017)

    MathSciNet  Google Scholar 

  36. Holzapfel, G.A., Ogden, R.W.: Constitutive modelling of passive myocardium: a structurally based framework for material characterization. Math. Phys. Eng. Sci. 367, 3445–3475 (2009)

    MathSciNet  MATH  Google Scholar 

  37. Huyghe, J.M., Van Campen, D.H.: Finite deformation theory of hierarchically arranged porous solids-i. Balance of mass and momentum. Int. J. Eng. Sci. 33(13), 1861–1871 (1995)

    MATH  Google Scholar 

  38. Jianhai, Z., Dapeng, C., Shengquan, Z.: Ale finite element analysis of the opening and closing process of the artificial mechanical valve. Appl. Math. Mech. 17(5), 403–412 (1996)

    MATH  Google Scholar 

  39. Jung, A., Gsell, M., Augustin, C., Plank, G.: An integrated workflow for building digital twins of cardiac electromechanics-a multi-fidelity approach for personalising active mechanics. Mathematics 10(5), 823 (2022)

  40. Karabelas, E., Longobardi, S., Fuchsberger, J., Razeghi, O., Rodero, C., Strocchi, M., Rajani, R., Haase, G., Plank, G., Niederer, S.: Global sensitivity analysis of four chamber heart hemodynamics using surrogate models. IEEE Trans. Biomed. Eng. 69(10), 3216–3223 (2022)

  41. Katz, A.M.: Physiology of the Heart. Lippincott Williams & Wilkins (2010)

    Google Scholar 

  42. Klabunde, R.: Cardiovascular Physiology Concepts. Lippincott Williams & Wilkins (2011)

    Google Scholar 

  43. Land, S., Niederer, S.A.: Influence of atrial contraction dynamics on cardiac function. Int. J. Numer. Methods Biomed. Eng. 34(3), e2931 (2018). https://doi.org/10.1002/cnm.2931

    Article  MathSciNet  Google Scholar 

  44. Mitchell, J.R., Wang, J.J.: Expanding application of the Wiggers diagram to teach cardiovascular physiology. Adv. Physiol. Educ. 38(2), 170–175 (2014)

    Google Scholar 

  45. Mittal, R., Seo, J.H., Vedula, V., Choi, Y.J., Liu, H., Huang, H.H., Jain, S., Younes, L., Abraham, T., George, R.T.: Computational modeling of cardiac hemodynamics: current status and future outlook. J. Comput. Phys. 305, 1065–1082 (2016). https://doi.org/10.1016/j.jcp.2015.11.022

    Article  MathSciNet  MATH  Google Scholar 

  46. Niederer, S.A., Lumens, J., Trayanova, N.A.: Computational models in cardiology. Nat. Rev. Cardiol. 16(2), 100–111 (2019). https://doi.org/10.1038/s41569-018-0104-y

    Article  Google Scholar 

  47. 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)

    Google Scholar 

  48. Ogden, R.: Non-linear Elastic Deformations. Dover Publications (1997)

    Google Scholar 

  49. Pagani, S., Dede’, L., Frontera, A., Salvador, M., Limite, L., et al.: A computational study of the electrophysiological substrate in patients suffering from atrial fibrillation. Front. Physiol. 12, 673612 (2021)

    Google Scholar 

  50. Peirlinck, M., Yao, J., Sahli Costabal, F., Kuhl, E.: How drugs modulate the performance of the human heart. Comput. Mech. (2022). https://doi.org/10.1007/s10237-021-01421-z

    Article  MathSciNet  MATH  Google Scholar 

  51. Peskin, C.S.: Flow patterns around heart valves: a numerical method. J. Comput. Phys. 10(2), 252–271 (1972)

    MathSciNet  MATH  Google Scholar 

  52. 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). https://doi.org/10.1007/s10237-018-1098-4

    Article  Google Scholar 

  53. Piersanti, R., Africa, P.C., Fedele, M., Vergara, C., Dedè, L., Corno, A.F., Quarteroni, A.: Modeling cardiac muscle fibers in ventricular and atrial electrophysiology simulations. Comput. Methods Appl. Mech. Eng. 373, 113468 (2021)

    MathSciNet  MATH  Google Scholar 

  54. Quarteroni, A.: Numerical Models for Differential Problems. Springer (2017)

    MATH  Google Scholar 

  55. Quarteroni, A., Dede’, L., Manzoni, A., Vergara, C.: Mathematical Modelling of the Human Cardiovascular System: Data, Numerical Approximation, Clinical Applications. Cambridge University Press (2019)

    MATH  Google Scholar 

  56. 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)

    MathSciNet  MATH  Google Scholar 

  57. Regazzoni, F., Dedè, L., Quarteroni, A.: Biophysically detailed mathematical models of multiscale cardiac active mechanics. PLoS Comput. Biol. 16(10), e1008294 (2020). https://doi.org/10.1371/journal.pcbi.1008294

    Article  Google Scholar 

  58. 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. 49, 87–118 (2021)

    MathSciNet  MATH  Google Scholar 

  59. Regazzoni, F., Salvador, M., Africa, P., Fedele, M., Dedè, L., Quarteroni, A.: A cardiac electromechanical model coupled with a lumped-parameter model for closed-loop blood circulation. J. Comput. Phys. 457, 111083 (2022). https://doi.org/10.1016/j.jcp.2022.111083

    Article  MathSciNet  MATH  Google Scholar 

  60. Regazzoni, F., Salvador, M., Dedè, L., Quarteroni, A.: A machine learning method for real-time numerical simulations of cardiac electromechanics. Comput. Methods Appl. Mech. Eng. 393, 114825 (2022)

    MathSciNet  MATH  Google Scholar 

  61. Salvador, M., Regazzoni, F., Pagani, S., Dede, L., Trayanova, N., Quarteroni, A.: The role of mechano-electric feedbacks and hemodynamic coupling in scar-related ventricular tachycardia. Comput. Biol. Med. 142, 105203 (2022)

  62. Santiago, A., Aguado-Sierra, J., Zavala-Aké, M., Doste-Beltran, R., Gómez, S., Arís, R., Cajas, J.C., Casoni, E., Vázquez, M.: Fully coupled fluid-electro-mechanical model of the human heart for supercomputers. Int. J. Numer. Methods Biomed. Eng. 34(12), e3140 (2018). https://doi.org/10.1002/cnm.3140

    Article  MathSciNet  Google Scholar 

  63. Smith, N., Nickerson, D., Crampin, E., Hunter, P.: Multiscale computational modelling of the heart. Acta Numer. 13, 371–431 (2004)

    MathSciNet  MATH  Google Scholar 

  64. Smith, N.P., Nickerson, D.P., Crampin, E.J., et al.: Multiscale computational modelling of the heart. Acta Numer. 13, 371–431 (2004)

    MathSciNet  MATH  Google Scholar 

  65. Stella, S., Vergara, C., Maines, M., Catanzariti, D., Africa, P.C., Demattè, C., Centonze, M., Nobile, F., Del Greco, M., Quarteroni, A.: Integration of activation maps of epicardial veins in computational cardiac electrophysiology. Comput. Biol. Med. 127, 104047 (2020)

    Google Scholar 

  66. Sugiura, S., Okada, J., Washio, T., Hisada, T.: UT-Heart: A Finite Element Model Designed for the Multiscale and Multiphysics Integration of our Knowledge on the Human Heart, pp. 221–245. Springer, US, New York (2022)

    Google Scholar 

  67. Tomek, J., Bueno-Orovio, A., Passini, E., Zhou, X., Minchole, A., Britton, O., Bartolucci, C., Severi, S., Shrier, A., Virag, L., Varro, A., Rodriguez, B.: Development, calibration, and validation of a novel human ventricular myocyte model in health, disease, and drug block. eLife 8, e48890 (2019)

    Google Scholar 

  68. Trayanova, N., Rice, J.: Cardiac electromechanical models: from cell to organ. Front. Physiol. 2, 43 (2011)

    Google Scholar 

  69. Trayanova, N.A.: Computational cardiology: the heart of the matter. International Scholarly Research Notices (2012). https://doi.org/10.5402/2012/269680

  70. Vergara, C., Stella, S., Maines, M., Africa, P.C., Catanzariti, D., Demattè, C., Centonze, M., Nobile, F., Quarteroni, A., Del Greco, M.: Computational electrophysiology of the coronary sinus branches based on electro-anatomical mapping for the prediction of the latest activated region. Med. Biol. Eng. Comput. 60(8), 2307–2319 (2022)

    Google Scholar 

  71. Verzicco, R.: Electro-fluid-mechanics of the heart. J. Fluid Mech. 941 (2022)

  72. Viola, F., Meschini, V., Verzicco, R.: Fluid-structure-electrophysiology interaction (fsei) in the left-heart: a multi-way coupled computational model. Eur. J. Mech.-B/Fluids 79, 212–232 (2020). https://doi.org/10.1016/j.euromechflu.2019.09.006

    Article  MathSciNet  MATH  Google Scholar 

  73. Washio, T., Okada, Ji., Takahashi, A., Yoneda, K., Kadooka, Y., Sugiura, S., Hisada, T.: Multiscale heart simulation with cooperative stochastic cross-bridge dynamics and cellular structures. Multiscale Modeling & Simulation 11(4), 965–999 (2013)

  74. Zingaro, A., Fumagalli, I., Dede, L., Fedele, M., Africa, P.C., Corno, A.F., Quarteroni, A.: A geometric multiscale model for the numerical simulation of blood flow in the human left heart. Discrete and Continuous Dynamical Systems-S 15(8), 2391–2427 (2022)

    MathSciNet  MATH  Google Scholar 

  75. Zingaro, A., Menghini, F., Quarteroni, A., et al.: Hemodynamics of the heart’s left atrium based on a variational multiscale-les numerical method. European Journal of Mechanics-B/Fluids 89, 380–400 (2021)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

We would like to thank Professor Antonio Corno of the Children’s Heart Institute, Children’s Memorial Hermann Hospital and McGovern Medical School in Houston, Professor Paolo Della Bella of I.R.C.C.S. San Raffaele Hospital in Milan and Dr. Antonio Frontera of Humanitas Research Hospital, Dr. Roberto Scrofani of the Policlinico Ca’ Granda Hospital in Milan, Drs. Gianluca Pontone and Laura Fusini of the Monzino Cardiology Center in Milan, Drs. Maurizio Del Greco and Domenico Catanzariti of the S. Maria del Carmine Hospital in Rovereto.

Funding

This work was funded by Ministero dell’Istruzione, dell’Università e della Ricerca with Grant no. PRIN17 2017AXL54F and by Gruppo Nazionale per il Calcolo Scientifico with Grant no. CUP_E55F22000270001.

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Authors and Affiliations

  1. MOX, Dipartimento di Matematica, Politecnico di Milano, P.zza Leonardo da Vinci 32, 20133, Milan, Italy

    Alfio Quarteroni, Luca Dede’ & Francesco Regazzoni

  2. Professor Emeritus, Mathematics Institute, École Polytechnique Fédérale de Lausanne (EPFL), Av. Piccard, 1015, Lausanne, Switzerland

    Alfio Quarteroni

  3. LABS, Dipartimento di Chimica, Materiali e Ingegneria Chimica “Giulio Natta”, Politecnico di Milano, P.zza Leonardo da Vinci 32, 20133, Milan, Italy

    Christian Vergara

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  1. Alfio Quarteroni
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Correspondence to Francesco Regazzoni.

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This work has been supported by the Italian research project MIUR PRIN17 2017AXL54F “Modeling the heart across the scales: from cardiac cells to the whole organ” and by the GNCS, “Gruppo Nazionale per il Calcolo Scientifico” (National Group for Scientific Computing) under the INdAM GNCS Project CUP_E55F22000270001.

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Quarteroni, A., Dede’, L., Regazzoni, F. et al. A mathematical model of the human heart suitable to address clinical problems. Japan J. Indust. Appl. Math. 40, 1547–1567 (2023). https://doi.org/10.1007/s13160-023-00579-6

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