Analysis of Systematic Errors in the Computation of Exposure Distributions from Brachytherapy Sources.

Systematic errors in the basic physical constants and algorithms used in calculating the exposure from brachytherapy sources have been investigated. The specific gamma ray constant for radium filtered by 0.5 mm of platinum, 10% irridium, is the fundamental constant in this field. It has been measured a number of times over the years, with estimated bounds for the systematic error given for each measurement. Using modern techniques for the statistical analysis of systematic errors of different measurements it was found that the true value for the specific gamma ray constant for radium is within the range of 8.26 (+OR-) 0.05 R/mg hr at 1 cm with a probability of 97%. The exposure from cylindrical brachytherapy sources has generally been calculated by assuming a line source and then applying corrections for self absorption and effective wall thickness calculated in the cross sectional plane. Using these approximations several authors have calculated the dose distribution around different types of brachytherapy sources. Systematic error of up to 17% were uncovered in some of these published dose tables through the procedure of recalculating all of them using a single algorithm. In each case the specific cause of the discrepancy was uncovered by judiciously varying the parameters until agreement was achieved. It was necessary to verify which set of parameters was actually used by the author of each dose table since agreement was achieved with several different combinations of parameters. The author has eliminated the systematic errors produced by these simplifying assumptions by evaluating the complete three dimensional (3D) integral numerically. Enough terms in the series expansion were integrated in closed form to yield a formula that agrees with the numeric integration to within 0.2% for a typical source. The different terms in the series expansion of the 3D integral were shown to include the conventional line integral, self absorption, additional attenuation due to wall curvature, and geometrical effects as well as interactions between each of these effects. This formalism was extended to take into account the preferential attenuation of lower energy photons by self absorption before calculating the attenuation of polyenergetic radiation in the encapsulation. The 3D integral also introduces some systematic error since it assumes a straight line path from each element of source material to a given dose point. In order to account for all the different attenuation and scattering processes the attenuation in the source and encapsulation were defined as polynomials with parameters to be found by fitting techniques. The data used to find the best fit was obtained by Monte Carlo modeling since no measurements of the exposure distribution have been performed with the required accuracy. The Monte Carlo data for self absorption of cesium gamma rays in a cylinder of ceramic microspheres was slightly less than the total attenuation. The Monte Carlo data for absorption of cesium gamma rays in a half millimeter thick encapsulation of platinum-irridium was close to the real attenuation for small slant thicknesses and significantly less than the real attenuation for slant thicknesses greater than two centimeters. The cause of this anamoulus behavior was found to be due to the streaming of photons down the long dimension of the low density source material and then scattering and penetrating through the encapsulation in the transverse direction. Fortunately a good fit could be found to the data for less than two and a half centimeters which corresponds to the slant thickness encountered in the longest sources used clinically.