An orthotropic electro-viscoelastic model for the heart with stress-assisted diffusion

Abstract

We propose and analyse the properties of a new class of models for the electromechanics of cardiac tissue. The set of governing equations consists of nonlinear elasticity using a viscoelastic and orthotropic exponential constitutive law (this is so for both active stress and active strain formulations of active mechanics) coupled with a four-variable phenomenological model for human cardiac cell electrophysiology, which produces an accurate description of the action potential. The conductivities in the model of electric propagation are modified according to stress, inducing an additional degree of nonlinearity and anisotropy in the coupling mechanisms; and the activation model assumes a simplified stretch-calcium interaction generating active tension or active strain. The influence of the new terms in the electromechanical model is evaluated through a sensitivity analysis, and we provide numerical validation through a set of computational tests using a novel mixed-primal finite element scheme.

Precision medicine in human heart modeling

“…We adopt a quasi-static approach and postulate that inertia and damping effects are negligible ( Baillargeon et al 2014 ). To characterize the response during passive filling, we adopt an orthotropic invariant-based constitutive model ( Holzapfel and Ogden 2009 ; Propp et al 2020 ). To describe the response during active contraction in the ventricles, the atria, and the chordae tendineae, we adopt the concept of time-varying elastance ( Walker et al 2005 ) and introduce the active stress as a function of the regional action potential and the cardiomyocyte stretch according to Frank-Starling’s law ( Peirlinck et al 2019 ).…”

Mixed Kirchhoff stress–displacement–pressure formulations for incompressible hyperelasticity

“…The motivation for using three-field elasticity formulations is to avoid volumetric locking [47], and to provide direct approximation of a variable of importance [62,56], nonetheless at a higher computational cost. As mentioned in [24], capturing stress concentrations with the guarantee of stress convergence under mesh refinement is a key requirement that is very difficult to achieve in a point-wise manner simply by standard displacement-based formulations.…”

“…We employ the Holzapfel-Ogden material law with the constitutive parameters used in Example 1. The active contraction of the ventricle is incorporated through the so-called active strain approach (see, e.g., [56]) where the activation function ξ is incremented together with the endocardial pressure up to a maximal activation of 12%. In order to achieve sufficient torsion and thickening of the ventricular wall, we also use an algebraic relation between the activation and the myocyte shortening that models sliding myofilaments of collagen and a transmurally heterogeneous activation that modulates different values of ξ in each direction (see details in [11,60])…”

“…More pertinent to the present case, formulations in terms of Kirchhoff stress, displacement and pressure were introduced in [23] and have been adapted to the study of cardiac electromechanics in [56,62]. Recasting the three-field formulation in this particular set of unknowns is very useful for modelling the physiological responses of cardiac tissue, as the Kirchhoff stress tensor acts as the coupling field with the equations describing electrical propagation in stress-assisted diffusion models.…”

On the Role of Ionic Modeling on the Signature of Cardiac Arrhythmias for Healthy and Diseased Hearts

“…Moreover, continuous increases in computational power have led to whole-heart electrical models that have shown promising translational outcomes, improving the general understanding of heart function, its pathologies, and therapies [4][5][6][7][8][9]. Still, critical modeling challenges arise when local heterogeneities at different spatio-temporal scales are taken into account [10][11][12][13][14][15][16].…”