Modeling solid tumor growth in complex, dynamic geometries

In the last ten years, nonlinear continuum models have been used to study the effects of shape instabilities on avascular, angiogenic and vascular solid tumor growth. Shape instabilities are important because the mechanisms that control the tumor morphology also control the ability of the tumor to be locally invasive, and local invasiveness is believed to be a precursor of metastasis. In this thesis, I use a multiphase mixture model, taking into account homotype adhesion between tumor cells and heterotype adhesion between the cells and a basement membrane. I simplified the description of membrane elasticity to penalize global stretching and bending. The governing equations are derived using a variational approach that ensures the themodynamic consistency of the model. To solve the governing equations efficiently, I develop an adaptive energy-stable nonlinear multigrid finite difference method, which enables the use of large time steps and efficient numerical solution of the equation. A series of numerical simulations in two and three dimensions are performed that demonstrate the accuracy of the numerical method and illustrate the shape instabilities and nonlinear effects of the membrane resistive forces on tumor growth, including the build-up stress in the tissue and membrane buckling.;In this thesis, I also develope a mathematical model of tumor growth in complex, dynamic and elastic geometries. I consider cell-membrane interactions by introducing a wall free energy (Jacqmin, 1999), which takes into account differences in relative strengths of the cell-cell and cell-membrane adhesive forces. I represent the complex, dynamic membrane domain using a phase-field function, and apply the diffuse domain approach to formulate the governing equations of tumor growth. This method allows a straightforward implementation using the adaptive energy-stable numerical methodology described above. Two and three dimensional simulations are performed where the adhesion between tumor cells and a deformable basement membrane is varied. The membrane's resistance to bending is also modeled. The results demonstrated the nontrivial dependence of the growing tumor on the adhesion of cells and flexibility of the basement membrane.;Since cancer cells have the ability to invade the local tissue. I also extend the model to consider local invasion into the stroma. In particular, I model the ability of tumor to secrete matrix degrading enzymes, which degrade the extracellular matrix and the basal membrane. Two and three dimensional simulations are performed to characterize the behavior as a function of cell-membrane adhesiveness and stiffness of the membrane.