The ability of tumors to metastasize is preceded by morphological instabilities such as chains or fingers that invade the host environment.... Show moreThe ability of tumors to metastasize is preceded by morphological instabilities such as chains or fingers that invade the host environment. Parameters that control tumor morphology may also contribute to its invasive ability. In this thesis, we investigate tumor growth using a two-phase Stokes model. We first examine the morphological changes using the surface energy of the tumor-host interface and investigate its nonlinear dynamics using a boundary integral method. In an effort to understand the interface stiffness, we then model the tumor-host interface as an elastic membrane governed by the Helfrich bending energy. Using an energy variation approach, we derive a modified Young-Laplace condition for the stress jump across the interface, and perform a linear stability analysis to evaluate the effects of viscosity, bending rigidity, and apoptosis on tumor morphology. Results show that increased bending rigidity versus mitosis rate contributes to a more stable growth. On the other hand, increased tumor viscosity or apoptosis may lead to an invasive fingering morphology. Comparison with experimental data on glioblastoma spheroids shows good agreement especially for tumors with high adhesion and low proliferation. Next, we evaluate tumor regression during cancer therapy by a combined modality involving chemotherapy and radiotherapy. The goal is to address the complexities of a vascular tumor (e.g. apoptosis and vascularization) during treatment. We introduce an apoptotic time delay and study its impact on tumor regression using numerical and asymptotic techniques. In particular, we implement the linear-quadratic model and identify two extreme sets of parameter data, namely the slow, and fast tumor response to therapy. Numerical simulations for the slow response set show good agreements with data representing non-small cell lung carcinoma. Using the evolution equation for tumor radius with time delay, we find that tumors with shorter time interval to the onset of apoptosis shrink faster. Ph.D. in Applied Mathematics, May 2016 Show less
