Conversion coefficients for H′(3;Ω) for photons

In this work, conversion coefficients for the operational quantity H′(3;Ω) have been calculated for both mono-energetic photons from 2 keV to 50 MeV for angles of incidence from 0° up to 180° in steps of 15° (to complement ICRU 57) as well as for photon reference radiation qualities (to complement ISO 4037). Finally, parameters necessary to determine the influence of the air density on the conversion coefficients have been determined.


Introduction
In its report number 57 on conversion coefficients for use in radiological protection [1], the International Commission on Radiation Units and Measurements (ICRU) has published conversion coefficients for the operational quantities in radiation protection for mono-energetic particles (photons, electrons, and neutrons). However, data for the quantity directional dose equivalent at 3 mm depth, H′(3;Ω), for photons are lacking in that report. Consequently, the standard on photon reference radiation qualities, ISO 4037-3 [2], from the International Organization for Standardization (ISO) does not contain corresponding data for the radiation qualities used for calibrations. Therefore, in this work, those data for mono-energetic photons have been calculated and folded with the spectra of the X and gamma radiation qualities defined in ISO 4037 [2][3][4][5]. Finally, the influence of the air density on the spectral distributions and consequently on the conversion coefficients was determined by applying the exponential attenuation law for photons to the spectra. Subsequently, the conversion coefficients were calculated for air densities from ρ=0.96 kg m −3 to ρ=1.32 kg m −3 to obtain the corresponding correction factors for the conversion coefficients that were calculated [6].

Geometry and calculations
The operational quantity H′(3;Ω) at a point in a radiation field is defined as 'the dose equivalent that would be produced by the corresponding expanded field in the ICRU sphere at a depth of 3 mm on a radius in a specified direction, Ω′ [7,8]. An expanded radiation field is defined as a hypothetical radiation field in which the fluence and its angular and energy distributions have the same value throughout the volume of interest as in the actual field at the point of [7]. The ICRU sphere is a tissue-equivalent sphere, 30 cm in diameter, with a density of 1.0 g cm −3 and a mass composition of 76.2% oxygen, 11.1% carbon, 10.1% hydrogen and 2.6% nitrogen [9], i.e. '4-element ICRU tissue'. Calculations have been performed in accordance with this definition for a homogeneous and unidirectional radiation field in vacuum impinging on the ICRU sphere, i.e. a circular radiation field with a diameter of 30.2 cm. This calculation method is equivalent to that used for the data for H′ (0.07;Ω) in ICRU 57 [1]. In the following, the direction of radiation incidence is called α: 'In the particular case of a unidirectional field, the direction can be specified in terms of the angle, α, between the radius opposing the incident field and the specified direction.', see section 2.6.3 of ICRU 57 [1]. This nomenclature is in line with ISO 4037-3 [2]. Accordingly, H′(3;α) will be used throughout the rest of this paper instead of H′(3;Ω). Figure 1 illustrates the scoring regions at 3 mm depth inside the sphere: 13 regions representing angles of incidence from 0°up to 180°in steps of 15°; the regions extend from a radius of 14.699 cm to 14.701 cm (i.e. Δr=20 μm in thickness) and Δζ=±2°around their nominal angle (i.e. an extent of about 10 mm). With this geometry, one calculation per photon energy simultaneously yielded results for all 13 angles of incidence (due to the symmetry of the sphere and the radiation field). Calculations were performed for monoenergetic photons from 2 keV up to 50 MeV for several energy values. In earlier calculations for conversion coefficients for operational quantities kerma approximation had been used (ICRU 57 [1]), therefore, it was also used for the calculations in this paper, i.e. the electron transport was switched off, resulting in charged particle equilibrium. All calculations were performed using the EGSnrc code package, version V4-r2-4-0 (version as of 2013; Fortran based code including the C++ class library extension) [10,11] and the resulting absorbed dose at 3 mm depth, D(3 mm), per incident fluence, Φ, was determined. As the quality factor, Q, for photons is 1 Sv Gy −1 [8], the conversion coefficient for mono-energetic photons of energy E results in By division with the kerma factor, K a /Φ, the kerma-to-dose conversion coefficient is obtained: Figure 2. Ratio of the values in ICRU 57 and this work for h′ K (0.07;E;α) and h′ K (10;E; α) for different angular directions, α.
where K a is the collision air kerma and μ en /ρ is the energy absorption coefficient [12] for mono-energetic photons of energy E. Data for the latter are listed in In order to make sure the simulations are reliable, additional scoring regions around 0.07 and 10 mm depth were used to determine h′ K (0.07;E; α) and h′ K (10;E;α), respectively. Figure 2 shows a comparison of these data with the corresponding values in ICRU 57 [1], see tables A.23 and A.22, respectively, therein. No uncertainties are stated in ICRU 57, the statistical uncertainties of the values in this work are smaller than the symbols in figure 2 and, therefore, not shown. As the calculations for ICRU 57 had been performed with limited computer power it is assumed that the 'outliers', e.g. at d=0.07 mm, α=60°, E<10 keV and at d=10 mm, α=180°, E=100 keV, have their reason in rather large statistical uncertainties of the data in ICRU 57. This assumption is supported by the fact that these 'outliers' are not systematic. In contrasts, the systematic deviations are present at energies below about 50 keV and at angular directions around 90°. The reason probably is that the values in ICRU 57 were determined using different dose scoring volumes: for the h′ K (10;E;α) data Δr=2 mm and Δζ=±5°was used instead of 0.02 mm and ±2°, respectively, in this work. This has already been acknowledged by Grosswendt and Hohlfeld [13], which is the reference to table A.22 in ICRU 57: 'The use of such ring shaped finite volume elements in calculating a point quantity such as dose equivalent may lead to some deviations in the angular distribution especially for angles near 90°because of the resulting grazing incidence of photons into the corresponding volume elements of the sphere. This is particularly true in the case of photons with energies of some tens of keV due to the low penetration in matter which causes a large gradient in the photon fluence even in the case of small penetration depths' [13]. To make sure the calculations from this work do not suffer from similar problems, a second method of validation was employed.
2.2.2. Use of different calculation geometries. In addition to the geometry shown in figure 1, a different geometry was used with the incident radiation field rotated by 90°about the top to bottom axis in figure 1. The resulting geometry is shown in figure 3. In that geometry, the former 0°scoring region becomes the 90°region. Calculations were performed for several energies below 100 keV. The results for 90°are consistent within the statistical uncertainties (usually below 1%) with the result from the calculations according to figure 1. Thus, it is concluded that the calculations in this work do not suffer from systematic errors due to the ring shaped scoring elements. As outlined above, the scoring regions have a maximum extent of about 10 mm (but 20 μm in depth). To make sure the results do not suffer from this, additional calculations with much smaller scoring regions having a maximum extent of 5 μm were performed at several energies, especially below 20 keV. In the results, no systematic deviations from the original calculations were observed in the reference depth of 3 mm.

Results for mono-energetic photons
The resulting values for h′ K (3;E;α) are shown in figure 4 and listed in table A2 in the appendix. The statistical uncertainties are usually much smaller than 1%, except for h′ K (3;E; α) <10 −4 Sv Gy -1 where they are larger. For comparison, in figure 5 the values for h′ K (0.07;E;α) and h′ K (10;E;α) are shown. The following features can be seen: • Above about 1 MeV, h′ K (d;E;α) is quite similar for all three scoring depths of d=0.07, 3, and 10 mm, as photon absorption within the first 10 mm of the ICRU sphere is not significant. In addition, as secondary electrons were not transported, dose build-up is completed in the whole sphere and, therefore, at all scoring depths. • The maximum of h′ K (d;E;α) at 70 keV and at small angles of incidence increases with increasing depth from about 1.6 Sv Gy −1 up to about 1.8 Sv Gy −1 at 0.07-10 mm, respectively. This small 'photon dose build-up' effect is probably due to the increasing contribution of photons scattered in the sphere in front of the scoring volume into the forward direction. • At α=90°and around 70-100 keV, h′ K (d;E;α) decreases with increasing depth from about 1.3 Sv Gy −1 down to about 0.9 Sv Gy −1 at 0.07-10 mm, respectively. This is due to the increasing path length in the sphere from its surface to the scoring volume: for d=0.07, 3, and 10 mm, the path length at α=90°is 4.6, 30, and 54 mm, respectively. For α>90°, the path length difference becomes less significant resulting in rather similar values of h′ K (d;E;α) for all three d. • Below about 40 keV, photon absorption in the sphere is significant resulting in smaller values of h′ K (d;E;α) for larger values of the scoring depth d.
Also for comparison, in figure 6 the ratio of h′ K (3;E;α) from this work and h pK (3;E; α) cylinder from Vanhavere et al [14], the latter for the cylinder phantom, is shown. The ratio gives the difference between the operational quantities for area and individual monitoring to estimate the dose to the lens of the eye. h′ K (3;E;α) is smaller than h pK (3;E;α) cylinder as it was calculated in the ICRU sphere with a diameter of 30 cm while h pK (3;E;α) cylinder was calculated in a cylinder made of the same material (4-element ICRU tissue) but only 20 cm in diameter. Accordingly, the path length to the reference point at 3 mm depth through material is larger for the sphere than for the cylinder for oblique incidence with the difference getting larger for larger angles of incidence: for α=0°, there is no difference (i.e. 3 mm for both phantoms resulting in the same values for both phantoms) while for α=180°, the path length is 29.7 cm for the sphere and only 19.7 cm for the cylinder. Consequently, the difference between h′ K (3;E;α) and h pK (3;E;α) cylinder increases for larger values of α and as well for smaller photon energies as the absorption increases for smaller photon energies.

Calculation of spectrum averaged conversion coefficients
Spectrum averaged values of the conversion coefficients, h′ K (3;R;α), were obtained by multiplying the fluence spectra of photon reference radiation qualities, R, by the corresponding conversion coefficients for mono-energetic photons from table A1: where K a (R;E i ) is the spectral air kerma of the radiation quality R at photon energy E i and N is the number of energy channels of the spectrum. The sources of the X and gamma radiation spectra are given in table 1. The conversion coefficients for mono-energetic photons are given for discrete photon energies only. A linear interpolation was applied to obtain values of h′ K (3;E i ;α) for photon energies E i between the discrete values, except for energies between 50 and 100 keV and α75°; a natural cubic spline was used in order to better follow the local maxima of the values.

Results for photon spectra
The resulting values for h′ K (3;R;α) are listed in the appendix in tables A3(1) and (2) for a distance between the radiation source and the point of test of 1.0 m and 2.5 m, respectively. In table A3(2), i.e. for a distance of 2.5 m, values are only given in case they deviate from those in 1.0 m by more than 0.5%. This is also the case for the mean energies. The deviations occur because the additional air path of 1.5 m results in scattering and absorption of-especially low energy-photons. This, in turn, hardens the photon spectra, i.e. the mean energy increases. Significant deviations occur at small energies and at large angles of incidence. This is due to the fact that here the conversion coefficient h′ K (3;E;α) strongly depends on the photon energy.
The shape of the x-ray spectra affects the actual value of the conversion coefficient (especially below 30 kV tube voltage). Therefore, before applying the values, it should be ensured that the spectra produced are similar to the ones used in this work [15].

Method
The spectrum of low energy photon radiation qualities depends on the air density during an irradiation as, for example, a larger air density results in more absorption and scattering Table 1. Sources for the spectra used.

Type of radiation quality
Radiation qualities and abbreviation a Source of spectra X radiation qualities with high voltage up to 300 kV Low air kerma rate series: L-10 up to L-240 Narrow spectrum series: N-10 up to N-300 Wide spectrum series: W-60 up to W-300 High air kerma rate series: H-10 up to H-300 Catalog of x-ray spectra [15] X radiation qualities with high voltage above 300 kV Narrow spectrum series: N-350 up to N-400 High air kerma rate series: H-350 up to H-400 Ankerhold [16] Gamma radiation qualities from radioactive sources Photons from 137 Cs and 60 Co: S-Cs and S-Co EGSnrc code package [17] Gamma radiation qualities from nuclear reactions Photons from the de-excitation of 12 C and 16 O: R-C and R-F Behrens et al [18] a Where available, the abbreviations are taken from ISO 4037-1 [3], otherwise from the corresponding reference given in col. 3. between the radiation source (usually an x-ray tube) and the point of test. Therefore, the spectra of all radiation qualities with a mean energy below 40 keV were calculated for air densities from ρ=0.96 kg m −3 to ρ=1.32 kg m −3 by applying the exponential attenuation law for photons. As basis, the spectra at reference conditions (see table 1), i.e. at ρ ref =1.1974 g cm −3 , were used. From the resulting spectra at different air densities the conversion coefficients h(ρ) have been calculated. The corresponding correction factor is given by where ρ is the considered air density, ρ ref is the reference air density, and h(ρ ref ) is the conversion coefficient calculated in the previous section, i.e. given in table A3. The corresponding correction factor for the quantity air kerma is The dependence of the conversion coefficients on the air density is approximately linear resulting in is the slope for an air path d air between the source and the point of test. For the quantity air kerma, K a , the following equation applies where d MC is the distance between the source and the monitor chamber to determine the air kerma during an actual irradiation, see ISO 4037-2 [4].
for the conversion coefficients h, and to (by inserting equation (6) into equation for the air kerma K a . The approximation via the slopes m(1.0 m) and m d results in errors no larger than 1% for the ranges of air densities specified in the tables of results, see below.
As the operational quantities are given by the product H=K a ·h, the corresponding correction factor is given by the product of the two contributions: The dose during an irradiation finally results from the dose under reference conditions in for the operational quantity H. Further details and examples for the calculation of the correction factors are outlined in a previous publication [6].

Results for the correction factors for air density
The parameters m(1.0 m) and m d are given for low energy photon reference radiation qualities in the appendix • in tables A4 (1) and (2), respectively, for h′(3;R;α); and, as the following values have not been published earlier, • in tables A5 (1) and (2), respectively, for h p,K (3;R;α) cylinder -defined in the 4-element ICRU tissue cylinder, 20 cm in diameter and 20 cm high [20]; and • in tables A6 (1) and (2), respectively, for K a ,h * K (10;R), and h p,K (10;R;α) slab -the latter defined in the 30×30×15 cm 3 4-element ICRU tissue slab [1]. Where available, the values were taken from an earlier publication [6].
The parameters are given for those radiation qualities only, for which the conversion coefficient itself is at least 0.0001 Sv Gy −1 .

Summary
In this work, the complete data set necessary to perform accurate photon irradiations in terms of H′(3;Ω) is presented: • conversion coefficients for mono-energetic photons from air kerma, K a , to H′(3;Ω): h′(3; E;α), ready for adoption by ICRU; • conversion coefficients for photon reference radiation qualities, h′(3;R;α), ready for adoption in ISO 4037-3 [2]; and • correction factors for the conversion coefficients h′(3;R;α)-and for several other quantities such as h p,K (3;R;α) cylinder , h * K (10;R), h p,K (10;R;α) slab , and K a -to account for the actual air density during an irradiation, ready for adoption in ISO 4037-4 [5].
All data for h′(3;α) are given for angles of radiation incidence, α, from 0°up to 180°in steps of 15°.

Acknowledgments
The author wishes to thank Oliver Hupe (PTB) for the motivation to undertake this work and Peter Ambrosi (formerly at PTB) for valuable contributions to the manuscript. Table A1 provides kerma coefficients used for the conversion from dose per fluence to dose per air kerma, see equation (2). The values for (μ en /ρ) were taken from Hubbell and Seltzer and E in MeV was used. The number 160.21 follows from the conversion of MeV cm 2 g −1 to pGy cm 2 . Table A2 provides data for h′ K (3;E;α) for mono-energetic photons of energy E. Interpolation between values at given energies should be performed as stated in section 3.1. Table A3 provides data for h′ K (3;R;α) for reference radiation qualities, R. In addition, the fluence weighted mean energies are given.

Appendix. Tabulated results
Tables A4-A6 provide values for m(1 m) and m d for the determination of the air density correction, see equation (6) and corresponding text for details.    - -       (1) Parameter m(1.0 m) for the simple approximation of m(d air ) given in equation (6) for air densities from ρ=0.96 kg m −3 to ρ=1.32 kg m −3 for h pK (3;R;α) cylinder -defined in the 4-element ICRU tissue cylinder, 20 cm in diameter and 20 cm high [20].