Seminars
The St Andrews Mathematical Biology group runs a seminar series, with talks occurring on Mondays at 1pm. Talks are in person, in Maths Tutorial Room 1A unless otherwise specified.
Upcoming Seminars:
Discrete and Continuum methods to describe invasion processes
The ability of biological cell populations to correlate their movement and proliferation processes can result in collective migration and evolving spatio-temporal patterns at the cellular population level. These collective mechanisms play an important role in the formation and growth of solid tumours. We have formulated agent-based models of cancer invasion wherein the infiltrating cancer cells may occupy a spectrum of states in phenotype space, for example ranging from `fully mesenchymal’ to `fully epithelial’. The more mesenchymal cells are those that display more migratory phenotypes, where we examine directed cell movement such as haptotaxis or pressure-dependent movement. However, as a trade-off, they have lower proliferative capacity than the more epithelial cells. We have then formally derived the corresponding continuum models, which takes the form of partial integro-differential equations for the local cell population density function. Despite the intricacy of these models, for certain parameter regimes it is possible to carry out detailed travelling wave analyses and obtain invading fronts with spatial structuring of phenotypes. As such, the models recapitulate similar observations into the structures of invading waves into leader-type and follower-type cells, witnessed in an increasing number of experimental studies over recent years.
Title: Mechanical Models of Pattern and Form in Biological Tissues: The Role of Stress–Strain Constitutive Equations
Abstract: Mechanical and mechanochemical models of pattern formation in biological tissues have been used to study a variety of biomedical systems, particularly in developmental biology, and describe the physical interactions between cells and their local surroundings. These models in their original form consist of a balance equation for the cell density, a balance equation for the density of the extracellular matrix (ECM), and a force-balance equation describing the mechanical equilibrium of the cell-ECM system. Under the assumption that the cell-ECM system can be regarded as an isotropic linear viscoelastic material, the force-balance equation is often defined using the Kelvin–Voigt model of linear viscoelasticity to represent the stress–strain relation of the ECM. However, due to the multifaceted bio-physical nature of the ECM constituents, there are rheological aspects that cannot be effectively captured by this model and, therefore, depending on the pattern formation process and the type of biological tissue considered, other constitutive models of linear viscoelasticity may be better suited. In this talk, we systematically assess the pattern formation potential of different stress–strain constitutive equations for the ECM within a mechanical model of pattern formation in biological tissues. The results obtained through linear stability analysis and the dispersion relations derived therefrom support the idea that fluid-like constitutive models, such as the Maxwell model and the Jeffrey model, have a pattern formation potential much higher than solid-like models, such as the Kelvin–Voigt model and the standard linear solid model. This is confirmed by the results of numerical simulations, which demonstrate that, all else being equal, spatial patterns emerge in the case where the Maxwell model is used to represent the stress–strain relation of the ECM, while no patterns are observed when the Kelvin–Voigt model is employed. Our findings suggest that further empirical work is required to acquire detailed quantitative information on the mechanical properties of components of the ECM in different biological tissues in order to furnish mechanical and mechanochemical models of pattern formation with stress–strain constitutive equations for the ECM that provide a more faithful representation of the underlying tissue rheology.