Fractionation in normal tissues: the (alpha/beta)(eff) concept can account for dose heterogeneity and volume effects

The simple Linear-Quadratic (LQ)-based Withers iso-effect formula (WIF) is widely used in external-beam radiotherapy to derive a new tumour dose prescription such that there is normal-tissue (NT) iso-effect when changing the fraction size and/or number. However, as conventionally applied, the WIF is invalid unless the normal-tissue response is solely determined by the tumour dose. We propose a generalized WIF (gWIF) which retains the tumour prescription dose, but replaces the intrinsic fractionation sensitivity measure (alpha/beta) by a new concept, the normal-tissue effective fractionation sensitivity, (alpha/beta)(eff)(NT), which takes into account both the dose heterogeneity in, and the volume effect of, the late-responding normal-tissue in question. Closed-form analytical expressions for (alpha/beta)(eff)(NT) ensuring exact normal-tissue iso-effect are derived for: (i) uniform dose, and (ii) arbitrary dose distributions with volume-effect parameter n = 1 from the normal-tissue dose-volume histogram. For arbitrary dose distributions and arbitrary n, a numerical solution for (alpha/beta)(eff)(NT) exhibits a weak dependence on the number of fractions. As n is increased, (alpha/beta)(eff)(NT) increases from its intrinsic value at n = 0 (100% serial normal-tissue) to values close to or even exceeding the tumour (alpha/beta) at n = 1 (100% parallel normal-tissue), with the highest values of (alpha/beta)(eff)(NT) corresponding to the most conformal dose distributions. Applications of this new concept to inverse planning and to highly conformal modalities are discussed, as is the effect of possible deviations from LQ behaviour at large fraction sizes.