A simple analytic derivation suggests that prostate cancer α/β ratio is low

Abstract

Since the publication of Brenner and Hall’s recent paper (1) suggesting that the a/b ratio for prostate cancer is 1.5 Gy, a debate has been propagated through the radiotherapy community (2–5). This surprisingly low value carries implications of more efficient (and perhaps less costly) regimens of external beam radiotherapy (i.e., hypofractionated schedules) and optimal forms of brachytherapy (i.e., high dose rate). Should it prove to be true, such a low ratio would negate the therapeutic advantage of conventional fractionation regimens and call out for clinical trials. There is certainly indirect evidence for a low a/b ratio. The very small proportion of proliferating cells (6, 7), the very long observed doubling times (8, 9), the slow response to irradiation, and the recent interim results of high-dose-rate brachytherapy of varying fractionation (10) are all are consistent with such a low value. The surprise is that this value is as low as, if not even lower than, normal tissue response to irradiation. Can one apply the linear-quadratic (LQ) equation to outcomes of a population of clinically localized prostate cancer after radiotherapy? The LQ equation, although purely phenomenological, has certainly proven to be of great use, both in the study of cultured tumor cells irradiated in vitro, in comparing regimens of different fractionation, and in the study of normal tissue response to radiation. Although issues of interand intratumor heterogeneity are not directly accounted for in the standard LQ formalism, these issues might not necessarily affect the a/b ratio when applied to changes in fraction size (4). Whereas the a term, a direct measure of radiosensitivity, does depend upon heterogeneity (3) and is intimately tied to the number of clonogens, the ratio a/b might not. We present a simple analytical derivation of a/b that is a logical conclusion from the observation that, for clinically localized prostate cancer, external beam and permanent brachytherapy appear to achieve equivalent outcomes. This derivation makes no assumptions and does not attempt to fit models to specific clinical data. For fractionated external-beam irradiation to a total dose De, with dose per fraction d, the LQ survival fraction of clonogens can be written as follows: