## 4 Discussion  
The MC simulation seems to represent the most valuable dosimetric tool in TRT of low-density strongly heterogeneous tissues such as lung tissue, especially for radionuclides with a low-energy decay spectrum. The results of the MC simulations with GATE/Geant4 code have been validated in previously published works for \({}^{90}\)Y [9, 43]. Moreover, the results for \({}^{166}\)Ho were validated for both LT and ST in homogeneous lungs (with LS = 10% and a density of 0.296 g/cm\({}^{3}\)) and liver (with LS = 0% and density of 1.04 g/cm\({}^{3}\)), respectively, proving the consistency of the MC simulations with respect to the VSV convolution (see Section 3.1). A more robust validation, but beyond the scope of the present work, to assess the reliability of the MC simulation of strongly heterogeneous low-density lungs, would involve a comparison of different MC simulation codes. To ensure that resampling from the original CT resolution to cubic voxels did not introduce significant density changes leading to notable variations in AD, an additional MC simulation was performed exclusively on the heterogeneous lung tissue. This simulation maintained the original CT resolution (\(1.37\times 1.37\times 3.27\,\mathrm{mm}^{3}\)) while assuming a uniform activity distribution and an LS of 10%. The results were compared with the same heterogeneous lungs resampled using the procedure outlined in Section 2.1. The comparison showed an RD below 1% with respect to the simulation using the original CT resolution. The analysis of the S-values plot over the source-voxel distance (Figure 1) allows us to test the main radiation-matter interaction features of \({}^{166}\)Ho in ST and LT, primarily the particle range in Continuous Slowing Down Approximation (CSDA), corresponding to the abrupt slope change in the shown plots. The CSDA range for the most energetic end-point electrons of the \(\beta^{-}\) spectrum is approximately 9 mm in ST and 30 mm in LT, well correlated with the corresponding values for the slope change of about 9.7 mm and 29.8 mm, respectively. As for \({}^{90}\)Y [9], the use of the lung kernel brings the necessity of an image crop of the activity map to extrapolate the lungs' activity only to overcome an unwanted cross-irradiation contribution to the lungs' AD from the activity in the liver region. Comparison of the \"classical\" dosimetric approaches (MIRD, kST, kST\({}_{L}\), kLT, kST\({}_{221}\) and kST\({}_{L}\)) with the reference MC simulations using \({}^{166}\)Ho show a remarkable underesti  
Figure 6: \(\overline{AD}\) NTCP model for RP incidence in partial lung irradiation treatments from EBRT as reported in QUANTEC [42]. The black solid line is the logistic model according to Equation (6) with parameters \(b_{0}=-3.87\) and \(b_{1}=0.126\) Gy\({}^{-1}\), the blue dots refer to \({}^{90}\)Y cases, and the yellow ones to \({}^{166}\)Ho cases, each for the LS = 10% case and for the labeled T\({}_{\mu}\). mation of the lungs' \(\overline{AD}\). The largest discrepancies arise for kST, resulting in \(-93\%\) RD, and an analogous effect is obtained for \({}^{90}\)Y [9], due to the higher density of the soft tissue used in the VSV calculations compared to the lung density. On the other hand, the best performance was obtained by the convolution with local density corrections for both ST and LT, resulting in approximately \(-56\%\) for kST\({}_{L}\) and \(-60\%\) for kLT\({}_{L}\). Notably, the use of a specific kernel for lung tissue with an ICRU tissue description does not yield better results in terms of \(\overline{AD}\), which results in an \(\overline{AD}\) about 12% higher with respect to the kST\({}_{L}\). This last result is remarkably higher than the analogous previously analyzed results on \({}^{90}\)Y [9], strengthening the hypothesis that, for highly heterogeneous low-density tissues, the description and characterization of the tissue's density distribution is essential for accurate lung dosimetry in MRT, especially when the radionuclide involved has abundant low-energy decay emissions.