An analytical method for the calculation of Voxel S-Values for $^{177}Lu$

Voxel S-Values (VSVs) are commonly employed to perform 3D internal dosimetry for non-uniform activity distributions of radionuclides, but are generally calculated and made available for a limited number of voxel sizes. In this study we present an analytical method developed to easily calculate VSVs in any cubic voxel dimension of practical utility in the case of $^{177}Lu$, a $\beta$-emitter widely used in radionuclide therapies. For this purpose, we evaluated via GAMOS GEANT4-based Monte Carlo simulations the $^{177}Lu$ VSVs in homogeneous soft-tissue geometries made of $15 \times$ 15 $\times 15$ cubic voxels of sides $d$ within the range ({2 $mm$, 6 ${mm}$), in steps of $0.5 {mm}$. For each $d$, the VSVs as a function of the normalized radius $R_{n} = R/d$ ($R$ being the distance from the center of the source voxel) were fitted with an appropriately built function. The trends of the fit parameters as a function of $d$ were consequently deduced, and through them it was finally made it possible to calculate the VSVs for any $d$ within the mentioned range, by simply exploiting a dedicated electronic spreadsheet.