Compatibility of the repairable-conditionally repairable, multi-target and linear-quadratic models in converting hypofractionated radiation doses to single doses

Abstract

We investigated the applicability of the repairable-conditionally repairable (RCR) model and the multi-target (MT) model to dose conversion in high-dose-per-fraction radiotherapy in comparison with the linear-quadratic (LQ) model. Cell survival data of V79 and EMT6 single cells receiving single doses of 2-12 Gy or 2 or 3 fractions of 4 or 5 Gy each, and that of V79 spheroids receiving single doses of 5-26 Gy or 2-5 fractions of 5-12 Gy, were analyzed. Single and fractionated doses to actually reduce cell survival to the same level were determined by a colony assay. Single doses used in the experiments and surviving fractions at the doses were substituted into equations of the RCR, MT and LQ models in the calculation software Mathematica, and each parameter coefficient was computed. Thereafter, using the coefficients and the three models, equivalent single doses for the hypofractionated doses were calculated. They were then compared with actually-determined equivalent single doses for the hypofractionated doses. The equivalent single doses calculated using the RCR, MT and LQ models tended to be lower than the actually determined equivalent single doses. The LQ model seemed to fit relatively well at doses of 5 Gy or less. At 6 Gy or higher doses, the RCR and MT models seemed to be more reliable than the LQ model. In hypofractionated stereotactic radiotherapy, the LQ model should not be used, and conversion models incorporating the concept of the RCR or MT models, such as the generalized linear-quadratic models, appear to be more suitable.

Figure 1.

Dose-survival data for V79 single cells. Open circle = single fraction; filled triangle = 2 fractions (4 Gy × 2, 5 Gy × 2); filled square = 3 fractions (4 Gy × 3, 5 Gy × 3); X = equivalent single doses for the hypofractionated doses using the linear-quadratic model (measured dose). Bars represent the standard error of four experiments. An approximate equation of the linear-quadratic model for the survival curve is S= exp(–0.027D²– 0.14D), giving an α/β of 5.1 Gy. Approximate equations of the repairable-conditionally repairable and multi-target models for the survival curve are S= exp(–19D) + 1.6Dexp(–0.69D) and S= exp(–0.17D) [1 – (1 – exp(–0.47D))^9.8], respectively.

Figure 2.

Dose-survival data for EMT6 single cells. Open circle = single fraction; filled triangle = 2 fractions (4 Gy × 2, 5 Gy × 2); filled square = 3 fractions (4 Gy × 3, 5 Gy × 3); X = equivalent single doses for the hypofractionated doses using the linear-quadratic model (measured dose). Bars represent the standard error of three experiments. An approximate equation of the linear-quadratic model for the survival curve is S= exp(–0.056D²– 0.020D), giving an α/β of 0.36 Gy. Approximate equations of the repairable-conditionally repairable and multi-target models for the survival curve are S= exp(–18D) + 5.1Dexp(–0.99D), and S = exp(–0.14D) [1 – (1 – exp(–0.89D))⁸¹], respectively.