Tumor growth modeling in the framework of porous media mechanics

Mathematical investigation of tumor growth is becoming increasingly popular in scientific literature. New experiments and novel test techniques provide extensive sets of data that can be analyzed by mathematical models and allow for their validation. Here we extend multiphase porous media mechanics to model tumor evolution, enforcing governing equations obtained by the Thermodynamically Constrained Averaging Theory (TCAT). In our approach, the tumor mass is considered as a multiphase medium, constituted by an extracellular matrix (ECM); tumor cells (TCs), that may become necrotic upon exposure to low levels of nutrients; healthy cells (HC); and an interstitial fluid (IF) that accounts for the transport of nutrients and growth factors. The resulting equations are solved by the finite element method, and predict the growth rate of the tumor mass as a function of initial tumor radius, nutrient concentration, cell adhesion and geometry. The numerical results for the growth of a tumor embedded in a host tissue are presented. In detail, we analyze the influence of nutrient dynamics and their impact on the tumor configuration. The model predicts the formation of a necrotic core at the center of the growing mass, and the presence of a viable rim, constituted by proliferating cells, at its border. Furthermore, we investigate the effect of the tumor microenvironment, for different degrees of TCs and HCs adhesion and for different values of the model parameters. A particular focus is set on the mechanical properties of the ECM, and its influence on the development of the malignant mass. Also, we consider the case of a growing melanoma in the avascular stage. In the case of lower ECM stiffness, the growing tumor deforms the ECM in an appreciable manner, producing a slight elevation on the skin surface. The model captures the importance of the basement membrane: the tumor mass grows symmetrically in the epidermis and, when it reaches the membrane border, we report a change in the growth pattern towards a more complex configuration. The modular structure of the mathematical model allows for the inclusion of additional phases and species, enhancing the level of complexity that could be obtained, and enabling the investigation of new phenomena. Interesting future developments are the inclusion of transport of therapeutic agents and the modeling of the first stages of tumor angiogenesis. We hope that this approach could be a valuable contribution in providing new insights for the development of novel therapeutic strategies, with the ultimate goal of improving the quality of life for patients.