| The hallmark of malignant tumors is their spread into neighboring tissue and metastasis to distant organs, which can lead to life threatening consequences. One of the defining characteristics of aggressive tumors is an unstable morphology, the formation of invasive fingers and protrusions observed both in vitro and in vivo. In spite of extensive biological, clinical and modeling study and research at physical scales ranging from the molecular to the tissue, the driving dynamics of tumor invasiveness are not completely understood, partly because it is challenging to observe and study cancer as a multi-scale system. Mathematical modelling has been applied to provide further insights into these complex invasive and metastatic behaviors. In this thesis, we employ reaction-diffusion models, free boundary models, analyses and nonlinear simulations to investigate the dynamics of tumor invasiveness. In the context of glioma multiforme, a very aggressive form of brain tumor, cell-cell adhesion is very low and the cancer cells invade the brain in a diffusive manner. We use a Go-or-Grow reactiondiffusion system to investigate the dynamics of a population of glioma cells in which there is a switch from a migratory to a proliferating phenotype (and vice-versa) that depends locally on the cell density. We demonstrate the ability of the density-dependent go-or-grow mechanism to produce complicated dynamics associated with tumor heterogeneity and invasion. When cell-cell adhesion is stronger, as in breast, prostate, colon and other epithelial tumors, we use a free-boundary model in which there is a clear delineation between the tumor and host. We examine different constitutive laws for the tumor cell velocity and evaluate the consistency between theoretical model predictions and experimental data from in vitro three-dimensional multicellular tumor spheroids. Using linear morphological stability analysis, we find that among the Darcy's law, Stokes' law and a combined Darcy-Stokes law, the Stokes model is most consistent with experimental observations. We then investigate the effects of nonlinearity on the dynamics of tumors under the Stokes law using boundary integral methods. Simulations reveal the importance of the biophysical properties of the tumor and microenvironment on tumor progression and provide insights in predicting treatment response. |
