Existence and uniqueness results on biphasic mixture model for an in-vivo tumor. Issue 15 (13th October 2022)

• Record Type: Journal Article
• Authors: Alam, Meraj; Byrne, H. M.; Raja Sekhar, G. P.
• Is Part Of: Applicable analysis. Volume 101:Issue 15(2022)
• Page Start: 5442
• Page End: 5468
• Publication Date: 2022-10-13
• Subjects: In-vivo tumor -- biphasic mixture theory -- weak formulation -- Galerkin method -- weak convergence; 76Zxx -- 74F10 -- 35Q74 -- 35J66 -- 35D30; Mathematical analysis -- Periodicals; 515
• DOI: 10.1080/00036811.2021.1895122
• Languages: English
• ISSNs: 0003-6811

Abstract

Abstract : In this article, we propose a mathematical model that describes hydrodynamics and deformation mechanics within a solid tumor which is embedded in or adjacent to a healthy (normal) tissue. The tumor and normal tissue regions are assumed to be deformable and the theory of mixtures is adapted to mass and momentum balance equations for fluid flow and tissue deformation mechanics in each region. The momentum balance equations are coupled via forces that interact between the phases (fluid and solid). Continuity of normal velocities, displacements, and normal stresses along with the Beaver–Joseph–Saffman condition are imposed at the interface between the tumor and tissue regions. The physiological transport parameters (such as hydraulic resistivity or permeability) are assumed to be heterogeneous and deformation dependent which makes the model nonlinear. We establish the existence of a weak solution using Galerkin and weak convergence methods. We show further that the solution is unique and depends continuously on the given data.