Thursday, 23 May 2013

Travelling wave analysis of a mathematical model of glioblastoma growth

Spurred by recent discussions with +Jacob Scott about preprints in biology and fed up with the slow review process of some journals I've decided to upload my most recent paper on brain tumour modelling on arXiv (and continue to do so with future papers).

This paper is quite technical (at least by my standards) and contains the mathematical analysis of a model of glioblastoma growth that was published last year in PLoS Computational Biology. In this model the cancer cells switch between a proliferative and migratory phenotype, and it was previously shown that the dynamics of the cell-based model can be captured by two coupled partial differential equations, that exhibit (like the Fisher equation) travelling wave solutions. In this paper I have analysed this PDE-system and shown the following things:

1. With a couple of assumptions on model parameters one can obtain an analytical estimate of the wave speed.
2. In the limit of large and equal switching rates the wave speed equals that of the Fisher equation (which is what you'd expect).
3. Using perturbation techniques one can obtain an approximate solution to the shape of the expanding tumour.
4. In the Fisher equation the wave speed and the slope of the front are one to one (faster wave <--> less steep front). This property does not hold for our system.

Here's a link to the submission at arXiv.
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