Mathematical modelling and analysis of unsteady poroelastohydrodynamics for an in-vivo type solid tumour

This study presents a mathematical model that describes the unsteady interstitial fluid percolation through a solid tumour and its surrounding healthy tissue, as well as the deformation of the cellular phase of the solid tumour and healthy tissue. The tumour and its healthy host are assumed to be connected via a smooth, fixed interface. Each of these tissue regions comprises interstitial fluid and solid constituents (i.e., tumour cells and extracellular matrix). The general mixture theory equations are adopted to represent conservation of mass and momentum in each tissue region. The fluid phase is modelled as an incompressible Newtonian fluid, and the solid phase as an isotropic deformable porous material. The governing equations are of mixed parabolic-hyperbolic type. We assume continuity of the interface fluid velocity (IFV), the solid-phase displacement (SPD), and the normal stress at the host-tumour interface, along with the Beavers-Joseph-Saffman condition. We establish well-posedness in a weak sense for the unsteady governing system using a Galerkin method and weak convergence. We then focus on calculating the system energy using the velocity fields of the fluid and solid components of the tumour and its host. The energy estimates in the context of well-posedness yield the maximum system energy (MASE), and the minimum system energy (MISE) is computed from the definitions of the $L^2$ and $H^1$ norms using the 1D solution of the governing equations. The system energy assists in ranking the viability of five types of tumours associated with five distinct carcinomas.