Dose timing in tumor radiotherapy: considerations of cell number stochasticity.

A typical tumor radiotherapy regimen using external beam X rays consists of doses on weekdays for 4-7 weeks. During the final weeks, the tumor may contain only a few cells capable of regenerating the tumor and may be growing exponentially between doses. Stochastic fluctuations of the cell number can influence the optimal time pattern of dose delivery. If the total dose is fixed, a deterministic model of exponential tumor growth, neglecting stochastic effects, predicts that the way the radiation dose is spread out in time does not affect the average number of tumor cells at the end. However, we here show, within the framework of a birth-death model, that when stochastics are taken into account, the earlier the dose is given (consistent with other constraints imposed by quite different considerations), the better. The proof uses a transformation that simplifies the characteristic equation of the partial differential equation governing the probability generating function for a birth-death process with time-dependent rates. The theorem that earlier is better holds for any statistical distribution of cell number from patient to patient at the start of the exponential growth phase and for virtually any cell-killing model. Numerical results indicate the stochastic effects, although not dominant, are not negligible.