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 X 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) I-125] and [(0.036,2.56) Pd-103], or that quoted by AAPM TG64, (c,n)=[(0.014,2.05) I-125] 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. When radioactivity is clumped as discrete seeds, higher activity is needed because of "wastage" in two aspects: (a) Dose cold-spots at intersource spaces, (b) hot-spots around the sources. Thus in theory, use of lower activity seeds will require less total activity to deliver a prescribed dose. Based on our study, Pd-103 delivers a higher therapeutic ratio and a lower integral dose to the patient compared to I-125.