Nonlinear biphasic mixture model: existence and uniqueness results

TL;DR

This paper develops a nonlinear biphasic mixture model for tumor tissue mechanics, incorporating anisotropic and heterogeneous hydraulic resistivity, and proves existence and uniqueness of solutions using fixed-point theory.

Contribution

It introduces a novel nonlinear biphasic mixture model with anisotropic, heterogeneous parameters and establishes mathematical existence and uniqueness results.

Findings

Model accounts for anisotropic, heterogeneous hydraulic resistivity.

Proves existence and uniqueness of solutions for the model.

Uses fixed-point theorems and Galerkin method for analysis.

Abstract

This paper is concerned with the development and analysis of a mathematical model that is motivated by interstitial hydrodynamics and tissue deformation mechanics (poro-elasto-hydrodynamics) within an in-vitro solid tumor. The classical mixture theory is adopted for mass and momentum balance equations for a two-phase system. A main contribution of this study, we treat the physiological transport parameter (i.e., hydraulic resistivity) as anisotropic and heterogeneous, thus the governing system is strongly coupled and nonlinear. We derived a weak formulation and then formulated the equivalent fixed-point problem. This enabled us to use the Galerkin method, and the classical results on monotone operators combined with the well-known Schauder and Banach fixed point theorems to prove the existence and uniqueness results.