## Abstract

Modeling of the heart ventricles is one of the most challenging tasks in soft tissue mechanics because cardiac tissue is a strongly anisotropic incompressible material with an active component of stress. In most current approaches with active force, the number of degrees of freedom (DOF) is limited by the direct method of solution of linear systems of equations. We develop a new approach for high-resolution heart models with large numbers of DOF by: (1) developing a hex-dominant finite element mixed formulation and (2) developing a Krylov subspace iterative method that is able to solve the system of linearized equations for saddle-point problems with active stress. In our approach, passive cardiac tissue is modeled as a hyperelastic, incompressible material with orthotropic properties, and mixed pressure-displacement finite elements are used to enforce incompressibility. Active stress is generated by a model with force dependence on length and velocity of muscle shortening. The ventricles are coupled to a lumped circulatory model. For efficient solution of linear systems, we use Flexible GMRES with a nonlinear preconditioner based on block matrix decomposition involving the Schur complement. Three methods for approximating the inverse of the Schur complement are evaluated: inverse of the pressure mass matrix; least squares commutators; and sparse approximate inverse. The sub-matrix corresponding to the displacement variables is preconditioned by a V-cycle of hybrid geometric–algebraic multigrid followed by correction with several iterations of GMRES preconditioned by sparse approximate inverse. The overall solver is demonstrated on a high-resolution two ventricle mesh based on a human anatomy with roughly 130 K elements and 1.7 M displacement DOF. Effectiveness of the numerical method for active contraction is shown. To the best of our knowledge, this solver is the first to efficiently model ventricular contraction using an iterative linear solver for the mesh size demonstrated and therefore opens the possibility for future very high-resolution models. In addition, several relatively simple benchmark problems are designed for a verification exercise to show that the solver is functioning properly and correctly solves the underlying mathematical model. Here, the output of the newly designed solver is compared to that of the mechanics component of Chaste (‘Cancer, Heart and Soft Tissue Environment’). These benchmark tests may be used by other researchers to verify their newly developed methods and codes.

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## Notes

- 1.
We are using the \(l^2\) norm. Residual entries have weights 1 kPa\(^{-1}\)mm\(^{-2}\) and 1 mm\(^{-3}\) for the displacement and pressure equations, respectively. For the volume constraints, the residual weight was \(\frac{1}{V_\mathrm{u}}\), where \(V_\mathrm{u}\) is the unloaded volume of the ventricle.

- 2.
Hypre implements right preconditioning for GMRES.

- 3.
For instance, for edge nodes \(n_i=2\), for brick face nodes \(n_i=4\), for the interior node of the brick element \(n_i=8\).

- 4.
See Chaste documentation, e.g., https://chaste.cs.ox.ac.uk/trac/wiki/ChasteGuides/FiniteElementImplementations.

- 5.
Tetrahedral meshes were generated using the mesh generator Tetgen http://wias-berlin.de/software/tetgen/.

- 6.
SuperLU was used to invert the sparse matrix \(A_{\mathrm{pc}}\), and LAPACK was used to invert the dense matrix \(S_{\mathrm{pc}}\).

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## Acknowledgments

We thank Tzanio Kolev and Ulrike Meier Yang from the Hypre team at Lawrence Livermore National Laboratory for very informative discussions. The work of J.-L. Fattebert and D. F. Richards was performed under the auspices of the US Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.

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The mention of commercial products, their sources, or their use in connection with material reported herein is not to be construed as either an actual or implied endorsement of such products by the Department of Health and Human Services.

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V. Gurev and P. Pathmanathan contributed equally to this work.

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Gurev, V., Pathmanathan, P., Fattebert, JL. *et al.* A high-resolution computational model of the deforming human heart.
*Biomech Model Mechanobiol* **14, **829–849 (2015). https://doi.org/10.1007/s10237-014-0639-8

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### Keywords

- Cardiac mechanics
- Soft tissue mechanics
- Parallel computations
- Ventricular model