## 4 Discussion  
Using a standard desktop computer with \(10^{9}\) primaries and no variance reduction techniques, \(MC\) simulations on the Reference phantom took about 6 days each. The relative uncertainties affecting \(AD\) values (reported in Table 1) ranged from 1 to 3% for \(LS\geq 10\%\). Given these benefits, the \(MC\) calculation method could be adapted for clinical applications with relatively minimal effort, especially when compared to the 2-3 days at most required for inverse planning in EBRT. The validation of the presented results is supported by previously published work that internally validated the GATE code for soft tissue with controlled tissue heterogeneities. This prior research demonstrated good agreement between the \(MC\)\(AD\) distributions and those obtained using the MIRD and VSV method with a soft-tissue kernel [12]. Additionally, a second validation step was performed on homogeneous lung tissue with an \(LS\) of 20%. This step demonstrated the internal consistency of the \(MC\) simulation results in comparison to the VSV method using a lung tissue kernel (see Section 3.2). A more extensive validation to assess the reliability of the \(MC\) simulation for strongly heterogeneous lung tissue would involve comparing different \(MC\) codes. However, such an evaluation was beyond the scope of the present work. Comparisons between the VSV kernels in Figure 2 highlight the specific properties of radiation transport in lung tissue compared to ST. In the lungs, the \(AD\) increases considerably for any given source-target distance relative to ST, as does the radiation penetration in the medium. The most abrupt slope change in the lung VSV trend occurs at approximately 38 mm. This is well-correlated with the maximum penetration of the \({}^{90}\)Y emitted beta radiation in the lung, whose maximum CSDA range is approximately 38.6 mm [31] for the \(\beta^{-}\) end-point energy of \({}^{90}\)Y; beyond that distance, the only contributions to the absorbed dose are due to bremsstrahlung radiation. Data scattering increases with the source-target distance. This is due to growing statistical uncertainty, as fewer and fewer interaction events are recorded when approaching and exceeding the maximum penetration range of beta radiation. The use of the lung kernel on the entire patient's activity map equates to the assumption that all the patient's tissues have the same composition as the lung. For these reasons, cross-irradiation from the liver to the lungs due to the activity retained in the liver resulted in values three times higher than those obtained by \(MC\) simulation. Therefore, when using VSV kernels, it is crucial to apply activity map cropping for approximated \(AD\) calculations in scenarios where organs with highly differing densities are contiguous. This approach is particularly important for estimating the \(AD\) to tissues with lower densities. Some new approaches [32] have been proposed, introducing a convolution approach using mixed kernels for regions defined as soft tissue or lungs. These approaches report good agreement with reference \(MC\) simulations on patient data. Comparisons of the \"classical\" dosimetric approaches (MIRD, LED, and VSVs with soft-tissue kernel) with reference \(MC\) simulations show unacceptable underestimation of the lungs' \(\overline{AD}\). The MIRD and LED methodologies inherently assume a direct proportionality between the \(\overline{AD}\) and the \(LS\). However, the contribution of cross-irradiation from the hepatic region, as indicated by the \(MC\) simulations, varies depending on the activity in the liver. This results in \(RD\) ranging from \(-42\%\) to \(-40\%\) for SVOX_ST and from \(-26.4\%\) to \(-27.4\%\) for LED. When using the MIRD approach with a lung mass of 1 kg instead of the personalized mass, the relative differences compared to using the personalized mass are about \(-28\%\), worsening the \(RD\) with respect to the reference data.