The LQ model is increasingly being used to predict control probability (TCP) and normal tissue complication probability (NTCP) using logistic models, for instance for radiobiological treatment planning [79,80,81]. In this study we summarized published values for the LQ parameters α, β and α/β of human tumours, for as many tumour sites and tumour histologies as possible. This overview shows a large study heterogeneity in reported values of LQ parameters, which indicates substantial clinical and methodological differences between studies. Despite study heterogeneity, some relevant patterns could be identified.

Commonly, α/β values are categorized by tumour site [4, 82], implicitly assuming that tumour site is the most important factor determining radiobiological behaviour. The rationale for categorization by tumour site is that clinical radioresponsiveness would predominantly be determined by the tumour environment (e.g. hypoxia). However, Fertil and Malaise [83] already showed in 1985 that radiosensitivity is (at least partly) intrinsic to the histology of the tumour. Our data support that both tumour site and histology independently determine radioresponsiveness. The first idea, that tumour site is important, is supported by the fact that prostate tumours seem to have even lower α/β (±1–2 Gy) than breast tumours (±2–4.5 Gy), even though both are adenocarcinoma. The second idea, that tumour histology is an independent important factor, is supported by the observation that similar histologies show consistently similar α/β values, regardless of tumour site. Adenocarcinomas, both in prostate and breast cancer, overall display a high fractionation sensitivity (low α/β, see Additional file 1: Figure S3.1). On the other hand, epithelial histologies such as squamous cell carcinoma, transitional cell carcinoma, basal cell carcinoma and non small cell lung carcinoma all exhibit low fractionation sensitivity (high α/β). Finally, some tumour sites (e.g. skin and central nervous system tumours) exhibit very mixed fractionation sensitivities that correlate well with the different histologies occurring at those sites. In summary, both site and histology are important factors for α/β. Therefore, it has been suggested that for tumour sites at which multiple histologies occur (e.g. squamous-cell carcinoma and adenocarcinoma in oesophageal cancer), LQ parameters should be reported separately for each histology [14], which enables estimation of separate α/β values for each histology. This finding may be relevant for LQ calculations in radiotherapy practice, for instance in a patient with cancer of unknown origin, or for a patient with a tumour in a site with more histologies (i.e. lung, oesophagus, cervix uteri), for whom we recommend to choose an α/β based on the tumour histology.

Apart from tumour site and histology, the type of LQ model used in an analysis may affect the values estimated for α, β and a/β and thus partially explains study heterogeneity. For example, in the study by Suwinski et al. [48] a higher α/β was reported when a time factor was included (α/β = 11.1 Gy) than without time factor (α/β = 5.1 Gy). This can be explained by the fact that high dose-per-fraction treatment schedules are often shorter than low dose-per-fraction schedules. Therefore, when using a time factor to account for repopulation, part of the efficacy of a high-dose-per fraction schedule is attributed to a shortened overall treatment time, and not to the higher fraction dose. Then, the inclusion of a time factor will result in a higher estimate for α/β. Another example is that the estimates for α and β are higher when intratumour heterogeneity is accounted for in the LQ model [20]. This is because these values represent the mean radiosensitivity, while the tumour control is mostly determined by the most radioresistant (i.e. low α and β) tumour cells within the tumour.

Due to statistical variation, some studies find small, negative β estimates. As a result, large, negative values are calculated for α/β (e.g. [65]). This is merely a statistical effect: regardless of the sign, a small absolute value β (and correspondingly large absolute value of α/β) indicates that the tumour has a very low sensitivity to the effects of fractionation. Although from radiobiological point of view negative values of the α/β ratio are not realistic, it is not advised to constrain negative values in radiobiological analyses. When parameters are constrained, aggregate estimates do not converge to the true value. Furthermore, constraining parameters in e.g. maximum likelihood regression results in inaccurate estimates of the confidence intervals. Withers et al. [65] suggested to use β/α instead of α/β [65], since β/α has better balanced statistical properties. While statistical variation could still result in negative β/α estimates, these now have a more intuitive interpretation: all tumours with β/α close to zero have a low fractionation sensitivity, while tumours with a large β/α are sensitive to fractionation. Nevertheless, the α/β-ratio remained the standard LQ parameter for fractionation sensitivity.

Prior to this study, Qi et al. [33] did a meta-analysis on breast cancer, and Vogelius et al. [59] on prostate cancer. Vogelius et al. [59] determined α/β based on five prostate cancer studies (including 1965 patients), and showed that within their data no heterogeneity was present (I2 = 0%). Qi et al. [33] determined both α/β and α based on seven aggregated studies (including 8269 patients). They did not calculate study heterogeneity, but all data required for heterogeneity calculation were reported. For α/β, no heterogeneity was present (I2 = 0%), while for α heterogeneity was substantial (I2 = 58%). For the majority of tumour sites in our study, study heterogeneity in α/β was substantially higher than what was found in these two studies. This difference is most likely due to the specific design of these studies, which excluded several potential sources of heterogeneity. For example, Vogelius et al. only included studies in which external radiotherapy was the primary treatment for prostate cancer (i.e. no brachytherapy or prior prostatectomy), thereby excluding these potential sources of clinical heterogeneity. Furthermore, rather than aggregating available radiobiological parameters, these two meta-analyses used local control and biochemical control of PSA from fractionation trials to derive LQ parameters for each individual trial. As a result, the LQ parameter estimates were derived using exactly the same statistical analysis, excluding potential sources of methodological heterogeneity. This approach is unfortunately only feasible for those tumour sites where many fractionation trials have been performed. Moreover, due to the strict inclusion criteria of Vogelius et al., their results are only applicable to a very specific patient group, whereas our study aimed to present a complete overview of the available data.

Qi et al. [33] and Vogelius et al. [59] previously reported meta-analyses of LQ parameters for uniformly treated patients with breast cancer and prostate cancer respectively, both yielding relatively low α/β values for tumour, and low study heterogeneities. Their results are only applicable to two specific patient groups. We chose to aggregate LQ parameter estimates for as many sites and histologies as possible, at the cost of a higher study heterogeneity. To select radiobiological parameters from this overview, one should try to select parameters from a study that matches the situation of interest (both clinically and methodologically) as close as possible (see Additional file 1: Table S4). We recommend this elaborate approach for the selection of LQ parameters in the design of a radiobiological treatment planning system.

However, for LQ calculations in daily radiation practice, we recommend to use a range of plausible α/β values rather than a single value. A plausible range can be found in Figs. 1, 2, 3. For example, when selecting an α/β for breast tumours, the radiobiological calculation could be performed with α/β = 2 Gy, 3.5 Gy and 5 Gy respectively. If a consistent conclusion can be drawn for the whole range of plausible values (e.g. one schedule is more effective than another, for all three α/β values), this conclusion may be considered robust to the uncertainty in the selection of appropriate parameters. This approach is valid irrespective of the estimated heterogeneity, although the range of plausible values is likely to be larger for tumour sites with larger heterogeneity.