## 4 Discussion
The largest discrepancies arise from using the SVOX methodology with the ST VSV kernel, resulting in about \(-88\%\)\(RD\). This is mainly due to the higher density of the soft tissue used in the VSV calculations compared to the lung density, not counterbalanced by the increase in energy deposition, leading to lower \(AD\) values. Conversely, lower differences relative to \(MC\) are observed with the SVOX methodology when local density corrections are applied or with the LED approach (which inherently accounts for local density variations). Notably, after applying local density corrections, the corrected SVOX data (named SVOX_L) become almost identical to the LED data. This is due to the limited penetration range of the \({}^{90}\)Y emitted radiation in ST, compared to the voxel size or spatial resolution. In this work, activity maps were modeled without imaging data, so the nominal spatial resolution is determined by the voxel size (2.21 mm). However, the abrupt decrease in the S-value means that contributions to the target voxels from distant voxels are negligible, and the main dosimetric contributions are coming from the nearest neighbors. Therefore, for \({}^{90}\)Y, using the ST VSV kernel for convolution calculations with local density corrections intrinsically approximates LED conditions. Moreover, the initial evidence of the inaccuracy of post-convolution local density corrections is highlighted by the VSV comparison. When considerable radiation transport occurs, as expected for lung tissue, a reliable \(AD\) correction would require correction factors that vary with the source-target distance for each contribution to the target voxel. These issues cannot be addressed with a single correction factor based on a density ratio, as clearly demonstrated by the VSV kernel comparison. This corrective method was initially proposed for correcting voxel \(AD\) for slight density variations and small penetration ranges (as seen in ST). However, it has been well established that local density correction only accounts for the fluency correction of primary particles without affecting the contribution of secondary particles to the \(\overline{AD}\)[15]. Therefore, this correction becomes increasingly imprecise with greater radiation penetration ranges and larger density variations in the medium. The use of a lung VSV kernel yielded the results reported in Figure 5. The \(RD\) obtained using a VSV kernel specifically calculated for a lung medium density of 0.296 g/cm\({}^{3}\) (ICRU lung tissue [21; 22; 28]) varied between \(-63.8\%\) and \(-62\%\) compared to the reference \(MC\) simulation. Although the relative difference shows a slight improvement over the data obtained with the ST kernel, the agreement with the \(MC\) reference data remains poor. Despite the VSVs being calculated using a medium density considerably closer to the Reference phantom's mean lung density (0.221 g/cm\({}^{3}\)), the significant discrepancy cannot be easily attributed to the density differences between the Reference phantom and the ICRU mean lung density. A global density correction based on the average density of 0.221 g/cm\({}^{3}\) resulted in a still high \(RD\) (from \(-51.5\%\) to \(-49.1\%\)). Even with local density corrections, the \(RD\) remained unacceptable (varying from \(-38.1\%\) to \(-35.4\%\)). Since the \(\overline{AD}\) computed using the lung VSV kernel was in good agreement with the reference data for a homogeneous lung medium and provided activity map cropping, the observed differences should be correlated with the significant tissue heterogeneities of the lungs. If this is the case, an \(MC\) simulation of the reference phantom with uniform density lungs (corresponding to the mean density value of the phantom's lungs) should result in a smaller \(\overline{AD}\) compared to an \(MC\) simulation on the same phantom with heterogeneous lung density and the same mean density. Indeed, the \(RD\) between the two simulations under the stated density conditions resulted in approximately \(-70\%\), clearly indicating that:  
1.