## 4 Discussion  
This work fits into the \({}^{90}\)Y dosimetry studies by evaluating the differences in absorbed dose when different approaches are applied at the voxel level. The aim was to compare the dosimetric results obtained from two dedicated software packages implementing water-equivalent dose calculations and fast MC simulations, respectively. There are several methods to calculate voxel-based absorbed doses for \({}^{90}\)Y-TARE, but there are not so many works in the literature that compare different techniques and methodologies, especially at the lung level. To achieve this objective, a first phantom analysis was performed, followed by a verification on a selected cohort of patients with the same histological disease (HCC). The results of the phantom analysis show that when the dose is calculated within uniform water-equivalent density regions, the differences between the values obtained using the VSV method and those obtained using the MC method are very small (1.2% and 0.5% for _setup 1_ and _setup 2_, respectively). These values are very similar to those reported by [PERSON] et al. (2018) and [PERSON] et al. (2019), as well as for differences evaluated in air-equivalent density regions. [PERSON] et al. reported an underestimation of the MAD in the right lung of about \(-60\%\) and about \(-70\%\) for the absorbed dose deep in the lung when a soft tissue kernel (i.e., VSV method) is used with respect to the MC. This lung dose underestimation behavior is confirmed for both phantom simulations and patient analysis, with mean percentage differences for lung mean dose of \(-56\%\) and \(-67\%\) in the phantom simulations and \(-61.1\%\) in the patient study. Statistical analysis through the Passing-Bablok regression method showed that no systematic differences (confidence intervals of the intercept reported in Table 3 contain the value zero) between MC and VSV for almost all structures except for the H-L appear to be present. Moreover, at least a proportional difference (confidence intervals of the slope reported in Table 3 do not contain the value one) between the two methods for all structures is evident. These proportional differences are small for water-equivalent structures (slopes very close to one), and are more significant for air-equivalent structures, where a greater discrepancy between the two methods is observed, in line with the dosimetric evaluations. Lung dosimetry plays an important role in \({}^{90}\)Y-TARE, and, to our knowledge, very few works have investigated the importance of a correct dosimetric model and its implications at the lung level (including the liver-lungs interface) ([PERSON] et al., 2018; [PERSON] et al., 2019). There have been major advances in liver and tumor dosimetry models that allow clinicians to plan treatments prospectively with higher administered activities to increase the likelihood of local tumor control after radioembolization. Unfortunately, lung dosimetry and dose constraints are very rudimentary ([PERSON] et al., 2019). The clinical data that justified lung dose limits of 30 and 50 Gy are based on studies dating back to 25 years ago ([PERSON] et al., 2019) on planar imaging data for a standard lungs mass of 1 kg ([PERSON] et al., 2019). A reliable state-of-the-art lungs-absorbed dose limit for radioembolization is not yet available. Using dedicated software that implements a MC code would be important, especially when there are significant density differences and motion artefact