Why RBE must be a variable and not a constant in proton therapy

Abstract

Objective: This article considered why the proton therapy (PT) relative biological effect (RBE) should be a variable rather than a constant.

Methods: The reasons for a variable proton RBE are enumerated, with qualitative and quantitative arguments. The heterogeneous data sets collated by Paganetti et al (2002) and the more homogeneous data of Britten et al (2013) are further analyzed using linear regression fitting and RBE-inclusive adaptations of the linear-quadratic (LQ) radiation model.

Results: The in vitro data show RBE increasing as dose per fraction is lowered. In the Paganetti et al (2002) data sets, the differences between observed and expected effects are smaller when the LQ model is used, but with such data heterogeneity, firm statistical conclusions cannot be obtained. The more homogeneous data set shows an unequivocal variation in RBE with dose per faction. The in vivo data are inappropriate for assessments of late normal tissue effects in radiotherapy. Also, if there is the same degree of uncertainty in an RBE of 1.1 or in an RBE of 2-3 for C ions, the fractional and biological effective doses can vary considerably and be greater in the proton case. So, errors in RBE assignment are important for protons, just as with C ions.

Conclusion: Further experimental programmes are proposed, including late normal tissue end points. Better RBE allocations might further improve PT outcomes.

Advances in knowledge: This study provides a rigorous critique of the 1.1 RBE used for protons, from theoretical and practical standpoints. Data analysis shows that the LQ model is more appropriate than simple linear regression. Comprehensive research programmes are suggested.

Figure 1.

Further analysis of data of Britten et al for (a) α radiosensitivity in Hep-2 cells with changes in linear energy transfer (LET), for two different values of the initial incident energy. (b) Relative biological effect (RBE) plots with dose per fraction calculated using Equation (A7) in the Appendix A, constructed from the α and β radiosensitivities found at various positions in spread-out Bragg peaks and obtained using two different incident energies. The coded letters (a–g) refer respectively to different irradiation parameters given in the table.

Figure 2.

Displays of relative biological effect (RBE) and dose data from Paganetti et al, with modelled plots of dose per fraction and RBE plots using parameters obtained from the weighted least squares fitted 1.35 – 0.02d linear regression model (the straight line) and the linear–quadratic (LQ) model (for varying α/β ratios), which provides curves for in vitro data with RBEs > 1; (a) a magnified view of plotted models and (b) with superimposed data over a larger scale using the same codes. Owing to frequent overlapping, error bars are omitted in order to improve visual inspection, but can be seen in the original reference. The LQ RBE plots with dose per fraction are constructed using Equation (A7) in the Appendix A.

Figure 3.

Dose per fraction and relative biological effect (RBE) plots using parameters obtained from the linear fit of 1.12 + 0.d (shown as the straight line) and the linear–quadratic (LQ) model (for varying α/β ratios), which provides curves, for in vivo data with RBEs > 1: (a) a magnified view of plotted models and (b) with superimposed data over a larger scale using the same codes. Error bars are omitted in order to improve visual inspection, but can be seen in the original reference. The LQ RBE plots with dose per fraction are constructed using Equation (A7) in the Appendix A.