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Details of Grant 

EPSRC Reference: EP/T03131X/1
Title: Modelling and analysis of cancer invasion with flux-saturation mechanisms
Principal Investigator: Zhigun, Dr A
Other Investigators:
Researcher Co-Investigators:
Project Partners:
Department: Sch of Mathematics and Physics
Organisation: Queen's University of Belfast
Scheme: New Investigator Award
Starts: 01 July 2021 Ends: 30 September 2023 Value (£): 199,738
EPSRC Research Topic Classifications:
Cells Mathematical Analysis
EPSRC Industrial Sector Classifications:
Related Grants:
Panel History:
Panel DatePanel NameOutcome
01 Jun 2020 EPSRC Mathematical Sciences Prioritisation Panel June 2020 Announced
Summary on Grant Application Form
Invasion is an essential stage of cancer progress. Once initiated, a malignant tumour can grow and make its way through the surrounding tissue. Cancer cells can penetrate blood vessels and, after transportation across the vasculature, they colonise distant sites forming further neoplasms. This process is termed metastasis. Cancer invasion results from the interplay of many effects, including migration and proliferation. Mathematical models of invasion and their analytical and numerical study can improve our understanding of the involved biological phenomena. They enable predictions about the development and the extent of a tumour and testing of therapy strategies in silico.

Reaction-diffusion-transport (RDT) systems are the most popular tools in this context. They describe the evolution of a tumour in terms of macroscopic densities depending only upon time and position in space, which enables comparisons with information gained by standard biomedical imaging techniques, e.g. MRI and CT. Another advantage is the availability of well-developed mathematical analysis tools and efficient numerical methods. Individual and subcellular level processes are, however, greatly influencing the overall development of a tumour. These processes are often modelled with ODEs for additional state variables on which the cell density distributions depend. Unfortunately, the resulting multiscale settings are difficult to resolve numerically. Hence, a suitable upscaling is usually performed yielding an RDT system which still contains some essential lower-level information. If successfully studied analytically and simulated numerically, using parameters determined from experiments, such RDT offer a more careful description of cell migration than, e.g. a system obtained through standard balance of macroscopic fluxes. It often involves challenging nonstandard terms. Prominent examples include anisotropic diffusion and flux-saturated diffusion and chemotaxis.

Chemotaxis is a directed movement of cells or organisms in response to chemical cues. It plays a crucial role in many self-organisation processes such as slime mold fruiting body formation, embryonic morphogenesis, wound healing, cancer invasion, etc. and leads to formation of aggregates. Equation systems modelling chemotaxis have been enjoying great popularity in recent decades. The best-known among them is the classical Keller-Segel model introduced in the 1970's. However, experiments show that this well-studied system is incapable of reproducing the cell behaviour observed in certain biological processes. For instance, it has no built in control upon the maximum propagation speed which in reality is an intrinsic quality of a population. The resulting propagation is often too fast and may even lead to infinite densities due to chemotaxis-driven aggregation.

In 2010 a new model able to deal with this and other criticism was introduced by Bellomo et al. It features a flux-saturated degenerate diffusion and a flux-saturated novel chemotaxis term. Some studies showed that typical solutions such as travelling waves had realistic patterns evolving with controlled speed. However, the general existence of solutions has remained an open question.

In this project we will prove the first result on the global existence for the above model from 2010. Moreover, we will construct novel, more realistic than previously, multiscale systems based on kinetic transport equations modelling cancer cell propagation on the mesoscale, scale them up to the macroscopic level formally and, wherever possible, also rigorously, and finally study some of the resulting macroscopic systems analytically. We will pay particular attention to preserving flux-saturation mechanisms as well as including other relevant highly nonlinear effects.
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Organisation Website: http://www.qub.ac.uk