Mathematical modeling and analysis of hydroelastodynamics inside a solid tumor containing deformable tissue. Issue 5 (7th February 2019)

• Record Type: Journal Article
• Authors: Alam, Meraj; Dey, Bibaswan; Sekhar, G. P. Raja
• Is Part Of: Zeitschrift für angewandte Mathematik und Mechanik. Volume 99:Issue 5(2019)
• Page Start: n/a
• Page End: n/a
• Publication Date: 2019-02-07
• Subjects: biphasic mixture theory -- isolated tumor -- stress fields -- system energy -- weak formulation -- 76Txx -- 76Zxx -- 35Q74 -- 35D30 -- 35C10; Mathematics -- Periodicals; Mechanics, Applied -- Periodicals; Engineering -- Periodicals; 519
• DOI: 10.1002/zamm.201800223
• Languages: English
• ISSNs: 0044-2267

Abstract

Abstract: Present work reports a mathematical modeling for the interstitial hydrodynamics and mechanical behavior of the solid phase inside a solid tumor. A tumor tissue is a visco‐poroelastic deformable living biomaterial with cellular phase and extracellular matrix (ECM) as the solid phase (also small volume of blood vessels) and physiological extracellular fluid as the fluid phase. The intravascular fluid or blood and the interstitial fluid form a single fluid phase. We write down the mass and momentum balance equations for both the phases. The momentum equations are coupled due to the relative interaction (or drag) force between the phases. This study shows the well‐posedness of poroelastohydrodynamics in the weak sense under following assumptions (i) motion of interstitial fluid flow and solid phase deformation are slow and (ii) nutrient proliferation rate is much faster than the tumor cell growth. Subsequently, we use the semi‐discrete Galerkin method to establish the well‐posedness. Further, we simulate some analytical results corresponding to the one‐dimensional spherical symmetry model. Our results on the unsteady poroelastohydrodynamic model would give an idea about the time required for the necrosis formation from the initial stage of perfusion based on the system energy which can be computed using L 2 and H 1 norms. Abstract : Present work reports a mathematical modeling for the interstitial hydrodynamics and mechanical behavior of the solid phase inside a solid tumor. A tumor tissue is a visco‐poroelastic deformable living biomaterial with cellular phase and extracellular matrix (ECM) as the solid phase (also small volume of blood vessels) and physiological extracellular fluid as the fluid phase. The intravascular fluid or blood and the interstitial fluid form a single fluid phase. We write down the mass and momentum balance equations for both the phases. The momentum equations are coupled due to the relative interaction (or drag) force between the phases. This study shows the well‐posedness of poroelastohydrodynamics in the weak sense under following assumptions (i) motion of interstitial fluid flow and solid phase deformation are slow and (ii) nutrient proliferation rate is much faster than the tumor cell growth. Subsequently, we use the semi‐discrete Galerkin method to establish the well‐posedness. Further, we simulate some analytical results corresponding to the one‐dimensional spherical symmetry model. Our results on the unsteady poroelastohydrodynamic model would give an idea about the time required for the necrosis formation from the initial stage of perfusion based on the system energy which can be computed using L 2 and H 1 norms.