Modelling Spatial Cancer Heterogeneity

Lead Research Organisation: University of Oxford
Department Name: Statistics

Abstract

Context and Impact: Cancer is one of the leading causes of death worldwide, see e.g. [SMJ19], and its severity often translates into the grief of the
affected, their relatives and wider social circles.
Modern mathematical research views cancer as a population of malignant cells which interact, mutate and evolve in a complex manner. A major obstruction to producing
a tractable mathematical model of a growing tumour is the heterogeneous nature of cancerous tissue; see [BGL16] for several manifestations of this problem. This results in
particular challenges when one attempts to incorporate spatial structure into a model. Motivated by the challenges of cancer modelling, but with wider applications in biology,
our work will aim to provide a framework through which to model the expansion of heterogeneous populations.
Novelty of the Research Methodology: The ultimate goal is to describe the genealogical relationships between cells in a growing tumour. Previous work in this direction includes
[DF15] and [Dur18], both of which build on a 'discrete' mathematical model (the Williams-Bjerknes, or biased voter model). 'Discrete' here means that locally there can only be
a finite number of cells, but in fact both papers rely on a 'high-density' approximation to derive their results. In this step they encounter a very familiar challenge, well known
in the population genetics literature in the context of what is often referred to as 'the pain in the torus'. Some of the limitations of their approach stem from taking the high
intensity limit.
There are various ways to bypass the pain in the torus. One is to resolutely stick with modelling populations restricted to live in a discrete space. Another, which motivates
us here, is to replace the discrete individual-based model by an 'event-based' one. The resulting model (or really framework for modelling) is known as the 'Spatial Lambda
Fleming-Viot' process (SLFV), first introduced in [Eth08] and formally constructed in [BEV10]. We shall develop the SLFV framework to incorporate features relevant to the
growth and spatial heterogeneity of cancerous tissue.
Aims and Objectives: Much of the research done in the case of the Williams-Bjerknes model is yet to be replicated in the SLFV framework. As a first step, we need to quantify
the speed of tumour growth. We shall investigate the classical derivations [BG80], [BG81] and work to derive similar results in the SLFV case. It is worth remarking that the bulk
of the work on the SLFV to date has concentrated on populations distributed in a two dimensional space, whereas now we must move to three dimensions, where the behaviour
will be qualitatively different.
After quantifying the speed of growth, we shall turn to establishing the genealogical relationships between individuals in the population -cancer cells in the application that
we have in mind- depending on their physical location within the tumour.
Alignment to EPSRC's Strategies and Research Areas: This project spans the EPSRC Statistics and Applied Probability, Mathematical Biology and Mathematical Analysis research areas.
References:
[BEV10] N. H. Barton, A. M. Etheridge, and A. Véber. A new model for evolution in a spatial continuum. Electron. J. Probab., 15:no. 7, 162-216, 2010.
[BG80] Maury Bramson and David Griffeath. On the Williams-Bjerknes tumour growth model. II. Math. Proc. Cambridge Philos. Soc., 88(2):339-357, 1980.
[BG81] Maury Bramson and David Griffeath. On the Williams-Bjerknes tumour growth model. I. Ann. Probab., 9(2):173-185, 1981.
[BGL16] Niko Beerenwinkel, Chris D. Greenman, and Jens Lagergren. Computational cancer biology: An evolutionary perspective. PLOS Computational Biology,12(2):1-12, 02 2016.
[DF15] Rick Durrett and Wai-Tong Louis Fan. Genealogies in Expanding Populations. arXiv e-prints, page arXiv:1507.00918, July 2015.
[Dur18] Rick Durrett. Genealogies in growing solid tumors. bioRxiv, 2018.
[Eth08] Alison M. Etheridge. Drift, draft and st

Publications

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Studentship Projects

Project Reference Relationship Related To Start End Student Name
EP/R513295/1 01/10/2018 30/09/2023
2443958 Studentship EP/R513295/1 01/10/2020 30/09/2023 Adrian Martini