Usually tumors are treated with a fractionated course of irradiation that allows cell proliferation to occur during the intervals between successive fractions of the radiation dose. How well is the number of clonogens surviving such a regimen of radiotherapy described by a Poisson distribution? The problem was posed exactly in this formulation by Tucker et al. (1). I think this is a very interesting and potentially important paper, because until it was published there had been no doubt that the Poisson statistic is appropriate for the description of the number of clonogens in treated tumors (2-6). To highlight the problem formulated above, the authors conducted computer simulations. To describe briefly the design of their study, we will use the following notation: No = initial number of clonogens (assumed to be nonrandom); p = probability of clonogen survival after one fraction of the radiation dose; n = total number of dose fractions separated by equal intervals; a = probability of cell division during the interval between two successive fractions. Tucker et al. proceeded from the assumption that the probability p is the same for all clonogenic cells and for each fractional dose, and that it is also independent of the radiation dose given before. The same was assumed for the division probability a. The asynchronous transition of cells to the proliferative state and the temporal organization of the mitotic cycle were disregarded in the simple version of the model used by Tucker et al. In other words, cell proliferation activity was determined solely by the value of a. No significant discrepancies were observed when the results were compared with those obtained using a more complicated model in which an age-dependent probability of cell division was taken into account. Therefore, we will dwell on the first part of this simulation study. A large number of tumors, each containing the same initial number, No, of clonogenic cells, were simulated; the regimen of fractionated irradiation was uniform, i.e., D, = D2 = ... = D,. A typical output of these simulations was the frequency of tumor cure as a function of the total dose given in equal fractions. In the first series of simulation exp riments, studying the case where a = 0 (no cell proliferation during treatment), the authors observed an excellent agreement with the predictions based on Poisson distribution. We shall show that this result is, in fact, a general one and can be obtained easily without computer assistance. In what follows we shall use the methods and terminology of the theory of branching stochastic processes. A reader who is not familiar with the fundamentals of this theory is referred to Chapter 2 of the book by Jagers (7) or Chapter 1 of the book by Yakovlev and Yanev (8). Let Z, be he number of clonogens surviving t successive fractions of radiation dose. The random variable z, may be thought of as a Galton-Watson branching process with probabilitygenerating function 4(s) for the number of offspring in every generation t. The function 4(s) is also called the reproduction-generating function. If we proceed from the same assumptions as those used in the simulation model, the reproduction-generating function for t = 1, ., n is specified by the following expression: