Analysis of Stabilized Finite Element Methods for a Morpho-Poroelastic Model Applied to Tumor Growth

Abstract

The theory of morpho-poroelasticity is applied to tumor growth. This allows for the modeling of permanent deformation in the tissue as a result of the presence of tumor cells. The work done in this project can be divided into two parts. In the first part we review existing models for elasticity and build corresponding finite element models. The aim of the first part is to gain understanding in the behaviour of morphoelasticity and poroelasticity. In morphoelasticity the deformation tensor is decomposed into an elastic and plastic component, allowing for permanent deformations to occur in the tissue. The theory of morphoelasticity describes the interplay between a fluid in a porous tissue and the tissue's elastic properties. A common problem in poroelastic models is the occurence of non-physical oscillations in the pressure. The second part of this project contains novel work. First, a rigorous mathematical derivation is presented of the tuning parameter found in the diffusive stabilization method for poroelastic systems. Secondly, we present a finite element model for the novel combination of morpho- and poroelasticity. The derivation concerning the diffusive stabilization is also applicable to the morpho-poroelastic finite element model. We obtain a promising tumor growth model by combining a biochemcial tumor growth model with morpho-poroelasticity. This biochemical model is based on a nutrient transport equation and a tumor cell density evolution equation. Some example output of the tumor growth finite element model is shown. It can be used as a starting point for more elaborate tumor growth models.