On Similarity Solutions and Interface Reactions for a Vector-Valued Stefan Problem

In this paper first it is shown for several geometries that classical similarity solutions for particle growth exist if and only if the Stefan problem is well-posed in the sense of being mass conserving. The extension of the similarity solutions to multicomponent alloys, which makes the problem nonlinear, is illustrated by the application to a hypothetic alloy with realistic input values. The similarity solutions are based on the assumption of local equilibrium at the interface. In the second part, the assumption of local equilibrium is relaxed using a first-order interface reaction. The influence of the interface reaction on the movement of the interface and on the interface concentrations is evaluated using Finite Difference calculations. A Newton scheme is used to solve the nonlinear problem

A Simplified Model for Growth Factor Induced Healing of Wounds

A mathematical model is developed for the rate of healing of a circular or elliptic wound. In this paper the regeneration, decay and transport of a generic \u27growth factor\u27, which induces the healing of the wound, is taken into account. Further, an equation of motion is derived for radial healing of a circular wound. The expressions for the equation of motion and the distribution of the growth factor are related in such a way that no healing occurs if the growth factor concentration at the wound edge is below a threshold value. In this paper we investigate the influence of the behaviour of the thickness of the active layer, in which the growth factor is produced, on the healing process. Also a correction is made to a result in earlier work. © 2006 Elsevier Ltd. All rights reserved

Review on some Stefan Problems for Particle Dissolution in Solid Metallic Alloys

This paper is a review of a suite of mathematical models of increasing complexity on particle dissolution in metallic alloys. This work deals with models for multi-component particle dissolution in multi-component alloys, where various chemical species diffuse simultaneously, and a two-dimensional model incorporating interfacial reactions as in the model of Nolfi [1]. The work is mathematically rigorous where asymptotic solutions and solution bounds are derived but is also of a practical nature as particle dissolution kinetics is modelled for industrially relevant conditions

A pilot study of a phenomenological model of adipogenesis in maturing adipocytes using Cahn–Hilliard theory

We consider the accumulation and formation of lipid droplets in an adipocyte cell. The process incorporates adipose nucleation (adipogenesis) and growth. At later stages, there will be merging of droplets and growth of larger droplets at the expense of the smaller droplets, which will essentially undergo lipolysis. The process is modeled by the use of the Cahn–Hilliard equation, which is mass-conserving and allows the formation of secondary phases in the context of spinodal decomposition. The volume of fluid (VOF) method is used to determine the total area that is occupied by the lipids in a given cross section. Further, we present an algorithm, applicable to all kinds of grids (structured or unstructured) in two spatial dimensions, to count the number of lipid droplets and the portion of the domain of computation that is occupied by the lipid droplets as a function of time during the process. The results are preliminary and are validated from a qualitative point using experiments carried out on cell cultures. It turns out that the Cahn–Hilliard theory can model many of the features during adipogenesis qualitatively

Computational modelling of wound healing insights to develop new treatments

About 1% of the population will suffer a severe wound during their life. Thus, it is really important to develop new techniques in order to properly treat these injuries due to the high socioeconomically impact they suppose. Skin substitutes and pressure based therapies are currently the most promising techniques to heal these injuries. Nevertheless, we are still far from finding a definitive skin substitute for the treatment of all chronic wounds. As a first step in developing new tissue engineering tools and treatment techniques for wound healing, in silico models could help in understanding the mechanisms and factors implicated in wound healing. Here, we review mathematical models of wound healing. These models include different tissue and cell types involved in healing, as well as biochemical and mechanical factors which determine this process. Special attention is paid to the contraction mechanism of cells as an answer to the tissue mechanical state. Other cell processes such as differentiation and proliferation are also included in the models together with extracellular matrix production. The results obtained show the dependency of the success of wound healing on tissue composition and the importance of the different biomechanical and biochemical factors. This could help to individuate the adequate concentration of growth factors to accelerate healing and also the best mechanical properties of the new skin substitute depending on the wound location in the body and its size and shape. Thus, the feedback loop of computational models, experimental works and tissue engineering could help to identify the key features in the design of new treatments to heal severe wounds