A viscoactive constitutive modeling framework with variational updates for the myocardium

Abstract

We present a constitutive modeling framework for contractile cardiac mechanics by formulating a single variational principle from which incremental stress–strain relations and kinetic rate equations for active contraction and relaxation can all be derived. The variational framework seamlessly incorporates the hyperelastic behavior of the relaxed and contracted tissue along with the rate – and length – dependent generation of contractile force. We describe a three-element, Hill-type model that unifies the active tension and active deformation approaches. As in the latter approach, we multiplicatively decompose the total deformation gradient into active and elastic parts, with the active deformation parametrizing the contractile Hill element. We adopt as internal variables the fiber, cross-fiber, and sheet normal stretch ratios. The kinetics of these internal variables are modeled via definition of a kinetic potential function derived from experimental force–velocity relations. Additionally, we account for dissipation during tissue deformation by adding a Newtonian viscous potential. To model the force activation, the kinetic equations are coupled with the calcium transient obtained from a cardiomyocyte electrophysiology model. We first analyze our model at the material point level using stress and strain versus time curves for different viscosity values. Subsequently, we couple our constitutive framework with the finite element method (FEM) and study the deformation of three-dimensional tissue slabs with varying cardiac myocyte orientation. Finally, we simulate the contraction and relaxation of an ellipsoidal left ventricular model and record common kinematic measures, such as ejection fraction, and myocardial tissue volume changes.

The importance of mechano-electrical feedback and inertia in cardiac electromechanics

“…Nonetheless, we plan to incorporate our findings into a four-chamber, high resolution heart simulator [4] and explore the influence of inertia and mechano-electrical currents on clinically relevant variables such as electrocardiograms [34, 35, 26, 49] and pressure-volume loops [16]. We also plan to include a more sophisticated material model and study the effects of viscosity [22, 44]. Additionally, we would like to include the effect of stretch and velocity in the active contraction model [21, 47, 50, 38], to simulate relevant macroscopic characteristics of the heart, such as the Frank-Starling effect.…”

Kinematic boundary conditions substantially impact in silico ventricular function

“…Despite the wide range of (bi) ventricular modeling studies found in literature, no consistent approach in incorporating the effect of external tissue support into these models could be found (we considered a non‐exhaustive subset of (bi) ventricular modeling studies (published over the last decade) where generic kinematic BCs were used:). Overall, most studies adopted the practice to apply Neumann BCs on the endocardial surface and to constrain basal out‐of‐plane motion by constraining the basal nodes not to move along the basal surface's normal direction (be it using an exact or penalty‐method based Dirichlet BC).…”

A transmurally heterogeneous orthotropic activation model for ventricular contraction and its numerical validation

“…Note that the requirement is a constitutive assumption and this constraint is not namely required for the overall incompressibility of the tissue, which is instead assured by the nearly incompressible formulation. However, it allows to simplify the model, since in general, one must specify the evolution of the whole active deformation gradient F A …”

“…Such deformation is often missing even in recent heart models. 11 In some instances, the models may even yield an opposite behavior of what expected, thus resulting in apex-to-base lengthening and unrealistic rotations. 12 This shows why the mathematical modeling of the active contraction of the heart is particularly challenging.…”

“…The left ventricular apex‐to‐base shortening, due to the downward motion of the aortic valve ring during systole, is the most important feature of cardiac contraction, indeed allowing the heart to work efficiently. Such deformation is often missing even in recent heart models . In some instances, the models may even yield an opposite behavior of what expected, thus resulting in apex‐to‐base lengthening and unrealistic rotations .…”