Abstract

Many tumours invade in complex and heterogeneous fashion, creating significant difficulties for therapy. An important determinant of invasion is the extracellular environment of the cells which, for many tissues, may be highly anisotropic. In this talk I will discuss the modelling of cell migration in anisotropic tissue environments. Starting from a velocity-jump random walk model for cell movement, I will derive transport and anisotropic PDE models that can be used to describe the macroscopic spread of a population. I will show how the interactions between cells and their environment can lead to complex pattern formation processes and, via exploiting data from DTI imaging, show how the model can be applied to predict the invasion patterns of malignant gliomas, a brain tumour with very poor prognosis.

Abstract

Cancer invasion is a complex process involving interactions between cancer cells, between cells and extracellular matrix, and between cancer cells and various types of immune cells that can be found inside the tumour microenvironment. These interactions are facilitated by different molecules (e.g., cytokines, chemokines), which can be produced by the cancer cells themselves, or by other cells in the microenvironment. Here we introduce a class of nonlocal mathematical models for cancer proliferation and invasion (models described by hyperbolic and/or parabolic PDEs), and investigate analytically and numerically the spatial and spatio-temporal patterns exhibited by these nonlocal models.

Abstract

In recent years nonlocal alternatives to the traditional reaction-diffusion-taxis systems for cancer invasion have been proposed in which a local transport flux is replaced by an integral of signal-dependent quantity over a suitable set, typically a ball of a so-called sensing radius r > 0. Terms of this sort allow to describe long-range effects such as cell-cell and cell-tissue adhesions. Some heuristic analysis via local Taylor expansions was performed and led to expect that as r->0 the solutions of these nonlocal models converge to solutions of the corresponding local models. However, to the best of our knowledge no rigorous analytical study of such convergence has as yet been performed. In this talk I will demonstrate a way to close this gap. It relies on a novel reformulation of the involved nonlocality in terms of an integral operator applied directly to the gradient of a signal-dependent quantity. I will show on concrete examples the advantages of the new reformulation. Finally, some numerical simulations in 1D will provide an illustration of the theoretical findings.

The talk is based on a joint work with Maria Krasnianski, Kevin Painter, and Christina Surulescu.