## 2 Materials and Methods  
### Anthropomorphic Voxelized Phantom and Common Dosimetric Methods  
The same phantom described and used in a previously published paper [9] was employed in the present study, following the same rationale. Here is a brief summary of its characteristics: the CT scan (voxel size of 1.37 \(\times\) 1.37 \(\times\) 3.27 mm\({}^{3}\)) of a healthy, normotensive patient (weighing 72 kg and 166 cm tall) was segmented with ITK-Snap software (version 3.8.0) [21] into six tissue classes (air, lungs, adipose tissue, soft tissue, liver, and bone) and resampled to cubic voxels with a 2.21 mm side (converting a volume of about 6.1 mm\({}^{3}\) to 10.8 mm\({}^{3}\)) using 3D-Slicer software (version 5.2.2)'s [22] built-in module \"Resample Scalar Volume\" with the \"Nearest Neighbor\" algorithm [23]. Three spherical lesions with a density higher than the liver parenchyma (1.2 g/cm\({}^{3}\) and 1.04 g/cm\({}^{3}\), respectively [24; 25]) were introduced with diameters of 10, 20, and 30 mm. For lung tissue, the actual spatial density distribution of the patient was assumed, based on the HU-density calibration of the CT scanner. Table 1 summarizes the main density features of the lungs. Five activity distributions were considered, with specific LS conditions (0%, 10%, 20%, 30%, 40%) and a fixed activity concentration ratio between lesions and healthy liver of 5:1, assuming a uniform distribution of microspheres within each compartment. Numerical values representing the activity concentration in each voxel were determined using the \"relative calibration\" method [12]. The use of an anthropomorphic phantom created from a CT scan allows us to define a realistic and controlled framework to test several approaches of internal dosimetry. Several common dosimetric methods were tested in this study (Table 2), namely the mono-compartmental organ-level model of the MIRD Committee (hereinafter referred to as MIRD), convolution with voxel S-values (VSVs) using the soft tissue (ST) (kST) and the lung tissue (LT) kernel (kLT), both computed with dedicated MC simulations (further details in the following section). The MIRD model [7; 8; 26; 27] is a widely used approach for dosimetry at the organ level with beta-emitter radiopharmaceuticals. It allows the computation of the mean absorbed dose within a specific 3D ROI, based on the activity in that ROI and the ROI's mass, under the assumption of a uniform radionuclide distribution in a homogeneous medium with complete energy deposition within the compartment. While the MIRD model is routinely employed for liver dosimetry, it has also been proposed for lung dosimetry with additional approximations, such as assuming a lung mass of 1 kg and neglecting the contribution from the hepatic dome to the homolateral lung. In this study, the MIRD approach was applied using the specific mass (727 g) of the reference phantom. The convolution with the VSV [28; 29; 30] employs a 3D kernel of values representing the absorbed dose per decay in a target voxel with its centroid at position \((i,j,k)\), due to the source voxel positioned at \((0,0,0)\). The VSV kernel is computed using dedicated Monte Carlo (MC) simulations of electron and photon transport in a lattice of uniform biological tissue (e.g., soft tissue or lung tissue) with well-defined characteristics. This kernel is then convolved with the time-integrated activity distribution map to derive the absorbed dose map. In this study, VSV kernels for both soft tissue and lung tissue were considered for \({}^{166}\)Ho. The absorbed dose was subsequently corrected for local density variations relative to the uniform tissue density assumed in the VSV calculations, with rescaling based on the CT-derived voxel density. Local energy deposition (LED) was not considered suitable for dosimetry with Ho-MS due to the presence of \(\gamma\) emissions, which result in unavoidable radiation transport outside the source voxels.