Mathematical analysis of hydrodynamics and tissue deformation inside an isolated solid tumor

In this article, we present a biphasic mixture theory based mathematical model for the hydrodynamics of interstitial fluid motion and mechanical behavior of the solid phase inside a solid tumor. The tumor tissue considered here is an isolated deformable biological medium. The solid phase of the tumor is constituted by vasculature, tumor cells, and extracellular matrix, which are wet by a physiological extracellular fluid. Since the tumor is deformable in nature, the mass and momentum equations for both the phases are presented. The momentum equations are coupled due to the interaction (or drag) force term. These governing equations reduce to a one-way coupled system under the assumption of infinitesimal deformation of the solid phase. The well-posedness of this model is shown in the weak sense by using the inf-sup (Babuska–Brezzi) condition and Lax–Milgram theorem in 2D and 3D. Further, we discuss a one-dimensional spherical symmetry model and present some results on the stress fields and energy of the system based on $L^2$ and Sobolev norms. We discuss the so-called phenomena of “necrosis” inside a solid tumor using the energy of the system.