## 2 Materials and Methods
### Evaluation of Voxel-Based Dosimetric Approaches  
Each voxel-based dosimetric approach was evaluated in terms of \(\overline{AD}\) in the lung region, as shown in Equation (1). \[\overline{AD}=\frac{1}{N_{\text{Lungs}}}\sum_{x,y,z}AD(x,y,z) \tag{1}\]  
where the sum is taken over all voxels within the lung region of interest (ROI), \(N_{\text{Lungs}}\) is the total number of lung voxels, and \(AD(x,y,z)\) is the absorbed dose in the voxel with its centroid at the position \((x,y,z)\). The relative difference (RD) with respect to the reference MC simulation was evaluated as defined in Equation (2). Correlation plots compare the reference MC \(\overline{AD}\) on the x-axis with respect to the tested method's \(\overline{AD}\) on the y-axis for each LS condition. \[RD(\%)=\frac{\text{Tested}-\text{MC}}{\text{MC}}\times 100 \tag{2}\]  
As regards the dosimetric comparison between \({}^{90}\)Y and \({}^{166}\)Ho, MC simulations of both radionuclides have been evaluated as AD rate (ADr) maps for LS = 10% only. The ADr maps were compared using cumulative dose-volume histograms (DVH) and homogeneity index (Equation (3)) as defined by [PERSON] et al. [34]. \[HI=\frac{D_{5\%}}{D_{95\%}} \tag{3}\]  
In targeted radiotherapy (TRT), radiation is delivered continuously with a decreasing dose rate, causing both lethal and sub-lethal damage, the latter competing with the tissue repair mechanisms. Since the treatment extends over hours or days, the radiation-induced damage and the repair of sub-lethal damage occur simultaneously. The biological effective dose (BED) in the Dale formulation [35; 36] quantifies the biological effects of the radiation delivery patterns taking into account the repair of sub-lethal damage as a competitive process with the radiation damage. The Dale BED formulation depends on two key parameters: the tissue's \(\alpha/\beta\) ratio, which describes tissue sensitivity to radiation, and the sub-lethal damage repair rate (\(\mu\)), related to the rate at which single-strand DNA breaks are repaired. Repair half-times (T\({}_{\mu}\)) of typical organ tissues range from 0.5 to 3 h [36; 37], and the repair rate is given by \(\mu=\log(2)/T_{\mu}\). The BED for continuous radiation delivery with an exponentially decreasing dose rate (Equation (4)) depends on the initial dose rate (\(R_{0}\)), the radionuclide's decay constant (\(\lambda=\log(2)/T_{1/2}\), where \(T_{1/2}\) is the physical half-life), the repair rate (\(\mu\)), and the tissue's \(\alpha/\beta\) ratio [36]. \[BED_{TRT}=\frac{R_{0}}{\lambda}\cdot\left(1+\frac{R_{0}}{(\mu+\lambda)\cdot( \alpha/\beta)}\right) \tag{4}\]Assuming an administered activity of \({}^{90}\)Y and \({}^{166}\)Ho so that the \(\overline{AD}\) in the whole liver region was 60 Gy [38], resulting from the mono-compartmental model for the liver region (\(A_{max}\)(GBq) = \(\frac{60\,\text{Gy}\cdot M_{linear}}{\text{DF}}\), where \(M_{linear}\) is the mass of the whole liver in kg and DF is a dosimetric factor that numerically is \(15.87\,\,\text{Gy}\cdot\text{kg}/\text{GBq}\) for \({}^{166}\)Ho [27] and \(49.87\,\,\text{Gy}\cdot\text{kg}/\text{GBq}\) for \({}^{90}\)Y [7]), \(1.8\,\,\text{GBq}\) and \(5.7\,\,\text{GBq}\) were obtained, respectively. The BED maps were computed by converting the MC voxel AD values using Equation (4). \(T_{\mu}\) of 0.5 and 1 h for LT were considered [37], along with an \(\alpha/\beta\) ratio of 3 Gy [39] and the radionuclide-specific physical half-life (\(T_{1/2}\)). From the BED maps, the corresponding \(EQD_{2}\) maps and the DVHs were calculated using the linear quadratic model (Equation (5)) [40], considering the values of \(T_{\mu}\) for LT.  
\[EQD_{2}=\frac{BED}{1+\frac{2}{\alpha/\beta}} \tag{5}\]  
Additionally, the volumetric lung absorbed dose constraints from external beam radiotherapy (EBRT) [41], listed in Table 4, have been evaluated on the \(EQD_{2}\) DVHs for the LS = 10% case.