Acid-mediated tumor invasion: How does vasculature affect the growth   characteristics?
  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. 
Bindu. S. Govindan, W. B. Spillman, Jr., J. L. Robertson and W. R.   Huckle (Virginia Tech)