## 4 Discussion
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Structures** & **Fit Parameters** & **Value** & \
\begin{tabular}{c} **Lower Confidence** \\ **Interval** \\ \end{tabular} &
\begin{tabular}{c} **Upper Confidence** \\ **Interval** \\ \end{tabular} \\ \hline \multirow{2}{*}{Tumor} & Intercept & \(-0.289\) & \(-1.382\) & 1.621 \\  & Slope & \(1.073\) & \(1.046\) & 1.082 \\ \hline \multirow{2}{*}{Lung R} & Intercept & \(0.012\) & \(-0.301\) & 0.308 \\  & Slope & \(0.382\) & \(0.339\) & 0.469 \\ \hline \multirow{2}{*}{Lung L} & Intercept & \(-0.010\) & \(-0.036\) & 0.030 \\  & Slope & \(0.400\) & \(0.300\) & 0.445 \\ \hline \multirow{2}{*}{Lungs} & Intercept & \(0.004\) & \(-0.216\) & 0.171 \\  & Slope & \(0.380\) & \(0.332\) & 0.475 \\ \hline \multirow{2}{*}{Liver–lungs} & Intercept & \(0.085\) & \(-0.314\) & 0.620 \\  & Slope & \(0.486\) & \(0.448\) & 0.532 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: _Cont._issues. In fact, when the correct medium density is applied, the activity at the liver-lungs interface has a significant impact on the lungs' MAD, and locally at the L-L Edge region. Nevertheless, the phantom study showed that the liver MAD is not so much influenced by the possible presence of activity in the lung: by adding a representative percentage (10%) of activity, the liver MAD changed from 61.40 Gy to 62.50 Gy (an increase of about 1.8%), and from 62.11 Gy to 62.79 Gy (an increase of about 1.1%) when using the MC and VSV methods, respectively. Obviously, the patient's respiratory motion could potentially affect the definition of the liver-lungs boundary region and its effective activity distribution. However, the small difference between the MC and VSV methods in the liver region confirms that VSV can provide accurate dose estimation in cases where the lung shunt is limited and motion artefacts are not so evident. The use of a free-breathing CT scan within the PET/CT could be a potential limitation as well as the accuracy of \({}^{90}\)Y-PET [39]. It is well known that the extremely low positron yield of 32 \(\times\) 10\({}^{-6}\) per \({}^{90}\)Y disintegration [40] has been a unique challenge in \({}^{90}\)Y PET imaging. This limitation is overcome in the phantom study by manually setting the activity concentration in each voxel of the PET image series, but remains in the patient analysis. Furthermore, this work confirmed that at the lungs level the MAD values could potentially vary significantly by using a MC instead of a soft-tissue dosimetric model (i.e., VSV), but no clinical implications were explored and future works must be conducted to include evaluations on the lungs' dose-response constraint, both in the pre-treatment evaluation with \({}^{99\mathrm{m}}\)Tc-MAA and in the post-treatment evaluation with \({}^{90}\)Y microspheres. In fact, the results of this study highlight that the actual dose constraints of 30 Gy and 50 Gy need to be reviewed and updated with new retrospective studies based on MC calculations, given the differences in the lungs' dose when using an MC instead of a soft-tissue dosimetric model. MC methods are considered the gold standard in dose assessment and the increase in modified versions of MC codes operating on GPUs represents a significant improvement.