A biphasic model for avascular tumor growth and drug response

Tumor spheroids provide an effective in vitro tool to study the early stages of cancer growth. The nutrient and proliferation gradients typical of the avascular stage of tumor development can be reproduced in the laboratory, with a strict control of the environmental conditions. Moreover, current techniques allow to tune the mechanical milieu in which the spheroids are immersed, and evaluate the effects of mechanical stress on tumor development. Remarkably, several studies report a reduction of cell proliferation as a function of increasingly applied stress on the surface of the spheroids. Recently, a computational model based on porous media theory has been proposed to predict tumor growth and its interaction with the host tissue. Inspired by this work, here we report on a specialization of the original model, adapted for tumor spheroids. Starting from standard balance equations for a multiphase system, we close the model with the aid of constitutive relations that are formulated on the basis of experimental observations. In particular, we introduce mathematical expressions describing the mass exchange terms between the different components of the system, and enforce a constitutive relation for the stresses that accounts for the microscopic interactions between the cancer cells. Also, we study the spatiotemporal evolution of a nutrient and of a chemotherapeutic agent over the tumor domain, evaluating their coupled effects on cell proliferation. The spatial discretization of the coupled problem is carried out via the Finite Element Method, and the resulting system is solved with an implicit scheme in COMSOL Multiphysics. A set of experiments is performed to validate the model, investigating the growth of U-87MG spheroids both freely growing in the culture medium and subject to an externally controlled mechanical pressure. Model results are compared to experimental data, showing a good agreement for both the experimental settings. We present a new mathematical law describing the inhibitory effect of mechanical compression on cell proliferation. The new law is validated against the experimental data, providing better results when compared to other expressions in literature.