Short Review of Computational Models for Single Cell Deformation and Migration

This short review communication aims at enumerating several modeling efforts that have been performed to model cell migration and deformation. To optimize and improve medical treatments against diseases like cancer, ischemic wounds or pressure ulcers, it is of vital importance to understand the underlying biological mechanisms. Such biological mechanisms also take place on a cellular scale, where cells are known to migrate, proliferate, and differentiate and to die.Delft Institute of Applied MathematicsElectrical Engineering, Mathematics and Computer Scienc

Zener solutions for particle growth in multi-component alloys

In this paper the Zener theory on precipitate growth in supersaturated alloys for planar, cylindrical and spherical geometries is extended to multi-component alloys. The obtained solutions can be used to check the results from numerical simulations under simplified conditions. Further, the multi-component solutions are used to derive the quasi-binary diffusion coefficient for planar, cylindrical and spherical geometries of the growing particle in a multi-component alloy. In the illustrations, hypothetic data are used.Electrical Engineering, Mathematics and Computer Scienc

Particle methods to solve modelling problems in wound healing and tumor growth

The paper deals with a compilation of several of our modelling studies on particle methods used for simulation of wound healing and tumor growth processes. The paper serves as an introduction of our modelling approaches to researchers with interest in biological cell-based models that use particle-based modelling approaches. The particles that we consider in the present models mimic either cells or points on cell boundaries that are allowed to migrate as a result of several chemical and mechanical factors. A distinct feature of our modelling frameworks with respect to conventional particle models, is that cells, mimicked by particles, are allowed to divide, differentiate and to die as a result of apoptosis or any causes for cell death. The paper is merely descriptive, rather than written in full mathematical rigor, and will show some of the potentials of the applied modelling.Delft Institute of Applied MathematicsElectrical Engineering, Mathematics and Computer Scienc

Mathematical models for particle dissolution in extrudable aluminium alloys

Applied Science

Solution of vector Stefan problems with cross-diffusion

A general model for the dissolution of particles in multi-component alloys is proposed and analyzed. The model is based on diffusion equations with cross-terms for the several species, combined with a Stefan condition as the equation of motion of the interface between the particle and diffusant phase. To facilitate the analysis we use a diagonalization argument or Jordan factorization for the diffusion matrix. Self-similar solutions with the Boltzmann transformation are derived to get insight into qualitative behavior of the solution and for comparison with numerical solutions. Several numerical schemes for the solution of the Stefan problem are proposed and compared. It turns out that diagonalization is usefull for numerical purposes too. However, for the case of position dependent diffusion coefficients or a non diagonalizable diffusion matrix, one has to use a different scheme. Here, we analyze stability and workload of several time integrations.Electrical Engineering, Mathematics and Computer Scienc

A moving finite element framework for fast infiltration in nonlinear poroelastic media

Poroelasticity theory can be used to analyse the coupled interaction between fluid flow and porous media (matrix) deformation. The classical theory of linear poroelasticity captures this coupling by combining Terzaghi’s effective stress with a linear continuity equation. Linear poroelasticity is a good model for very small deformations; however, it becomes less accurate for moderate to large deformations. On the other hand, the theory of large-deformation poroelasticity combines Terzaghi’s effective stress with a nonlinear continuity equation. In this paper, we present a finite element solver for linear and nonlinear poroelasticity problems on triangular meshes based on the displacement-pressure two-field model. We then compare the predictions of linear poroelasticity with those of large-deformation poroelasticity in the context of a two-dimensional model problem where flow through elastic, saturated porous media, under applied mechanical oscillations, is considered. In addition, the impact of introducing a deformation-dependent permeability according to the Kozeny-Carman equation is explored. We computationally show that the errors in the displacement and pressure fields that are obtained using the linear poroelasticity are primarily due to the lack of the kinematic nonlinearity. Furthermore, the error in the pressure field is amplified by incorporating a constant permeability rather than a deformation-dependent permeability.Numerical Analysi

Numerical Methods to Solve Elasticity Problems with Point Sources

Numerical Analysi

Point forces in elasticity equation and their alternatives in multi dimensions

Deep dermal wounds induce skin contraction as a result of the traction forcing exerted by (myo)fibroblasts on their immediate environment. These (myo)fibroblasts are skin cells that are responsible for the regeneration of collagen that is necessary for the integrity of skin We consider several mathematical issues regarding models that simulate traction forces exerted by (myo)fibroblasts. Since the size of cells (e.g. (myo)fibroblasts) is much smaller than the size of the domain of computation, one often considers point forces, modelled by Dirac Delta distributions on boundary segments of cells to simulate the traction forces exerted by the skin cells. In the current paper, we treat the forces that are directed normal to the cell boundary and toward the cell centre. Since it can be shown that there exists no smooth solution, at least not in H1 for solutions to the governing momentum balance equation, we analyse the convergence and quality of approximation. Furthermore, the expected finite element problems that we get necessitate to scrutinize alternative model formulations, such as the use of smoothed Dirac Delta distributions, or the so-called smoothed particle approach as well as the so-called ‘hole’ approach where cellular forces are modelled through the use of (natural) boundary conditions. In this paper, we investigate and attempt to quantify the conditions for consistency between the various approaches. This has resulted into error analyses in the L2-norm of the numerical solution based on Galerkin principles that entail Lagrangian basis functions. The paper also addresses well-posedness in terms of existence and uniqueness. The current analysis has been performed for the linear steady-state (hence neglecting inertia and damping) momentum equations under the assumption of Hooke's law.Numerical Analysi

Self-similar solutions for the foam drainage equation

The travelling wave solutions of the equation for foam drainage in porous media are developed taking into account the mass conservation criterion. The existence of traveling wave solutions is also discussed. Finally, numerical solutions are obtained using a finite difference scheme together with the Van Leer flux limiter, to reduce numerical dispersion. An excellent match is obtained between the analytical and the numerical solutions.Electrical Engineering, Mathematics and Computer Scienc

Point Forces and Their Alternatives in Cell-Based Models for Skin Contraction in Two Dimensions

Deep tissue injury is often followed by contraction of the scar tissue. This contraction occurs as a result of pulling forces that are exerted by fibroblasts (skin cells). We consider a cell-based approach to simulate the contraction behavior of the skin. Since the cells are much smaller than the wound region, we model cellular forces by means of Dirac Delta distributions. Since Dirac Delta distributions cause a singularity of the solution in terms of loss of smoothness, we study alternative approaches where smoothed forces are considered. We prove convergence and consistency between the various approaches, and we also show computational consistency between the approaches. Numerical Analysi