Gissell Estrada, Treatment-induced shrinking of tumour aggregates
We propose a novel mechano-transduction mechanism where the introduction of a chemotherapeutic treatment induces mechanical changes at the cell level. We analyse the influence of these individual mechanical changes on the properties of the aggregates obtained at the population level. We employ a nonlinear volume-filling chemotactic system of PDEs, where the elastic properties of the cells are taken into account through the so-called squeezing probability, which depends on the concentration of the treatment in the extracellular microenvironment. Moreover, we provide numerical simulations in 1D and 2D that illustrate the shrinking of the aggregates due to the presence of the treatment.
Markus Schmidten, On the Incompressible Limit for a Tumour Growth Model Incorporating Convective Effects
We present a porous medium model with applications to tissue movement and tumour growth. The model is based on the standard fluid mechanics approach to living tissues. We extend the analysis proposed in 2014 by Perthame, Quirós, and Vázquez, by incorporating the advective effects caused, for instance, by the presence of nutrients, oxygen, or a chemo-attractant. Passing to the singular limit for a stiff pressure law (incompressible limit), it is possible to connect a density-based model and a free-boundary problem of Hele-Shaw type. Our result extends known results due to weaker assumptions and a more general setting. In particular, we are able to recover the so-called complementarity relation, which allows to derive the pressure through an elliptic equation. To this end, we prove the strong compactness of the pressure gradient, blending two different techniques : an extension of the usual Aronson-Bénilan estimate in an L3-setting and an L4-uniform bound of the pressure gradient. Based on joint work with Noemi David.
Ariane Trescases, Models for chemotaxis with local sensing
We present some models for chemotaxis that are based on local sensing, that is, the cells respond to a certain concentration of chemoattractant perceived locally (as opposed to gradient sensing, when the cells are able to perceive a gradient of concentration). We will observe the specificity of these models, compared for example to the minimal Keller-Segel system, through the study of the well-posedness and the long-time behaviour.
Havva Yoldas, Long-time behaviour of the run and tumble equation for chemotaxis
In this talk, I will present recent results on the long-time behaviour of the run and tumble equation which is a kinetic-transport equation modelling the bacteria movement under the effect of chemical stimulus. We show the exponential convergence to unique stationary state for the linear run and tumble equation. This result is an improvement of a recent work by Mischler & Weng, KRM, 2017. We also consider a weakly nonlinear equation with a nonlocal coupling on the chemoattractant concentration. We construct a unique stationary solution for the weakly non-linear equation and show the exponential convergence towards it. I will also mention how this result give insights of tackling the higher nonlinearities with more physically relevant couplings.
This talk is based on joint works with Josephine Evans (University of Warwick).