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Purpose To examine the equivalent uniform dose (EUD) formalism using the universal survival curve (USC) applicable to high-dose stereotactic body radiotherapy (SBRT). Materials and methods For nine non-small-cell carcinoma cell (NSCLC) lines, the linear-quadratic (LQ) and USC models were used to calculate the EUD of a set of hypothetical two-compartment tumor dose–volume histogram (DVH) models. The dose was varied by ±5%, ±10%, and ±20% about the prescription dose (60 Gy/3 fractions) to the first compartment, with fraction volume varying from 1% and 5% to 30%. Clinical DVHs of 21 SBRT treatments of NSCLC prescribed to the 70–83% isodose lines were also considered. The EUD of non-standard SBRT dose fractionation (EUDSBRT) was further converted to standard fractionation of 2 Gy (EUDCFRT) using the LQ and USC models to facilitate comparisons between different SBRT dose fractionations. Tumor control probability (TCP) was then estimated from the LQ- and USC-EUDCFRT. Results For non-standard SBRT fractionation, the deviation of the USC- from the LQ-EUDSBRT is largely limited to 5% in the presence of dose variation up to ±20% to fractional tumor volume up to 30% in all NSCLC cell lines. Linear regression with zero constant yielded USC-EUDSBRT = 0.96 × LQ-EUDSBRT (r2 = 0.99) for the clinical DVHs. Converting EUDSBRT into standard 2‑Gy fractions by the LQ formalism produced significantly larger EUDCFRT than the USC formalism, particularly for low $$\alpha /\beta$$ α / β ratios and large fraction dose. Simplified two-compartment DVH models illustrated that both the LQ- and USC-EUDCFRT values were sensitive to cold spot below the prescription dose with little volume dependence. Their deviations were almost constant for up to 30% dose increase above the prescription. Linear regression with zero constant yielded USC-EUDCFRT = 1.56 × LQ-EUDCFRT (r2 = 0.99) for the clinical DVHs. The clinical LQ-EUDCFRT resulted in median TCP of almost 100% vs. 93.8% with USC-EUDCFRT. Conclusion A uniform formalism of EUD should be defined among the SBRT community in order to apply it as a single metric for dose reporting and dose–response modeling in high-dose-gradient SBRT because its value depends on the underlying cell survival model and the model parameters. Further investigations of the optimal formalism to derive the EUD through clinical correlations are warranted.

In this paper, we compare two radiation effect models: the average surviving fraction (ASF) model and the integral biologically effective dose (IBED) model for deriving the optimal irradiation scheme and show the superiority of ASF. Minimizing the effect on an organ at risk (OAR) is important in radiotherapy. The biologically effective dose (BED) model is widely used to estimate the effect on the tumor or on the OAR, for a fixed value of dose. However, this is not always appropriate because the dose is not a single value but is distributed. The IBED and ASF models are proposed under the assumption that the irradiation is distributed. Although the IBED and ASF models are essentially equivalent for deriving the optimal irradiation scheme in the case of uniform distribution, they are not equivalent in the case of non-uniform distribution. We evaluate the differences between them for two types of cancers: high α/β ratio cancer (e.g. lung) and low α/β ratio cancer (e.g. prostate), and for various distributions i.e. various dose–volume histograms. When we adopt the IBED model, the optimal number of fractions for low α/β ratio cancers is reasonable, but for high α/β ratio cancers or for some DVHs it is extremely large. However, for the ASF model, the results keep within the range used in clinical practice for both low and high α/β ratio cancers and for most DVHs. These results indicate that the ASF model is more robust for constructing the optimal irradiation regimen than the IBED model.