Power output and efficiency of beta-emitting microspheres

• Range-energy relationship for the beta particles in yttrium-90 is calculated.
• Formalism for the semi-analytical calculation of self-absorption coefficients.
• Energy-dependent self-absorption coefficient calculated for yttrium-90.
• Flux rate of beta particles from a self-attenuating radioactive sphere is shown.
• The efficiency of beta particle emitting radioactive microspheres is calculated.

Current standard methods to calculate the dose of radiation emitted during medical applications by beta-minus emitting microspheres rely on an over-simplistic formalism. This formalism is a function of the average activity of the radioisotope used and the physiological dimensions of the patient only. It neglects the variation in energy of the emitted beta particle due to self-attenuation, or self-absorption, effects related to the finite size of the sphere. Here it is assumed the sphere is comprised of a pure radioisotope with beta particles being emitted isotropically throughout the material. The full initial possible kinetic energy distribution of a beta particle is taken into account as well as the energy losses due to scattering by other atoms in the microsphere and bremsstrahlung radiation. By combining Longmire’s theory of the mean forward range of charged particles and the Rayleigh distribution to take into account the statistical nature of scattering and energy straggling, the linear attenuation, or self-absorption, coefficient for beta-emitting radioisotopes has been deduced. By analogy with gamma radiation transport in spheres, this result was used to calculate the rate of energy emitted by a beta-emitting microsphere and its efficiency. Comparisons to standard point dose kernel formulations generated using Monte Carlo data show the efficacy of the proposed method. Yttrium-90 is used as a specific example throughout, as a medically significant radioisotope, frequently used in radiation therapy for treating cancer.