  We study the growth of an implanted a-vascular tumor seed in two-dimensions
based on a model where the mechanism of invasion is centered on tumor-induced
acidification of the micro-environment and consequent death of normal cells.
The spatial distribution of the acid density around the tumor is found using
mean-field analysis. By assuming that the viability of both normal and tumor
cells falls sharply below certain threshold values of the local pH, we
determine the conditions for the formation of a necrotic core at the center, as
well as its radius as a function of the tumor radius. We show that the mean
micro-vessel density (MVD) plays a pivotal role in determining the growth
characteristics of the tumor. When the MVD is sufficiently small, accumulation
of excess acid inside the tumor leads to the formation of a necrotic core,
which occupies a significant fraction of the total area in large tumors.
However, necrosis is reduced when the mean MVD inside the tumor is larger than
outside because of the more efficient removal of excess acid. At sufficiently
high MVD, necrosis is absent in the tumor, or confined to small regions mostly
devoid of micro-vessels. Quantitative estimates of MVD for these different
phases of growth are obtained, and verified using explicit cellular automaton
simulations. Recent experimental studies on the correlation between necrosis
and MVD support our main conclusions.
