16570

16570

بررسي عددي دسته اي از مدل هاي رياضي نفوذ سلول هاي سرطاني در مغز

كارشناسي ارشد

رياضي كاربردي - آناليز عددي

شهريور 1395

دكتر مرتضي گرشاسبي

دكتر جواد وحيدي

رياضي

1395/11/19

1/1/1900 12:00:00 AM

Abstract: In this thesis, we numerically solve an equation modeling the evolution of the density of glioma in the brain- a mathematical modelling of evolution of the density of Glioma is investigated numerically. 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 an​d that the coefficient of diffusion can vary according to the gray an​d white matter composition of the brain at that location. The analysis provided in this thesis 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 an​d investigate numerically the behavior of the evolution of tumor concentration of the glioma, we calculated an​d plotted the number of tumor cells, the average mean radial distance, an​d the speed of the tumor cells along with charting the effects of net dispersal rate an​d net proliferation rate terms versus time for different center position values of Gaussian initial profile for each zone (gray an​d white matter tissues). Two numerical methods is used: the implicit backward Euler an​d the averaging in time an​d 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 an​d mesh-discretization. The Crank–Nicolson implicit method is shown to be the better choice to solve the problem. Keywords: Cancer tumor, Diffusion coefficient, Mathematical models, Numerical methods, Crank-Nicolson method, Implicit backward Euler method.