A sharp-front moving boundary model for malignant invasion

We analyse a novel mathematical model of malignant invasion which takes the form of a two-phase moving boundary problem describing the invasion of a population of malignant cells into a population of background tissue, such as skin. Cells in both populations undergo diffusive migration and logistic proliferation. The interface between the two populations moves according to a two-phase Stefan condition. Unlike many reaction-diffusion models of malignant invasion, the moving boundary model explicitly describes the motion of the sharp front between the cancer and surrounding tissues without needing to introduce degenerate nonlinear diffusion. Numerical simulations suggest the model gives rise to very interesting travelling wave solutions that move with speed $c$, and the model supports both malignant invasion and malignant retreat, where the travelling wave can move in either the positive or negative $x$-directions. Unlike the well-studied Fisher-Kolmogorov and Porous-Fisher models where travelling waves move with a minimum wave speed $c \ge c^* > 0$, the moving boundary model leads to travelling wave solutions with $|c| < c^{**}$. We interpret these travelling wave solutions in the phase plane and show that they are associated with several features of the classical Fisher-Kolmogorov phase plane that are often disregarded as being nonphysical. We show, numerically, that the phase plane analysis compares well with long time solutions from the full partial differential equation model as well as providing accurate perturbation approximations for the shape of the travelling waves.Comment: 48 pages, 21 figure

Invading and receding sharp-fronted travelling waves

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.Comment: 47 pages, 13 figure

Exact sharp-fronted travelling wave solutions of the Fisher-KPP equation

A family of travelling wave solutions to the Fisher-KPP equation with speeds $c=\pm 5/\sqrt{6}$ can be expressed exactly using Weierstrass elliptic functions. The well-known solution for $c=5/\sqrt{6}$, which decays to zero in the far-field, is exceptional in the sense that it can be written simply in terms of an exponential function. This solution has the property that the phase-plane trajectory is a heteroclinic orbit beginning at a saddle point and ends at the origin. For $c=-5/\sqrt{6}$, there is also a trajectory that begins at the saddle point, but this solution is normally disregarded as being unphysical as it blows up for finite $z$. We reinterpret this special trajectory as an exact sharp-fronted travelling solution to a \textit{Fisher-Stefan} type moving boundary problem, where the population is receding from, instead of advancing into, an empty space. By simulating the full moving boundary problem numerically, we demonstrate how time-dependent solutions evolve to this exact travelling solution for large time. The relevance of such receding travelling waves to mathematical models for cell migration and cell proliferation is also discussed

Mathematical models of biological invasion

This thesis studies mathematical models of a population of cells invading the surrounding environment or another living population. A classical single-species model is reformulated using a moving boundary to track the position of the moving front of the invading population. The moving boundary is also used to separate two populations. Other models studied are coupled partial differential equations to describe the interaction of a population with another one. Different types of interaction are represented: the degradation of healthy skin by cancer and the growth of bone tissue on substrate

Non-vanishing sharp-fronted travelling wave solutions of the Fisher-Kolmogorov model

The Fisher-Kolmogorov-Petrovsky-Piskunov (KPP) model, and generalizations thereof, involves simple reaction-diffusion equations for biological invasion that assume individuals in the population undergo linear diffusion with diffusivity D, and logistic proliferation with rate λ. For the Fisher-KPP model, biologically relevant initial conditions lead to long-time travelling wave solutions that move with speed c=2√λD. Despite these attractive features, there are several biological limitations of travelling wave solutions of the Fisher-KPP model. First, these travelling wave solutions do not predict a well-defined invasion front. Second, biologically relevant initial conditions lead to travelling waves that move with speed c=2√λD > 0. This means that, for biologically relevant initial data, the Fisher-KPP model cannot be used to study invasion with c ≠ 2√λD, or retreating travelling waves with c < 0. Here, we reformulate the Fisher-KPP model as a moving boundary problem and show that this reformulated model alleviates the key limitations of the Fisher-KPP model. Travelling wave solutions of the moving boundary problem predict a well-defined front that can propagate with any wave speed, -∞ < c < ∞. Here, we establish these results using a combination of high-accuracy numerical simulations of the time-dependent partial differential equation, phase plane analysis and perturbation methods. All software required to replicate this work is available on GitHub.</p

Travelling wave analysis of cellular invasion into surrounding tissues

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 Browning et al. (2019). 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. Matlab software to implement all calculations is available at GitHub.</p

Traveling waves, blow-up, and extinction in the Fisher–Stefan model

While there is a long history of employing moving boundary problems in physics, in particular via Stefan problems for heat conduction accompanied by a change of phase, more recently such approaches have been adapted to study biological invasion. For example, when a logistic growth term is added to the governing partial differential equation in a Stefan problem, one arrives at the Fisher–Stefan model, a generalization of the well-known Fisher–KPP model, characterized by a leakage coefficient (Formula presented.) which relates the speed of the moving boundary to the flux of population there. This Fisher–Stefan model overcomes one of the well-known limitations of the Fisher–KPP model, since time-dependent solutions of the Fisher–Stefan model involve a well-defined front which is more natural in terms of mathematical modeling. Almost all of the existing analysis of the standard Fisher–Stefan model involves setting (Formula presented.), which can lead to either invading traveling wave solutions or complete extinction of the population. Here, we demonstrate how setting (Formula presented.) leads to retreating traveling waves and an interesting transition to finite-time blow-up. For certain initial conditions, population extinction is also observed. Our approach involves studying time-dependent solutions of the governing equations, phase plane, and asymptotic analysis, leading to new insight into the possibilities of traveling waves, blow-up, and extinction for this moving boundary problem. MATLAB software used to generate the results in this work is available on Github.</p