arXiv subjects

Rebecca M. Crossley

Publications and source records attributed to Rebecca M. Crossley.

12 recordsLinked to original sources

arXiv2026

Speed and stability of segregated waves in a pressure-based model of heterogeneous cell populations

We consider a minimal pressure-based model of heterogeneous cell populations consisting of proliferative and non-proliferative cells with different mobilities. The model is formulated as a system of reaction--cross--diffusion equations describing the spatio-temporal dynamics of the cell densities. The model is known to admit one-dimensional travelling wave solutions with strictly segregated components: non-proliferative cells occupy a finite region at the leading edge, while proliferative cells remain at the rear. However, the speed, parameter dependence, and stability of these waves remain poorly understood. In this work, we derive an almost explicit variational bound on the wave speed by reformulating the problem as a free-boundary problem for a generalised porous--Fisher equation. The estimates we obtain apply to general pressure laws and growth kinetics, agree closely with the results of numerical simulations, and become sharp in the incompressible limit, where we formally recover a fully explicit characterisation of the wave speed. We then analyse the stability of the waves to show that segregated waves are stable only when non-proliferative cells are more mobile than proliferative cells. Finally, motivated by numerical observations of finger-like protrusions, we investigate the stability of incompressible segregated circular waves through asymptotic shape-perturbation analysis. This yields explicit expressions for the pressure, interface velocity, and growth rates of angular modes, thereby making evident the destabilisation mechanisms that may lead to the emergence of fingering instability. Interestingly, we find that, in contrast with the one-dimensional case, the stability of such circular waves is not determined solely by the relative value of the mobility coefficients, and thus instabilities may arise irrespective of which cell type has the larger mobility.

math.AP↗

arXiv2026

Reliable mechanistic operator recovery with biologically-informed neural networks: principles for architecture and optimisation design

Many biological processes are governed by complex dynamical mechanisms that remain incompletely understood despite increasing volumes of experimental data. Biologically-informed neural networks (BINNs) seek to address this challenge by embedding mechanistic differential equations into neural network training, enabling interpretable constitutive operators to be recovered directly from sparse and noisy observations. However, reliable operator recovery depends sensitively on network architecture, optimisation strategy, and data informativeness. Here, we present a systematic empirical study of how these factors influence mechanistic inference using BINNs applied to canonical one-dimensional advection-diffusion-reaction partial differential equation models. Across a suite of benchmark problems, we investigate how network expressivity, learning rate, loss weighting, and batch size influence optimisation behaviour and operator recovery. We show that successful mechanistic inference depends on balancing competing objectives rather than maximising any single aspect of the model or optimisation. Moderately expressive architectures outperform overly complex networks, intermediate learning rates improve optimisation stability, balanced data and PDE losses are essential for accurate operator recovery, and intermediate batch sizes provide the best compromise between computational efficiency and reproducibility. We further identify practical diagnostics for recognising common failure modes, including over-fitting, unstable optimisation, and poor mechanistic recovery when the ground truth is unavailable. Together, these findings provide evidence-based guidelines for deploying BINNs as credible tools for biological model discovery.

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arXiv2026

A likelihood-based framework for simultaneously learning both noise and growth dynamics using biologically-informed neural networks

In recent years, neural ordinary differential equation frameworks such as Biologically-Informed Neural Networks (BINNs) have shown promise for learning mechanistic laws from sparse data. However, most existing approaches implicitly assume homoscedastic Gaussian noise, and therefore do not account for potentially meaningful structure in biological variability. Here, we present an extension to the existing BINNs framework that includes a learnable noise model, allowing discovery of the noise model directly from data. Using population growth as an example, we demonstrate that the framework accurately recovers the underlying noise structure and improves predictions of the underlying growth laws compared to existing approaches. As such, this work establishes a general likelihood-based framework for jointly learning dynamics and heteroscedastic noise within mechanistic neural network approaches.

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arXiv2026

An optimal control approach to nonlinear wave speed selection in reaction-diffusion equations

Travelling wave solutions of reaction-diffusion equations are widely used to model the spatial spread of populations and other phenomena in biology and physics. In this article, we reinterpret the classical variational principle approach through an optimal control formulation, in order to obtain a lower bound on the invasion speed of travelling wave solutions in systems of nonlinear partial differential equations. We begin by analysing single-species models, where the evolution of the density is governed by a scalar equation with a density-dependent diffusion term and a nonlinear reaction term. We show that for any admissible test function, maximising with respect to the parameter of interest yields a bound on the travelling wave speed. We apply this framework to several examples, including the porous-Fisher equation, and examine when nonlinear selection mechanisms dominate over the classical linear marginal stability criterion. Extending this approach, we then consider multi-species systems of reaction-diffusion equations and, reframed as Pontryagin-type optimality systems, we derive analogous bounds on the travelling wave speed using a variational framework under weak coupling. Finally, we employ numerical simulations to confirm the accuracy of the predicted wave speeds across a range of illustrative examples.

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arXiv2025

Modelling collective cell migration in a data-rich age: challenges and opportunities for data-driven modelling

Mathematical modelling has a long history in the context of collective cell migration, with applications throughout development, disease and regenerative medicine. The aim of modelling in this context is to provide a framework in which to mathematically encode experimentally derived mechanistic hypotheses, and then to test and validate them to provide new insights and understanding. Traditionally, mathematical models have consisted of systems of partial differential equations that model the evolution of cell density over time, together with the dynamics of any associated biochemical signals or the underlying substrate. The various terms in the model are usually chosen to provide simplified, phenomenological descriptions of the underlying biology, and follow long-standing conventions in the field. However, with the recent development of a plethora of new experimental technologies that provide quantitative data on collective cell migration processes, we now have the opportunity to leverage statistical and machine learning tools to determine mathematical models directly from the data. This perspectives article aims to provide an overview of recently developed data-driven modelling approaches, outlining the main methodologies and the challenges involved in using them to interrogate real-world data relating to collective cell migration.

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arXiv2025

Modelling the impact of phenotypic heterogeneity on cell migration: a continuum framework derived from individual-based principles

Collective cell migration plays a crucial role in numerous biological processes, including tumour growth, wound healing, and the immune response. Often, the migrating population consists of cells with various different phenotypes. This study derives a general mathematical framework for modelling cell migration in the local environment, which is coarse-grained from an underlying individual-based model that captures the dynamics of cell migration that are influenced by the phenotype of the cell, such as random movement, proliferation, phenotypic transitions, and interactions with the local environment. The resulting, flexible, and general model provides a continuum, macroscopic description of cell invasion, which represents the phenotype of the cell as a continuous variable and is much more amenable to simulation and analysis than its individual-based counterpart when considering a large number of phenotypes. We showcase the utility of the generalised framework in three biological scenarios: range expansion; cell invasion into the extracellular matrix; and T cell exhaustion. The results highlight how phenotypic structuring impacts the spatial and temporal dynamics of cell populations, demonstrating that different environmental pressures and phenotypic transition mechanisms significantly influence migration patterns, a phenomenon that would be computationally very expensive to explore using an individual-based model alone. This framework provides a versatile and robust tool for understanding the role of phenotypic heterogeneity in collective cell migration, with potential applications in optimising therapeutic strategies for diseases involving cell migration.

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arXiv2025

Optimal control in combination therapy for heterogeneous cell populations with drug synergies

Cell heterogeneity plays an important role in patient responses to drug treatments. In many cancers, it is associated with poor treatment outcomes. Many modern drug combination therapies aim to exploit cell heterogeneity, but determining how to optimise responses from heterogeneous cell populations while accounting for multi-drug synergies remains a challenge. In this work, we introduce and analyse a general optimal control framework that can be used to model the treatment response of multiple cell populations that are treated with multiple drugs that mutually interact. In this framework, we model the effect of multiple drugs on the cell populations using a system of coupled semi-linear ordinary differential equations and derive general results for the optimal solutions. We then apply this framework to three canonical examples and discuss the wider question of how to relate mathematical optimality to clinically observable outcomes, introducing a systematic approach to propose qualitatively different classes of drug dosing inspired by optimal control.

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arXiv2025

Existence of weak solutions for a volume-filling model of cell invasion into extracellular matrix

We study the existence of weak solutions for a model of cell invasion into the extracellular matrix (ECM), which consists of a non-linear partial differential equation for the density of cells, coupled with an ordinary differential equation (ODE) describing the ECM density. The model contains cross-species density-dependent diffusion and proliferation terms that capture the role of the ECM in providing structural support for the cells during invasion while also preventing growth via volume-filling effects. Furthermore, the model includes ECM degradation by the cells. We present an existence result for weak solutions which is based on carefully exploiting the partial gradient flow structure of the problem which allows us to overcome the non-regularising nature of the ODE involved. In addition, we present simulations based on a finite difference scheme that illustrate that the system exhibits travelling wave solutions, and we investigate numerically the asymptotic behaviour as the ECM degradation rate tends to infinity.

math.AP↗

arXiv2024

Phenotypic switching mechanisms determine the structure of cell migration into extracellular matrix under the `go-or-grow' hypothesis

A fundamental feature of collective cell migration is phenotypic heterogeneity which, for example, influences tumour progression and relapse. While current mathematical models often consider discrete phenotypic structuring of the cell population, in-line with the `go-or-grow' hypothesis \cite{hatzikirou2012go, stepien2018traveling}, they regularly overlook the role that the environment may play in determining the cells' phenotype during migration. Comparing a previously studied volume-filling model for a homogeneous population of generalist cells that can proliferate, move and degrade extracellular matrix (ECM) \cite{crossley2023travelling} to a novel model for a heterogeneous population comprising two distinct sub-populations of specialist cells that can either move and degrade ECM or proliferate, this study explores how different hypothetical phenotypic switching mechanisms affect the speed and structure of the invading cell populations. Through a continuum model derived from its individual-based counterpart, insights into the influence of the ECM and the impact of phenotypic switching on migrating cell populations emerge. Notably, specialist cell populations that cannot switch phenotype show reduced invasiveness compared to generalist cell populations, while implementing different forms of switching significantly alters the structure of migrating cell fronts. This key result suggests that the structure of an invading cell population could be used to infer the underlying mechanisms governing phenotypic switching.

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arXiv2024

Travelling waves in a minimal go-or-grow model of cell invasion

We consider a minimal go-or-grow model of cell invasion, whereby cells can either proliferate, following logistic growth, or move, via linear diffusion, and phenotypic switching between these two states is density-dependent. Formal analysis in the fast switching regime shows that the total cell density in the two-population go-or-grow model can be described in terms of a single reaction-diffusion equation with density-dependent diffusion and proliferation. Using the connection to single-population models, we study travelling wave solutions, showing that the wave speed in the go-or-grow model is always bounded by the wave speed corresponding to the well-known Fisher-KPP equation.

math.AP↗

arXiv2024

Modeling the extracellular matrix in cell migration and morphogenesis: A guide for the curious biologist

The extracellular matrix (ECM) is a highly complex structure through which biochemical and mechanical signals are transmitted. In processes of cell migration, the ECM also acts as a scaffold, providing structural support to cells as well as points of potential attachment. Although the ECM is a well-studied structure, its role in many biological processes remains difficult to investigate comprehensively due to its complexity and structural variation within an organism. In tandem with experiments, mathematical models are helpful in refining and testing hypotheses, generating predictions, and exploring conditions outside the scope of experiments. Such models can be combined and calibrated with in vivo and in vitro data to identify critical cell-ECM interactions that drive developmental and homeostatic processes, or the progression of diseases. In this review, we focus on mathematical and computational models of the ECM in processes such as cell migration including cancer metastasis, and in tissue structure and morphogenesis. By highlighting the predictive power of these models, we aim to help bridge the gap between experimental and computational approaches to studying the ECM and to provide guidance on selecting an appropriate model framework to complement corresponding experimental studies.

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arXiv2023

Travelling waves in a coarse-grained model of volume-filling cell invasion: Simulations and comparisons

Many reaction-diffusion models produce travelling wave solutions that can be interpreted as waves of invasion in biological scenarios such as wound healing or tumour growth. These partial differential equation models have since been adapted to describe the interactions between cells and extracellular matrix (ECM), using a variety of different underlying assumptions. In this work, we derive a system of reaction-diffusion equations, with cross-species density-dependent diffusion, by coarse-graining an agent-based, volume-filling model of cell invasion into ECM. We study the resulting travelling wave solutions both numerically and analytically across various parameter regimes. Subsequently, we perform a systematic comparison between the behaviours observed in this model and those predicted by simpler models in the literature that do not take into account volume-filling effects in the same way. Our study justifies the use of some of these simpler, more analytically tractable models in reproducing the qualitative properties of the solutions in some parameter regimes, but it also reveals some interesting properties arising from the introduction of cell and ECM volume-filling effects, where standard model simplifications might not be appropriate.

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