A mathematical model for self-limiting brain tumors.

It is puzzling that certain brain tumors exhibit arrested exponential growth. We have observed in pediatric low-grade astrocytomas (LGA) at a certain volume approximately 100-150 cm(3) that the tumor ceases to grow. This observation led us to develop a macroscopic mathematical model for LGA growth kinetics that assumes the flow through the surface of the astrocytoma of a triggering agent or "promoter" that is uniformly distributed throughout the tumor, thereby providing relatively homogeneous growth. The model relates the transport of the promoter by the electrochemical potential associated with the tumor and diffusion effects through the surface of the growth and its consumption throughout the tumor volume via a pair of ordinary differential equations. The model assumes a constant growth rate, if the promoter density is above some threshold, and is zero otherwise. We also develop equations for an electrochemical (Nernst) transport mechanism for the promoter, and describes the microscopic basis for the macroscopic evolution to the equilibrium state at a well-defined and universal size. The latter description is unstable to asymmetric perturbations and provides a "star-like" shape for emergent tumors and a spheroidal shape for fully developed ones. The underlying assumption in our hypothesis would also result in tumor growth remission beginning from the periphery and proceeding inwards, a feature that has now been validated clinically.