One of the fundamental tools in radiation biology is a formalism describing time-dose relationships. For example, there is a need for reliable predictions of radiotherapeutic isoeffect doses when the temporal exposure pattern is changed. The most commonly used tool is now the linear-quadratic (LQ) formalism, which describes fractionation and dose-protraction effects through a particular functional form, the generalized Lea-Catcheside time factor, G. We investigate the relationship of the LQ formalism to those describing other commonly discussed radiobiological models in terms of their predicted time-dose relationships. We show that a broad range of radiobiological models are described by formalisms in which a perturbation calculation produces the standard LQ relationship for dose fractionation/protraction, including the same generalized time factor, G. This approximate equivalence holds not only for the formalisms describing binary misrepair models, which are conceptually similar to LQ, but also for formalisms describing models embodying a very different explanation for time-dose effects, namely saturation of repair capacity. In terms of applications to radiotherapy, we show that a typical saturable repair formalism predicts practically the same dependences for protraction effects as does the LQ formalism, at clinically relevant doses per fraction. For low-dose-rate exposure, the same equivalence between predictions holds for early-responding end points such as tumor control, but less so for late-responding end points. Overall, use of the LQ formalism to predict dose-time relationships is a notably robust procedure, depending less than previously thought on knowledge of detailed biophysical mechanisms, since various conceptually different biophysical models lead, in a reasonable approximation, to the LQ relationship including the standard form of the generalized time factor, G.

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