F.J. Vermolen

A mathematical analysis of physiological and morphological aspects of wound closure

A computational algorithm to study the evolution of complex wound morphologies is developed based on a model of wound closure by cell mitosis and migration due to Adam [Math Comput Model 30(5–6):23–32, 1999]. A detailed analysis of the model provides estimated values for the incubation and healing times. Furthermore, a set of inequalities are defined which demarcate conditions of complete, partial and non-healing. Numerical results show a significant delay in the healing progress whenever diffusion of the epidermic growth factor responsible for cell mitosis is slower than cell migration. Results for general wound morphologies show that healing is always initiated at regions with high curvatures and that the evolution of the wound is very sensitive to physiological parameters.

A semi-stochastic cell-based formalism to model the dynamics of migration of cells in colonies

We consider the movement and viability of individual cells in cell colonies. Cell movement is assumed to take place as a result of sensing the strain energy density as a mechanical stimulus. The model is based on tracking the displacement and viability of each individual cell in a cell colony. Several applications are shown, such as the dynamics of filling a gap within a fibroblast colony and the invasion of a cell colony. Though based on simple principles, the model is qualitatively validated by experiments on living fibroblasts on a flat substrate.

A finite-element model for healing of cutaneous wounds combining contraction, angiogenesis and closure

A simplified finite-element model for wound healing is proposed. The model takes into account the sequential steps of dermal regeneration, wound contraction, angiogenesis and wound closure. An innovation in the present study is the combination of the aforementioned partially overlapping processes, which can be used to deliver novel insights into the process of wound healing, such as geometry related influences, as well as the influence of coupling between the various existing subprocesses on the actual healing behavior. The model confirms the clinical observation that epidermal closure proceeds by a crawling and climbing mechanism at the early stages, and by a stratification process in layers parallel to the skin surface at the later stages. The local epidermal oxygen content may play an important role here. The model can also be used to investigate the influence of local injection of hormones that stimulate partial processes occurring during wound healing. These insights can be used to improve wound healing treatments.

A mathematical model for the dissolution kinetics of Mg2Si-phases in Al–Mg–Si alloys during homogenisation under industrial conditions

A numerical analysis of the homogenisation treatment of aluminium alloys under industrial circumstances is presented. The basis of this study is a mathematical model which is applicable to the dissolution of stoichiometric multicomponent phases in both finite and infinite ternary media. It handles both complete and incomplete particle dissolution as well as the subsequent homogenisation of the matrix. The precipitate volume fraction and matrix homogeneity are followed during the entire homogenisation treatment. First, the influence of the metallurgical parameters, such as particle size distribution, initial matrix concentration profile and particle geometry on the dissolution- and matrix homogeneity kinetics is analysed. Then, the impact of the heating-rate and local temperature on the homogenisation kinetics is investigated. Conclusions for an optimal homogenisation treatment of aluminium alloys may be drawn. The model presented is general but the calculations were performed for the system Al‐Mg‐Si with an Al-rich matrix and Mg2Si-precipitates.

A phenomenological model for chemico-mechanically induced cell shape changes during migration and cell–cell contacts

A phenomenological model for the evolution of shape transition of cells is considered. These transitions are determined by the emission of growth-factors, as well as mechanical interaction if cells are subjected to hard impingement. The originality of this model necessitates a formal treatment of the mathematical model, as well as the presentation of elementary cases in order to illustrate the consistence of the model. We will also show some small-scale relevant applications.

A numerical model for the dissolution of spherical particles in binary alloys under mixed mode control

A general numerical model is described for the dissolution kinetics of spherical particles in binary systems for any combination of first order reactions at the particle-matrix interface and long distance diffusion within the matrix. The model is applicable to both finite and infinite media and handles both complete and partial particle dissolution. It is shown that interfacial reactions can have a strong effect on the dissolution kinetics, the solute concentration at the particle-matrix interface and the solute concentration profile in the matrix.

Dissolution of β particles in an AlMgSi alloy during DSC runs

Abstract A DSC experimental study of the dissolution kinetics of β precipitates in an AlMgSi alloy has been carried out in order to test the validity of a recent particle dissolution model [F.J. Vermolen, K. Vuik, S. van der Zwaag, Mater. Sci. Eng. A246 (1998) 93–103; F.J. Vermolen, K. Vuik, S. van der Zwaag, Mater. Sci. Eng. A254 (1998) 13–32; F.J. Vermolen, Ph.D Thesis, Delft University, Press, The Netherlands]. The results demonstrate that the model is successful in simulating particle dissolution during linear heating. The concentrations of both alloying elements at the particle–matrix interface during particle dissolution are decided by both the solubility product and the diffusion coefficients. The existence of a size distribution of particles was found to have a significant effect on the dissolution kinetics and therefore should be included in any model if an accurate prediction of the dissolution kinetics is to be achieved.

A mathematical model for the dissolution of particles in multi-component alloys

Dissolution of stoichiometric multi-component particles is an important process occurring during the heat treatment of as-cast aluminum alloys prior to hot extrusion. A mathematical model is proposed to describe such a process. In this model equations are given to determine the position of the particle interface in time, using a number of diffusion equations which are coupled by nonlinear boundary conditions at the interface. This problem is known as a vector valued Stefan problem. A necessary condition for existence of a solution of the moving boundary problem is proposed and investigated using the maximum principle for the parabolic partial differential equation. Furthermore, for an unbounded domain and planar co-ordinates an asymptotic approximation based on self-similarity is derived. The asymptotic approximation is used to gain insight into the influence of all components on the dissolution. Subsequently, a numerical treatment of the vector valued Stefan problem is described. The numerical solution is compared with solutions obtained by the analytical methods. Finally, an example is shown.

A numerical method to compute the dissolution of second phases in ternary alloys

Abstract Dissolution of stoichiometric multi-component particles in ternary alloys is an important process occurring during the heat treatment of as-cast aluminium alloys prior to hot extrusion. A mathematical model is proposed to describe such a process. In this model an equation is given to determine the position of the particle interface in time, using two diffusion equations which are coupled by nonlinear boundary conditions at the interface. Some results concerning existence, uniqueness, and monotonicity are given. Furthermore, for an unbounded domain an analytical approximation is derived. The main part of this work is the development of a numerical solution method. Finite differences are used on a grid which changes in time. The discretization of the boundary conditions is important to obtain an accurate solution. The resulting nonlinear algebraic system is solved by the Newton-Raphson method. Numerical experiments illustrate the accuracy of the numerical method. The numerical solution is compared with the analytical approximation.

Control of flow through porous media using polymer gels

We examine the effect of a dynamic stress on the reduction of flow in porous media using polymer gels formed in situ. To develop the theory for the response of the gel, we consider three dominant factors: (a) compressive (elastic) deformation of the gel and porous medium, (b) microscopic flow in this system, and (c) gel displacement. The latter occurs when the stress p is larger than a certain critical value pc, satisfying pcR2=constant (R=effective pore radius), where the constant is an increasing function of elastic modulus of the gel and its cross-linking energy. The expulsion of the gel above pc is reminiscent of growing Saffman-Taylor instabilities. To derive analytic expressions for the macroscopic saturation profiles we use the formalism for fully miscible two-phase flow. The equation of evolution of the pressure, established by mass balance arguments, was solved analytically. For p<pc, the pressure obeys an exponential saturation function while for p<pc, it first increases, reaches a maximum value...