The microdosimetric one-hit detector model for calculating the relative efficiency of the alanine pellet dosimeter in low energy x-ray beams

6 The alanine pellet dosimeter is a widely used reference dosimeter in both medical and indus-trial dosimetry across a wide range of beam qualities. A challenge when using alanine in low 8 energy x-ray beams is its strong energy dependence; a signiﬁcant decline is observed in the alanine response per dose-to-water relative to irradiations in a cobalt-60 reference ﬁeld. This decrease 10 is caused by the physical difference in alanine to water dose ratios combined with a radiochemical decrease in the intrinsic detector efﬁciency but is difﬁcult to characterize mainly due to experimen- 12 tal uncertainties. Here we have applied a microdosimetric one-hit detector model to characterize the intrinsic 14 detector efﬁciency of the alanine pellet dosimeter in low energy x-ray beams. Microdosimetric distributions were estimated from track structure calculations using the Geant4-DNA Monte Carlo 16 software, where literature data was used to determine free model parameters. The model was applied to two sets of x-ray spectra with low (40kV to 170kV) and medium 18 (100kV to 300kV) tube potential, covering a wide range of beam qualities. A relative detector efﬁciency of 0.937 was obtained for the low energy set with variations between − 1.0% and 1.5%, 20 whereas the efﬁciency varied between approximately 0.925 and 0.985 for the medium energy set, with a strong correlation to the half-value layer of the beam. 22 It is concluded, that the tube potential and half-value layer of an x-ray beam is not sufﬁcient characterization to uniquely determine the relative efﬁciency of an alanine pellet dosimeter. How- 24 ever, the variation in relative efﬁciency with respect to the half-value layer is small. x-ray ﬁelds to the general dependence of the detector efﬁciency to different beam characteristics.

where R is the detector response and D w is the dose to water. H Q,Q 0 is the ratio of dose to detector material D dos to D w in the photon quality Q relative to the reference quality Q 0 and G Q,Q 0 is the relative detector efficiency The EPR response of the alanine pellet dosimeter irradiated at low energy x-ray qualities relative 52 to cobalt-60 is energy dependent. Recently efforts has been made in characterizing this intrinsic energy dependence from experiments (Zeng and McCaffrey, 2005;Waldeland et al., 2010;Anton and 54 Büermann, 2015; Khoury et al., 2015;Hjørringgaard et al., 2020). All experimental characterizations of the relative efficiency are carried out in well defined x-ray fields. It is not obvious how the transfer 56 from reference conditions to non-reference conditions, such as small self-shielded x-ray emitters, affects the relative efficiency of the alanine pellet dosimeter. Direct measurements of the relative 58 efficiency in these kind of geometries are difficult at best, and other approaches for determining the relative efficiency are desired.
detector efficiency of the alanine pellet dosimeter in kV x-ray fields. This is achieved by applying a microdosimetric one-hit detector model to a wide range of x-ray beam qualities. calculations are available for a selection of beam parameters and materials, e.g. Geant4-DNA (Incerti et al., 2010b), PHITS (Sato et al., 2018) etc. In microdosimetry a target volume in the material is con-76 sidered. An ionizing particle passing through the target volume, producing at least one ionization in the target volume, is called a single event. Single events leading to a production of detector signal 78 (stable free radical formation in alanine) is called a hit. Since the transfer of energy occurs as discrete events (ionizations and excitations) the energy deposited in the target volumes is not uniform but 80 constitutes a characteristic microdosimetric distribution.
The energy ε deposited in a target volume for a single event is related to the number of ioniza-82 tions j within the target volume by ε = j · W, where W is the mean energy required to produce an ion pair in the material. The energy deposited normalized to the mass m of the the target volume is the 84 specific energy z = ε/m in the target volume -the microdosimetric analogue of dose. The stochastic nature of ionizations and the related specific energy motivates the consideration of the frequency 86 distributions f 1 (z). For a frequency distribution normalized to one event ( ∞ 0 f 1 (z) dz = 1) the first moment (mean specific energy) is The subscript 1 refers to a single event. Microdosimetric distributions of ionizations are defined in an analogous way. Single event distributions are independent of the dose, but do depend on track 90 and target volume characteristics, such as size and shape.

The microdosimetric one-hit detector model 92
The microdosimetric one-hit detector model is based on the multi-hit model which can describe inactivation of microorganisms. In multi-hit theory it is assumed that the detector contains a type 94 of target. The target can tolerate a certain amount of hits, however after n hits the target is affected (e.g. cell death, radical formation, trapping of electron in TLD, etc.). It is assumed that the hits occur 96 independently of each other and thus can be described by Poisson statistics. The probability of no effect occurring S as a function of dose D is then (Kellerer, 1987) 98 3 J o u r n a l P r e -p r o o f

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This is the probability that n hits does not lead to an effect in the target.
One-hit detectors are a special case of Equation (6) where the probability of no effect occurring 100 is purely exponential (n = 1, see Figure 7) so that where α is a saturation parameter. In the one-hit model this probability can be expressed in terms 102 of microdosimetric quantities as (Zaider, 1990;Olko, 2002Olko, , 2006) Here z F is the average dose deposited in the target volume by single events, the ratio D/z F is thus 104 the average number of events occurring in the target volume after irradiation with dose D. The function r(z) = 1 − exp(−αz) represents the probability of an effect occurring after irradiation of 106 specific energy z, and is called the response function for a one-hit detector. The integral is then the average probability that the effect takes place in the target volume given a frequency distribution of 108 specific energy f 1 (z).
The normalized detector response R after irradiation with dose D is the complement probability of the probability of no effect occurring By setting the characteristic dose D 0 equal to the detector response of Equation (10) can be simplified as which is the characteristic response function for one-hit detectors (see Figure 7).

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For low doses D z F Equation (10) reduces to and since the relative efficiency (see Equation (1)) is the relative efficiency can be calculated with the microdosimetric one-hit detector model as 4 J o u r n a l P r e -p r o o f Journal Pre-proof parameter α and the target diameter d (assuming spherical volume target). The latter does not appear directly in Equation (15), but the frequency distribution of specific energy is dependent on 118 this parameter.

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The general procedure for implementing the microdosimetric one-hit detector model is outlined below 122 1. A set of monoenergetic electron tracks in water with energy E ranging from 1 keV to 1400 keV is produced. Positions of ionization produced by both the primary electrons and the produced 124 secondaries are scored.

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3. Energy distribution of initial Compton-and photoelectrons produced in, or entering, the detector region for a specific primary photon spectrum is scored. Only the electrons produced 130 directly by the primary photon beam, or entering the detector from a different region, is scored in order to avoid double counting in regard to Step 1. 5. The relative detector efficiency for the primary photon spectra at the irradiation conditions 136 used to obtain the secondary electron spectra is calculated according to Equation (15).
Steps 1 and 3 are performed using MC calculation, while steps 2, 4, and 5 are obtained from post-138 processing of the MC calculated results. An overview of the MC calculations is given in Table 1.

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This general procedure will be applied in different contexts. First, to estimate the free parameters  1 The initial energy distribution of secondary electrons will henceforth be referred to as the secondary electron spectrum.

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J o u r n a l P r e -p r o o f Transport parameters MD: Geant4-DNA physics processes. Specifically G4DNABornIonisationModel for ionizations.
Variance reduction technique (VRT) No VRT was applied.
Scored quantities MD: (x, y, z)-coordinates of ionizations in water.

SES:
Energy distribution of Compton-and photoelectrons produced in or entering alanine pellet.
No. histories MD: Varying significantly to obtain desired number of ionizations.
Statistical methods MD: Electron tracks are calculated until ∼ 2 × 10 5 ionization positions are scored.

SES:
The batch method was used to evaluate statistical uncertainty on energy distribution.
Post-processing Microdosimetric frequency distributions for primary photon spectra are calculated by weighting of the monoenergetic frequency distributions of the secondary electron spectra (see Equation (18)).

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Alanine pellet dosimeters
Since different alanine pellet dosimeters are commercially available, pellets of different size and 148 composition are used throughout the literature. Two different versions of the alanine pellet dosimeter are used for calculations in the present work, and a brief description is given here.

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For calculations where data from Waldeland et al. (2010) is used, the alanine pellets investigated in their work are considered. These pellets consist of 96 % alanine and 4 % unspecified binder, with 152 a height of 3 mm and diameter 4.8 mm. For the calculations the binder was assumed to be paraffin wax and a bulk density of 1.2 g cm −3 was used.

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For calculations concerning irradiations in a cobalt-60 Gammacell, alanine pellets obtained from Harwell Dosimeters were considered. These pellets have height 2.7 mm and diameter 4.8 mm, with a 156 composition of 91 % alanine and 9 % paraffin wax. Calculations for these dosimeters were performed using a dosimeter material composition with bulk density 1.23 g cm −3 .

Microdosimetric distributions for monoenergetic electrons
A set of monoenergetic electron tracks in water was calculated using the Geant4-DNA MC code 160 (Incerti et al., 2010b,a;Bernal et al., 2015;Incerti et al., 2018). The energies range from 1 keV to 1400 keV in steps of 1 keV. The electron tracks consist of (x, y, z)-coordinates for all ionizations 162 produced by the primary electron and the subsequent secondaries. Water (or water vapor scaled to unity density) is typically used for track structure calculations because of the lack of appropriate 164 cross section data for other materials at very low energies.
The number of electron tracks at each energy was chosen such that the total number of ioniza-166 tions produced was of the order 2 × 10 5 , however each individual track was analyzed independently.
The frequency distribution of ionizations for the individual electron energies f 1 (j, E) for target 168 diameters ranging from 5 nm to 30 nm was calculated using the weighted sampling procedure described in Kellerer and Chmelevsky (1975), Kellerer et al. (1985), and Rossi and Zaider (1996). By 170 this method the dose distribution of ionizations d 1 (j) is obtained, and the frequency distribution is calculated according to The frequency distribution of specific energy f 1 (z, E) was obtained by multiplying the number of ionizations j by a chosen W value of 30 eV per ion pair, identical to what was used by Olko (2002), 174 to obtain the deposited energy ε, and normalizing this to the mass of the target volume. In total the specific energy is calculated by where ρ target is the density of the target material, which for the calculation in water is 1.0 g cm −3 .
Examples of the obtained frequency distribution of specific energy for monoenergetic electrons in

Microdosimetric distributions for photon spectra
To illustrate the process of calculating the microdosimetric distributions from primary photon spec-188 tra the following section will be based on a cobalt-60 reference beam. The microdosimetric distributions for the reference beam is required (according to Equation (15)) for calculation of the relative 190 efficiency. The reference beam quality is a Nordion GC220 Gammacell located at Risø High Dose Reference Laboratory (HDRL). The spectral distribution of photons at the central position in the 192 Gammacell was calculated using the FLURZnrc usercode of the EGSnrc MC software (Kawrakow et al., 2017), based on published information on material and geometry of the Gammacell (Hefne,194 2000; Rodrigues et al., 2009).
To obtain the microdosimetric distributions, first the initial energy distribution of secondary 196 electrons produced in the detector material by primary photons must be calculated. The secondary electron spectra were calculated using the Geant4 MC toolkit (Agostinelli et al., 2003). Electrons pro-  The microdosimetric dose distribution of specific energy for the photon spectra were then calculated by folding the monoenergetic electron frequency distributions (see Figure 1) over the secondary Here the superscript Q 0 imply that the equation is valid for the reference cobalt-60 quality, how-212 ever the same equation is applicable for an arbitrary photon beam quality Q. The resulting dose distribution of specific energy is shown in Figure 3 for different target volumes. The shape of the 214 microdosimetric distributions is governed by the ionization density as well as the size of the target volume. For an increase in target size more ionizations may occur within the target volume, 216 however since the target volume increase the specific energy will typically be shifted towards lower  The same process for calculating the microdosimetric distribution of specific energy described in this section for the cobalt-60 reference beam is applied for all x-ray and gamma spectra analyzed in 220 the present study.

Fixing free parameters of the microdosimetric one-hit detector model
Literature values of the relative efficiency of the alanine pellet dosimeter are used to determine 224 the value of the two free parameters of the microdosimetric one-hit detector model α and d. The data used for the analysis is obtained from Waldeland et al. (2010). In their work they used spec-226 tra of the x-ray beam qualities calculated using SpekCalc (Poludniowski and Evans, 2007;Poludniowski, 2007). These spectra are reproduced here using the detailed information about their input 228 for SpekCalc. A list of the beam modalities from Waldeland et al. (2010) is shown in Table 2. For the calculation of x-ray spectra the input voltage was adjusted (on average roughly 14 %) to make 230 the calculated HVL match the measured HVL as done in Waldeland et al. (2010).
Other literature values of the relative efficiency of the alanine pellet dosimeter in this energy 232 range exist, however available information about the beam qualities are insufficient for reproduction using SpekCalc.

Secondary electron spectra
The secondary electron spectrum was calculated for all beam modalities listed in Table 2 based on the 236 source, material, and geometrical information available in Waldeland et al. (2010). The irradiation geometry was implemented in Geant4 with simulation parameters given in Table 1. The alanine 238 pellet dosimeter was placed at 2 cm depth in a water phantom at 50 cm source to surface distance.
The calculated energy distribution of Compton-and photoelectrons produced in, or entering, the 240 alanine dosimeter volume is shown in Figure 4 for the 50 kV and 200 kV beam modalities in Table 2.
It is evident that the secondary electron spectra for the 50 kV beam is heavily dominated by low  Table 2 as well as the cobalt-60 reference (green), also shown in Figure

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The microdosimetric frequency distribution of specific energy is calculated for target diameters in the range 5 nm to 30 nm according to Equation (18). Figure Table 2). The optimal set of model parameters is determined by minimizing the relative least squares of the calculated and 268 experimental values where the sum is over all beam qualities in the set.

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The obtained values of the parameter M as a function of the saturation parameter α is shown in Figure 6. The optimal saturation parameter, for each target diameter, is then obtained by locating the  278 As an additional constraint on the choice of free parameters, the predictive ability of the model is

Linearity index of cobalt-60 reference
Here the reference dose point is chosen to be D w = 1 kGy, corresponding to D ala = 0.97 kGy.
284 Figure 7 show the calculated response and linearity curves for the free parameter sets obtained in Figure 6 as well as the response curve for pellets irradiated in a cobalt-60 Gammacell, normalized to 286 the saturation response. The relative difference between model and experimental values calculated where X is the physical quantity investigated. The comparison of model and experimental data in  it is adopted for further use in the model.
This comparison of experimental response curves normalized to saturation response with re-294 sponse curves calculated using the microdosimetric one-hit detector model represents an assumption that the saturation of response is only due to a saturation in stable free radical production at 296 high doses. This assumption implies that the EPR-readout, in this case the peak-to-peak height in the first derivative of the EPR-spectrum, is proportional to the concentration of stable free radicals,  (Wieser and Girzikowsky, 1996;Malinen et al., 2003). Here the comparison is used to pick out a set of model parameters, from a list 302 of parameter sets which all, according to Figure 6, reproduce literature values at kV x-ray energies reasonably well.

Uncertainty considerations
In the following section the considerations and handling of uncertainties for the model is described.

Monte Carlo calculations
For MC calculations of radiation transport uncertainties typically include components from trans-308 lation from laboratory conditions to MC geometry and materials, cross-section data, input physics, and statistical effects. In the present study the relative detector efficiency is calculated according 310 to Equation (15), where z F and f 1 (z) are obtained through MC calculations. Since Equation (15) is expressed as a ratio with these parameters appearing in both the numerator and denominator, and 312 these parameters are obtained using the same MC code and input physics, it is assumed that the contribution to the final uncertainty of the relative efficiency is negligible compared the contribution 314 from the applied experimental data.

Literature values 316
A significant contribution to the overall uncertainty comes from the use of literature data based on measurements to fix the free parameters of the model (see Table 2). The effect of experimen-318 tal uncertainties on the determined set model parameters is examined by performing a bootstrap analysis. This analysis was performed keeping the target diameter fixed at d = 10 nm and scoring 320 the corresponding optimal saturation parameter for 1 × 10 5 random samples, with replacement and including uncertainties, of the data points.

322
The distribution of optimal saturation parameters obtained from this analysis was best described by a log-normal distribution. By fitting a log-normal distribution to the collection of optimal satura-324 tion parameters an upper and lower limit, defined by the 2.5% and 97.5% percentile corresponding to two standard deviations for a normal distributed parameter, was estimated. The saturation pa-326 rameter obtained by this analysis was α = (2.86 +3.11 −1.65 ) × 10 −5 Gy −1 .

Characterization of the alanine pellet dosimeter 328
The parameter values obtained in Section 4.1 can now be applied for relevant primary x-ray spectra in order to calculate the relative efficiency of the alanine dosimeter in that particular beam quality. In The calculation of secondary electron spectra was performed using the same geometry, and alanine pellet size and composition, as was used for fixing the model parameters in Section 4.1, 334 based on the information from Waldeland et al. (2010).

Monoenergetic photons 336
The secondary electron spectrum was calculated with the Geant4 MC toolkit for a range of monoenergetic primary photons. Using Equation (15)

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The bottom part of Figure 8 show the corresponding mean specific energy calculated according to Equation (5) using target diameter d = 10 nm. The mean specific energy is a measure for how 344 localized the dose deposition is. It is worth noting that the local minimum and maximum apparent in the energy dependence of the relative efficiency -at 20 keV and 50 keV respectively -directly 346 corresponds to the opposite extrema in the mean specific energy. The bump in the curve for mean specific energy occurs as the fraction of secondary electrons produced by Compton scattering in-348 creases, since the low energy electrons produced through Compton scattering deposit their energy more locally. The capabilities of the model for calculating the relative efficiency of the alanine pellet dosimeter 352 in low energy x-ray fields was tested by comparison with literature data. In Figure 8 the relative efficiency measured in Zeng and McCaffrey (2005), Waldeland et al. (2010), Anton and Büermann 354 (2015), and Hjørringgaard et al. (2020) is presented as a function of the effective energy of the respective x-ray beams. The data on relative efficiency from Anton and Büermann (2015) is obtained 356 by taking the ratio of relative response to MC calculated dose ratios listed in Table 6 and Table 7 of their paper.

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By visual inspection of Figure 8 good agreement is observed between model calculated relative efficiency for monoenergetic x-rays and experimental data characterized by the effective energy of 360 the x-ray beam. The overall trend with local maximum and minimum around 20 keV and 50 keV appear to be present in the experimental data as well.

362
Statistical examination of the model calculations was based on the null hypothesis that the relative efficiency is unity over the energy range covered by literature data. This choice of null hypoth-364 esis is based on the fact that the relative efficiency converges to unity for higher energies. Due to the small sample size in the literature data, the analysis was performed using bootstrapping includ-366 ing the experimental uncertainties. For each set of resampled data the mean residual of both the unity model of the null hypothesis and the microdosimetric model to the bootstrapped sample was 368 scored. The difference between the two models was determined to be statistically significant, and it is thereby assumed that the microdosimetric model is a better representation of the experimental 370 data.
It is of course not obvious that effective energy as a beam qualifier is sufficient classification of 372 beams to determine the relative efficiency. As such a direct comparison of relative efficiency for monoenergetic photons to effective energy of composite x-ray fields may not be optimal. Lack of 374 knowledge about the spectral distribution for the literature data does however make this comparison the most reasonable. How the relative efficiency depends on other beam characteristics than just the 376 effective energy can now be investigated using the microdosimetric one-hit detector model.

378
Differences in x-ray tube geometry -filtration material and thickness, target material and angle, etc.
-entails a wide range in HVL values for the same tube potential. From a survey on the status of 380 clinical x-ray dosimetry in North American clinics the variation in HVL for x-ray tubes with tube potential 10 kV to 300 kV is obtained (see Figure 2 of Ma et al., 2001). This variation gives an indication 382 of the x-ray beam quality range for which the dosimeter properties should be characterized. Based on the model parameters determined in this study, the general dependence of the relative 384 efficiency of the alanine pellet dosimeter on different x-ray beam qualifiers is investigated. A set of primary x-ray spectra was generated using the SpekCalc software. The x-ray spectra was of varying 386 high voltage (HV), 40 kV to 300 kV, and HVL, the latter obtained by varying the external filtration.

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J o u r n a l P r e -p r o o f Journal Pre-proof and anode angle 30 • . The variation in HVL is obtained by changing the external aluminum or copper filtration. The range in input parameters for the calculation of x-ray spectra was chosen such 390 that the generated spectra cover the range of beam qualities listed in Ma et al. (2001).
The relative detector efficiency for each beam quality was calculated in the same manner as 392 described in previous sections. Figure 9a show the calculated relative efficiency calculated for the set of x-ray spectra with tube potential 40 kV to 170 kV and HVL ≈ 0.2 mm Al to 14 mm Al.

Discussion
In the present study the microdosimetric one-hit detector model was applied to the alanine pellet 408 dosimeter to calculate the relative efficiency in low energy x-ray beams with respect to a cobalt-60 reference field. Literature values obtained from Waldeland et al. (2010) were used to fix the 410 free parameters of the model, the target diameter d and the saturation parameter α. Using only 18 J o u r n a l P r e -p r o o f determine an optimal parameter set, therefore an additional constraint concerning the prediction of response curves for pellets irradiated in a Cobalt-60 Gammacell was included. The optimal 414 set of model parameters determined in this manner was obtained to be d = 10 nm in water and α = 2.86 × 10 −5 Gy −1 . A different value for the mean energy required to produce an ion pair W 416 chosen for the analysis might result in different sets of model parameters. Olko (2002) arrived at a target diameter of d ala = 6 nm corresponding to 8-9 nm in water, converting by density ratio.

418
As discussed by Olko (2002) this order of target diameter is much larger than the size of an alanine molecule implying the saturation effect at high doses are not due to a lack of unionized 420 alanine molecules. Olko (2002) argue that other effects of ionizing radiation -cross linking, coiling of chains -can trap the free radicals, preventing recombination at normal dose ranges. At high doses, 422 or high Linear Energy Transfer (LET), these structures are destroyed, leading to recombination and a reduced detector efficiency. In this case the target size is interpreted as the effective range of 424 recombination of the free radicals.
The comparison between calculated response, and linearity, curves with measured response 426 curves for alanine pellet dosimeters irradiated in a cobalt-60 Gammacell include an underlying assumption that the measured EPR response (peak-to-peak height of the first derivative of the EPR 428 spectrum) is directly proportional to the concentration of stable free radicals. This assumption implies that the observed saturation in the EPR response is only due to saturation in the production 430 of stable free radicals. Effects of the readout procedure, such as choice of microwave power and modulation amplitude, may influence the relative spectrometer sensitivity for low and high radical 432 concentrations, and thus affect the dose level at which saturation occurs. This effect would need to be characterized for consistency. In the present study the comparison is applied to pick a set of 434 model parameters from a list of model parameter sets which all reproduce the kV x-ray literature data for relative detector response reasonably well.

436
The agreement between model calculated relative efficiency of the alanine pellet dosimeter for monoenergetic photons in the energy range 5 keV to 1000 keV and experimental data was assessed 438 by both visual inspection and by performing a bootstrapping analysis. This analysis showed the difference between the model calculations and the null hypothesis of a unity relative efficiency was 440 statistically significant. The use of effective energy as the single beam qualifier may however be an oversimplification as shown in Section 4.3.2.

442
Applying the microdosimetric one-hit detector model for a set of monoenergetic primary photon beams showed a similar anomalous dependency of the relative detector efficiency on photon energy.

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The energy dependence showed a local extremum at 25 keV and 50 keV. The characteristic anomalous shape occurs as with increasing photon energy the fraction of secondary electrons produced 446 by Compton effect increases. The same anomalous shape is obtained by Olko (1999Olko ( , 2002, however they report local maximum and minimum at 40 keV and 80 keV respectively. This difference may be 448 due to the difference in applied model parameters as well as the choice of alanine detector material. Olko (2002) use a mixture of 90 % alanine and 10 % paraffin wax in an unspecified geometry for 450 19 J o u r n a l P r e -p r o o f calculating the secondary electron spectra, whereas a mixture of a mixture of 96 % alanine and 4 % paraffin wax has been used in the present study.

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The microdosimetric one-hit detector model has further been applied to two sets of generated x-ray spectra covering a wide range of x-ray beam qualities used in clinics. The low energy set was 454 generated with HV in the range 40 kV to 170 kV and HVL from 0.2 mm Al to 14 mm Al, and the medium energy set consist of HVs from 100 kV to 300 kV with HVL from 0.2 mm Cu to 5 mm Cu.

456
This study was performed to explore the usefulness of different beam qualifiers for characterization of the relative detector efficiency. The relative detector efficiency in the low energy set varies within 458 −1.0 % and 1.5 % of the average value of G Q,Q 0 = 0.937 for the investigated beam quality range.
This indicates that the tube potential may be sufficient for practical use in choosing a literature 460 value of the relative detector efficiency to apply to measurements. However further investigation of the influence of e.g. phantom material, alanine pellet position in phantom, etc. should be explored 462 in detail. For the medium energy set of generated x-ray spectra a significant dependence of the relative detector efficiency on the HVL was observed. For HVL > 3 mm Cu the relative detector 464 efficiency appear to be independent on the x-ray tube potential. The reason for this is probably that the fraction of low energy electrons generated in the detector by these hard filtered beams is quite 466 low for all the investigated tube potentials making variations between spectra insignificant.
Several aspects of the model calculations which may impact the final model prediction of the 468 relative efficiency have not been considered in the present study. MC calculation at very low energies should be interpreted with reservations regarding interaction cross-sections. Different ionization 470 cross-section models for very low energies are available in the Geant4-DNA MC toolkit (Bernal et al., 2015). The effect on the model predicted relative efficiency of the alanine pellet dosimeter from using 472 different low energy ionization cross-section models has not been explored in the present study.
Only cross sections for water are available for the calculation of the microdosimetric distributions, 474 which is a major limitation in the application of this model.
The recombination of free radicals has been shown to be dependent on the beam quality (Hansen 476 and Olsen, 1989). For heavy charged particles high LET beams show significantly greater fading compared to lower LET beams. The same effect could be present for x-rays, where low energy x-rays 478 have greater LET (by secondaries) relative to high energy x-rays. This effect may be of importance when assessing the detector response, but have not been investigated here.

Conclusion
The microdosimetric one-hit detector model was applied to calculate the relative efficiency of the 482 alanine pellet dosimeter. The free parameters of the model was determined by comparison of model calculations with literature data (Waldeland et al., 2010) on the relative efficiency.

484
The model was then applied to a set of monoenergetic photon beams, showing a characteristic relationship between the relative efficiency and monoenergetic photon energy. The relative efficiency 486 was found to be overall increasing with photon energy, but with a local maximum and minimum at 20 J o u r n a l P r e -p r o o f Lastly, the model was applied to a wide set of x-ray beam qualities. The results showed that the effective energy (or HVL) of an x-ray beam is not necessarily sufficient uniquely to determine the 490 relative efficiency of the alanine pellet dosimeter.