## 4 Discussion  
The validation of MCID platform is demonstrated at both organ and voxel level. In particular, for the UL scenario, the comparison between mean absorbed doses to liver assessed with the MIRD approach at the organ level and with the MC-based TPS showed a very good agreement (RD = 0.27%). This result highlights the efficiency of the developed  
Figure 7: Absorbed dose profile for the NUL-b case and comparison. (**a**) Absorbed dose image obtained with the MIRD schema and absorbed dose profile selection; (**b**) absorbed dose image obtained with GATE and absorbed dose profile selection; (**c**) relative difference (RD) image for the whole slice; (**d**) comparison between absorbed dose profiles; (**e**) plot of the absorbed dose relative differences RD for each voxel of the profile. -dosimetric routine: under the hypothesis of homogeneous density and uniform activity distribution in each tissue, both methods are equivalent, as expected. It is interesting to point out that D\({}_{\text{mird}}\) is a merely theoretical quantity, while D\({}_{\text{gate}}\) depends on the image quality, e.g., partial volume effects (PVEs), which in this first case appear negligible due to the big size of the observed object (i.e., the whole liver). The effects due to image blurring become relevant instead when dealing with smaller objects, as in the SR case. This scenario allowed a comparison between the average absorbed doses to each sphere. While the BS and the MS present a RD < 8% (absolute value) between the two methods, the SS shows a dramatic RD of \(-69.2\%\), caused by the PVEs affecting the SPECT simulation with SIMIND. One contribution to PVEs rises from the matrix resampling (512 \(\times\) 512 to 128 \(\times\) 128), an additional contribution derives from the impulsive response of the imaging system, in the SIMIND simulation: spill-out effects affecting voxels cause a change in the activity quantified by imaging, only partially recovered with CDR corrections during the tomographic reconstruction. D\({}_{\text{gate}}\) depends on the activity estimated from the image, while D\({}_{\text{mird}}\) is only related to the theoretical initial activity value. The drastic impact of PVEs is evidenced by the results obtained without the SIMIND simulation: for the BS and the MS, RD reduced to 2% and 3% (absolute values), respectively, while for the SS, RD reduced to 9% (absolute value). Remaining discrepancies could be related to some differences between the MC codes used for calculating the OLINDA-EXM sphere S-factors and the updated code used in this work, and to possible mismatching in source description. Relative differences for the BS and the MS are in agreement with the results shown in [41], where the authors presented an analogous comparison for various sphere diameters: their smallest sphere had a 27 mm diameter, resulting in RD = \(-5\%\), so the RD here obtained for SS (20 mm diameter, see Table 6) also appears reasonable and comparable. Dosimetry in lesions with size equal (or less) to the maximum \({}^{90}\)Y \(\beta^{-}\) range (around 12 mm in soft tissue) should be treated carefully, also due to limited resolving power of SPECT imaging. An analysis of dosimetry in small lesions and a correction factor for the MIRD standard equation are proposed in [42]. A validation of the TPS at the voxel level was presented in the NUL-a case. The differences between the two methods (MC simulation and S-voxel convolution) for each absorbed dose profile were always within 3%. Finally, the NUL-b scenario shows the importance of a personalized dosimetry in heterogeneous tissues, accounting for both activity distribution and density inhomogeneities, which are often overlooked in present internal dosimetry evaluations.