Analytic solution for source distributions in brachytherapy to obtain uniform dose rates in spherical tumor models

Interstitial brachytherapy involves implanting many small radioactive sources into a tumor, with the goal of delivering a uniform radiation dose to the target volume. As a guide for the optimal placement of these sources, we assumed a spherical tumor irradiated by a continuously distributed radiation source. The solution of the ensuing integral equation shows that the source density is very low near the center of the sphere, increases rapidly toward the surface and becomes infinite at the surface. Integration of the source density over a given spherical sub-volume shows that only about 6% of the total activity is contained in the central core up to 50% of the tumor radius, while about one-half of the activity has to be placed in the outer spherical shell having a thickness of one-tenth of the tumor radius. Since attenuation is not taken into account, the results are applicable to highly penetrating radiation of isotopes such as 192Ir and 137Cs and tumor radii of a few cm. This situation is approximated in the high dose rate (HDR) treatment of the prostate using 192Ir. The results are in good agreement with the recommendations given in the traditional Paterson-Parker tables for radium and cesium treatments and a numerical solution to the problem.