A theoretical derivation of the nomograms for permanent prostate brachytherapy

This study calculates the required minimum radioactivity to deliver a prescribed dose of radiation to a target using radioisotopes in permanent prostate brachytherapy. Assuming the radioactivity to be in a continuous form, an integral equation - Fredholm equation of the first kind, can be formulated with the radioactivity density used as the variable. The density distribution to produce a uniform volume dose rate is determined using a quadrature method and the radial profile behaves smoothly from the zero radius, and peaks sharply approaching the volume boundary. The density for Pd-103 is about 1.5 times that of I-125 due to its higher spatial attenuation. A nomogram is the relationship between the total activity per unit dose (A) and the dimension of the volume (d). Expressing the nomogram as A = c×dn U/Gy, then (c,n) = [(0.0098, 2.09) I-125] and [(0.031, 2.25) Pd-103]. Compared with the Memorial nomogram, (c,n)=[(0.011,2.2) I125] and [(0.036,2.56) Pd-103], or that quoted by AAPM TG64, (c,n)=[(0.014,2.05) I125] and [(0.056,2.22) Pd-103], our calculation determined an average 33% and 35% decrease for I-125, and 89% and 77% decrease for Pd-103, respectively. Two reasons for the extra total activity found in the Memorial and AAPM nomograms are: (a) An imperfect clinical situation limited by the restraints of implant techniques (e.g., use of templates) associated with the presence of adjacent normal organs, and (b) source discretization into seeds.