Nonlinear ghost waves accelerate the progression of high-grade brain tumors (original) (raw)
Related papers
Waves of cells with an unstable phenotype accelerate the progression of high-grade brain tumors
2014
In this paper we study a reduced continuous model describing the local evolution of high grade gliomas-a lethal type of primary brain tumorthrough the interplay of different cellular phenotypes. We show how hypoxic events, even sporadic and/or limited in space may have a crucial role on the acceleration of the growth speed of high grade gliomas. Our modeling approach is based on two cellular phenotypes one of them being more migratory and the second one more proliferative with transitions between them being driven by the local oxygen values, assumed in this simple model to be uniform. Surprisingly even acute hypoxia events (i.e. very localized in time) leading to the appearance of migratory populations have the potential of accelerating the invasion speed of the proliferative phenotype up to speeds close to those of the migratory phenotype. The high invasion speed of the tumor persists for times much longer than the lifetime of the hypoxic event and the phenomenon is observed both when the migratory cells form a persistent wave of cells located on the invasion front and when they form a evanecent wave dissapearing after a short time by decay into the more proliferative phenotype. Our findings are obtained through numerical simulations of the model equations. We also provide a deeper mathematical analysis of some aspects of the problem such as the conditions for the existence of persistent waves of cells with a more migratory phenotype.
Traveling Waves of a Go-or-Grow Model of Glioma Growth
SIAM Journal on Applied Mathematics
Glioblastoma multiforme is a deadly brain cancer in which tumor cells excessively proliferate and migrate. The first mathematical models of the spread of gliomas featured reactiondiffusion equations, and later an idea emerged through experimental study called the "Go or Grow" hypothesis in which glioma cells have a dichotomous behavior: a cell either primarily proliferates or primarily migrates. We analytically investigate an extreme form of the "Go or Grow" hypothesis where tumor cell motility and cell proliferation are considered as separate processes. Different solution types are examined via approximate solution of traveling wave equations, and we determine conditions for various wave front forms.
Density-Dependent Diffusion Model of Glioblastoma Growth
2014
Glioblastoma multiforme is an aggressive brain cancer that is extremely fatal. It is characterized by both proliferation and large amounts of migration, which contributes to the difficulty of treatment. Previous models of this type of cancer growth often include two separate equations to model proliferation or migration. We propose a single equation which uses densitydependent diffusion to capture the behavior of both proliferation and migration. We analyze the model to determine the existence of traveling wave solutions. To prove the viability of the density-dependent diffusion function chosen, we compare our model with well-known in vitro experimental data.
Mathematical Biosciences and Engineering, 2020
Mathematical modeling for cancerous disease has attracted increasing attention from the researchers around the world. Being an effective tool, it helps to describe the processes that happen to the tumour as the diverse treatment scenarios. In this paper, a density-dependent reaction-diffusion equation is applied to the most aggressive type of brain cancer, Glioblastoma multiforme. The model contains the terms responsible for the growth, migration and proliferation of the malignant tumour. The traveling wave solution used is justified by stability analysis. Numerical simulation of the model is provided and the results are compared with the experimental data obtained from the reference papers.
A data-motivated density-dependent diffusion model of in vitro glioblastoma growth
Mathematical Biosciences and Engineering, 2015
Glioblastoma multiforme is an aggressive brain cancer that is extremely fatal. It is characterized by both proliferation and large amounts of migration, which contributes to the difficulty of treatment. Previous models of this type of cancer growth often include two separate equations to model proliferation or migration. We propose a single equation which uses densitydependent diffusion to capture the behavior of both proliferation and migration. We analyze the model to determine the existence of traveling wave solutions. To prove the viability of the density-dependent diffusion function chosen, we compare our model with well-known in vitro experimental data.
The Evolution of Mathematical Modeling of Glioma Proliferation and Invasion
Journal of Neuropathology and Experimental Neurology, 2007
Gliomas are well known for their potential for aggressive proliferation as well as their diffuse invasion of the normalappearing parenchyma peripheral to the bulk lesion. This review presents a history of the use of mathematical modeling in the study of the proliferative-invasive growth of gliomas, illustrating the progress made in understanding the in vivo dynamics of invasion and proliferation of tumor cells. Mathematical modeling is based on a sequence of observation, speculation, development of hypotheses to be tested, and comparisons between theory and reality. These mathematical investigations, iteratively compared with experimental and clinical work, demonstrate the essential relationship between experimental and theoretical approaches. Together, these efforts have extended our knowledge and insight into in vivo brain tumor growth dynamics that should enhance current diagnoses and treatments.
Mathematical Biosciences and Engineering, 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.
The biology and mathematical modelling of glioma invasion: a review
Adult gliomas are aggressive brain tumours associated with low patient survival rates and limited life expectancy. The most important hallmark of this type of tumour is its invasive behaviour, characterized by a markedly pheno-typic plasticity, infiltrative tumour morphologies and the ability of malignant progression from low-to high-grade tumour types. Indeed, the widespread infiltration of healthy brain tissue by glioma cells is largely responsible for poor prognosis and the difficulty of finding curative therapies. Meanwhile, mathematical models have been established to analyse potential mechanisms of glioma invasion. In this review, we start with a brief introduction to current biological knowledge about glioma invasion, and then critically review and highlight future challenges for mathematical models of glioma invasion.
RESEARCH Open Access Mathematical modelling of spatio-temporal glioma evolution
2014
Full list of author information is available at the end of the article Background: Gliomas are the most common types of brain cancer, well known for their aggressive proliferation and the invasive behavior leading to a high mortality rate. Several mathematical models have been developed for identifying the interactions between glioma cells and tissue microenvironment, which play an important role in the mechanism of the tumor formation and progression. Methods: Building and expanding on existing approaches, this paper develops a continuous three-dimensional model of avascular glioma spatio-temporal evolution. The proposed spherical model incorporates the interactions between the populations of four different glioma cell phenotypes (proliferative, hypoxic, hypoglychemic and necrotic) and their tissue microenvironment, in order to investigate how they affect tumor growth and invasion in an isotropic and homogeneous medium. The model includes two key variables involved in the prolifera...