A stabilized finite element method for a Morpho-Visco-Poroelastic Model

Studying the structure of soft tissues is important and relevant in biology, particularly in some diseases, such as tumor growth and dermal contraction after burn injury. Based on the complicated characteristics of the tissue and for the sake of a better understanding of the underlying biomechanics, we propose a mathematical model that combines elastic, viscous, and porous effects with growth or shrinkage due to microstructural changes. The framework is referred to as morpho-visco-poroelasticity. Although the existence results of the solution to the problem are not given in this study, we assess the stability of the equilibria for both the continuous and semi-discrete versions of the model, and the key features of this modelling framework have been discussed. To obtain reliable numerical solutions, a stabilized finite element (FE) scheme is proposed for the morpho-visco-poroelasticity equations to avoid spurious oscillations in the pressure profile; the success of this FE scheme is verified by numerical simulations and convergence investigation in both spatial and temporal aspects. For a more quantitative assessment, the total variation of the pressure profile is evaluated as a function of the stabilization parameter.