Tumor Control Probability Models

Abstract

Cancer is one of the major causes of death in the world. In the field of Oncology, clinical trials form the crux of medical effort to find better treatment schedules. These trials are expensive, time consuming, and carry great risks for the patients involved. Mathematical models provide a complimentary, non-invasive tool in the development of improved treatments. Examples of such modeling efforts are the tumor control probability (TCP), used to measure the probability of tumor cell eradication; the cumulative radiation effect (CRE) and the normal tissue complication probability (NTCP) model, used for quantifying normal tissue complication. In this thesis, I begin with a simple Poisson TCP based on mean cell population dynamics. Optimal treatment schedules are obtained by maximizing this TCP while constraining the CRE under a given threshold. Some of the optimal results suggest the usage of hyperfractionated treatments, which are applied in the treatment of prostate cancer. A TCP derived from a birth-death process is obtained to include stochastic effects. The Poisson TCP is suitable for larger tumors whereas new TCP is preferable for smaller ones. Furthermore, I also derive a NTCP model from a birth-death process. The calculation of this NTCP model provides an alternative proof to a formula derived by Hanin (Hanin, 2004) to compute the probability distribution of the tumor size from its generating function. My formula is computationally more efficient, compared to Hanin’s. Inspired by Ecology, I also studied a third TCP model derived from the first passage time problem. This problem has been used in animal movement to find the mean time for a predator to target a motionless prey. I applied this idea to the radiation treatment of tumor to find the mean time to reduce the tumor size to zero.