Modelling the interaction of keratinocytes and fibroblasts during normal and abnormal wound healing processes

The crosstalk between fibroblasts and keratinocytes is a vital component of the wound healing process, and involves the activity of a number of growth factors and cytokines. In this work, we develop a mathematical model of this crosstalk in order to elucidate the effects of these interactions on the regeneration of collagen in a wound that heals by second intention. We consider the role of four components that strongly affect this process: transforming growth factor-beta, platelet-derived growth factor, interleukin-1 and keratinocyte growth factor. The impact of this network of interactions on the degradation of an initial fibrin clot, as well as its subsequent replacement by a matrix that is mainly comprised of collagen, is described through an eight-component system of nonlinear partial differential equations. Numerical results, obtained in a two-dimensional domain, highlight key aspects of this multifarious process such as reepithelialisation. The model is shown to reproduce many of the important features of normal wound healing. In addition, we use the model to simulate the treatment of two pathological cases: chronic hypoxia, which can lead to chronic wounds; and prolonged inflammation, which has been shown to lead to hypertrophic scarring. We find that our model predictions are qualitatively in agreement with previously reported observations, and provide an alternative pathway for gaining insight into this complex biological process.

Double diffusive magneto-convection fluid flow in a strong cross magnetic field with uniform surface heat and mass flux

In this study, magnetohydrodynamic natural convection boundary layer flow of an electrically conducting and viscous incompressible fluid along a heated vertical flat plate with uniform heat and mass flux in the presence of strong cross magnetic field has been investigated. For smooth integrations the boundary layer equations are transformed in to a convenient dimensionless form by using stream function formulation as well as the free variable formulation. The nonsimilar parabolic partial differential equations are integrated numerically for Pr ≪1 that is appropriate for liquid metals against the local Hartmann parameter ξ . Further, asymptotic solutions are obtained near the leading edge using regular perturbation method for smaller values of ξ . Solutions for values of ξ ≫ 1 are also obtained by employing the matched asymptotic technique. The results obtained for small, large and all ξ regimes are examined in terms of shear stress, τw, rate of heat transfer, qw, and rate of mass transfer, mw, for important physical parameter. Attention has been given to the influence of Schmidt number, Sc, buoyancy ratio parameter, N and local Hartmann parameter, ξ on velocity, temperature and concentration distributions and noted that velocity and temperature of the fluid achieve their asymptotic profiles for Sc ≥ 10:0.

Mixed convection flow of micropolar fluid in an open ended arc-shape cavity

Numerical simulations for mixed convection of micropolar fluid in an open ended arc-shape cavity have been carried out in this study. Computation is performed using the Alternate Direct Implicit (ADI) method together with the Successive Over Relaxation (SOR) technique for the solution of governing partial differential equations. The flow phenomenon is examined for a range of values of Rayleigh number, 102 ≤ Ra ≤ 106, Prandtl number, 7 ≤ Pr ≤ 50, and Reynolds number, 10 ≤ Re ≤ 100. The study is mainly focused on how the micropolar fluid parameters affect the fluid properties in the flow domain. It was found that despite the reduction of flow in the core region, the heat transfer rate increases, whereas the skin friction and microrotation decrease with the increase in the vortex viscosity parameter, Δ.

Anomalous subdiffusion equations by the meshless collocation method

This paper aims to develop an implicit meshless collocation technique based on the moving least squares approximation for numerical simulation of the anomalous subdiffusion equation(ASDE). The discrete system of equations is obtained by using the MLS meshless shape functions and the meshless collocation formulation. The stability and convergence of this meshless approach related to the time discretization are investigated theoretically and numerically. The numerical examples with regular and irregular nodal distributions are used to the newly developed meshless formulation. It is concluded that the present meshless formulation is very effective for the modeling of ASDEs.

A boundary preserving numerical algorithm for the Wright-Fisher model with mutation

The Wright-Fisher model is an Itô stochastic differential equation that was originally introduced to model genetic drift within finite populations and has recently been used as an approximation to ion channel dynamics within cardiac and neuronal cells. While analytic solutions to this equation remain within the interval [0,1], current numerical methods are unable to preserve such boundaries in the approximation. We present a new numerical method that guarantees approximations to a form of Wright-Fisher model, which includes mutation, remain within [0,1] for all time with probability one. Strong convergence of the method is proved and numerical experiments suggest that this new scheme converges with strong order 1/2. Extending this method to a multidimensional case, numerical tests suggest that the algorithm still converges strongly with order 1/2. Finally, numerical solutions obtained using this new method are compared to those obtained using the Euler-Maruyama method where the Wiener increment is resampled to ensure solutions remain within [0,1].

An implicit RBF meshless method for a fractal mobile/immobile transport model

Fractional differential equation is used to describe a fractal model of mobile/immobile transport with a power law memory function. This equation is the limiting equation that governs continuous time random walks with heavy tailed random waiting times. In this paper, we firstly propose a finite difference method to discretize the time variable and obtain a semi-discrete scheme. Then we discuss its stability and convergence. Secondly we consider a meshless method based on radial basis functions (RBF) to discretize the space variable. By contrast to conventional FDM and FEM, the meshless method is demonstrated to have distinct advantages: calculations can be performed independent of a mesh, it is more accurate and it can be used to solve complex problems. Finally the convergence order is verified from a numerical example is presented to describe the fractal model of mobile/immobile transport process with different problem domains. The numerical results indicate that the present meshless approach is very effective for modeling and simulating of fractional differential equations, and it has good potential in development of a robust simulation tool for problems in engineering and science that are governed by various types of fractional differential equations.

Analytical solutions for the multi-term time-space Caputo-Riesz fractional advection-diffusion equations on a finite domain

Generalized fractional partial differential equations have now found wide application for describing important physical phenomena, such as subdiffusive and superdiffusive processes. However, studies of generalized multi-term time and space fractional partial differential equations are still under development. In this paper, the multi-term time-space Caputo-Riesz fractional advection diffusion equations (MT-TSCR-FADE) with Dirichlet nonhomogeneous boundary conditions are considered. The multi-term time-fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals [0, 1], [1, 2] and [0, 2], respectively. These are called respectively the multi-term time-fractional diffusion terms, the multi-term time-fractional wave terms and the multi-term time-fractional mixed diffusion-wave terms. The space fractional derivatives are defined as Riesz fractional derivatives. Analytical solutions of three types of the MT-TSCR-FADE are derived with Dirichlet boundary conditions. By using Luchko's Theorem (Acta Math. Vietnam., 1999), we proposed some new techniques, such as a spectral representation of the fractional Laplacian operator and the equivalent relationship between fractional Laplacian operator and Riesz fractional derivative, that enabled the derivation of the analytical solutions for the multi-term time-space Caputo-Riesz fractional advection-diffusion equations. © 2012.

Velocity jump processes with proliferation

Cell invasion involves a population of cells that migrate along a substrate and proliferate to a carrying capacity density. These two processes, combined, lead to invasion fronts that move into unoccupied tissues. Traditional modelling approaches based on reaction–diffusion equations cannot incorporate individual–level observations of cell velocity, as information propagates with infinite velocity according to these parabolic models. In contrast, velocity jump processes allow us to explicitly incorporate individual–level observations of cell velocity, thus providing an alternative framework for modelling cell invasion. Here, we introduce proliferation into a standard velocity–jump process and show that the standard model does not support invasion fronts. Instead, we find that crowding effects must be explicitly incorporated into a proliferative velocity–jump process before invasion fronts can be observed. Our observations are supported by numerical and analytical solutions of a novel coupled system of partial differential equations, including travelling wave solutions, and associated random walk simulations.

A note on a reaction-diffusion model describing the bone morphogen protein gradient in Drosophila embryonic patterning

In this article, we consider the Eldar model [3] from embryology in which a bone morphogenic protein, a short gastrulation protein, and their compound react and diffuse. We carry out a perturbation analysis in the limit of small diffusivity of the bone morphogenic protein. This analysis establishes conditions under which some elementary results of [3] are valid.

Mixed convection along a vertical flat plate in a non-absorbing medium

Here mixed convection boundary layer flow of a viscous fluid along a heated vertical semi-infinite plate is investigated in a non-absorbing medium. The relationship between convection and thermal radiation is established via boundary condition of second kind on the thermally radiating vertical surface. The governing boundary layer equations are transformed into dimensionless parabolic partial differential equations with the help of appropriate transformations and the resultant system is solved numerically by applying straightforward finite difference method along with Gaussian elimination technique. It is worthy to note that Prandlt number, Pr, is taken to be small (<< 1) which is appropriate for liquid metals. Moreover, the numerical results are demonstrated graphically by showing the effects of important physical parameters, namely, the modified Richardson number (or mixed convection parameter), Ri*, and surface radiation parameter, R, in terms of local skin friction and local Nusselt number coefficients.

Natural convection flow in a strong cross magnetic field with radiation

The problem of MHD natural convection boundary layer flow of an electrically conducting and optically dense gray viscous fluid along a heated vertical plate is analyzed in the presence of strong cross magnetic field with radiative heat transfer. In the analysis radiative heat flux is considered by adopting optically thick radiation limit. Attempt is made to obtain the solutions valid for liquid metals by taking Pr≪1. Boundary layer equations are transformed in to a convenient dimensionless form by using stream function formulation (SFF) and primitive variable formulation (PVF). Non-similar equations obtained from SFF are then simulated by implicit finite difference (Keller-box) method whereas parabolic partial differential equations obtained from PVF are integrated numerically by hiring direct finite difference method over the entire range of local Hartmann parameter, $xi$ . Further, asymptotic solutions are also obtained for large and small values of local Hartmann parameter $xi$ . A favorable agreement is found between the results for small, large and all values of $xi$ . Numerical results are also demonstrated graphically by showing the effect of various physical parameters on shear stress, rate of heat transfer, velocity and temperature.

Natural convection flow with surface radiation along a vertical wavy surface

In this study, natural convection boundary layer flow of thermally radiating fluid along a heated vertical wavy surface is analyzed. Here, the radiative component of heat flux emulates the surface temperature. Governing equations are reduced to dimensionless form, subject to the appropriate transformation. Resulting dimensionless equations are transformed to a set of parabolic partial differential equations by using primitive variable formulation, which are then integrated numerically via iterative finite difference scheme. Emphasis has been given to low Prandtl number fluid. The numerical results obtained for the physical parameters, such as, surface radiation parameter, R, and radiative length parameter, ξ, are discussed in terms of local skin friction and Nusselt number coefficients. Comprehensive interpretation of velocity distribution is also given in the form of streamlines.

Analytical Solution of a Walters’ Liquid B Flow Over a Linear Stretching Sheet in a Porous Medium

This chapter represents the analytical solution of two-dimensional linear stretching sheet problem involving a non-Newtonian liquid and suction by (a) invoking the boundary layer approximation and (b) using this result to solve the stretching sheet problem without using boundary layer approximation. The basic boundary layer equations for momentum, which are non-linear partial differential equations, are converted into non-linear ordinary differential equations by means of similarity transformation. The results reveal a new analytical procedure for solving the boundary layer equations arising in a linear stretching sheet problem involving a non-Newtonian liquid (Walters’ liquid B). The present study throws light on the analytical solution of a class of boundary layer equations arising in the stretching sheet problem.

Exponential integrators and a dual-scale model for wood drying

For the timber industry, the ability to simulate the drying of wood is invaluable for manufacturing high quality wood products. Mathematically, however, modelling the drying of a wet porous material, such as wood, is a diffcult task due to its heterogeneous and anisotropic nature, and the complex geometry of the underlying pore structure. The well{ developed macroscopic modelling approach involves writing down classical conservation equations at a length scale where physical quantities (e.g., porosity) can be interpreted as averaged values over a small volume (typically containing hundreds or thousands of pores). This averaging procedure produces balance equations that resemble those of a continuum with the exception that effective coeffcients appear in their deffnitions. Exponential integrators are numerical schemes for initial value problems involving a system of ordinary differential equations. These methods differ from popular Newton{Krylov implicit methods (i.e., those based on the backward differentiation formulae (BDF)) in that they do not require the solution of a system of nonlinear equations at each time step but rather they require computation of matrix{vector products involving the exponential of the Jacobian matrix. Although originally appearing in the 1960s, exponential integrators have recently experienced a resurgence in interest due to a greater undertaking of research in Krylov subspace methods for matrix function approximation. One of the simplest examples of an exponential integrator is the exponential Euler method (EEM), which requires, at each time step, approximation of φ(A)b, where φ(z) = (ez - 1)/z, A E Rnxn and b E Rn. For drying in porous media, the most comprehensive macroscopic formulation is TransPore [Perre and Turner, Chem. Eng. J., 86: 117-131, 2002], which features three coupled, nonlinear partial differential equations. The focus of the first part of this thesis is the use of the exponential Euler method (EEM) for performing the time integration of the macroscopic set of equations featured in TransPore. In particular, a new variable{ stepsize algorithm for EEM is presented within a Krylov subspace framework, which allows control of the error during the integration process. The performance of the new algorithm highlights the great potential of exponential integrators not only for drying applications but across all disciplines of transport phenomena. For example, when applied to well{ known benchmark problems involving single{phase liquid ow in heterogeneous soils, the proposed algorithm requires half the number of function evaluations than that required for an equivalent (sophisticated) Newton{Krylov BDF implementation. Furthermore for all drying configurations tested, the new algorithm always produces, in less computational time, a solution of higher accuracy than the existing backward Euler module featured in TransPore. Some new results relating to Krylov subspace approximation of '(A)b are also developed in this thesis. Most notably, an alternative derivation of the approximation error estimate of Hochbruck, Lubich and Selhofer [SIAM J. Sci. Comput., 19(5): 1552{1574, 1998] is provided, which reveals why it performs well in the error control procedure. Two of the main drawbacks of the macroscopic approach outlined above include the effective coefficients must be supplied to the model, and it fails for some drying configurations, where typical dual{scale mechanisms occur. In the second part of this thesis, a new dual{scale approach for simulating wood drying is proposed that couples the porous medium (macroscale) with the underlying pore structure (microscale). The proposed model is applied to the convective drying of softwood at low temperatures and is valid in the so{called hygroscopic range, where hygroscopically held liquid water is present in the solid phase and water exits only as vapour in the pores. Coupling between scales is achieved by imposing the macroscopic gradient on the microscopic field using suitably defined periodic boundary conditions, which allows the macroscopic ux to be defined as an average of the microscopic ux over the unit cell. This formulation provides a first step for moving from the macroscopic formulation featured in TransPore to a comprehensive dual{scale formulation capable of addressing any drying configuration. Simulation results reported for a sample of spruce highlight the potential and flexibility of the new dual{scale approach. In particular, for a given unit cell configuration it is not necessary to supply the effective coefficients prior to each simulation.

Two novel numerical methods for solving the two-dimensional fractional Fitzhugh-Nagumo-monodomain model

Fractional mathematical models represent a new approach to modelling complex spatial problems in which there is heterogeneity at many spatial and temporal scales. In this paper, a two-dimensional fractional Fitzhugh-Nagumo-monodomain model with zero Dirichlet boundary conditions is considered. The model consists of a coupled space fractional diffusion equation (SFDE) and an ordinary differential equation. For the SFDE, we first consider the numerical solution of the Riesz fractional nonlinear reaction-diffusion model and compare it to the solution of a fractional in space nonlinear reaction-diffusion model. We present two novel numerical methods for the two-dimensional fractional Fitzhugh-Nagumo-monodomain model using the shifted Grunwald-Letnikov method and the matrix transform method, respectively. Finally, some numerical examples are given to exhibit the consistency of our computational solution methodologies. The numerical results demonstrate the effectiveness of the methods.