Tumor cell and microvessel densities during the growth of a brain tumor: A theoretical study

Abstract

Mathematical model for the tumor growth incorporating energy supply and requirement, angiogenesis efficiency and effect of elasticity of adjacent normal tissue to understand tumor biology and predict saturation status is rare to find. This study is conducted to address these issues. We propose mathematical expressions to explain alterations of tumor cell density (nT), microvessel density (MVD), and growth rate(r) during the development of brain tumors. We assume that nT increases during the growth of the tumor due to the increase of external pressure from the initial cell density (nT0); nT0 is same as the external normal tissue. The rate of increase in tumor cells (dNT/dt) depends on the rate of energy available for the creation of new cells and the energy required for a single cell division(γ). Due to the increase of tumor cell density, hypoxia is developed, which up-regulates the secretion of vascular endothelial growth factor (VEGF) and new capillaries are generated. Therefore, the surface area density of capillaries (Acs) in tumors increases. Hence, we consider that Acs(t) ∞ nT(t). A modified logistic equation is developed. Temporal variations of nT(t), Acs(t), r(t) and tumor cell population ‘NT(t)’ are examined. The expressions of saturated cell density(nTM), saturated microvessel surface area density (AcsM) and tumor saturation time(Ts) are formulated. An important feature, tumor saturation factor ‘fTS’ is determined. When fTS<1, a tumor will saturate at Ts, and nTM depends solely on fTS.