Recent models for organism and tumor growth yield simple scaling laws based on conservation of energy. Here, we extend such a model to include spatial dependence to model necrotic core formation. We adopt the allometric equation for tumor volume with a reaction-diffusion equation for nutrient concentration. In addition, we assume that the total metabolic energy and average cellular metabolic rate depend on nutrient concentration in a Michaelis-Menten-like manner. From experimental results, we relate the necrotic volume to nutrient consumption and estimate both the time and nutrient concentration at necrotic core formation. Based on experimental results, we demand that the necrotic core radius varies linearly with tumor radius after core formation and extend the equations for tumor volume and nutrient concentration to the postnecrotic core regime. In particular, we obtain excellent agreement with experimental data and the final steady-state viable rim thickness.
Peer Reviewed
http://deepblue.lib.umich.edu/bitstream/2027.42/87333/2/124701_1.pdf