In this thesis we investigate the modeling, analysis and computation of tumor growth.The sharp interface model we considered is to understand... Show moreIn this thesis we investigate the modeling, analysis and computation of tumor growth.The sharp interface model we considered is to understand how the two key factors of (1) the mechanical interaction between the tumor cells and their surroundings, and (2) the biochemical reactions in the microenvironment of tumor cells can influence the dynamics of tumor growth. From this general model we give its energy formulation and solve it numerically using the boundary integral methods and the small-scale decomposition under three different scenarios.The first application is the two-phase Stokes model, in which tumor cells and the extracellular matrix are both assumed to behave like viscous fluids. We compared the effect of membrane elasticity on the tumor interface and the curvature-weakening one and found the latter would promote the development of branching patterns.The second application is the two-phase nutrient model under complex far-field geometries, which represents the heterogeneous vascular distribution. Our nonlinear simulations reveal that vascular heterogeneity plays an important role in the development of morphological instabilities that range from fingering and chain-like morphologies to compact,plate-like shapes in two-dimensions.The third application is for the effect of angiogenesis, chemotaxis and the control of necrosis. Our nonlinear simulations reveal the stabilizing effects of angiogenesis and the destabilizing ones of chemotaxisand necrosis in the development of tumor morphological instabilities if the necrotic core is fixed. We also perform the bifurcation analysis for this model.In the end, as a future work, we propose new models through Energetic Variational Approach (EnVarA) to shed light on the modeling issues. Show less
