A high-resolution computational model of the deforming human heart

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.

Organ‐level validation of a cross‐bridge cycling descriptor in a left ventricular finite element model: effects of ventricular loading on myocardial strains

“…Finite element models simulating the LV mechanical behavior during a cardiac cycle have been developed using a variety of active contraction models (Kerckhoffs et al 2007;Niederer and Smith 2009;Gurev et al 2010Gurev et al , 2015. To the best of our knowledge, however, there have been no rigorous attempts to show that the models are able to simultaneously reproduce key organ-level physiology measured in the intact heart, specifically, (1) linear, load-independent ESPVRs generated from PV loops of different loading conditions, (2) a linear, load-independent MVO 2 -PVA relationship, and (3) a consistent strain-time profile.…”

Integrated Heart—Coupling multiscale and multiphysics models for the simulation of the cardiac function

“…At the same time, high-order spatial approximations are actively studied [8,9,266] as an alternative to the low-order finite-element discretizations that lead to problem sizes in the hundreds of millions, and naturally provide a unified spatial approximation framework for electromechanics simulations. Recent multiscale models for the mechanics of cardiac tissue include [41,111].…”

Influence of atrial contraction dynamics on cardiac function

“…Therefore, alternative methods based on iterative solvers were developed. We use an approach similar to the one proposed by Gurev et al [13]. The nonlinear system to be solved with each time step is r(v) = 0, where the unknowns v = (x, p h , p c ) are nodal positions x, hydrostatic pressure p h and cavity pressures p c .…”

“…To solve this system, we apply flexible GMRES with the preconditioner: pc [13]. To approximate A −1 pc x we apply an algebraic multigrid preconditioner for A, specifically the Boomer-AMG implementation available in PETSc [2,14], with options -apc hypre boomeramg relax type all SOR/Jacobi -apc hypre boomeramg coarsen type CLJP -apc hypre boomeramg strong threshold 0.3 -apc hypre boomeramg max row sum 1.0 -apc hypre boomeramg interp type FF1.…”