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Mathematical modeling and analysis of hydroelastodynamics inside a solid tumor containing deformable tissue
Author(s) -
Alam Meraj,
Dey Bibaswan,
Sekhar G. P. Raja
Publication year - 2019
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/zamm.201800223
Subject(s) - interstitial fluid , poromechanics , drag , mechanics , fluid dynamics , phase (matter) , extracellular fluid , extracellular matrix , work (physics) , physics , galerkin method , diffusion , momentum (technical analysis) , mixture theory , materials science , classical mechanics , porous medium , finite element method , chemistry , mathematics , thermodynamics , porosity , extracellular , medicine , composite material , mixture model , biochemistry , quantum mechanics , statistics , finance , economics
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.