Density-Dependent Diffusion Model of Glioblastoma Growth (original) (raw)
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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.
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.
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.
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.
An Explicit Boundary Condition Treatment of a Diffusion – Based Glioblastoma Tumor Growth Model
A careful boundary condition handling is a sine qua non prerequisite for a reliable diffusion based solution to the problem of clinical tumor growth and in particular glioma progression. However, to the best of our knowledge no explicit treatment of the numerical application of boundary conditions in this context has appeared in the literature as yet. Therefore, the aim of this paper is to outline a detailed numerical handling of the boundary conditions imposed by the presence of the skull in the case of glioblastoma multiforme (GBM), a highly diffusive and invasive brain tumor.
Acta Neurologica Belgica, 2018
Glioblastoma is known to be among one of the deadliest brain tumors in the world today. There have been major improvements in the detection of cancerous cells in the twenty-first century. However, the threshold of detection of these cancerous cells varies in different scanning techniques such as magnetic resonance imaging (MRI) and computed tomography (CT). The growth of these tumors and different treatments have been modeled to assist medical experts in better predictions of the related tumor growth and in the selection of more accurate treatments. In clinical terms the tumor consisted of two parts known as the visible part, which is the part of the tumor that is above the threshold of the detecting device and the invisible part, which is below the detecting threshold. In this study, the common reaction-diffusion model of tumor growth is used to simulate the growth of the glioblastoma tumor. Also resection and radiotherapy have been modeled as methods to prevent the growth of the tumor. The results demonstrate that although the selected treatments were effective in reducing the number of cancerous cells to under the threshold of detection, they did not eliminate all cancerous cells and if no further treatments were applied, the cancerous cells would spread and become malignant again. Although previous studies have suggested that the ratio of proliferation to diffusion could describe the malignancy of the tumor, this study in addition shows the importance of each of the coefficients regarding the malignancy of the tumor.
In this article, we numerically solve an equation modeling the evolution of the density of glioma in the brain-the most malignant form of brain tumor quantified in terms of net rates of proliferation and invasion. We employ a non-linear heterogeneous diffusion logistic density model. This model assumes that glioma cell invasion throughout the brain is a reaction-diffusion process and that the coefficient of diffusion can vary according to the gray and white matter composition of the brain at that location. The analysis provided in this article demonstrates that using the correct finite difference scheme can overcome the stability issues caused by the discontinuities of the diffusion coefficient. We also observe that at the steady-state these discontinuities vanish. To visualize and investigate numerically the behavior of the evolution of tumor concentration of the glioma, we calculated and plotted the number of tumor cells, the average mean radial distance, and the speed of the tumor cells along with charting the effects of net dispersal rate and net proliferation rate terms versus time for different center position values of Gaussian initial profile for each zone (gray and white matter tissues). We have proposed two numerical methods, the implicit backward Euler and the averaging in time and forward differences in space (the Crank-Nicolson scheme), both in combination with Newton's method for solving the governing equations. These methods are compared in terms of their performance in varying time-step and mesh-discretization. The Crank-Nicolson implicit method is shown to be the better choice to solve the equation.
Existence of Traveling Wave Solutions for a Model of Tumor Invasion
Siam Journal on Applied Dynamical Systems, 2014
The existence of travelling wave solutions to a haptotaxis dominated model is analysed. A version of this model has been derived in Perumpanani et al. (1999) to describe tumour invasion, where diffusion is neglected as it is assumed to play only a small role in the cell migration. By instead allowing diffusion to be small, we reformulate the model as a singular perturbation problem, which can then be analysed using geometric singular perturbation theory. We prove the existence of three types of physically realistic travelling wave solutions in the case of small diffusion. These solutions reduce to the no diffusion solutions in the singular limit as diffusion as is taken to zero. A fourth travelling wave solution is also shown to exist, but that is physically unrealistic as it has a component with negative cell population. The numerical stability, in particular the wavespeed of the travelling wave solutions is also discussed.