Providing solid angle formalism for skyshine calculations.

We detail, derive and correct the technical use of the solid angle variable identified in formal guidance that relates skyshine calculations to dose-equivalent rate. We further recommend it for use with all National Council on Radiation Protection and Measurements (NCRP), Institute of Physics and Engineering in Medicine (IPEM) and similar reports documented. In general, for beams of identical width which have different resulting areas, within ± 1.0 % maximum deviation the analytical pyramidal solution is 1.27 times greater than a misapplied analytical conical solution through all field sizes up to 40 × 40 cm². Therefore, we recommend determining the exact results with the analytical pyramidal solution for square beams and the analytical conical solution for circular beams.

I. INTRODUCTION

The determination of skyshine involves the calculation of the solid angle subtended by a linac radiation beam of known field size and square shape. Previous research has been documented where analytical conical expressions were used.( – ) Cone‐based equations are more appropriate for circular collimation such as from round apertures of cerrobend blocks. An analytical pyramidal equation should be used when medical accelerators are involved, since square‐shaped apertures result in an inverted pyramid shaped beam. We detail, derive and correct the technical use of the solid angle variable identified in formal guidance that relates skyshine calculations to dose‐equivalent rate. We further recommend it for use with all NCRP, IPEM and similar reports documented.( – – )

II. MATERIALS AND METHODS

McGinley( ) has shown that the skyshine measured dose‐equivalent rate is directly dependent on the transmission through the barrier. The form of the equation is shown in Eq.1: where is the X‐ray absorbed dose‐rate at 1 m from the target, (m) is the vertical distance from the target to a point 2 m above the roof, (m) is the lateral distance from the isocenter to a point outside the barrier where measurements are to be taken, Ω is the solid angle formed by the radiation beam in steradians and is given by Eq. 2: In this equation, the angle Θ (degrees) is the angle subtended between the central axis and the edge of the beam as radiation projects away from the source. Thus, Eq. 2 represents the analytical conical expression, where the beam pointing upward would resemble an inverted cone.

Equation 2 represents the correct form of the solid angle equation, but only for circular fields such as from those formed by cerrobend blocks. It is rarely considered how the effect of cerrobend blocking with circular apertures affects radiation levels outside the vault. It is nearly universal that accelerators include moving jaw systems or even multileaf collimators, which may be completely opened to a square dimension as large as defined at 1m from the machine isocenter. It is these square apertures that are used for radiation protection purposes. For discussion here, we define the distance h as the height of the beam for which the solid angle is determined, commonly used as the 100 cm source‐axis‐distance (SAD) for accelerators, where the field size is defined. Given this geometry, the radiation beam will no longer resemble a cone, but rather an inverted pyramid with SAD‐projected field size denoted a. The form of the resulting equation for the solid angle, which should be used for all medical accelerator skyshine calculations, is expressed in Eq. 3:

III. RESULTS & DISCUSSION

We present an accurate solid angle equation necessary to considering skyshine mathematically. A derivation is presented in the Appendix to this research for the correct form of the solid angle. An analysis of the numerical change expected while using it instead of the conical expression is also discussed, to prevent further inconsistencies found elsewhere in literature.

For the purposes of shielding, only perfectly square fields should be used. A resulting solid angle for the field size ) defined at a distance of 100 cm () is thus steradians. It is noteworthy for approximating results that, for a typical square clinical aperture, the analytical conical solution will give good approximations (within 0.2% of Eq. 3) for the solid angle, but only when Θ is determined such that the area of the circular base is equal to the square aperture area. Therefore, an approximated solution, though inexact, is achievable by comparing the resulting square area to the circular area given as πr2. The resulting equivalent circular radius denoted r (cm), which is also equivalent to the quotient α/2 representing half the length of the field, is then found to be cm or 22.6 cm. The conical projection with radians (12.7°) for this geometry results in a solid angle of 1steradians. This is very comparable to the pyramidal solution, although inexact.

Still, with the solid angle dependence varying as , skyshine dose‐equivalent rate error increases dramatically when equations are misused. Thus, it is very important to simplify calculations whenever possible. One cannot assume the same base width (diameter) for the circular beam as for the square beam. In general, for beams of identical width which have different resulting areas, within maximum deviation the analytical pyramidal solution is 1.27 times greater than the analytical conical solution through all field sizes up to . Therefore, we recommend determining the exact results with the analytical pyramidal solution for square beams and the analytical conical solution for circular beams. Results for 6 MV and 18 MV X‐rays are already published.( , )