Comparison of fractionation schedules in the large heterogeneity limit

The author develops an analytical formula that relates total doses in different fractionation schemes using the linear-quadratic (LQ) model in the limit when the interpatient variation in radiosensitivity is large. We use published formulas for tumor control probability and parameter dependence developed for the large heterogeneity limit. We consider the isoeffect equation and uses simple algebraic manipulation to obtain the new fractionation formula valid for the large heterogeneity limit. We study the case of going from a dose per fraction of 1.8 Gy to a dose per fraction of 3 Gy, which is relevant for current hypofractionation schemes for prostate and breast cancer. We find that the total dose for the new fractionation depends not only on the alpha/beta ratio r but also on the ratio of the corresponding standard deviations sigma(alpha)/sigma(beta) and on the ratio between the original dose D1 and D50. In some special cases, notably when D1 = D50(1), the standard LQ model is recovered. In general, when the relative widths of the distributions for alpha and beta are equal or similar, the deviation from the standard LQ model is small, mostly within 5% for D1/D50(1) from 0.4 up to 1.6. When the width of the distribution for alpha is much larger than the one for beta, larger deviations from the standard LQ model are found for small values of r but they fall within 10% for r larger than 4 Gy and within 5% for r larger than 10 Gy. The most unforgiving case is when the width of the distribution for alpha is much smaller than the one for beta, where the largest differences with the standard LQ model are found. While, in general, the formula can give a significantly different result than the standard LQ model equation, we find a range of parameters for which the difference is surprisingly small.