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.

The theory of active contours is applied to microscopic images of cells. A model is developed that approximates cell borders by dynamic curves. This model is based on gradient vector flow (GVF), an external force that acts on the contours. Both active contours and the GVF force field are defined as functions that minimize certain functionals. The corresponding Euler-Lagrange equations are derived and analyzed theoretically. A number of auxiliary algorithms are designed to aid the performance of the main snake algorithm, including a preprocessing algorithm, a method for detecting cell centers and an algorithm that detects areas devoid of cells. Results of the snake algorithm are presented, along with practical considerations regarding parameter choice. Finally, statistical methods are applied to the results to demonstrate their usefulness.