Modelling biological invasions: Individual to population scales at interfaces

Abstract

We explore the influence of spatial heterogeneity in biological systems. We consider how local transport mechanisms impact behaviour near interfaces. We study cell movement between white and grey matter in the brain, as an example. Profound differences in population behaviour emerge in the presence of interface. The commonly used Fickian diffusion transport model cannot predict this dynamics. a r t i c l e i n f o b s t r a c tExtracting the population level behaviour of biological systems from that of the individual is critical in understanding dynamics across multiple scales and thus has been the subject of numerous investigations. Here, the influence of spatial heterogeneity in such contexts is explored for interfaces with a separation of the length scales characterising the individual and the interface, a situation that can arise in applications involving cellular modelling. As an illustrative example, we consider cell movement between white and grey matter in the brain which may be relevant in considering the invasive dynamics of glioma. We show that while one can safely neglect intrinsic noise, at least when considering glioma cell invasion, profound differences in population behaviours emerge in the presence of interfaces with only subtle alterations in the dynamics at the individual level. Transport driven by local cell sensing generates predictions of cell accumulations along interfaces where cell motility changes. This behaviour is not predicted with the commonly used Fickian diffusion transport model, but can be extracted from preliminary observations of specific cell lines in recent, novel, cryo-imaging. Consequently, these findings suggest a need to consider the impact of individual behaviour, spatial heterogeneity and especially interfaces in experimental and modelling frameworks of cellular dynamics, for instance in the characterisation of glioma cell motility.

Edge effects in game-theoretic dynamics of spatially structured tumours

“…In each of these two scenarios, tumour cells express the motile phenotype, but only when against the physical structure in question. Another recent result suggesting the importance of effects of changes in local architecture is the work of Belmonte-Beitia et al [53], in which the authors show that the dynamics of motility change drastically at the grey-white interface in glioblastoma in ways that cannot be predicted by simply comparing the behaviour in each zone individually.…”

Direct and inverse analysis of diffusive logistic population evolution with time delay and impulsive culling via integral transforms and hybrid optimization

“…The elements of the Jacobian matrix J, defined in Eq. (11), are called the sensitivity coefficients. In order to obtain good estimates, within reasonable confidence intervals, it is required the sensitivity coefficients to be relatively high and, when two or more unknowns are simultaneously estimated, their sensitivity coefficients must be linearly independent, which graphically means that they should not present the same slope in absolute value.…”

“…Since the pioneering works of Holling [1] and Segel and Jackson [2] many works have been proposed towards the modeling of populations growth [3][4][5][6][7][8][9][10][11][12][13], most of them considering Fickian dispersion and logistic like growth rates, seeking to take into account different aspects that play some role in the population dynamics, such as the migration to less populated areas, presence of predators and preys, toxicant, culling sites and several other parameters that affect the birth and death rates. Most of these papers are concerned with the proposition of different models and their stability analysis, presenting some analytical solution results to some oversimplified models or numerical solutions obtained with classical numerical methods.…”

Boundedness in a parabolic‐elliptic chemotaxis system with nonlinear diffusion and sensitivity and logistic source

“…Moreover, we assume that α ≤1 and that the function χ generalises the standard chemotactic sensitivity The mathematical formulation of this problem describes phenomena tied to chemotaxis models, which indicate the movement of a cell population at a point x of an environment, identified as Ω, and at an instant t (ie, u = u ( x , t )) when stimulated by a chemical signal (ie, v = v ( x , t )), which is also distributed in the space and produced by the cells. In addition, the zero‐flux boundary conditions on both u and v (the homogeneous Neumann boundary conditions) express that the interaction takes places in a totally insulated domain …”

“…In addition, the zero-flux boundary conditions on both u and v (the homogeneous Neumann boundary conditions) express that the interaction takes places in a totally insulated domain. 1 Problem (1) is one of several generalisations of the following landmark model proposed by Keller and Segel in 1970 (see Keller and Segel 2 and the more recent reviews 3,4 ),…”