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

Combined therapies of antithrombotics and antioxidants delay in silico brain tumor progression

Alicia Martínez-González, Mario Durán-Prado, Gabriel F. Calvo, Francisco J. Alcaín, Luis A. Pérez-Romasanta, Víctor M. Pérez-García

(Submitted on 10 Jan 2014)

Glioblastoma multiforme, the most frequent type of primary brain tumor, is a rapidly evolving and spatially heterogeneous high-grade astrocytoma that presents areas of necrosis, hypercellularity and microvascular hyperplasia. The aberrant vasculature leads to hypoxic areas and results in an increase of the oxidative stress selecting for more invasive tumor cell phenotypes. In our study we assay in silico different therapeutic approaches which combine antithrombotics, antioxidants and standard radiotherapy. To do so, we have developed a biocomputational model of glioblastoma multiforme that incorporates the spatio-temporal interplay among two glioma cell phenotypes corresponding to oxygenated and hypoxic cells, a necrotic core and the local vasculature whose response evolves with tumor progression. Our numerical simulations predict that suitable combinations of antithrombotics and antioxidants may diminish, in a synergetic way, oxidative stress and the subsequent hypoxic response. This novel therapeutical strategy, with potentially low or no toxicity, might reduce tumor invasion and further sensitize glioblastoma multiforme to conventional radiotherapy or other cytotoxic agents, hopefully increasing median patient overall survival time.

http://arxiv.org/abs/1401.2397

Tumour Control Probability in Cancer Stem Cells Hypothesis

Tumour Control Probability in Cancer Stem 

Cells Hypothesis

A. Dhawan, M. Kohandel, R. P. Hill, S. Sivaloganathan

(Submitted on 19 Dec 2013)

The tumour control probability (TCP) is a formalism derived to compare various treatment regimens of radiation therapy, defined as the probability that given a prescribed dose of radiation, a tumour has been eradicated or controlled. In the traditional view of cancer, all cells share the ability to divide without limit and thus have the potential to generate a malignant tumour. However, an emerging notion is that only a sub-population of cells, the so-called cancer stem cells (CSCs), are responsible for the initiation and maintenance of the tumour. A key implication of the CSC hypothesis is that these cells must be eradicated to achieve cures, thus we define TCP_S as the probability of eradicating CSCs for a given dose of radiation. A cell surface protein expression profile, such as CD44high/CD24low for breast cancer, is often used as a biomarker to monitor CSCs enrichment. However, it is increasingly recognized that not all cells bearing this expression profile are necessarily CSCs, and in particular early generations of progenitor cells may share the same phenotype. Thus, due to the lack of a perfect biomarker for CSCs, we also define a novel measurable TCP_CD+, that is the probability of eliminating or controlling biomarker positive cells. Based on these definitions, we use stochastic methods and numerical simulations to compare the theoretical TCP_S and the measurable TCP_CD+. We also use the measurable TCP to compare the effect of various radiation protocols.

link: http://arxiv.org/abs/1312.5582

Radiotherapy planning for glioblastoma based on a tumor growth model: Improving target volume delineation

Jan Unkelbach, Bjoern H. Menze, Ender Konukoglu, Florian Dittmann, Matthieu Le, Nicholas Ayache, Helen A. Shih

Glioblastoma di ffer from many other tumors in the sense that they grow in filtratively into the brain tissue instead of forming a solid tumor mass with a de fined boundary. Only the part of the tumor with high tumor cell density can be localized through imaging directly. In contrast, brain tissue infi ltrated by tumor cells at low density appears normal on current imaging modalities. In current clinical practice, a uniform margin, typically two centimeters, is applied to account for microscopic spread of disease that is not directly assessable through imaging. The current treatment planning procedure can potentially be improved by accounting for the anisotropy of tumor growth, which arises from di fferent factors: Anatomical barriers such as the falx cerebri represent boundaries for migrating tumor cells. In addition, tumor cells primarily spread in white matter and in lfiltrate gray matter at lower rate. We investigate the use of a phenomenological tumor growth model for treatment planning. The model is based on the Fisher-Kolmogorov equation, which formalizes these growth characteristics and estimates the spatial distribution of tumor cells in normal appearing regions of the brain. The target volume for radiotherapy planning can be defi ned as an isoline of the simulated tumor cell density. This paper analyzes the model with respect to implications for target volume de finition and identifi es its most critical components. A retrospective study involving 10 glioblastoma patients treated at our institution has been performed.

To illustrate the main fi ndings of the study, a detailed case study is presented for a glioblastoma located close to the falx. In this situation, the falx represents a boundary for migrating tumor cells, whereas the corpus callosum provides a route for the tumor to spread to the contralateral hemisphere. We further discuss the sensitivity of the model with respect to the input parameters. Correct segmentation of the brain appears to be the most crucial model input. We conclude that the tumor growth model provides a method to account for anisotropic growth patterns of glioma, and may therefore provide a tool to make target delineation more objective and automated.

Link: http://arxiv.org/pdf/1311.5902.pdf

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.