## 4 Discussion
Radiation transport is strongly influenced by the tissue heterogeneities of the lungs, which substantially affect the absorbed dose. 2. A lung tissue with a uniform density corresponding to the average density of the case under study is not an accurate descriptor of the real tissue. Lung density variations in this phantom range from 0 to 1.06 g/cm\({}^{3}\), with a positively skewed distribution. A heterogeneous density pattern is generally observed in bone tissue. However, in that case, the density distribution is shifted towards a higher value, resulting in local energy deposition. This contrasts with lung tissue, for which the \(MC\) simulations of radiation transport are essential to achieve an accurate description of energy deposition. The choice of this reference phantom is not representative of the entire population but is intended to address the physical nature of the problem under discussion. Density distribution varies significantly with age, individual conditions, and comorbidities [28]. Therefore, it is crucial to evaluate on a case-by-case basis the differences that arise from using different computational approaches in a realistic case. To maintain accuracy, it is essential to use a _MC_ radiation transport approach tailored to the specific density distribution being treated. Two studies [13; 14] have investigated the role of lung density in the dosimetric evaluation of _LS_ in radioembolization using different dosimetric approaches. [PERSON] et al. [14] conducted a two-step study comparing the performance of a fast-MC Graphic Processing Unit (GPU)-driven code versus the VSV method using a soft-tissue kernel. This comparison was performed on a reference lung-liver phantom to simulate a 10% _LS_ environment and was also applied to data from 24 patients. In this study, the _MC_ showed an approximate RD of \(-60\%\) for the VSV \(\overline{AD}\) compared to the _MC_\(\overline{AD}\) on the reference phantom. Additionally, the VSV method resulted in a global underestimation of the \(\overline{AD}\) of lungs in the patients' dataset, which is consistent with the findings of our study. [PERSON] et al. [13] conducted a general analysis of lung \(\overline{AD}\) for the homolateral lung using patient data. They compared _MC_ simulations performed with the Electron Gamma Shower of the National Research Council of Canada (EGSnrc) against classical approaches, including LED and the VSV approach with a soft-tissue kernel. Their analysis included comparisons with and without local density corrections. In this study, the VSV approach showed a \(-60\%\)_RD_ with respect to MC, while the VSV approach corrected for local density and LED reported a 20% and 17% _RD_, respectively, with a global underestimation of the _MC_\(\overline{AD}\) for the first method and a global overestimation of the \(\overline{AD}\) for the last two. This latter result contrasts with our analysis, which showed a global overestimation of the \(\overline{AD}\) for the same dosimetric approaches across all _LS_ conditions in our reference phantom. This difference may be attributed to variations in patient geometry modeling in the EGSnrc simulation, such as differences in voxel density definitions and the poor image resolution of the post-therapy activity images obtained by \({}^{90}\)Y bremsstrahlung SPECT/CT. Unlike previously published papers, this work included all commonly used dosimetric approaches for lung dosimetry in radioembolization and compared them with an _MC_ simulation using a reference phantom representing the same complexity as real patients but with known lung composition and density. Additionally, our study introduces an optimized alternative method for comparison: VSV with lung tissue kernel. Regarding the potential clinical impact of this work, Table 6 illustrates that the primary factor determining \(AD\) in the lungs is the activity present in the lungs themselves. For the case considered, to avoid exceeding the dose limit of 30 Gy, the lung activity should be limited to approximately 260 MBq.