Tracking bifurcating solutions of a model biological pattern generator

We study heterogeneous steady-state solutions of a cell-chemotaxis model for generating biological spatial patterns in two-dimensional domains with zero flux boundary conditions. We use the finite-element package ENTWIFE to investigate bifurcation from the uniform solution as the chemotactic parameter varies and as the domain scale and geometry change. We show that this simple cell-chemotaxis model can produce a remarkably wide and surprising range of complex spatial patterns

Multiscale modelling of intestinal crypt organization and carcinogenesis

Colorectal cancers are the third most common type of cancer. They originate from intestinal crypts, glands that descend from the intestinal lumen into the underlying connective tissue. Normal crypts are thought to exist in a dynamic equilibrium where the rate of cell production at the base of a crypt is matched by that of loss at the top. Understanding how genetic alterations accumulate and proceed to disrupt this dynamic equilibrium is fundamental to understanding the origins of colorectal cancer. Colorectal cancer emerges from the interaction of biological processes that span several spatial scales, from mutations that cause inappropriate intracellular responses to changes at the cell/tissue level, such as uncontrolled proliferation and altered motility and adhesion. Multiscale mathematical modelling can provide insight into the spatiotemporal organisation of such a complex, highly regulated and dynamic system. Moreover, the aforementioned challenges are inherent to the multiscale modelling of biological tissue more generally. In this review we describe the mathematical approaches that have been applied to investigate multiscale aspects of crypt behavior, highlighting a number of model predictions that have since been validated experimentally. We also discuss some of the key mathematical and computational challenges associated with the multiscale modelling approach. We conclude by discussing recent efforts to derive coarse-grained descriptions of such models, which may offer one way of reducing the computational cost of simulation by leveraging well-established tools of mathematical analysis to address key problems in multiscale modelling

Erosion resistance of laser clad Ti-6Al-4V/WC composite for waterjet tooling

AbstractIn waterjet operations, milled surfaces are left with some undesirable dimensional artefacts, thus the use of abrasion resistant mask has been proposed to improve the surface quality of machined components. In this study, the erosion performance of laser clad Ti-6Al-4V/WC composite coating subjected to plain water jet (PWJ) and abrasive water jet (AWJ) impacts to evaluate its potentials for use as waterjet impact resistant mask material and coating on components was investigated. Results showed that composite with 76wt.% WC composition subjected to PWJ and AWJ impacts offered resistance to erosion up to 13 and 8 times that of wrought Ti-6Al-4V respectively. Scanning electron microscopy (SEM) examination of the eroded composite surfaces showed that the erosion mechanism under PWJ impacts is based on the formation of erosion pits, tunnels and deep cavities especially in the interface between the WC particles and the composite matrix owing to lateral outflow jetting and hydraulic penetration. Composite suffered ploughing of the composite matrix, lateral cracking and chipping of embedded WC particles and WC pull-out under AWJ impacts. The composite performance is attributed to the embedded WC particles and the uniformly distributed nano-sized reaction products (TiC and W) reinforcing the ductile β-Ti composite matrix, with its mean hardness enhanced to 6.1GPa. The capability of the Ti-6Al-4V/WC composite coating was demonstrated by effective replication of a pattern on a composite mask to an aluminium plate subjected to selective milling by PWJ with an overall depth of 344μm. Thus, composite cladding for tooling purpose would make it possible to enhance the lifetime of jigs and fixtures and promote rapid machining using the water jet technique

Boundary-driven instability

We analyse a reaction-diffusion system and show that complex spatial patterns can be generated by imposing Dirichlet boundary conditions on one or more of the reactant concentrations. This pattern persists even when the homogeneous steady state with Neumann conditions is stable

Estimation of effective vaccination rate for pertussis in New Zealand as a case study

In some cases vaccination is unreliable. For example vaccination against pertussis has comparatively high level of primary and secondary failures. To evaluate efficiency of vaccination we introduce the idea of effective vaccination rate and suggest an approach to estimate it. We consider pertussis in New Zealand as a case study. The results indicate that the level of immunity failure for pertussis is considerably higher than was anticipated

Bifurcating spatial patterns arising from travelling waves in a tissue interaction model

We analyse a tissue interaction model recently proposed to account for pattern formation in the morphogenesis of skin organ primordia. We show that the model can exhibit travelling wave solutions which leave in their wake a spatially nonuniform, steady state solution

Scale-invariant moving finite elements for nonlinear partial differential equations in two dimensions

A scale-invariant moving finite element method is proposed for the adaptive solution of nonlinear partial differential equations. The mesh movement is based on a finite element discretisation of a scale-invariant conservation principle incorporating a monitor function, while the time discretisation of the resulting system of ordinary differential equations is carried out using a scale-invariant time-stepping which yields uniform local accuracy in time. The accuracy and reliability of the algorithm are successfully tested against exact self-similar solutions where available, and otherwise against a state-of-the-art h-refinement scheme for solutions of a two-dimensional porous medium equation problem with a moving boundary. The monitor functions used are the dependent variable and a monitor related to the surface area of the solution manifold

Analysis of tumor as an inverse problem provides a novel theoretical framework for understanding tumor biology and therapy

We use a novel “inverse problem” technique to construct a basic mathematical model of the interacting populations at the tumor-host interface. This approach assumes that invasive cancer is a solution to the set of state equations that govern the interactions of transformed and normal cells. By considering the invading tumor edge as a traveling wave, the general form of the state equations can be inferred. The stability of this traveling wave solution imposes constraints on key biological quantities which appear as parameters in the model equations. Based on these constraints, we demonstrate the limitations of traditional therapeutic strategies in clinical oncology that focus solely on killing tumor cells or reducing their rate of proliferation. The results provide insights into fundamental mechanisms that may prevent these approaches from successfully eradicating most common cancers despite several decades of research. Alternative therapies directed at modifying the key parameters in the state equations to destabilize the propagating solution are proposed

Analysis of travelling waves associated with the modelling of aerosolised skin grafts

A previous model developed by the authors investigates the growth patterns of keratinocyte cell colonies after they have been applied to a burn site using a spray technique. In this paper, we investigate a simplified one-dimensional version of the model. This model yields travelling wave solutions and we analyse the behaviour of the travelling waves. Approximations for the rate of healing and maximum values for both the active healing and the healed cell densities are obtained

A moving mesh finite element algorithm for the adaptive solution of time-dependent partial differential equations with moving boundaries

A moving mesh finite element algorithm is proposed for the adaptive solution of nonlinear diffusion equations with moving boundaries in one and two dimensions. The moving mesh equations are based upon conserving a local proportion, within each patch of finite elements, of the total “mass” that is present in the projected initial data. The accuracy of the algorithm is carefully assessed through quantitative comparison with known similarity solutions, and its robustness is tested on more general problems. Applications are shown to a variety of problems involving time-dependent partial differential equations with moving boundaries. Problems which conserve mass, such as the porous medium equation and a fourth order nonlinear diffusion problem, can be treated by a simplified form of the method, while problems which do not conserve mass require the full theory