A THREE-DIMENSIONAL ANGIOGENESIS MODEL WITH TIME-DELAY

In this paper, we study the effect of time delays on the dynamics of angiogenesis tumor growth. The delays represent the time taken for the mitosis. The model is based on the reaction-diffusion equation described in the form of concentration of nutrients sigma and the distribution of internal pressure p caused by movement of cells. The vasculature supplies nutrients to the tumor, so that & part;sigma/& part; n +beta(sigma - (sigma) over bar ) = 0 holds on the boundary, where a positive constant beta is the rate of nutrient supply to the tumor and (sigma) over bar is the nutrient concentration outside the tumor. A parameter mu in the model expresses the "aggressiveness " of the tumor. It is proved that under non-radially symmetric perturbations, there exists a mu* > 0 such that the stationary solution is linearly stable for mu < mu*, and is linearly unstable for mu > mu*. Moreover, we also found that adding time delay to the model would lead to a large stationary tumor. The bigger the tumor proliferation intensity mu and angiogenesis intensity beta are, the greater the effect of time delay on tumor size.