Analysis of Differential Equations Modelling the Reactive Flow through a Deformable System of Cells (original) (raw)
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Analysis of Dierential Equations Modelling the Reactive Flow through a Deformable System of Cells
A system of model equations coupling fluid flow, deformation of solid structure and chemical reactions is formulated starting from pro- cesses in biological tissue. The main aim of this paper is to analyse this non-standard system, where the elasticity modules are functionals of a concentration and the diusion coecients of the chemical substances are functions of their concentrations. A new approach and new meth- ods are required adapted to these nonlinearities and the transmission conditions on the interface solid-fluid. Strong solutions for the initial and boundary value problem are constructed under suitable regularity assumptions on the data, and stability estimates of the solutions with respect to the initial and boundary values are proved. These estimates imply uniqueness directly.
SIAM Journal on Mathematical Analysis, 2011
In this article we obtain rigorously the homogenization limit for a fluid-structurereactive flow system. It consists of cell tissue and intercellular liquid, transporting solutes. The cell tissue is supposed linearly elastic and deforming with a viscous non-stationary flow. The elastic moduli of the tissue change with cumulative concentration value. In the limit, when the scale parameter goes to zero, we obtain the quasi-static Biot system, coupled with the upscaled reactive flow. Effective Biot's coefficients depend on the reactant concentration. Additionally to the weak two-scale convergence results, we prove convergence of the elastic and viscous energies. This then implies a strong two-scale convergence result for the fluid-structure variables. Next we establish the regularity of the solutions for the upscaled equations. In our knowledge, it is the only known study of the regularity of solutions to the quasi-static Biot system. The regularity is used to prove the uniqueness for the upscaled model.
Reaction-diffusion problems in cell tissues
Journal of Dynamics and Differential Equations, 1997
In this paper we consider reaction diffusion problems describing a reaction occurring in a planar domain (regarded as cell tissue). The diffusivity is assumed to be large except in the neighborhood of curves (regarded as membranes), around which it is assumed to be small. The subregions determined by the membranes and by the boundary of the domain (tissue wall) are regarded as cells. We assume that the tissue wall is a barrier through which no substance can pass. We prove that the dynamics is described by a system of ordinary differential equations that can be explicitly exhibited through the parameters defining the reaction diffusion problem, the length of the membranes and the area of the cells. The tools employed are a detailed analysis of an eigenvalue problem and the invariant manifold theory.
Modeling Large-deformation-induced Microflow in Soft Biological Tissues
Theoretical and Computational Fluid Dynamics, 2006
The homogenization approach to multiscale modeling of soft biological tissues is presented. The homogenized model describes the relationship between the macroscopic hereditary creep behavior and the microflow in a fluid-saturated dual-porous medium at the microscopic level. The micromodel is based on Biot's system for quasistatic deformation processes, modified for the updated Lagrangian formulation to account for coupling the fluid diffusion through a porous solid undergoing large deformation. Its microstructure is constituted by fluid-filled inclusions embedded in the porous matrix. The tangential stiffness coefficients and the retardation stress for the macromodel are derived for a time-stepping algorithm. Numerical examples are discussed, showing the strong potential of the model for simulations of deformation-driven physiological processes at the microscopic scale.
Formulation of a finite deformation model for the dynamic response of open cell biphasic media
Journal of The Mechanics and Physics of Solids, 2011
The paper illustrates a biphasic formulation which addresses the dynamic response of fluid saturated porous biphasic media at finite deformations with no restriction on the compressibility of the fluid and of the solid skeleton. The proposed model exploits four state fields of purely kinematic nature: the displacements of the solid phase, the velocity of the fluid, the density of the fluid and an additional macroscopic scalar field, termed effective Jacobian, associated with the effective volumetric deformation of the solid phase.
Reactive Flows in Deformable, Complex Media
Oberwolfach Reports, 2014
Many processes of highest actuality in the real life are described through systems of equations posed in complex domains. Of particular interest is the situation when the domain is changing in time, undergoing deformations that depend on the unknown quantities of the model. Such kind of problems are encountered as mathematical models in the subsurface, material science, or biological systems.The emerging mathematical models account for various processes at different scales, and the key issue is to integrate the domain deformation in the multi-scale context. The focus in this workshop was on novel techniques and ideas in the mathematical modelling, analysis, the numerical discretization and the upscaling of problems as described above.
A low-dimensional deformation model for cancer cells in flow
Physics of Fluids, 2012
A low-dimensional parametric deformation model of a cancer cell under shear flow is developed. The model is built around an experiment in which MDA-MB-231 adherent cells are subjected to flow with increasing shear. The cell surface deformation is imaged using differential interference contrast microscopy imaging techniques until the cell releases into the flow. We post-process the time sequence of images using an active shape model from which we obtain the principal components of deformation. These principal components are then used to obtain the parameters in an empirical constitutive equation determining the cell deformations as a function of the fluid normal and shear forces imparted. The cell surface is modeled as a 2D Gaussian interface which can be deformed with three active parameters: H (height), σ x (xwidth), and σ y (y-width). Fluid forces are calculated on the cell surface by discretizing the surface with regularized Stokeslets, and the flow is driven by a stochastically fluctuating pressure gradient. The Stokeslet strengths are obtained so that viscous boundary conditions are enforced on the surface of the cell and the surrounding plate. We show that the low-dimensional model is able to capture the principal deformations of the cell reasonably well and argue that active shape models can be exploited further as a useful tool to bridge the gap between experiments, models, and numerical simulations in this biological setting.
Diffusion of a fluid through an elastic solid undergoing large deformation
International journal of non-linear mechanics, 2004
This paper is concerned with the modeling of slow di usion of a uid into a swelling solid undergoing large deformation. Both the stress in the solid as well as the di usion rates are predicted. The approach presented here, based on the balance laws of a single continuum with mass di usion, overcomes the di culties inherent in the theory of mixtures in specifying boundary conditions. A "natural" boundary condition based upon the continuity of the chemical potential is derived by the use of a variational approach, based on maximizing the rate of dissipation. It is shown that, in the absence of inertial e ects, the di erential equations resulting from the use of mixture theory can be recast into a form that is identical to the equations obtained in our approach. The boundary value problem of the steady ow of a solvent through a gum rubber membrane is solved and the results show excellent agreement with the experimental data of Paul and Ebra-Lima (J. Appl. Polym. Sci. 14 (1970) 2201) for a variety of solvents. ?
Tissue dynamics with permeation
The European Physical Journal E, 2012
Animal tissues are complex assemblies of cells, extracellular matrix (ECM), and permeating interstitial fluid. Whereas key aspects of the multicellular dynamics can be captured by a one-component continuum description, cell division and apoptosis imply material turnover between different components that can lead to additional mechanical conditions on the tissue dynamics. We extend our previous description of tissues in order to account for a cell/ECM phase and the permeating interstitial fluid independently. In line with our earlier work, we consider the cell/ECM phase to behave as an elastic solid in the absence of cell division and apoptosis. In addition, we consider the interstitial fluid as ideal on the relevant length scales, i.e., we ignore viscous stresses in the interstitial fluid. Friction between the fluid and the cell/ECM phase leads to a Darcy-like relation for the interstitial fluid velocity and introduces a new characteristic length scale. We discuss the dynamics of a tissue confined in a chamber with a permeable piston close to the homeostatic state where cell division and apoptosis balance, and we calculate the rescaled effective diffusion coefficient for cells. For different mass densities of the cell/ECM component and the interstitial fluid, a treadmilling steady state due to gravitational forces can be found.
Modeling the dynamics of cell-sheet : From Fisher-KPP equation to bio-mechano-chemical systems
2016
This paper is devoted to study some predictions on injured cell-sheet based on reaction-diffusion equations. In the context of wound cell-sheet healing, we investigated the validity of the reaction-diffusion model of Fisher-KPP type for simulation of cell-sheet migration. In order to study the validity of this model, we performed experimental observations on the MDCK cell monolayers. The obtained videoscopies allow to obtain, after segmentation and binarization, the variations of area and of scar fronts profiles with good accuracy. We were interested in comparing the calculated variations of fronts to those experimental fronts, after a step of calibration parameters.