The application of probability-generating functions to linear–quadratic radiation survival curves

Abstract

Purpose : To illustrate how probability-generating functions (PGFs) can be employed to derive a simple probabilistic model for clonogenic survival after exposure to ionizing irradiation. Methods : Both repairable and irreparable radiation damage to DNA were assumed to occur by independent (Poisson) processes, at intensities proportional to the irradiation dose. Also, repairable damage was assumed to be either repaired or further (lethally) injured according to a third (Bernoulli) process, with the probability of lethal conversion being directly proportional to dose. Using the algebra of PGFs, these three processes were combined to yield a composite PGF that described the distribution of lethal DNA lesions in irradiated cells. Results : The composite PGF characterized a Poisson distribution with mean, alphaD + betaD 2, where D was dose and alpha and beta were radiobiological constants. This distribution yielded the conventional linear–quadratic survival equation. To test the composite model, the derived distribution was used to predict the frequencies of multiple chromosomal aberrations in irradiated human lymphocytes. The predictions agreed well with observation. This probabilistic model was consistent with single-hit mechanisms, but it was not consistent with binary misrepair mechanisms. Conclusions : A stochastic model for radiation survival has been constructed from elementary PGFs that exactly yields the linear–quadratic relationship. This approach can be used to investigate other simple probabilistic survival models.