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.
Keywords: linear-quadratic model, stereotactic radiotherapy, equivalent dose, repairable-conditionally repairable model, multi-target model
INTRODUCTION
With the worldwide distribution of gamma knife units and linac surgery systems, considerable amounts of clinical data on single-fraction radiosurgery are now available [1, 2]. On the other hand, hypofractionated stereotactic radiotherapy is attracting more attention owing to its efficacy and lower toxicity [2, 3]. Therefore, it would be useful if hypofractionated doses could be converted to single doses with reliable precision, using mathematical models, in order to compare single-fraction radiosurgery and hypofractionated stereotactic radiotherapy data.
The linear-quadratic (LQ) model is considered useful for dose conversion in conventionally fractionated radiation therapy [4, 5]. However, our previous studies showed that the LQ model is unreliable in converting hypofractionated doses to single doses [6–8]. The LQ model underestimated the effect of 2- to 5-fraction irradiation, and was considered to be inapplicable to stereotactic irradiation in the clinical setting. After the study, we attempted to apply computer calculation to other dose-conversion models, i.e. the repairable-conditionally repairable (RCR) model and the multi-target (MT) model, using mathematical software. Calculation with these models is not easy without mathematical software such as Mathematica (Wolfram Research, Inc., Champaign, IL, USA). In this study, therefore, we investigated the applicability of the RCR and MT models to dose conversion between single and hypofractionated doses, comparing them to the LQ model. We employed these conversion models because the RCR model has several advantages in its accurate description of cell survival at both low and high doses compared to the LQ model (by taking some biological parameters into consideration), and the MT model is a basic and classical linear model which fits the empirical data well at high doses compared to the LQ model [9–12].
MATERIALS AND METHODS
Experimental data
The experimental data used in this study are derived from our previous experiments [6]. Briefly, EMT6 mouse mammary sarcoma cells were used for single cell experiments, and V79 Chinese hamster lung fibroblasts were used for single cell and spheroid experiments, as described in detail previously [6]. V79 spheroids were used, grown to approximately 0.8 mm in diameter 14–18 days after spinner culture. Irradiation was carried out using a 210-kVp X-ray machine (10 mA with 2-mm Al filter; Chubu Medical Co., Matsusaka, Japan) at a dose rate of 1.8 Gy/min. In both experiments for single cells and spheroids, irradiation was given at appropriate intervals to allow potentially lethal damage repair (PLDR) and sublethal damage repair (SLDR) [6, 13, 14]. A standard colony assay was also used to determine cell survival. Both V79 and EMT6 single cells received single doses of 0–12 Gy or 2 or 3 fractions of 4 or 5 Gy each at 4-h intervals. V79 spheroids received single doses of 0–26 Gy or 2–5 fractions of 4–12 Gy at 2–4-h intervals. A colony assay was used to determine cell survival.
Evaluation of biological equivalence and statistical analyses
Dose-survival curves were obtained for single doses of 0–12 Gy in single cell experiments and 0–26 Gy in spheroid experiments. In the single cell experiments, single doses used in the experiments, and surviving fractions at those doses, were substituted into equations of the RCR 
models (where Sis the surviving fraction, Dis the dose in Gy, d1and d0are parameters that determine the initial and final slope of the survival curve, respectively, and nis the y-intercept of the asymptote) in the calculation software Mathematica. Then, each parameter coefficient was computed by carrying out the optimum convergence to approximate the survival curves. The software employs the minimum chi-square model to fit the curves. Cell surviving fractions after hypofractionated doses were superimposed on each dose-survival curve after single–fraction irradiation, and actually-determined equivalent single doses for the hypofractionated doses (measured doses), were estimated for these three models. The method is shown in Figs 1 and 2, the symbol X representing the measured equivalent single doses for the hypofractionated doses using the LQ model. Thereafter, cell survival data after hypofractionated doses were assigned to these three models with the parameter coefficients, and equivalent single doses for the hypofractionated doses were calculated (calculated doses). The calculated doses were then compared with the measured doses. In V79 spheroid experiments, the α/β ratio and parameter coefficients for single cells were used because there were no appropriate models to estimate the α/β ratio or parameter coefficients for spheroid cells [6, 15]. Moreover, since the surviving fractions in the high dose range for the V79 spheroid cells could be approximated by a linear line, the measured doses for these three models were calculated using the linear equations:
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for 2- or 3-fraction experiments and
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for 4- or 5-fraction experiments. Statistical analyses were carried out with StatView version 5 (SAS Institute Inc., Cary, NC, USA) and SPSS 11.0J (SPSS Japan Inc., Tokyo, Japan).
RESULTS
Figure 1shows results of the V79 single-cell experiments. The three lines represent the approximate cell survival curves using these three models. Surviving fractions after two or three fractions of 4 or 5 Gy are also plotted, and equivalent single doses for the hypofractionated doses using the LQ model (measured doses for the LQ model) are indicated. The equation of the LQ model for the dose-survival curve was
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for V79 single cells and
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for EMT6 cells. Therefore, the α/β ratio was 5.1 Gy (95% confidence interval [CI]: 2.5 – 7.8) for V79 single cells and 0.36 Gy (95% CI: –2.4 – 4.6) for EMT6 single cells. For the dose-survival curves of the RCR and MT models, the three parameter coefficients of the RCR model (a, band c) were estimated to be 19, 1.6 and 0.69, respectively, for V79, and 18, 5.1 and 0.99, respectively, for EMT6. Those of the MT model (d1, d0and n) were 0.17, 0.47 and 9.8, respectively, for V79, and 0.14, 0.89 and 81, respectively, for EMT6. Therefore, the equations for the RCR and MT models were
 |
and
 |
respectively, for V79 single cells, and
 |
and
 |
respectively, for EMT6 single cells. Figure 2shows results of the EMT6 single-cell experiments.
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.
Table 1shows correspondence between the measured and calculated equivalent single doses. For example, the effects of two fractions of 4 Gy corresponded to those of a single 7.1 Gy dose (95% CI: 6.6–7.5 Gy), 7.2 Gy (95% CI: 6.9–7.5 Gy) and 7.3 Gy (95% CI: 7.0–7.7 Gy) on the actual dose-survival curves for V79 approximated by the RCR, MT and LQ models, respectively, as shown in Fig. 1, but the equivalent doses calculated from the RCR, MT and LQ models were 5.9 Gy (95% CI: 5.5–6.2 Gy), 6.0 Gy (95% CI: 5.5–6.4 Gy) and 6.4 Gy (95% CI: 5.9–6.7 Gy), respectively. As shown in Table 1, equivalent single doses for hypofractionated doses calculated by the RCR, MT and LQ models tended to be lower than the actually-measured biologically-equivalent single doses by approximately 3–25% in V79 and EMT6 single cells (when the calculated doses were simply divided by the measured doses). Thus, the RCT, MT and LQ model calculations tended to underestimate the equivalent single dose in both single cells. In these experiments, using fractional doses of 5 Gy or lower, the calculated doses from the LQ model tended to be closer to the measured doses than those from the RCR and MT models.
Table 1.
Correspondence between measured and calculated equivalent single doses in V79 and EMT6 single cells after two or three fractions of 4 or 5 Gy
| Cell | Dose (Gy) × fractions | Equivalent single dose (Gy) |
Calculated dose/ Measured dose (%) (RCR, MT, LQ) | |||||
|---|---|---|---|---|---|---|---|---|
| Calculated |
Measured |
|||||||
| (RCR) | (MT) | (LQ)* | (RCR) | (MT) | (LQ) | |||
| V79 | ||||||||
| 4 × 2 | 5.9 (5.5–6.2) | 6.0 (5.5–6.4) | 6.4 (5.9–6.7) | 7.1 (6.6–7.5) | 7.2 (6.9–7.5) | 7.3 (7.0–7.7) | 83, 83, 88 | |
| 5 × 2 | 7.6 (7.2–7.9) | 7.5 (7.0–7.9) | 7.8 (7.3–8.3) | 8.6 (8.3–8.9) | 8.7 (8.4–9.0) | 8.9 (8.6–9.1) | 88, 86, 88 | |
| 4 × 3 | 7.5 (6.8–8.2) | 7.6 (6.8–8.3) | 8.2 (7.4–9.0) | 10.0 (9.1–10.8) | 10.0 (9.1–10.8) | 10.1 (9.3–10.8) | 75, 76, 81 | |
| 5 × 3 | 10.0 (9.3–10.6) | 9.6 (8.7–10.6) | 10.0 (9.1–10.9) | 12.3 (11.7–12.9) | 12.2 (11.7–12.7) | 12.0 (11.6–12.5) | 81, 79, 83 | |
| EMT6 | ||||||||
| 4 × 2 | 5.2 (4.7–5.7) | 5.2 (4.9–5.4) | 5.7 (4.8–6.6) | 5.5 (5.2–5.7) | 6.1 (5.7–6.4) | 6.0 (5.7–6.3) | 95, 85, 95 | |
| 5 × 2 | 7.1 (6.6–7.5) | 6.5 (5.9–7.0) | 7.1 (6.2–8.1) | 7.3 (6.9–7.7) | 7.7 (7.6–7.7) | 7.9 (7.7–8.1) | 97, 84, 90 | |
| 4 × 3 | 6.3 (5.5–7.2) | 6.0 (5.5–6.5) | 7.0 (5.4–8.7) | 7.7 (7.3–8.0) | 7.9 (7.8–8.1) | 8.2 (8.0–8.4) | 82, 76, 85 | |
| 5 × 3 | 9.0 (8.2–9.9) | 7.7 (6.5–8.9) | 8.7 (7.1–10.4) | 10.1 (9.5–10.8) | 10.1 (9.5–10.6) | 10.2 (9.8–10.7) | 89, 76, 85 | |
RCR = repairable-conditionally repairable, MT = multi-target, LQ = linear-quadratic. The values in the parentheses indicate the 95% confidence interval.
* Calculated by LQ model using an α/β of 5.1 and 0.36 Gy for V79 and EMT6, respectively.
Table 2shows correspondence between the measured and calculated equivalent single doses in V79 spheroids. The calculated doses from the RCR, MT and LQ models were also lower than the measured doses by approximately 6–30%. Thus, the RCR, MT and LQ model calculation also tended to underestimate the equivalent single dose in V79 spheroids. At 6 Gy or higher doses, the equivalent single doses for the hypofractionated doses calculated using the RCR, MT and LQ models were lower than the measured biologically-equivalent single doses by 6–22%, 6–26% and 18–30%, respectively (Table 2). The deviation tended to be more marked when the LQ model was applied to 6 Gy or higher doses.
Table 2.
Correspondence between measured and calculated equivalent single doses in V79 spheroids after 2–5 fractions of 4–12 Gy
| Dose (Gy) × fractions | Equivalent single dose (Gy) |
Calculated dose/ Measured dose (%) (RCR, MT, LQ) | |||
|---|---|---|---|---|---|
| Calculated |
Measured | ||||
| (RCR) | (MT) | (LQ)* | |||
| 8 × 2 | 13.0 (12.7–13.3) | 12.7 (12.1–13.4) | 12.2 (11.5–12.7) | 15.3 (14.3–16.3) | 85, 83, 80 |
| 10 × 2 | 16.7 (16.4–17.0) | 16.5 (15.8–17.3) | 15.0 (14.3–15.7) | 18.4 (17.4–19.4) | 91, 90, 82 |
| 12 × 2 | 20.5 (20.2–20.7) | 20.5 (19.7–21.2) | 17.9 (17.2–18.5) | 21.7 (21.0–22.4) | 94, 94, 82 |
| 6 × 3 | 12.5 (11.9–13.1) | 12.0 (10.9–13.1) | 11.8 (10.9–12.8) | 15.2 (13.3–17.0) | 82, 79, 78 |
| 7 × 3 | 15.1(14.5–15.7) | 14.6 (13.3–15.8) | 13.6 (12.6–14.6) | 17.8 (16.1–19.5) | 85, 82, 76 |
| 8 × 3 | 17.7 (17.2–18.3) | 17.3 (15.9–18.6) | 15.4 (14.3–16.4) | 20.0 (19.3–20.8) | 89, 87, 77 |
| 5 × 4 | 12.2 (11.9–12.5) | 11.7 (10.3–13.2) | 11.9 (10.6–13.1) | 16.1 (14.4–17.7) | 76, 73, 74 |
| 6 × 4 | 15.5 (15.2–15.8) | 14.8 (13.1–16.5) | 14.0 (13.1–14.7) | 20.0 (19.5–20.4) | 78, 74, 70 |
| 7 × 4 | 18.9 (18.7–19.1) | 18.2 (16.3–20.2) | 16.0 (14.6 –17.5) | 22.9 (21.4–24.5) | 83, 79, 70 |
| 4 × 5 | 10.6 (10.0–11.2) | 10.5 (9.0–12.0) | 11.2 (9.7–12.6) | 15.1 (14.1–16.2) | 70, 70, 74 |
| 5 × 5 | 14.4 (13.8–15.1) | 13.8 (11.8–15.7) | 13.5 (12.0–15.1) | 19.2 (17.9–20.6) | 75, 72, 70 |
| 6 × 5 | 18.5 (17.9–19.0) | 17.6 (15.3–19.9) | 15.9 (14.2–17.6) | 22.0 (21.4–22.6) | 84, 80, 72 |
RCR = repairable-conditionally repairable, MT = multi-target, LQ = linear-quadratic. The values in the parentheses indicate the 95% confidence interval.
* Calculated by LQ model using an α/β of 5.1 Gy.
DISCUSSION
In this study, equivalent single doses for hypofractionated doses calculated using the three models did not agree with the actually-determined equivalent single doses; all the models tended to underestimate the effects of hypofractionated irradiation. This implies that none of the three models can be used to convert hypofractionated doses to single doses. Alternatively, however, our experimental procedures may need reevaluation. In determining interfraction intervals for hypofractionated irradiation, we determined the time required for SLDR in two single cell lines and spheroids, and that for PLDR in the spheroids. The experiments indicated that SLDR was complete within 4 h in the single cells, and both SLDR and PLDR were completed within 2 h in the spheroids. Therefore, we used interfraction intervals of 4 h in the single cell experiments, and 2 or 4 h in the spheroids. It seemed undesirable to keep spheroids for a longer time on agar dishes, so an interval of 2–4 h appeared adequate. Indeed, there were no significant differences between the cell surviving fractions irradiated with the short intervals and those irradiated with longer intervals. On the other hand, a few studies indicated that completion of SLDR and PLDR took 6 to 9 h in other cell lines and tumors [16–18]. Therefore, if it is assumed that the SLDR and/or PLDR in our cell lines and spheroids were not 100% complete within the interfraction intervals of 2 or 4 h (although their magnitudes were so small that they could not be proven experimentally), the effects of hypofractionated irradiation would become greater than in those with longer intervals. If this is true, the measured equivalent single doses will tend to become higher, leading to discrepancies, as seen in this study, between the measured and calculated equivalent single doses. Taking such a possibility into account, the following discussions consider which model is the more appropriate in low- and high-dose ranges.
Originally, the LQ model was considered applicable to doses per fraction of 1–10 Gy or fraction numbers of 6 or more [4]. Fowler et al.[19] later stated that the LQ model may be applicable to doses up to 23 Gy. In a high-dose range, however, the LQ model predicts a continuously bending curve due to the β cell kill component, whereas experimental data usually show a linear relationship. Therefore, it has been shown that the actual data deviate from the LQ model in the higher dose range [20, 21]. Nevertheless, many clinicians have used the LQ model to convert hypofractionated doses to single doses and the biologically effective dose (BED) derived from the model to evaluate the efficacy of stereotactic irradiation [22–24]. A few clinical reports suggest that 30 Gy in three fractions is equivalent to a single dose of 20 Gy, assuming an α/β ratio of 10 Gy for tumors, and local control and survival rates are better with BED ≥ 100 Gy [23–26]. Their clinical data do not significantly deviate from those expected from the LQ model, but the data are limited and radiation treatments including treatment planning differ among institutions. Notably, other dose conversion models were not evaluated. Therefore, these clinical data do not necessarily indicate that the LQ model is the best fit for the high-dose data.
The RCR model distinguishes between the potentially and conditionally repairable damage. The potentially repairable damage may be lethal if unrepaired or misrepaired. The conditionally repairable damage may be repaired or may lead to apoptosis if not repaired correctly. This model imports the concept of biological parameters based on the interaction of two Poisson processes with a natural separation of cell damage into two distinct types, and it is considered to solve both the low and high dose-response problems [9, 10]. The RCR model takes the low-dose hypersensitivity [27] into account so that the fitted cell survival curves are biphasic at very low doses, as shown in Figs 1 and 2. The model introduces a hypothesis that the repair system is triggered when cells are irradiated by certain doses. Accordingly, the survival fractions after hypofractionated doses may not be simply expressed by the product of those after single doses calculated by the model. The repair system may be inactivated again during the interfraction intervals. This may be a limitation to the use of the RCR model for dose conversion. Nevertheless, the RCR model can be approximated by linear regression in the high-dose range. The MT model provides an alternative description of clonogenic survival as a function of radiation dose. Although this classical model also represents a straight line at high doses, which is not supported by the mechanism of the underlying radiobiological processes, the MT model is still valuable because it fits the empirical data well, especially in the high-dose range [11, 12]. However, conversion with the RCR and MT models is not easy in clinical practice since there are many parameters which generally cannot be determined. In this study, the RCR and MT models were able to be used for biological dose estimation using the Mathematica software. Although the equivalent single doses for the hypofractionated doses from all models tended to be lower than the measured equivalent single doses, the RCR and MT models tended to provide slightly better estimates than the LQ model at 6 Gy or higher single doses. These results may be due to the above-mentioned characteristics of the two models, that the data in the high-dose range can be approximated by linear regression. Thus, it seems better not to use the LQ model for hypofractionated stereotactic irradiation; the RCR and MT models appear more appropriate. Since slight changes in parameters due to the increase in the number of parameters may lead to large errors, more investigations are necessary to clarify the utility of these models in clinical practice.
Recently, other models which are suitable at high doses have been proposed. Park et al. [28] proposed the universal survival curve (USC) model. This model hybridizes the LQ model for low doses and the MT model for doses beyond a single transition dose (DT). They reported the availability of the USC model in an in vitrocell line where the DTwas 6.2 Gy. The DT(6.2 Gy) in their cell line almost agrees with the results of this study. Hence, the concept is relatively simple and is without consideration of biological parameters. In addition, the linear-quadratic-linear (LQL) model (or modified LQ model) [29, 30] and the generalized LQ (gLQ) [31] model have been proposed. The LQL model derives from a mechanism based on the lethal-potentially lethal model [32]. Although the equations for the LQL model are more complex, cell survival curves extend almost linearly at high doses, unlike those of the LQ model. Therefore, the applicability of the USC and LQL models to high-dose regions may be similar. The gLQ model takes SLDR and the conversion of sublethal damage to lethal damage during irradiation into account; the model is designed to cover any dose delivery pattern. Since the gLQ model can be approximated to the MT model at high doses, it is also similar to the USC model. All of these newer models seem to fit better than the LQ model in the high-dose range. Subsequently, it has been shown that the generalized LQ model incorporating the reciprocal time of SLDR fits better than the exponential repair model, and can be used to analyze the experimental and clinical data, where a slowing-down repair process appears during the course of radiation therapy [33]. However, it should be noted that none of these models takes the reoxygenation phenomenon, cell cycle effects, or host immune response into account. Particularly, reoxygenation is very important for the in vivotumor response to hypofractionated irradiation and this factor should be considered [34–36].
In conclusion, the RCR and MT models can be used for biological dose estimation using the software Mathematica. The LQ model may only be applicable to fractional doses of 5 Gy or less. The conversion models based on RCR, or MT models like the universal survival curve and generalized linear–quadratic models, seem to be more reliable than the LQ model at 6 Gy or higher doses. The LQ model should not be used in hypofractionated stereotactic irradiation. In the near future, it is desirable for a new, optimal and easy to use model to be established for clinical use in hypofractionated stereotactic radiotherapy, taking important biological parameters like reoxygenation into consideration.
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