Mu and colleagues have studied the effect of prolongation of fraction delivery time using Chinese hamster fibroblasts (V79-379-A) by subdividing the total dose per fraction into a number equal sub-fractions separated by fixed time intervals. We agree with their general conclusions that the prolongation of fraction time may spare tissues with a fast DNA repair and that there is a risk for sparing tumors. We also agree that this should be considered when IMRT is applied in the clinic. When we read their paper in greater detail we asked ourselves two questions: first, why do the theoretical predictions given in their paper not match the observed experimental data? Secondly, can we provide a theoretical model using the same assumptions about incomplete repair as Mu et al. that does indeed match the data? We believe we have come up with answers to both questions. Which are, first that Mu et al. have used an inappropriate theoretical model to describe their experiment (see Eq. (2) below), and secondly that if one uses an appropriate theoretical model (see Eq. (5) below) this model predicts the experimental results obtained by Mu et al. very well indeed for doses per fraction used in conventional radiotherapy. To show these points we start by stating the problem, the Linear Quadratic formalism including incomplete repair between fractions states that the surviving fraction, SFnd, of cells after n fractions of dose d can be described as follows (Eq. (3) in the Mu et al. paper):(1)SFnd=exp(−nαd−Gnβd2)where G is the correction for incomplete repair between fractions. In what follows, we will use for G the expressions given by Mu et al. (Eqs. (4) and (5)) to describe the incomplete repair between fractions. If one sets n=1 in Eq. (1)(2)SFd=exp(−ad−Gβd2),as one would think is applicable for one fraction of dose d and calculates using Mu et al.'s data (α=0.16Gy−1,β=0.016Gy−2, an acute surviving fraction for a fraction of 2Gy delivered continuously over 1 min SF2=0.68, and an acute surviving fraction for a fraction of 8Gy delivered continuously over 4 min SF8=0.10) then one obtains the ratios of surviving fractions of prolonged irradiation to the surviving fraction for acute irradiation shown in the fourth and fifth column of Table 1 which are the same as the theoretical values given in Table 2 of Mu et al.'s paper. However, this is incorrect, since in their experimental design Mu et al. have chosen to subdivide each treatment fraction into m subfractions of dose Δ separated by a time interval δT, such that the dose per fraction is given by d=mΔ and the overall treatment time is given by T=(m−1)δT. Now using Eq. (2) for a subfraction of dose Δ we find:(3)SFΔδT=exp(−αΔ−GβΔ2),Then the surviving fraction of cells after delivering m subfractions separated by time intervals δT is simply given by:(4)SFmΔδT=(SFΔδT)m=exp(−αmΔ−GβmΔ2),Using the fact that the dose per fraction is given by d=mΔ, (Eq. (4)) can be rewritten as follows:(5)SFdδT=exp−α(mΔ)−Gβm(mΔ)2=exp−αd−Gβmd2Eq. (5) instead of Eq. (2), as used by Mu et al., represents the correct theoretical model to describe their experiment. Now making use of Eq. (5) and either a mono-exponential repair model with T1/2=0.4h or a bi-exponential repair model with a 60% fast component, T1/2=0.4h, and a 40% slow component, T1/2=4h, one obtains the ratios of surviving fractions of prolonged irradiation to the surviving fraction for acute irradiation shown in the sixth, seventh, eighth, and ninth column of Table 1, respectively. As one can see from Table 1 using Eq. (5) yields excellent agreement between theoretical predicted and experimental observed ratios of surviving fractions of prolonged irradiation to the surviving fraction for acute irradiation for conventional doses per fraction and significantly over predicts for large doses per fraction using either a mono-exponential or bi-exponential repair model. Therefore, the real conclusions of the study by Mu et al. are that the tumorcidal effect of a radiotherapy treatment schedule can be adversely affected significantly by prolonging the delivery time per fraction to 40 min as could be possible for some complex IMRT plans, and furthermore that this effect is well predicted by current biological repair models in the range of conventional doses per fraction.
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Jan 1, 2008
Jian Z Wang + 4
Open Access