Solutions to the TG-43 Geometry Function for Ring, Disk, Sphere, and Dome Sources

Abstract

Purpose: A simple inverse square-law, IVS, calculation is sufficient for most brachytherapy calculations, particularly for distances far from the source. Closer to the source it becomes important to more accurately model the source geometry. The line source approximation is commonly used. This work provides solutions for the geometry function for several other interesting source geometries, including a ring, a disk, a sphere and a dome. Analytical solutions are presented, along with calculations for relevant clinical scenarios. Materials and Methods: TG-43 formalism incorporates a geometry function to isolate the inverse square-law component in dose calculations. The inverse-square law applies for radiation emitted outward radially from a point, where emitted radiation spreads out over the surface area of a sphere (4πr2) as distance from the source, r, increases. The amount of radiation passing through a given solid angle is inversely proportional to the square of the distance from the source. Hence the geometry function for a point source is simply 1/r2. The geometry function for more complicated source distributions may be calculated by integrating the contributions of many infinitesimal “point” sources. Traditional integration techniques are used to solve the geometry function for ring, disk, sphere, and dome source distributions. Sources are modeled as having uniform distribution of activity over the surface or volume. Coordinate system definitions are provided in Figure C. Results: Computation of the geometry function for ring, disk, and dome solutions provides an intuition for expected dose distributions. Figure A presents the geometry function along the transverse axis as a function of distance from the source: “Depth Dose” along transverse axis. For ring, disk, and sphere sources, distance = 0 corresponds to the center of the circle or sphere. For the dome source, distance = 0 corresponds to the apex of the dome. All distances in the plot are normalized by R (Figure C.). Dose falls off less rapidly for ring, disk, sphere, and dome sources than IVS. The dome result depends upon the ratio of the base of the dome and the radius of the sphere upon which the dome lies. Figure B presents the geometry function angular dependence for a 16-mm diameter eye plaque (dome) resting on a 24-mm diameter eye. Clinically relevant distances from the plaque are included. The geometry function decreases significantly as the measurement point rotates away from the transverse axis. The greatest change is observed for tissues nearest the plaque surface. Clinically this implies that, considering only geometric effects, the dose difference between the center of the plaque and the edge of the plaque, at fixed tissue depth, i.e. the penumbra, is sharpest near the plaque. Conclusions: We have obtained analytical solutions for several interesting source geometries. The solution for a dome of activity is particularly relevant for treatments using eye plaques, which ideally place a dome of activity distributed over a portion of the eye modeled as a sphere. These factors can be useful in simplified calculations using a spreadsheet, particularly for situations where IVS is an insufficient approximation.