## 2 Materials and Methods
### Anthropomorphic Voxelized Phantom
It enables the computation of the mean absorbed dose (\(\overline{AD}\)) to a specific 3D ROI, given the activity in that ROI (\(A_{ROI}\)) and the ROI's mass (\(M_{ROI}\)). The model assumes a uniform radionuclide distribution in a homogeneous medium, with all emitted radiation energy deposited within the same ROI. For \({}^{90}\)Y, using the mono-compartmental model and under the assumption (commonly used in radioembolization) of full radioactivity retention in tissues, the mean absorbed dose can be computed using the following Equation (2). \[\overline{AD}_{ROI}=49.38\times\frac{A_{ROI}}{M_{ROI}} \tag{2}\]  
The numerical factor in Equation (2) accounts for the physical properties of the radionuclide (i.e., the physical half-life and mean energies of the emitted \(\beta^{-}\) particle spectra as stated in the AAPM recommendations). It includes conversions between measurement units with \(\overline{AD}\) expressed in Gy, activity in GBq, and mass in kg. While the mono-compartmental model is commonly used for liver dosimetry, it has also been proposed for lung dosimetry with additional approximations. For lung dosimetry, the lung mass is set to 1 kg, and the contribution to the absorbed dose in the homolateral lung of the activity in the liver dome is considered negligible [2; 3]. In this study, the MIRD approach was applied according to Equation (2), using either the lungs' specific mass of the Reference phantom (724 g) or a constant mass of 1 kg. The local energy deposition approach [23] (hereinafter referred to as LED) assumes that the energy from the decay in a given voxel is completely absorbed within the same voxel. This assumption is reasonable when the range of \({}^{90}\)Y decay electrons in tissue is comparable to voxel sizes used in nuclear medicine tomographic imaging (typically 2-5 mm). For example, in soft tissue, the mean CSDA range of the emitted electrons is approximately 5 mm. The computation is straightforward using Equation (2), referring to the activity and mass of each voxel (the latter is estimated by a calibrated CT scanner). The convolution approach using VSV [25; 26; 27] (hereinafter referred to as SVOX) employs a 3D kernel of values that represent the absorbed dose per decay in a target voxel with its centroid positioned at \((i,j,k)\), due to a source voxel positioned at \((0,0,0)\). The VSV kernel is calculated using a dedicated \(MC\) simulation of electron and photon transport in a lattice of uniform medium consisting of biological tissue (e.g., soft tissue), then convoluted with the time-integrated activity distribution map to obtain the absorbed dose (\(AD\)) map. In this paper, a previously published VSV kernel for soft tissue (ST) was used [26] (methodology hereinafter referred to as SVOX_ST) to calculate the \(AD\) in the lungs as if they were composed of soft tissue. The \(AD\) was then corrected for local density variations relative to the uniform tissue density used for VSV calculations (methodology hereinafter referred to as SVOX_L), Equation (3), rescaling the voxel \(AD\) by the CT-derived lung voxel density [15]:  
\[AD_{SVOX}^{Lungs}(x,y,z)=\frac{AD_{SVOX}^{ST}(x,y,z)}{\rho(x,y,z)}\times\overline {\rho}_{ST} \tag{3}\]  
where \(AD_{SVOX}^{ST}\)\((x,y,z)\) is the \(AD\) to the voxel in the \((x,y,z)\) position computed with the ST VSV kernel; \(\overline{\rho}_{ST}\) is 1.04 g/cm\({}^{3}\) (the ST density used to compute the VSV); \(\rho(x,y,z)\) is the density of the voxel in the \((x,y,z)\) position; and \(AD_{SVOX}^{Lungs}\)\((x,y,z)\) is the \(AD\) in lungs for the same voxel, after applying the local density correction. \(\overline{AD}\) of lungs was then obtained as the mean value of the \(AD\) values. Given the large difference in density between ST and lungs, a lung VSV kernel was also computed in this study, assuming the International Commission on Radiation Units and Measurements (ICRU) lung tissue [21; 22; 28] (density of 0.296 g/cm\({}^{3}\)) as a medium (further details are provided in the following section).