1D mathematical model of the dynamics of a glioma with discontinuous diffusion coefficient and variable carrying capacity
DOI:
https://doi.org/10.35819/remat2020v6i2id4067Keywords:
Mathematical Oncology, Glioma, Diffusion-Reaction Equation, Variable Carrying Capacity, Numerical MethodsAbstract
In this work, we will numerically solve the equation that models the problem of the growth dynamics of a glioma, with carrying capacity that varies spatially. Due to the diffusive nature of the glioma, the problem is modeled by the Reaction-Diffusion Equation (RDE). We will study the one-dimensional case (1D). The RDE has a Gaussian profile, as an initial condition, and a boundary condition of the Neumman type. The tumor microenvironment is a portion of the brain, consisting mainly of glioma cells. It has three regions: two regions of gray matter, located in the extreme part of the microenvironment, and a region of white substance, located in the middle of the microenvironment. Two important facts characterize the modeling of this problem. First, the diffusion coefficient is a discontinuous function, and second, the carrying capacity, in the logistic growth model, is a Hill-type function that depends on the spatial variable. The problem is solved numerically by the Crank-Nicolson method, and the numerical results indicate a decrease in tumor growth when considering the variable carrying capacity.
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