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Comparison of fractionation schedules in the large heterogeneity limit
Author(s) -
Guerrero M.
Publication year - 2009
Publication title -
medical physics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.473
H-Index - 180
eISSN - 2473-4209
pISSN - 0094-2405
DOI - 10.1118/1.3096416
Subject(s) - standard deviation , fractionation , limit (mathematics) , mathematics , statistics , fraction (chemistry) , distribution (mathematics) , dosimetry , statistical physics , physics , nuclear medicine , mathematical analysis , chemistry , chromatography , medicine
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 α / β ratio r but also on the ratio of the corresponding standard deviationsσ α / σ βand on the ratio between the original doseD 1andD 50. In some special cases, notably whenD 1 = D 50 ( 1 ), the standard LQ model is recovered. In general, when the relative widths of the distributions for α and β are equal or similar, the deviation from the standard LQ model is small, mostly within 5% forD 1 / D 50 ( 1 )from 0.4 up to 1.6. When the width of the distribution for α is much larger than the one for β , 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 α is much smaller than the one for β , 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.