## 2 Materials and Methods  
### Anthropomorphic Voxelized Phantom  
A voxelized anthropomorphic phantom, based on the CT scan (voxel size of 1.37 \(\times\) 1.37 \(\times\) 3.27 mm\({}^{3}\)) of a healthy male patient, weighing 72 kg and 166 cm tall, was created using ITK-Snap software (version 3.8.0) [18]. The phantom was then resampled to a cubic voxel grid with a voxel size of 2.21 mm using the \"Resample Scalar Volume\" built-in module of 3D-Slicer software (version 5.2.2) [19], applying the \"nearest neighbor\" algorithm [20]. Three spherical lesions were artificially introduced into the phantom's liver, with diameters of 10, 20, and 30 mm. These lesions had a higher density than the liver parenchyma, i.e., 1.20 g/cm\({}^{3}\) for tumor tissue, compared to 1.04 g/cm\({}^{3}\) assigned to the surrounding tissue [21; 22]. For the lung tissue, the actual spatial density distribution of the patient was assumed for this phantom, based on the HU-density calibration of the CT scanner. The phantom's lung density distribution ranges from 0 to 1.06 g/cm\({}^{3}\), with a mean value of 0.221 g/cm\({}^{3}\) and a median of 0.179 g/cm\({}^{3}\). A visual representation is provided in Figure 1, which displays a histogram of the voxel density values within the lung Region of Interest (ROI). This phantom will be referred to as the reference phantom throughout the manuscript, with explicit notation when variations occur (e.g., reference phantom with uniform lung density in lungs of a specified value). Five count distribution maps (hereinafter referred to as \"activity maps\") were created for the reference phantom, assuming several lung shunt values (\(LS\) = 0%, 10%, 20%, 30%, 40%), a fixed activity concentration ratio of 5:1 for lesions with respect to the surrounding liver, and a uniform \({}^{90}\)Y activity distribution in each tissue. The total counts in the lungs were assigned using the definition of \(LS\) as in Equation (1)  
\[LS=\frac{C_{lungs}}{C_{lungs}+C_{licer}+\ C_{lesions}} \tag{1}\]  
where \(C_{lungs}\), \(C_{licer}\), and \(C_{lesions}\) are the total counts in lungs, liver parenchyma, and lesions, respectively. Numerical values representing voxel activity in each tissue, i.e., lung, lesion, and liver parenchyma, were calculated with the relative calibration method [23]. This involved multiplying each voxel count by a relative calibration factor defined as the total amount of activity in the ROI over the total counts detected in that ROI, while accounting for the \(LS\) value. The use of an anthropomorphic phantom aims to test the most common dosimetric approaches in RE within a framework of predefined controlled conditions that is as realistic as possible. In this study, dosimetric evaluations obtained by classical dosimetric approaches were compared with those obtained by \(MC\) simulations of radiation transport in the reference phantom. The classical approaches included the mono-compartmental organ-level method  
Figure 1: Density distribution in lungs ROI in g/cm\({}^{3}\). The distribution is characterized by a range of values from 0 to 1.06 g/cm\({}^{3}\), a mean value of 0.221 g/cm\({}^{3}\), and a median value of 0.179 g/cm\({}^{3}\). using the medical internal radiation dose (MIRD) scheme, local energy deposition (LED) at the voxel level, and convolution calculations with voxel S-values (VSV) kernels for soft and lung tissue. For the latter two methods, the specific lung voxel density (as obtained from the CT image) was incorporated into the calculations to account for lung heterogeneities as effectively as these methods allow. Classical Approaches: Mono-Compartmental Organ Level, Local Energy Deposition, and Convolution with Soft Tissue Voxel S-Values  
The mono-compartmental model of the MIRD Committee [2; 3; 24] (hereinafter denoted as MIRD) is a widely used approach for dosimetry at the organ level with beta-emitter radiopharmaceuticals.