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Mechanical factors play a major role in tumor development and response to treatment. This is more evident for tumors\ud grown in vivo, where cancer cells interact with the different components of the host tissue. Mathematical models are able\ud to characterize the mechanical response of the tumor and can provide a better understanding of these interactions. In\ud this work, we present a biphasic model for tumor growth based on the mechanics of fluid-saturated porous media. In our\ud model, the porous medium is identified with the tumor cells and the extracellular matrix, and represents the system’s\ud solid phase, whereas the interstitial fluid constitutes the liquid phase. A nutrient is transported by the fluid phase, thereby\ud supporting the growth of the tumor. The internal reorganization of the tissue in response to mechanical and chemical\ud stimuli is described by enforcing the multiplicative decomposition of the deformation gradient tensor associated with the\ud solid phase motion. In this way, we are able to distinguish the contributions of growth, rearrangement of cellular bonds,\ud and elastic distortions which occur during tumor evolution. Results are shown for three cases of biological interest,\ud addressing (i) the growth of a tumor spheroid in the culture medium, and (ii) the evolution of an avascular tumor\ud growing first in a soft host tissue and then (iii) in a three-dimensional heterogeneous region.We analyze the dependence\ud of tumor development on the mechanical environment, with particular focus on cell reorganization and its role in stress\ud relaxation

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We reformulate a model of avascular tumour growth in which the tumour tissue is studied as a biphasic medium featuring an interstitial fluid and a solid phase. The description of growth relies on two fundamental features: One of those is given by the mass transfer among the constituents of the phases, which is taken into account through source and sink terms; the other one is the multiplicative decomposition of the deformation gradient tensor of the solid phase, with the introduction of a growth tensor, which represents the growth-induced structural changes of the tumour. In general, such tensor is non-integrable, and it may allow to define a Levi-Civita connection with non-trivial curvature. Moreover, its evolution is related to the source and sink of mass of the solid phase through an evolution equation. Our goal is to study how growth can be influenced by the inhomogeneity of the growth tensor. To this end, we study the evolution of the latter, as predicted by two different models. In the first one, the dependence of the growth tensor on the tumour's material points is not explicitly considered in the evolution equation. In the second model, instead, the inhomogeneity of the growth tensor is resolved explicitly by introducing the curvature associated with it into the evolution equation. Through numerical simulations, we compare the results produced by these two models, and we evaluate a possible role of the material inhomogeneities on growth.

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