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.

Modeling the Dichotomy of the Immune Response to Cancer: Cytotoxic Effects and Tumor-Promoting Inflammation

Modeling the Dichotomy of the Immune Response to Cancer: Cytotoxic Effects and Tumor-Promoting Inflammation
Kathleen P. Wilkie, Philip Hahnfeldt
 (Wed, 15 May 2013 20:58:57)

Although the immune response is often regarded as acting to suppress tumor growth, it is now clear that it can be both stimulatory and inhibitory. The interplay between these competing influences has complex implications for tumor development and cancer dormancy. To study this biological phenomenon theoretically we construct a minimally parameterized framework that incorporates all aspects of the immune response. We combine the effects of all immune cell types, general principles of self-limited logistic growth, and the physical process of inflammation into one quantitative setting. Simulations suggest that while there are pro-tumor or antitumor immunogenic responses characterized by larger or smaller final tumor volumes, respectively, each response involves an initial period where tumor growth is stimulated beyond that of growth without an immune response. The mathematical description is non-identifiable which allows us to capture inherent biological variability in tumor growth that can significantly alter tumor-immune dynamics and thus treatment success rates. The ability of this model to predict immunomodulation of tumor growth may offer a template for the design of novel treatment approaches that exploit immune response to improve tumor suppression, including the potential attainment of an immune-induced dormant state.