چكيده به لاتين
Abstract
In this thesis, we study a mathematical model for the tumor vascularization
theory of tumor growth.Migration and proliferation of endothelial cells in
performed and newly formed blood vessels during tumor angiogenesis.In the
simplest version of this model, an avascular tumor secretes a tumor growth
factor (TGF) which is transported across an extracellular matrix (ECM) to
a neighboring vasculature where it stimulates endothelial cells to produce a
protease that acts as a catalyst to degrade the fibronectin of the capillary
wall and the ECM. The endothelial cells then move up the TGF gradient
back to the tumor, proliferating and forming a new capillary network. In the
model presented here, we include two mechanisms for the action of angiostatin.
In the first mechanism, substantiated experimentally, the angiostatin
acts as a protease inhibitor.A second mechanism for the production of protease
inhibitor from angiostatin by endothelial cells is proposed to be of
Michaelis–Menten type. Mathematically, this mechanism includes the former
as a subcase.
Our model is different from other attempts to model the process of tumor
angiogenesis in that it focuses (1) on the biochemistry of the process at the
level of the cell; (2) the movement of the cells is based on the theory of reinforced
random walks; (3) standard transport equations for the diffusion of
molecular species in porous media.
One consequence of our numerical simulations is that we obtain very good
computational agreement with the time of the oneset of vascularization and
the rate of capillary tip growth observed in rabbit cornea experiments.
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Finally an analysisis performed to find the numerical solution of a mathematical
model for capillary formationin tumor angiogenesis.Firstly, a time stepping
approach is employed for the time derivative,then a mesh free process
based on a global collocation method using the radial basis functions(RBFs)is
applied for solving the problem. Stability analysis of the method is investigated.
Because of non-availability of the exact solutions , efficiency and
accuracy of the method is demonstrated, by comparison with existing methods.
Also the method is successfully applied for solving the problem with high
values of the cell diffusion constant ,which many of the available methods are
not applicable for solving these cases.
Keywords: Tumor angiogenesis, Endothelial cells, Inhibitor, Radial basis
functions
