Photon skyshine from medical linear accelerators

Abstract A widely used formula for the prediction of photon skyshine has been shown to be very inaccurate by comparison with numerous measurements. Discrepancies of up to an order of magnitude have been observed. In addition to this, the formula does not predict the observed dependence on field size, nor the fact that skyshine dose rates exhibit a local maximum. A scaling formula is derived here, with a single fitting parameter, which properly accounts for these properties, provides physical insight into the skyshine phenomenon, and is more accurate. The location of the maximum dose rate depends on the ratio of the roof height above isocenter to the distance from the isocenter to the outer surface of the sidewall. For nominal linac room dimensions, the maximum dose occurs at a distance from the outer wall of approximately two times the height of the roof above the isocenter. The skyshine dose rate is proportional to the field area and not Ω1.3, as predicted by the standard formula, where Ω is the solid angle subtended by the beam. For lightly shielded roofs (concrete thickness less than about 0.5 m), the photon skyshine for 6 MV exceeds that for 18 MV. Evidence is presented that at intermediate distances the skyshine declines as one over the distance and not one over the distance squared. Predictions of skyshine dose rates depend critically on accurate knowledge of the roof transmission factor. If a roof is shielded so as to avoid designation as a “high radiation area,” photon skyshine will be negligible.

Skyshine parameters from some of the references cited above can be found in Table 1. The quantity d i is the distance from the target to a point 2 m above the top surface of the roof (see Fig. 2), h is the distance from the isocenter to the top surface of the roof and d max is the distance from the isocenter at which the skyshine dose rate has its maximum value. Figure 1 shows that _ H rises rapidly with distance d s just beyond the outside wall, reaches a peak, and then declines with increasing distance. This implies that survey measurements should not just be made at a distance of 30 cm beyond the outer barrier, as for radiation transmitted through the side wall, but at distances of up to 15 m. 1 It has been stated in the literature that the maximum dose rate occurs at a distance from the outer surface of the side barrier about equal to the height of the barrier. 1,5 The data in the last column of Table 1 show that, to a good approximation, the maximum actually occurs at a distance from the outer wall surface of approximately twice the distance from the isocenter to the roof surface.
Gossman, et al state that the value of d max depends on field size but this is not apparent in the Elder et al data. 2,4 It will be shown below that the location of the maximum dose is expected to depend on the ratio h/d w .
With the exception of the Gossman et al. data, the scaled 6 MV _ H is largest followed by 10 MV and then 18 MV. The physical interpretation of this will be discussed later in this paper. For the Gossman data, the calculated value of B xs has been used for scaling, whereas for the Elder data the value of B xs was actually measured. 2 Linac photon skyshine predictions have been based on a widely quoted empirical equation that is reproduced here for reference. [1][2][3][4][5][6] The geometry is shown in Fig. 2. The dose equivalent rate in nSv/h at a distance d s (in meters) from the isocenter is: where, B xs = roof shielding transmission factor for photons; Ω = the solid angle subtended by the beam (in steradians); _ D 0 = x-ray absorbed dose rate at a distance of 1.0 m from the target (Gy/h); d i = the vertical distance from the x-ray target to a point 2 m above the top of the roof (in meters); d s = distance from the isocenter (in meters).
The solid angle Ω subtended by a square field of side length a (at a distance of h i from the source) is: 7 This is evidence that _ H is directly proportional to field area. This is consistent with Fig. 2   This is acknowledged within NCRP 151 in Table 5 Table 1). It can be seen that Eq. (1) does not even reproduce the qualitative features of the measurements and is grossly in error at d max except for 18 MV.
In view of the poor predictive value of Eq. (1), it is desirable to find a simple, approximate scaling law for the dose rate for skyshine photons as measured at a distance d s (shown in Fig. 2) from the isocenter. 2 The number of photons scattered per unit time toward a detector subtending solid angle ΔΩ is given by: where n is the number of scattering centers, _ Φ is the fluence rate (number of incident photons per unit area per unit time) and dσ/dΩ is the differential cross section for Compton scattering. 8 Let us T A B L E 1 Skyshine parameters.

Authors
Beam energy (MV) Field size F 0 (cm 2 ) compute the contribution from a scattering volume element that is a cross section of the beam with thickness dz as shown in Fig. 2. In this case, n = ρ e F(z) dz, where ρ e is the electron density (per unit volume) of the air, F is the beam cross sectional area at distance z from the target.
where _ Φ 0 is the fluence rate at the isocenter (1.0 m from the source), z is measured in meters from the target and B xs is the transmission through the roof. The dose rate at the isocenter _ D 0 can be expressed in terms of the fluence rate as where we assume monoenergetic photons of energy E γ and mass-energy attenuation coefficient μ en =ρ ð Þfor water.
We assume that the transmission factor B xs is relatively high so that most of the photons passing through the roof are unscattered. It is also assumed that the air does not significantly contribute to attenuation of the primary beam. The photon fluence rate reaching the detector at point P in Fig. 2 due to scattering from the volume element of thickness dz is The dose rate at point P due to the fluence rate reaching point P We assume that the side walls of the structure are completely opaque to skyshine radiation. This seems valid given the low energy of these photons. With this assumption, the only scattered photons that can reach point P must originate at some minimum distance above the roof. The minimum scattering angle at a distance d s from isocenter is: The skyshine is calculated at the vertical height of the isocenter, which is usually 1.3 m. For the differential cross section dσ/dΩ, the scattering angle is between 90°and 180°, we assume that E γ >> m 0 c 2 even though this assumption is marginal at low energies. Putting all of the pieces together, the contribution to the dose equivalent rate d _ H from a scattering element of thickness dz is: The total instantaneous dose equivalent rate can be written in terms of the scattering angle θ: where r e is the classical electron radius. In principle, the mass attenuation coefficient for scatter, should remain inside the integral as it depends on the angle of scattering.

| RESULTS AND DISCUSSION
Carrying out the integration in Eq. (6), and scaling the field size to 20 × 20 = 400 cm 2 and the dose rate to 400 cGy/min at isocenter results in: where _ H is the instantaneous dose rate in units of nSv/s, k is an energy dependent proportionality constant, F 0 is the field area at isocenter expressed in cm 2 , _ D 0 is the dose rate at isocenter expressed in cGy/min, d s is the distance from the isocenter to the point of observation in meters and x = h/(d sd w ).
The rise in dose rate for points just beyond the side wall is primarily due to the fact that the distance r from the minimum observable altitude (of the point of scatter) to the observation point (see Fig. 2) is large at first and then drops rapidly with increasing d s (as d s → d w , r → ∞). As d s increases further, r begins to increase. This local maximum is not due to partial transmission through the roof or side wall or primarily to a higher probability of scatter at smaller angles. We have assumed that the side walls are opaque to photons scattered by air. In addition, the probability of scatter only rises very slowly with decreasing scatter angle for large angles.
Let us contrast Eq. (7) with Eq. (1). Equation (1) has a 1/d 2 s dependence. The dependence of Eq. (7) on d s is somewhat complex but in the limit that d s >> d w and h, the dependence is 1/d s and not 1/d 2 s . This is loosely analogous to the electric field around an infinite line charge in electrostatics, which is inversely proportional to the distance from the line. If the source of the scattered radiation is assumed to be a line source of length roughly equal to 1/μ, then for T A B L E 2 Skyshine fitting constants. Some of the data in Fig. 1 have been fit to equation (7) by finding values of the constant k that best reproduce the data. These values are listed in Table 2. Figure 3 shows the fit for the 40 × 40 cm 2 data from the paper by Elder et al. 4   "the solid angle between the source and the vertical wall." The only energy dependent terms in Eq. (7) are k and B xs . In the limit as B xs → 1 this predicts that a 6 MV beam will have approximately two times as much skyshine as an 18 MV beam (for equal field size and dose rate at isocenter). This can also be seen in Fig. 1 in which the transmission factor has been divided out. The maximum _ H in Fig. 1  The weekly dose equivalent corresponding to Eq. (7) is given by: where H w is in units of μSv/week, k should be taken from Table 2  Consider the following question. Given the scenario above, if the side wall barrier is adequately shielded so that the weekly equivalent dose rate is at an ALARA level of P = 10 μSv/week at a distance of 0.3 m from the side wall, will the skyshine, when added to this, exceed 20 μSv/week at any distance? Let us first consider the case in which the side wall is a primary barrier. In this instance  (7) and (8) may be used to predict instantaneous and weekly skyshine dose rates but caution is advised due to uncertainties in the values of the fitting constant k. The values of this parameter reported in Table 2, depend crucially on the accuracy of B xs . The values for 6 and 10 MV are based on measurements of B xs for a 10 × 10 cm 2 field and are presumably fairly accurate. The value of k F I G . 5. The solid curve shows the fit to Eq. (7) of the skyshine dose rates measured by Gossman et al. for 6 MV (40 × 40 cm 2 ) as a function of the distance from the isocenter. A value of B xs = 0.013 has been used for the fit. This is approximately 1/3 of the value quoted by Gossman, which is based only on the concrete in the roof of the facility. The presence of 5 cm of steel would reduce the reported transmission by more than a factor of three.
for 18 MV is based on a fit to measured 18 MV data for which B xs is variously reported as 1.0 (no roof) or 0.9. As little as 2 cm of steel implies B xs = 0.66 for 18 MV radiation. This would lead to a 50% error in the derived value of k. Monte Carlo calculations would be very helpful to "benchmark" Eq. (7) and provide more definitive values of the fitting constant k. Such calculations should concentrate on distances <20 m, typical values of h/d w , and common beam energies of 6, 10, 15, and 18 MV. Measurements for research purposes should include a direct measurement of B xs. . It is recommended that the roof be shielded so as to avoid designation as a "high radiation area" (100 mrem in any 1 h). For nominal parameters this will require B xs ≲ 0.02. This corresponds to about 1 m of concrete for 18 MV. With this level of roof shielding, the photon skyshine is totally negligible.
If the roof does not have added shielding, it may be prudent to assume that B xs = 1.0 as a worst case scenario and use Eq. (7) and (8) to predict the dose rates. In this instance, survey measurements should be made at the lowest beam energy as well as the highest because the dose rate may be highest at the lowest energy. Under these circumstances, if the side wall is shielded to an ALARA level of 10 μSv per week, it is unlikely that the total weekly dose, including skyshine, will exceed 20 μSv at any distance from the side wall.

ACKNOWLEDGMENTS
Thanks to Patton McGinley for some helpful correspondence.

CONF LICT OF I NTEREST
The author declares there is no conflict of interest to disclose.