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Segregated Algorithms for the Numerical Simulation of Cardiac Electromechanics in the Left Human Ventricle

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The Mathematics of Mechanobiology

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 2260))

Abstract

We propose and numerically assess three segregated (partitioned) algorithms for the numerical solution of the coupled electromechanics problem for the left human ventricle. We split the coupled problem into its core mathematical models and we proceed to their numerical approximation. Space and time discretizations of the core problems are carried out by means of the Finite Element Method and Backward Differentiation Formulas, respectively. In our mathematical model, electrophysiology is represented by the monodomain equation while the Holzapfel-Ogden strain energy function is used for the passive characterization of tissue mechanics. A transmurally variable active strain model is used for the active deformation of the fibers of the myocardium to couple the electrophysiology and the mechanics in the framework of the active strain model. In this work, we focus on the numerical strategy to deal with the solution of the coupled model, which is based on novel segregated algorithms that we propose. These also allow using different time discretization schemes for the core submodels, thus leading to the formulation of staggered algorithms, a feature that we systematically exploit to increase the efficiency of the overall computational procedure. By means of numerical tests we show that these staggered algorithms feature (at least) first order of accuracy. We take advantage of the efficiency of the segregated schemes to solve, in a High Performance Computing framework, the cardiac electromechanics problem for the human left ventricle, for both idealized and subject-specific configurations.

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Notes

  1. 1.

    https://cmcsforge.epfl.ch.

  2. 2.

    The MRI images are provided by Prof. J. Schwitter (Chief physician at the Centre Hospitalier Universitaire Vaudois CHUV, Lausanne) and Dr. P. Masci (CHUV) in the framework of the collaboration CMCS@EPFL–CHUV.

  3. 3.

    http://www.cscs.ch.

  4. 4.

    Unfortunately the maximum wall time allowed on the Piz Daint supercomputer is set to 24 h, thus making it impossible to complete a simulation of a full heartbeat in all cases—most notably for the (\({\mathcal {I}_{\mathrm {I}}\mathcal {E}_{\mathrm {I}}\mathcal {A}_{\mathrm {I}}\mathcal {M}_{\mathrm {I}}}\)) strategy, which is the most computational demanding. We hence run two sets of simulations: in the first case, we set T = 0.8 s thus obtaining the pressure-volume loops of Fig. 3.16; in the second one, we set T = 0.073 s (the maximum time reachable in 24 h with the (\({\mathcal {I}_{\mathrm {I}}\mathcal {E}_{\mathrm {I}}\mathcal {A}_{\mathrm {I}}\mathcal {M}_{\mathrm {I}}}\)) strategy) thus obtaining the results of Fig. 3.16.

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Acknowledgements

This research was partially supported by the Swiss Platform for Advanced Scientific Computing (PASC, project “Integrative HPC Framework for Coupled Cardiac Simulations”). We also gratefully acknowledge the Swiss National Supercomputing Center (CSCS) for providing the CPU resources for the numerical simulations under project IDs s635/s796.

We acknowledge Prof. J. Schwitter and Dr. P. Masci (CHUV, Lausanne) for providing the MRI images used in this work and for the enlightening discussions. We also thank Prof. P. Tozzi (CHUV, Lausanne) for the invaluable insights in the functioning of the human heart.

The authors acknowledge the ERC Advanced Grant iHEART, “An Integrated Heart Model for the simulation of the cardiac function”, 2017–2022, P.I. A. Quarteroni (ERC–2016–ADG, project ID: 740132).

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Authors and Affiliations

  1. MOX–Modeling and Scientific Computing, Mathematics Department, Politecnico di Milano, Milano, Italy

    L. Dede’, A. Gerbi & A. Quarteroni

  2. Institute of Mathematics, École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland

    A. Quarteroni

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  1. L. Dede’
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Correspondence to L. Dede’ .

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  1. Department of Mathematical Sciences, Polytechnic University of Turin, Torino, Italy

    Davide Ambrosi

  2. MOX, Dipartimento Di Matematica, Polytechnic University of Milan, Milano, Italy

    Pasquale Ciarletta

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Dede’, L., Gerbi, A., Quarteroni, A. (2020). Segregated Algorithms for the Numerical Simulation of Cardiac Electromechanics in the Left Human Ventricle. In: Ambrosi, D., Ciarletta, P. (eds) The Mathematics of Mechanobiology. Lecture Notes in Mathematics(), vol 2260. Springer, Cham. https://doi.org/10.1007/978-3-030-45197-4_3

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