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.

PACS number(s): 87.52.-g, 87.52.Df, 87.52.Tr, 87.53.-j, 87.53.Bn, 87.53.Dq, 87.66.-a, 89., 89.60.+x

Key words: scattering, shielding, skyshine, solid angle,

#### 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.(1-3) 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.(1,4-8)

#### II. Materials and Methods

McGinley(2) has shown that the skyshine measured dose-equivalent rate H(nSv h-1) is directly dependent on the transmission through the barrier. The form of the equation is shown in Eq.1:

where D (Gy/h) is the X-ray absorbed dose-rate at 1 m from the target, di (m) is the vertical distance from the target to a point 2 m above the roof, ds (m) is the lateral distance from the isocenter to a point outside the barrier where measurements are to be taken, O is the solid angle formed by the radiation beam in steradians and is given by Eq. 2:

In this equation, the angle T (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 40 × 40 cm2 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.

Michael Gossman, MS, DABR, RSO, is a Board Certified Qualified Expert Medical Physicist - Currently the Chief Medical Physicist & RSO of Radiation Oncology in Ashland, KY - a Medical Consultant to the U.S. Nuclear Regulatory Commission (U.S. NRC) - and an Accreditation Site Reviewer for the American College of Radiation Oncology (ACRO). He is the highest ranking scientist in the medical community. His expertise involves the safe, effective and precise delivery of radiation to achieve the therapeutic result prescribed in patient care by radiation oncologists.