On the biologically effective dose (BED)-using convolution for calculating the effects of repair: I. Analytical considerations

This work presents a new mathematical formulation of biologically effective dose (BED) for radiation therapy where effects of repair need to be considered. The formulation is based on the observation that the effects of repair, both during protracted irradiation and of incomplete repair between fractions, can be written using a convolution, i.e. BED(T) = integral(infinity)(-infinity) R-T (t)dt + 1/alpha/beta integral(infinity)(-infinity) R-T (t) [R-T (t) [R-T (t) I (t)] dt, where T is the total irradiation time, RT (t) is the absorbed dose rate as a function of time t and I(t) is the function describing repair. To validate this formulation, the previously published expressions for instant and protracted irradiation are first summarized. Then, by analytical derivation, it is shown that the new formulation gives identical results. The calculation of BED can thus be treated within one single mathematical framework, applicable in external beam therapy, brachytherapy, radionuclide therapy, or a combination of these treatment modalities. Moreover, the new formulation allows for a straightforward incorporation of different repair models and has the advantage of being numerically applicable.