Travelling wave solutions of the reaction-diffusion mathematical model of glioblastoma growth: An Abel equation based approach

Tiberiu Harko, M. K. Mak

(Submitted on 2 Sep 2014)

We consider quasi-stationary (travelling wave type) solutions to a nonlinear reaction-diffusion equation with arbitrary, autonomous coefficients, describing the evolution of glioblastomas, aggressive primary brain tumors that are characterized by extensive infiltration into the brain and are highly resistant to treatment. The second order nonlinear equation describing the glioblastoma growth through travelling waves can be reduced to a first order Abel type equation. By using the integrability conditions for the Abel equation several classes of exact travelling wave solutions of the general reaction-diffusion equation that describes glioblastoma growth are obtained, corresponding to different forms of the product of the diffusion and reaction functions. The solutions are obtained by using the Chiellini lemma and the Lemke transformation, respectively, and the corresponding equations represent generalizations of the classical Fisher--Kolmogorov equation. The biological implications of two classes of solutions are also investigated by using both numerical and semi-analytical methods for realistic values of the biological parameters.

link: http://arxiv.org/abs/1409.0605

The second preprint to appear on Warburg's Lens is by +Philip Gerlee, one of the contributors of the site. The paper contains a 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 the paper the PDE-system is analysed and the following things are shown:

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 this system.

Travelling wave analysis of a mathematical model of glioblastoma growth

Philip Gerlee, Sven Nelander

(Submitted on 22 May 2013)

In this paper we analyse a previously proposed cell-based model of glioblastoma (brain tumour) growth, which is based on the assumption that the cancer cells switch phenotypes between a proliferative and motile state (Gerlee and Nelander, PLoS Comp. Bio., 8(6) 2012). The dynamics of this model can be described by a system of partial differential equations, which exhibits travelling wave solutions whose wave speed depends crucially on the rates of phenotypic switching. We show that under certain conditions on the model parameters, a closed form expression of the wave speed can be obtained, and using singular perturbation methods we also derive an approximate expression of the wave front shape. These new analytical results agree with simulations of the cell-based model, and importantly show that the inverse relationship between wave front steepness and speed observed for the Fisher equation no longer holds when phenotypic switching is considered.

Originally posted on +Philip Gerlee 's personal blog.