Using Experimental Data and Information Criteria to Guide Model Selection for Reaction–Diffusion Problems in Mathematical Biology

Abstract

Reaction-diffusion models describing the movement, reproduction and death of individuals within a population are key mathematical modelling tools with widespread applications in mathematical biology. A diverse range of such continuum models have been applied in various biological contexts by choosing different flux and source terms in the reaction-diffusion framework. For example, to describe collective spreading of cell populations, the flux term may be chosen to reflect various movement mechanisms, such as random motion (diffusion), adhesion, haptotaxis, chemokinesis and chemotaxis. The choice of flux terms in specific applications, such as wound healing, is usually made heuristically, and rarely is it tested quantitatively against detailed cell density data. More generally, in mathematical biology, the questions of model validation and model selection have not received the same attention as the questions of model development and model analysis. Many studies do not consider model validation or model selection, and those that do often base the selection of the model on residual error criteria after model calibration is performed using nonlinear regression techniques. In this work, we present a model selection case study, in the context of cell invasion, with a very detailed experimental data set. Using Bayesian analysis and information criteria, we demonstrate that model selection and model validation should account for both residual errors and model complexity. These considerations are often overlooked in the mathematical biology literature. The results we present here provide a clear methodology that can be used to guide model selection across a range of applications. Furthermore, the case study we present provides a clear example where neglecting the role of model complexity can give rise to misleading outcomes.

A practical guide to pseudo-marginal methods for computational inference in systems biology

“…The ABC approach to likelihood-free inference is widely applicable and used extensively in practical applications (Browning et al, 2018;Johnston et al, 2016;Kursawe et al, 2018;Warne et al, 2019b;Wilkinson, 2011). For some applications, however, it can be difficult to determine a priori an appropriate discrepancy metric and acceptance threshold for reliable inference.…”

Modelling collective cell migration: neural crest as a model paradigm

“…Recent advances in imaging technology and single cell analyses now mean that, experimentally, we can track individual cell behaviours and interrogate gene expression changes to generate multiscale spatiotemporal data. One of the ongoing challenges in this field is to incorporate spatial statistics into modelling frameworks for parameter identification and model refinement (Warne et al 2018). This will allow more precise model predictions and validation.…”

Revisiting the Fisher-KPP equation to interpret the spreading-extinction dichotomy

“…for 0 < x ≤ L, with ∂U/∂x = 0 at x = 0 and U = 0 at x = L. In this approximate analysis we treat the domain length as a fixed quantity and this allows us to write down the exact solution of the linear equation (19) as…”

“…Alternatively, if the time dependent solution of Equations (9)- (12) evolves such that L(t) never exceeds π/2, the net loss of mass in the system means that the population will always go extinct, as in Figure 6(d)-(f). It is also worthwhile to note that since the result that L crit = π/2 is governed by a linearised model (19), this outcome will hold for any generalisation of the Fisher-Stefan model that can be linearised to give Equation (19). For example, if we extended the Fisher-Stefan model to consider a generalised logistic source term, u(1 − u) m , where m > 0 is some exponent [56,57,58,59], then the same L crit = π/2 would apply.…”

“…The Fisher-KPP model and its extensions have been used successfully in a wide range of applications including the study of spatial spreading of invasive species in ecology [11,12,13,14]. In cell biology, the spatial spreading of invasive cell populations has been modelled using the Fisher-KPP model and its extensions for a range of applications including in vitro cell biology experiments [15,16,17,18,19] and in vivo malignant spreading [20,21,22]. Other areas of application include combustion theory [23,24] and bushfire invasion [25].…”