A multiscale-in-time framework for simulation of maladaptive growth and remodeling (G&R) in the heart is presented. G&R is assumed to be driven by a deviation of mechanical stress or strain with respect to a homeostatic baseline state. Since ventricular loads vary on a much shorter time scale than processes of G&R occur, a staggered solution scheme discriminating between "small scale" heart beat dynamics and "large scale" G&R is chosen.
On the small scale, a coupled monolithic problem of 3D finite strain elasticity for the heart and 0D lumped-parameter flow is solved, using a closed-loop systemic and pulmonary circulation model to account for physiologic loading conditions on the myocardium. After computing a homeostatic reference state, the system is perturbed by introducing a cardiovascular disease (ie regurgitation of the mitral valve or aortic stenosis), eventually leading to a state of chronic volume or pressure overload for the ventricle.
On the large scale, the spatial field of fiber overstretch or tissue overstress is then imposed, and a pure solid mechanics problem of strain- or stress-mediated volumetric growth is solved together with a remodeling law that allows for change in elastic material parameters depending on the amount of growth.
Small and large time scales are mutually revisited until no further volume change occurs. Physiologically meaningful changes in ventricular pressure-volume relationship are obtained for ventricular volume and pressure overload and comply with general observations.
Nonlinear deformation-dependent growth requires local Newton updates at integration point level and is implemented in FEniCS by expressing growth residual and increments as forms at quadrature points. Inner virtual work is expressed explicitly with help of the fourth-order material tangent operator to account for all tangent contributions arising from the nonlinear G&R model.