A mathematical model of the immersion of a spherical tumor with a necrotic core into a nutrient bath

Abstract

Nutrients are essential for the development of tumors. Oxygen can be a determinant for necrosis and its absence can cause cell disintegration. In a similar manner, glucose plays a role in the development of necrosis, and also controls cellular viability. Tumors initially rely on diffusion for nutrient, oxygen, and waste transport. But after the growth of capillary sprouts that eventually penetrate the tumor (a process known as vascularization), it no longer depends on diffusion and grows uncontrollably. Eventually the nutrient concentration at the center of the tumor can fall below a critical level, resulting in a decrease in cell proliferation, slow growth rate and possibly even cell death. Thus, the tumor may develop what is known as a necrotic core, or inactive region in the center of the tumor. This paper focuses on the distribution of nutrient for a tumor with a necrotic core. Physically this models the effects of immersion of a tumor into a nutrient bath, or similarly the addition of nutrient to the tissue surrounding the tumor. The solution to the problem is analyzed in order to determine the most viable initial condition. The concentration as a function of time for fixed radial distance r will be investigated, as well as the concentration as a function of radial distance for fixed values of time t .