A simple model for calculating relative biological effectiveness of X-rays and gamma radiation in cell survival

Abstract

Objectives:

The relative biological effectiveness (RBE) of X-rays and γ radiation increases substantially with decreasing beam energy. This trend affects the efficacy of medical applications of this type of radiation. This study was designed to develop a model based on a survey of experimental data that can reliably predict this trend.

Methods:

In our model, parameters α and β of a cell survival curve are simple functions of the frequency-average linear energy transfer (L_F) of delta electrons. The choice of these functions was guided by a microdosimetry-based model. We calculated L_F by using an innovative algorithm in which L_F is associated with only those electrons that reach a sensitive-to-radiation volume (SV) within the cell. We determined model parameters by fitting the model to 139 measured (α,β) pairs.

Results:

We tested nine versions of the model. The best agreement was achieved with $\sqrt{\alpha }$ and β being linear functions of $L_F$ .The estimated SV diameter was 0.1–1 µm. We also found that α, β, and the α/β ratio increased with increasing $L_F$ .

Conclusions:

By combining an innovative method for calculating $L_F$ with a microdosimetric model, we developed a model that is consistent with extensive experimental data involving photon energies from 0.27 keV to 1.25 MeV.

Advances in knowledge:

We have developed a photon RBE model applicable to an energy range from ultra-soft X-rays to megaelectron volt γ radiation, including high-dose levels where the RBE cannot be calculated as the ratio of α values. In this model, the ionization density represented by $L_F$ determines the RBE for a given photon spectrum.

Introduction

X-rays and γ radiation are used extensively in medicine to diagnose and treat disease. Energy spectra of photons reaching the patient and the doses they deliver vary between various radiation devices and clinical procedures. The absorbed dose is usually known, but it does not fully determine the biological effects of radiation because the relative biological effectiveness (RBE) of photons increases considerably with decreasing beam energy.¹ A reliable model of this trend would help improve the balance between therapeutic and adverse effects, for example, in the following areas.

The use of X-rays for imaging is associated with a risk of radiation-induced cancers. These risks were comprehensively evaluated in the NCRP (National Council on Radiation Protection and Measurements) Report 181.¹ This report provides ample evidence of significant variations of RBE with photon energy. We did not consider these risks in our study. Radiation sources used in brachytherapy include radio isotopes that emit low-energy photons ¹⁰³Pd (21 keV) and ¹²⁵I (28 keV) and miniature X-ray generators that operate at 30–100 kV. The latter sources are also used in intraoperative radiotherapy.² In external beam radiotherapy, beam properties are highly variable. Corresponding variations in RBE are not negligible,³ with the RBE exceeding 1.2 in some cases. In research on tumor dose enhancement, several techniques are investigated: the use of nanoparticles, contrast agents, and photon-induced Auger effect.^4,5 Dose enhancement is caused by low-energy electrons produced by these agents. The RBE of these electrons enhances tumor damage.

Many different approaches have been developed previously to model the biological effects of photons. A full review of such models is not feasible within the scope of this article. We therefore summarize only the models most relevant to this study.

In the microdosimetric approach, parameter α of the linear quadratic model of cell survival is proportional to the dose average lineal energy $y_D,$ and the low-dose RBE, or RBE_M, is $y_D$ (test)/ $y_D$ (reference). Limitations of this approach were discussed in NCRP Report 181,¹ which points out that $y_D$ depends on the size of the sensitive-to-radiation volume (SV) representing a microscopic biological target within a cell. For a ⁶⁰Co source, reducing the SV diameter from 1 µm to 10 nm results in $y_D$ increasing by a factor of 9.0.¹ The variability of $y_D$ poses a difficulty because “there is no clear indication of the relevant site size for specific biological endpoints.”¹ Furthermore, the lineal energy $y$ is “indicative of the ionization density along the track” only for those particles whose range exceeds the SV size. For slower particles, $y$ underestimates the ionization density because of the volume averaging effect. By using equilibrium electron spectra,⁶ we found that for a 1 µm SV, the fraction of electrons whose range was less than the mean chord length of the SV was 0.043, 0.30, and 0.57 for a ⁶⁰Co source, 100 keV, and 10 keV photons, respectively. We add that this approach does not offer a model for parameter β of the linear quadratic model, which is required at therapeutic doses.

Microdosimetric modeling with much smaller SVs, however, has produced interesting results. Nikjoo et al⁷ reviewed experimental data for low energy electrons and photons on DNA double strand break (DSB) induction and RBE for cell survival. They compared these data with calculated dose and frequency average lineal energies, $y_D$ and $y_F$ , calculated for a 10×10 nm cylindrical SV. They found that the use of $y_D$ estimates RBE within a factor of 2–3 for ultra-soft X-rays. For electrons, they concluded that $y_D$ “seems to mimic the experimentally determined RBE values.” The study considered only RBE max defined as the ratio α(test)/α(reference).

Another approach combines Monte Carlo simulation of particle interactions with models of DNA damage and repair. A recent application of this approach to modeling of photon RBE⁸ produced accurate results for DNA DSBs but underestimated cell survival RBE for energies below 20 keV, with discrepancies reaching a factor of two for ultra soft X-rays. In contrast, another study⁹ using a Monte-Carlo–based approach reported agreement with experimental cell survival data for photons from “ultrasoft characteristic k-shell X-rays (0.25–4.55 keV) up to orthovoltage (200–500 kVp) X-rays.” That study provided “strong evidence supporting … hypothesis” that RBE for cell survival is equal to the RBE for DSB induction “for photons and electrons across a wide range of energies.” Other authors, however, have argued that the relationship between DSBs and cell survival is more complex. Liang et al⁸ had tested the above hypothesis and arrived at a different conclusion: “the average RBE max for soft x-rays from a large number of experiments is 20–50% higher than the relative yields of DSB and the disagreement continues to increase reaching 50–100% for ultrasoft x-rays.” Goodhead¹⁰ concluded from a broader review of experimental data that “the numbers of dsbs are not the predominant determinant of cellular effectiveness.” He wrote that the “higher RBEs,” for cell survival than RBEs for DSB production “are likely arise only from damage of somewhat greater complexity” (than DSB). Nikjoo and Lindborg⁷ have also commented on the difficulty of modeling this relationship: “in principle, double-strand breaks are considered as the initial damage leading to lesions which may initiate cell death and cancer, but no quantitative definitive measure exists for such links.”

Our approach is different. It is based on the non-Poisson multihit model (NPMH),^11,12 which is a revision of the multihit single target model of cell survival developed using a microdosimetric formalism. The NPMH model is also applicable to modeling single- and double-strand breaks. The model agreed¹¹ with previously reported data on DNA single- and double-strand breaks induced by protons and electrons. Based on those results we could, in principle, build a model of cell survival. We however chose a different approach that has been successful in modeling of proton RBE.^13–15 In this approach, no attempts are made to link cell survival to DSB induction data. Instead, the model is based directly on experimental cell survival data.

Methods and materials

Methodology overview

We limited our study to cell survival. We searched the literature for studies that reported both α and β, or data from which α and β could be derived. The search produced a data set that comprised 139 (α,β) pairs for photon sources, from ultrasoft X-rays to ⁶⁰Co γ rays. These cell survival data are summarized in Supplementary Material 1. We first applied our model to the entire data set and then considered separately three cell lines for which survival curves were measured for a sufficiently broad range of photon energies.

Our model was based on a method¹⁶ for calculating average LETs (Linear Energy Transfer) for electrons and photons for radiobiological applications. The method accounts for the fact that delta-electrons are produced primarily outside the SV. As they travel from the point of origin to the SV, they slow down and produce numerous low-energy electrons that can also reach the SV. We simulated all of these processes with Monte Carlo and calculated energy distributions only for those electrons that actually reached the SV.⁶ Using those spectra, we tabulated¹⁶ all the data needed to calculate the frequency- and dose- average LET, $L_F$ and $L_D$, respectively, for photon energies from 0.25 keV to 1.25 MeV. To calculate an average LET for a polyenergetic photon beam using these data, only the photon fluence spectrum is needed.

Our RBE model uses regression models for parameters α and β of the linear quadratic model with $L_F$ as an independent variable. In contrast to many previous studies^13–15 that used $L_D$ , in the NPMH model, $L_F$ is the variable characterizing the physics of X-ray and γ radiation beams. Our choice of regression models was guided by the NPMH model.^11,12 We did not apply the NPMH model in its rigorous form because that would make RBE calculations too complicated. Instead, we relied only on the model’s general predictions of the properties of α and β as functions of $L_F$ .

Calculation of photon spectra

For ultrasoft X-rays, we used the average photon energy. We calculated it as the average of characteristic K_α and, where appropriate, K_β x-ray energies weighted by their relative intensities. We took the necessary data from Kaye and Laby.¹⁷

For X-ray generators, we calculated the spectra with the MCNPX V.2.7.0 Monte Carlo code. Hernandez and Boone¹⁸ demonstrated the high accuracy of this code in modeling X-ray generators. The experimental data that we used lack detailed information on X-ray generators and irradiation geometry. We therefore used a generic model of a generator that consisted only of a target and a beryllium window. To model inherent filtration, we added 0.6–0.9 mm of aluminum, depending on beam energy.¹⁹ Based on the experiment description, we also added 1 mm of water or polystyrene to model photon absorption in cell culture or the dish base. In a few cases for which no information on added filtration was available, we used filters specified elsewhere for a similar source. For energies from 18 kV to 140 kV, in addition to Monte Carlo simulations, we calculated X-ray spectra with use of an online tool for calculating spectra of X-ray generators used in mammography and radiography (health.siemens.com/booneweb/index.html) developed by Siemens Healthcare GmbH (Erlangen, Germany). We verified the accuracy of this online tool by calculating several spectra using interpolating polynomials²⁰ and SPECTR V.3.0 software.²¹

For an ¹⁹²Ir brachytherapy source, Gamma Med 12i HDR,²² we used a spectrum calculated with Monte Carlo specifically for this source.²³

Model implementation

In the NPMH model, a discrete parameter, the number of hits $n$, determines how α and β depend on LET¹²:

• if $n=1$, then α decreases linearly with increasing LET and $\beta =0$;

• if $n=2$, then α increases linearly and β decreases linearly;

• if $n=3$, then α increases as a quadratic function of LET and β increases linearly.

We account for electrons that originated outside the SV as well as electrons produced by photons interacting within the SV. Relative probabilities of these two processes depend on the SV size, and so does $L_F$ . This dependence is much weaker for $L_F$ than it is for $y_D$ . For a ⁶⁰Co source, an increase in the SV size from 0 to 1 µm results in $L_F$ increasing only by 0.15%.

For monoenergetic photon beams, we derived the following formula for calculating $L_F$¹⁶:

$$L_{F,mono}=\frac{\left(S_e,L\right)+\left({\stackrel{-}{\mathrm{\Phi }}}_e,L\right)\bullet R/h}{1+R/h}$$ (1)

In this equation, $S_e$ is the source spectrum of electrons (i.e., energy distribution of initial electron energies at the point of origin), $R$ is the electron range, $h$ is the mean chord length of the SV, and ${\stackrel{-}{\mathrm{\Phi }}}_e$ is the electron fluence spectrum. $S_e$ and ${\stackrel{-}{\mathrm{\Phi }}}_e$ are normalized so that their integral over energy is 1. The parentheses denote integration over energy:

$$\left(f,g\right)\equiv \underset{0}{\overset{\infty }{\int }}f\left(E\right)g\left(E\right)dE.$$ (2)

We previously reported¹⁶ tables of $\left(S_e,L\right)$ , $\left({\stackrel{-}{\mathrm{\Phi }}}_e,L\right),$ and $R$ for monoenergetic photon beams with photon energies ranging from 0.25 keV to 1.25 MeV. To calculate $L_F$ for a polyenergetic photon beam, using these tables, we need only the photon fluence spectrum, ${\mathrm{\Phi }}_p\left(E\right)$ and the total cross-section of photon interactions that produce δ electrons ${\sigma }_p\left(E\right)$ :

$$L_F=\left({\sigma }_p{\mathrm{\Phi }}_p,L_{F,mono}\right)/\left({\sigma }_p{\mathrm{\Phi }}_p,1\right)$$ (3)

where ${\sigma }_p{\mathrm{\Phi }}_p$ is the photon collision density.²⁴

We have tested the models with $n=$ 1, 2, 3, and SV diameters 0, 0.1, 1 µm, against experimental cell survival data from our survey of the literature Supplementary Material 1. We chose only three well-separated SV sizes because of the weak dependence of $L_F$ on SV size.

Statistical methods

We used linear regression to model the relationship between β and $L_F$ and between α or $\sqrt{\alpha }$ and $L_F$ in the cell survival data. Diagnostic plots were used to check for trends in the residuals, non-normality of errors, and influential points. The Box-Cox method was used to assess whether a power transformation of α was appropriate. Analyses were done in Statistica and R v.3.4.0.

Results and discussion

Validation of calculated X-ray spectra

We compared average photon energies $E_{av}$ for several X-ray sources, from 10 to 250 kVp, calculated with Monte Carlo with measurements and with the Siemens online tool. The results are presented in Supplementary Material 1 (supplementary materials). Our Monte Carlo results agreed with the other data, in most cases within 1–2%. Only in one case (25 kVp) did Monte Carlo simulation agree poorly with both the measurement and the Siemens tool. This was caused by uncertainties in modeling L-shell characteristic X-rays. The spectrum produced by the Siemens tool was in better agreement with the experiment. Therefore, we used it for this case. For all other cases, we used Monte Carlo spectra.

Modeling β

All cell survival data that we used are reported in Supplementary Material 1. The tables also include properties of X-ray and γ-radiation sources used in the experiments and $L_F$ for each case. These data can be used in future studies for quick estimation of $L_F$ for a source that is within the range of our data. The number of distinct sources that we have considered is 31.

First, we determined whether β increases or decreases with increasing $L_F$. Increasing β supports $n=3$, whereas decreasing β supports $n=2$. A β close to zero may indicate either that $n=1$ or that β was too small to be measured accurately. Our test for $n=1$ was based on a variation of α with $L_F$. If α decreases with increasing $L_F$, then $n=1$. Otherwise, $n=2$ or 3 should be considered. In this preliminary analysis, our focus was on the slope of regression models. We tested the following linear model for β:

$$\beta =b_0+b_1{L}_F$$ (4)

We applied this model to all β data included in Supplementary Material 1, and then separately to three cell lines: Chinese hamster V79, human fibroblasts HF19, and mouse embryo fibroblasts C3H10T1/2.

One data point, β = 0.27 Gy⁻² (V79 cells²⁵) was highly influential. To show its impact, we reported data for two models: model one included all data and model two excluded β = 0.27 Gy⁻².

We summarized our β modeling results in Table 1.The adjusted $R$² values for the 0.1 and 1 µm diameters were similar in all cases, and they were lower for the zero diameter SV. The small $R$² for all cells (first row) reflects the heterogeneity of this dataset. The positive slope ($b_1\S amp;gt;0$) in this case was nevertheless significant for both models 1 and 2. That was also the case for V79 cells. The level of association for C3H10T1/2 cells was marginally significant ($p$≈0.07, excluding the zero-diameter SV). For HF19 cells, β was set to zero in all three studies that reported on this cell line.^25–27

Table 1.

The slope $b_1$ of the linear model for β given by Eq. (4) for three SV diameters. Model one included all data points; model two excluded β = 0.27 Gy⁻².

Cell line	Diameter=0			Diameter=0.1 µm			Diameter=1 µm
	$R^2$	$b_1$	$p$	$R^2$	$b_1$	$p$	$R^2$	$b_1$	$p$
All cells
Model 1	0.052	(>0)	0.004	0.12	(>0)	2·10−5	0.11	(>0)	2·10−5
Model 2	0.016	(>0)	0.08	0.024	(>0)	0.04	0.021	(>0)	0.05
V79
Model 1	0.24	(>0)	0.004	0.63	(>0)	2·10−7	0.61	(>0)	3·10−7
Model 2	0.22	(>0)	0.007	0.25	(>0)	0.004	0.28	(>0)	0.002
HF19	-	(=0)	-	-	(=0)	-	-	(=0)	-
C3H10T1/2	0.49	(>0)	0.1	0.63	(>0)	0.07	0.64	(>0)	0.07

Modeling α

A positive slope $b_1$ of the regression for β suggests that $n=3$, which means α is a quadratic function of $L_F$ . Accordingly, we used a linear regression for $\sqrt{\alpha }$ :

$$\sqrt{\alpha }=a_0+a_1{L}_F$$ (5)

We applied this model to our four datasets. The results are summarized in Table 2.

Table 2.

The slope $a_1$ of the linear model for $\sqrt{\alpha }$ given by Eq. (5) for three SV diameters

Cell line	Diameter=0			Diameter=0.1 µm			Diameter=1 µm
	$R^2$	$a_1$	$p$	$R^2$	$a_1$	$p$	$R^2$	$a_1$	$p$
All cells	0.15	(>0)	9·10−7	0.20	(>0)	1·10−8	0.21	(>0)	7·10−9
V79	0.37	(>0)	3·10−4	0.42	(>0)	8·10−5	0.46	(>0)	3·10−5
HF19	0.62	(>0)	0.1	0.995	(>0)	2·10−3	0.991	(>0)	3·10−3
C3H10T1/2	0.65	(>0)	0.06	0.90	(>0)	9·10−3	0.92	(>0)	6·10−3

The $R$²values indicated again that SV diameters of 0.1 and 1 µm were equally consistent with the experimental data. Using a zero diameter resulted in a worse fit. This suggests that the size of biological targets for cell survival is on the order of 0.1–1 µm. If we exclude the zero size SV, the positive slope $a_1$ for all cases in Table 2 is significant. This eliminated the single-hit model $n=1$.

For those cell types for which the sign of the slope, $b_1$ , was not significant or the slope was set to zero in the original studies^25–27 (HF19), we also tested the $n=2$ model. The regression formula in this case was

$$\alpha =a_0+a_1{L}_F$$ (6)

Fitting the model given by Eq.(6) to HF19 cell data produced $R$²= 0.50, 0.98, and 0.96 for the three SV diameters: 0, 0.1, and 1 µm, respectively. Hence, the three-hit model, Eq. (5) was in better agreement with the experiment than was the two-hit model, Eq.(6). However, the difference between the two models was small. For this reason, we also performed the Box-Cox procedure to determine whether a power transformation of the response variable α was appropriate. The results suggested that the power transformation $\alpha$→${\alpha }^{\lambda }$ was indeed appropriate, with λ≈0.42 maximizing the likelihood. The Box-Cox method is generally taken as a guide in identifying good transformations, and the exact value of λ can be guided by theoretical considerations. This result therefore supports the three-hit model, Eq. (5) where the square root $\sqrt{\alpha }$ represents a power transformation with λ = 0.5. In contrast, λ = 1,which corresponded to the two-hit model, Eq. (6) was outside the 95% CI for λ(−0.02, 0.79).

The final fit

Our analysis supports the three-hit model, $n=3$. This finding is consistent with our analysis of proton β,¹² in which data also pointed to the three-hit model. Models using zero-size SV performed poorly and can be excluded. Models using 0.1 µm or 1 µm diameters achieved similar accuracy, and neither of the diameters could be excluded. For brevity, we report the final regression parameters for the 0.1 µm SV only. Table 3 summarizes our final results for the four sets of experimental data. It also includes the α/β ratio for ⁶⁰Co predicted by the models. Our estimate of the SV size is consistent with previous studies: 0.05–0.1 µm²⁸, 0.84 µm,²⁹ and 0.45 µm¹². Our SV size estimate is also supported by more general arguments. The SV should be larger than 10 nm, because such a small volume cannot be hit by two independent particles at any reasonable doses,¹¹therefore a survival curve, ln S versus dose, would be a straight line (β = 0).¹² On the other hand, the SV size cannot exceed the size of a cell nucleus, and therefore it should be less than 10 µm.

Table 3.

Best-fit parameters for the three-hit ($n=3$) model, Eqs. (4)-(5), and the SV diameter of 0.1 µm.

Cell line	$\alpha$-fit parameters		$\beta$-fit parameters		$\alpha /\beta$ (std. dev.) for 60Co, Gy
	${\alpha }_1$(st.dev.), µm keV−1 Gy-1/2	${\alpha }_0$(st.dev.), Gy-1/2	${\beta }_1$(st.dev.), µm keV−1 Gy−2	${\beta }_0$(st.dev.), Gy−2
All cells
Model 1	0.061 (0.010)	0.384 (0.027)	0.0059 (0.0013)	0.0214 (0.0035)	7.0 (1.2)
Model 2	0.061 (0.010)	0.384 (0.027)	0.0025 (0.0012)	0.0258 (0.0030)	6.2 (1.8)
V79
Model 1	0.047 (0.010)	0.263 (0.031)	0.0167 (0.0024)	0.0041 (0.0073)	7.4 (5.2)
Model 2	0.047 (0.010)	0.263 (0.031)	0.0055 (0.0017)	0.0209 (0.0041)	3.44 (0.81)
HF19	0.0957 (0.0040)	0.721 (0.025)	0	0	∞
C3H10T1/2	0.0422 (0.0068)	0.405 (0.034)	0.0044 (0.0016)	0.0271 (0.0077)	6.2 (1.8)

All experimental data for β and linear regressions Eq. (4) for the two models, 1 and 2, are shown in Figure 1. All of the data for α and the model, Eq. (5), are shown in Figure 2. For V79 cells, the number of (α, β) pairs was 29. Experimental β and the model, Eq. (4), are shown in Figure 3. Similar to Figure 1, the model one in Figure 3 included all data, and model two excluded β = 0.27 Gy⁻². The experimental α for V79 cells and the model, Eq. (5), are shown in Figure 4. Previously, we pointed out that the $\alpha /\beta$ ratio should increase monotonically with increasing LET.¹² In Table 3, only model one for V79 cells did not meet this requirement. We therefore recommend using model two.

Figure 1.

Experimental β for all cell lines; two linear models, both are represented by Eq. (4). Model one included all of the data shown; model two excluded one point, β = 0.27 Gy⁻². Model parameters are shown in Table 3. Data are from multiple studies, and details are shown in Supplementary Material 1. Some data points are omitted for clarity.

Figure 2.

Experimental α for all cell lines; the model is represented by Eq.(5). Model parameters are shown in Table 3. Data are from multiple studies, and details are shown in Supplementary Material 1. Some data points are omitted for clarity.

Figure 3.

Experimental β for V79 cell lines; two models, both are represented by Eq. (4). Model one included all of the data shown; model two excluded β = 0.27 Gy⁻². Model parameters are shown in Table 3. Data are from several studies, and details are shown in Supplementary Material 1. A few data points are omitted for clarity.

Figure 4.

Experimental α for V79 cell lines; the model is represented by Eq.(5). Model parameters are shown in Table 3. Data are from several studies, and details are shown in Supplementary Material 1. A few data points are omitted for clarity.

For HF19 cells, we have four data points for α, and $\beta =0$ for all energies. The reported uncertainties for α were smaller than in other cases, only 2–3%. These data and the model, Eq. (5), are shown in Figure 5. Experimental data and models for C3H10T1/2 cells are shown in Figure 5 for α and in Figure 6 for β. The number of (α, β) pairs in this case was 5.

Figure 5.

Experimental α for HF19 and C3H10T1/2 cell lines; the model is represented by Eq. (5). Model parameters are shown in Table 3. Error bars for HF19 cells are approximately the size of the circles, except for the highest point, for which the uncertainty is not known. Data are from several studies, and details are shown in Supplementary Material 1.

Figure 6.

Experimental β for C3H10T1/2 cell lines; the model is represented by Eq. (4). Model parameters are shown in Table 3. Data are from four studies, and details are shown in Supplementary Material 1.

Finally, in Figure 7, we show the low-dose RBE_M$=\alpha /{\alpha }_{Co60}$ , for the four data sets that we have considered. The RBE was calculated using the best-fit parameters from Table 3. Grey areas in the figure are 95% CIs, and the vertical dashed lines show $L_F$ for several photon sources. In all four data sets, the increase in RBE exceeded model uncertainties for 250 kVp X-rays, and for all lower energy sources. The brachytherapy source ¹⁹²Ir was a borderline case.

Figure 7.

Low-dose $RBE=\alpha /{\alpha }_{Co60}$ for all cell lines, V79, HF19, and C3H10T1/2 cells. The best-fit curves (solid lines) were drawn using parameters from Table 3. The corresponding 95% CIs are also shown (grey area). The vertical dashed lines show the average LET ($L_F$) for selected photon sources.

To summarize, we tested nine models with the number of hits $n$= 1, 2, 3, and the SV diameters of 0, 0.1, and 1 µm. Of these, two models produced best agreement with the experiment: $n$=3 and SV diameters 0.1 and 1 µm. The data did not permit us to exclude either of the two diameters. For brevity, we reported the best-fit results for only the 0.1 µm SV. Agreement of our final model with the experiment was within experimental uncertainties. The model was simple: both α and β were derived from linear regressions, each requiring only two parameters, the slope and the intercept. The model was consistent with experimental data for photon energies from 0.27keV (carbon K_α X-rays) to1.25MeV(⁶⁰Co γ rays).

Our results support the statement that LET is useful for describing “energy deposition of radiation.”¹ We emphasize, however, that for calculating average LET, the spectrum of electrons that actually deposit energy in the SV must be used, which is very different from the spectrum of first-collision electrons. We previously proposed a method for calculating such “actual spectra.”¹⁶ The present study shows how to use the results that were reported in that article. We have also shown that α, β, and the α/β ratio all tend to increase with increasing $L_F$ . The full version of the NPMH model, however, predicts that in the high-LET limit, when $L_F$→∞, both α and β tend to 0, whereas the α/β ratio continues to increase.¹²

This study provides a simple method for RBE estimation and all the data needed for the calculation. The method does not require calculation of electron spectra and even calculation of photons spectra can often be avoided. Our supplementary data, Supplementary Material 1, cover a broad range of photon sources. Therefore, it is likely that for a given photon source a close match can be found in Supplementary Material 1, and the corresponding $L_F$ can be retrieved from the table. Once $L_F$ is known, the next step is to look up regression coefficients in Table 3, and calculate α and β. A limitation is that data for individual cell lines is very limited. Further measurements are highly desirable. Our study offers a methodology for planning such experiments and analyzing survival data. To use the model for medical applications, for example for optimization of brachytherapy treatment plans, the model must be comprehensively tested using clinical data. In its present form, the main medical application of our model is in research, such as retrospective analysis of treatment outcomes.

Conclusions

We constructed a simple model, supported by experimental data, for calculating cell survival RBE for X-rays and γ radiation. In our approach, the average ionization density within the SV determined the biological effect. This density was represented by the frequency average LET, $L_F.$ Guided by the NPMH model, we found expressions for parameters of cell survival α and β that involved only linear functions of the $L_F$ . This makes modeling experimental data remarkably simple. It involves only the standard linear regression analysis. In addition, we determined with a good level of confidence the properties of α and β describing their dependence on ionization density. Finally, we reported best-fit results that can be used to calculate the RBE for any photon source within the energy range that we covered in this study, and to plan further experiments that will reduce gaps and uncertainties in the data and will lead to more accurate RBE models.