This work is concerned with the development and analysis of a mathematical model motivated by interstitial hydrodynamics and tissue deformation mechanics (poroelastohydrodynamics) within an in-vitro solid tumour. The classical mixture theory is employed to derive the mass and momentum balance equations for a two-phase system. A key contribution of this study is the treatment of the physiological transport parameter, namely the hydraulic resistivity, as anisotropic and heterogeneous, which leads to a strongly coupled and nonlinear system.
We derive the corresponding weak formulation and reformulate the problem as an equivalent fixed-point problem. This framework enables the application of the Galerkin method together with classical results on monotone operators and the Schauder and Banach fixed-point theorems to establish the existence and uniqueness of solutions.