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Biological invasion, whereby populations of motile and proliferative individuals lead to moving fronts that invade into vacant regions, are routinely studied using partial differential equation (PDE) models based upon the classical Fisher--KPP model. While the Fisher--KPP model and extensions have been successfully used to model a range of invasive phenomena, including ecological and cellular invasion, an often--overlooked limitation of the Fisher--KPP model is that it cannot be used to model biological recession where the spatial extent of the population decreases with time. In this work we study the textit{Fisher--Stefan} model, which is a generalisation of the Fisher--KPP model obtained by reformulating the Fisher--KPP model as a moving boundary problem. The nondimensional Fisher--Stefan model involves just one single parameter, $kappa$, which relates the shape of the density front at the moving boundary to the speed of the associated travelling wave, $c$. Using numerical simulation, phase plane and perturbation analysis, we construct approximate solutions of the Fisher--Stefan model for both slowly invading and slowly receding travelling waves, as well as for rapidly receding travelling waves. These approximations allow us to determine the relationship between $c$ and $kappa$ so that commonly--reported experimental estimates of $c$ can be used to provide estimates of the unknown parameter $kappa$. Interestingly, when we reinterpret the Fisher--KPP model as a moving boundary problem, many disregarded features of the classical Fisher--KPP phase plane take on a new interpretation since travelling waves solutions with $c < 2$ are not normally considered. This means that our analysis of the Fisher--Stefan model has both practical value and an inherent mathematical value.
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Reaction-diffusion equations (RDEs) are often derived as continuum limits of lattice-based discrete models. Recently, a discrete model which allows the rates of movement, proliferation and death to depend upon whether the agents are isolated has been proposed, and this approach gives various RDEs where the diffusion term is convex and can become negative (Johnston et al., Sci. Rep. 7, 2017), i.e. forward-backward-forward diffusion. Numerical simulations suggest these RDEs support shock-fronted travelling waves when the reaction term includes an Allee effect. In this work we formalise these preliminary numerical observations by analysing the shock-fronted travelling waves through embedding the RDE into a larger class of higher order partial differential equations (PDEs). Subsequently, we use geometric singular perturbation theory to study this larger class of equations and prove the existence of these shock-fronted travelling waves. Most notable, we show that different embeddings yield shock-fronted travelling waves with different properties.
Single-species reaction-diffusion equations, such as the Fisher-KPP and Porous-Fisher equations, support travelling wave solutions that are often interpreted as simple mathematical models of biological invasion. Such travelling wave solutions are thought to play a role in various applications including development, wound healing and malignant invasion. One criticism of these single-species equations is that they do not explicitly describe interactions between the invading population and the surrounding environment. In this work we study a reaction-diffusion equation that describes malignant invasion which has been used to interpret experimental measurements describing the invasion of malignant melanoma cells into surrounding human skin tissues. This model explicitly describes how the population of cancer cells degrade the surrounding tissues, thereby creating free space into which the cancer cells migrate and proliferate to form an invasion wave of malignant tissue that is coupled to a retreating wave of skin tissue. We analyse travelling wave solutions of this model using a combination of numerical simulation, phase plane analysis and perturbation techniques. Our analysis shows that the travelling wave solutions involve a range of very interesting properties that resemble certain well-established features of both the Fisher-KPP and Porous-Fisher equations, as well as a range of novel properties that can be thought of as extensions of these well-studied single-species equations. Of particular interest is that travelling wave solutions of the invasion model are very well approximated by trajectories in the Fisher-KPP phase plane that are normally disregarded. This observation establishes a previously unnoticed link between coupled multi-species reaction diffusion models of invasion and a different class of models of invasion that involve moving boundary problems.
In this paper we analyse a previously proposed cell-based model of glioblastoma (brain tumour) growth, which is based on the assumption that the cancer cells switch phenotypes between a proliferative and motile state (Gerlee and Nelander, PLoS Comp. Bio., 8(6) 2012). The dynamics of this model can be described by a system of partial differential equations, which exhibits travelling wave solutions whose wave speed depends crucially on the rates of phenotypic switching. We show that under certain conditions on the model parameters, a closed form expression of the wave speed can be obtained, and using singular perturbation methods we also derive an approximate expression of the wave front shape. These new analytical results agree with simulations of the cell-based model, and importantly show that the inverse relationship between wave front steepness and speed observed for the Fisher equation no longer holds when phenotypic switching is considered.
Here, we investigated the influence of physicochemical characteristics of chondroitin sulfate (CS) on its in vitro absorption and anti-inflammatory activity. We used eight different synthetic and natural CS samples with a range of molecular weights (7-35 kDa) and sulfation patterns. Our studies indicate that the absorption of CS is moderately correlated to percentage of chondroitin-6-sulfate while the anti-inflammatory activity may be weakly related to the molecular weight and the amount of total sulfation in the samples. Our in vitro studies could provide helpful screening tools for quick and effective evaluation of CS samples as a preliminary step towards in vivo studies.
Several biological tissues undergo changes in their geometry and in their bulk material properties by modelling and remodelling processes. Modelling synthesises tissue in some regions and removes tissue in others. Remodelling overwrites old tissue material properties with newly formed, immature tissue properties. As a result, tissues are made up of different patches, i.e., adjacent tissue regions of different ages and different material properties, within evolving boundaries. In this paper, generalised equations governing the spatio-temporal evolution of such tissues are developed within the continuum model. These equations take into account nonconservative, discontinuous surface mass balance due to creation and destruction of material at moving interfaces, and bulk balance due to tissue maturation. These equations make it possible to model patchy tissue states and their evolution without explicitly maintaining a record of when/where resorption and formation processes occurred. The time evolution of spatially averaged tissue properties is derived systematically by integration. These spatially-averaged equations cannot be written in closed form as they retain traces that tissue destruction is localised at tissue boundaries. The formalism developed in this paper is applied to bone tissues, which exhibit strong material heterogeneities due to their slow mineralisation and remodelling processes. Evolution equations are proposed in particular for osteocyte density and bone mineral density. Effective average equations for bone mineral density (BMD) and tissue mineral density (TMD) are derived using a mean-field approximation. The error made by this approximation when remodelling patchy tissue is investigated. The specific time signatures of BMD or TMD during remodelling events may provide a way to detect these events occurring at lower, unseen spatial resolutions from microCT scans.
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