## 4 Discussion
It is noteworthy that the LED method is not applicable to \({}^{166}\)Ho due to the gamma emission of \({}^{166}\)Ho that makes the assumption of local deposition unreasonable. Examining the AD spatial distributions of the reference MC simulation in comparison to the two voxel-based dosimetric methods with stronger physical assumptions (kST\({}_{L}\) and kLT\({}_{L}\)) for LS = 10%, a pronounced heterogeneity can be easily observed, underscoring the fundamental importance of quantifying the correlation between the density distribution and the AD distribution. This is particularly crucial for radionuclides such as \({}^{166}\)Ho, which exhibit sufficiently abundant low-energy \(\beta^{-}\) decay electrons and internal conversion electrons, making the AD distribution more sensitive to morphological heterogeneities. It is necessary to consider that all results obtained in this paper should not be generalized as a global trend for data obtained by each method; thus, all results are specific to the simulated phantom, and the assumption of the same RD in clinical practice should be discouraged. However, the general performance of each method is representative of its limitations, with the major differences arising from the physical assumptions of each approach. Looking at the ADr spatial distribution differences in the MC simulations of \({}^{90}\)Y and \({}^{166}\)Ho for the LS = 10% case, it is clear that \({}^{166}\)Ho exhibits a more heterogeneous distribution than \({}^{90}\)Y, with a doubled HI. These remarkable spatial ADr differences, together with variations in the absolute ADr values, arise because the decay spectrum of \({}^{166}\)Ho includes a greater variety of low-energy emissions, leading to a highly inhomogeneous dose rate distribution that is strongly dependent on the anatomy of the individual patient. This makes the choice of methods that account for radiation transport particularly important, especially for settings like LT.  
To our knowledge, there are no prior studies on \({}^{166}\)Ho lung dosimetry; thus, this study serves as a groundbreaking MC-based investigation into lung dosimetry for \({}^{166}\)Ho-labeled microspheres liver radioembolization, highlighting the need and specific challenges of lung dosimetry for this therapeutic device. Studying the potential clinical impact of the results presented in this work, it is evident from Table 8, as previously demonstrated for \({}^{90}\)Y [9], that the primary determining factor for AD in the lungs is the activity present in the pulmonary region. For the examined case, to prevent the lungs from reaching the 30 Gy limit advised by international guidelines [7, 8], the maximum activity that lung tissue can tolerate is approximately 500 MBq, corresponding to a maximum administrable activity ranging from 4.88 GBq to 1.26 GBq along with increasing LS. The clinical impact becomes even more significant when attempting to simulate a treatment for both \({}^{166}\)Ho and \({}^{90}\)Y for the case of LS = 10%. Planning the treatment according to the guidelines of the Ho-MS manuals [15], it is necessary to administer an activity sufficient to achieve 60 Gy to the whole liver using the mono-compartmental MIRD model. Using the same criterion for both radionuclides, a treatment activity of 1.8 GBq for \({}^{90}\)Y and 5.7 GBq for \({}^{166}\)Ho was assumed. According to these prescriptions, the \(\overline{AD}\) to the lungs for \({}^{166}\)Ho exceeds the 30 Gy limit. Converting the AD map with these administrations into \(EQD_{2}\) by applying the corresponding BED values, the corresponding \(\overline{AD}\) in \(EQD_{2}\) again exceeds the constraints for EBRT listed in Table 4. Evaluating the volumetric distribution of \(EQD_{2}\) with the DVH for each radionuclide, the five volumetric constraints in Table 4 are mostly met for \({}^{90}\)Y, while for \({}^{166}\)Ho, almost all of them are violated, except for \(V_{40}\) depending on the considered T\({}_{\mu}\). However, in both cases, the constraints associated with low isodoses are violated for both devices.