The linear-quadratic transformation of dose-volume histograms in fractionated radiotherapy

Dose-volume histograms (DVHs) are often used in radiotherapy to provide representations of treatment dose distributions. DVHs are computed from physical dose and do not include radiobiological factors; therefore, the same DVH will be computed for a treatment plan whatever fractionation regimen is used. However, dose heterogeneity resulting from variation of daily treatment dose within the volume will have biological effects due to spatial heterogeneity of fraction size as well as total dose. The purpose of the paper is to present a radiobiological (LQ) transformation of the physical dose distribution which incorporates fraction size effects and may be better suited to the prediction of biological effects.

An analytic formula is derived for the linear-quadratic transformation of a normal distribution of dose to give the corresponding distribution of biologically equivalent dose given as 2 Gy fractions. This allows LQ-transformed DVHs to be computed from physical DVHs. The resultant LQ-DVH depends on the assumed value of the relevant alpha/beta ratio. It is a modified dose distribution (corrected for spatial heterogeneity of fraction size) but does not incorporate time factors or volume effects.

The analysis shows that the LQ-transformed distribution is always broader than the distribution of physical dose. Radiobiological 'hot spots' and 'cold spots' are further from the mean than physical distributions would indicate. The difference between conventional DVHs and LQ-transformed DVHs is dependent on the fractionation regimen used. LQ-DVHs for a single dose distribution (treatment plan) can be computed for different fractionation regimens with some simplifying assumptions (e.g. no time-factor-dependence of late effects). Regimens calculated to be radiobiologically equivalent at a single point nevertheless result in non-equivalent LQ-DVHs when spatial variation of daily treatment dose is included. The difference is especially important for tumour sites (such as breast and head and neck) for which considerable dose heterogeneity may occur and for which different treatment regimens are in use.

LQ-DVHs should be computed in parallel with conventional DVHs and used in the evaluation of treatment plans and fractionation regimens and in the analysis of high-dose side-effects in patients.