Mathematical modeling of avascular tumor growth

Cancer is an extremely complex disease, both in terms of its causes and consequences to the body. Cancer cells acquire the ability to proliferate without control, invade the surrounding tissues and eventually form metastases. It is becoming increasingly clear that a description of tumors that is uniquely based on molecular biology is not enough to understand thoroughly this illness. Quantitative sciences, such as physics, mathematics and engineering, can provide a valuable contribution to this field, suggesting new ways to examine the growth of the tumor and to investigate its interaction with the neighboring environment. In this dissertation, we deal with mathematical models for avascular tumor growth. We evaluate the effects of physiological parameters on tumor development, with a particular focus on the mechanical response of the tissue. We start from tumor spheroids, an effective three-dimensional cell culture, to investigate the first stages of tumor growth. These cell aggregates reproduce the nutrient and proliferation gradients found in the early stages of cancer and can be grown with a strict control of their environmental conditions. The equations of the model are derived in the framework of porous media theory, and constitutive relations for the mass transfer terms and the mechanical stress are formulated on the basis of experimental observations. The growth curves of the model are compared to the experimental data, with good agreement for the different experimental settings. A new mathematical law regulating the inhibitory effect of mechanical compression on cancer cell proliferation is also presented. Then, we perform a parametric analysis to identify the key parameters that drive the system response. We conclude this part by introducing governing equations for transport and uptake of a chemotherapeutic agent, designed to target cell proliferation. In particular, we investigate the combined effect of compressive stresses and drug action. Interestingly, we find that variation in tumor spheroid volume, due to the presence of a drug targeting cell proliferation, depends considerably on the compressive stress level of the cell aggregate. In the second part of the dissertation, we study a constitutive law describing the mechanical response of biological tissues. We introduce this relation in a biphasic model for tumor growth based on the mechanics of fluid-saturated porous media. The internal reorganization of the tissue in response to mechanical and chemical stimuli is described by enforcing the multiplicative decomposition of the deformation gradient tensor associated with the solid phase motion. In this way, we are able to distinguish the contributions of growth, rearrangement of cellular bonds, and elastic distortion, occurring during tumor evolution. Results are presented for a benchmark case and for three biological configurations. We analyze the dependence of tumor development on the mechanical environment, with particular focus on cell reorganization and its role in stress relaxation. Finally, we conclude with a summary of the results and with a discussion of possible future extensions.