## 2 Materials and Methods
### Phantom Studies
#### 2.1.3 Phantom Dose Calculation and Analysis  
AD computations rely on voxelized ADD derived from SPECT-MAA and PET-MS emission intensity maps [21; 22; 23]. The contribution of a source voxel to the AD of a target voxel is represented by a so-called voxel S-value, considering that both (i.e., source and target) voxels are in the same tissue or organ. In this work, we used the S-kernel convolution method to generate ADD from activity-voxelized distributions [24; 25]. The voxel S-values have been determined by several authors based on Monte Carlo simulations for several radionuclides, biological tissues, and voxel sizes [22; 24; 26]. We used a voxel S-kernel calculated for \({}^{90}\)Y, soft tissue (i.e., 1 g/mL), and 2.21 mm cubic voxels, expressed in units of mGy/(MBq.s) [24; 25].  
During the calculation of voxelized ADD, the SPECT and PET images need to be normalized and calibrated in activity. Normalization is undertaken so that the sum of the total intensity in the images equals the total activity (in MBq) contained in the _liver_box_ ROI. Then, the calibrated image is convolved with the 3D kernel expressed in mGy/(MBq.s). Finally, each voxel is multiplied by the time integral of \({}^{90}\)Y decay (i.e., from zero seconds to at least ten times the \({}^{90}\)Y half-life) so that the total number of decays expected for that voxel can be achieved.  
The maximization of the Pearson correlation coefficient (PCC) between the SPECT-MAA- and PET-MS-derived ADD was investigated. In addition, the intraclass correlation coefficient (ICC) and the \(\gamma\)-index agreement test were also computed and evaluated.  
The \(\gamma\)-index is a pass/fail type agreement test, which considers the simultaneous combination of two test conditions: the dose-difference (DD) and distance-to-agreement (DTA) [18; 27; 28]. A result of this test less than or equal to 1 concludes that the analyzed voxels are within the DD/DTA tolerance criterion, which means the test was successful. In the opposite case, where the test result is greater than 1, the test failed. The test outcome in each ROI is computed as the success rate of the ROI voxels referred to as the \(\gamma\)-index passing rate (PR). In this work, seven DD (\(\%\))/DTA (mm) test conditions (i.e., 5/5, 10/5, 5/10, 10/10, 15/10, 10/15, and 15/15) were considered in the computation of the \(\gamma\)-index voxelized distributions. These test conditions were chosen considering the 10 mm spatial resolution of the SPECT-MAA images and a reasonable AD deviation between the two distributions (i.e., MAA and MS). Based on data from the literature, we can assume that the NLV AD should not exceed 70 Gy [29; 30; 31]. In addition, it is also understandable that AD values to the PTV should be 80 Gy or higher [29; 30; 31]. From these values, we assumed a maximum percentage difference of around 15%. The resulting \(\gamma\)-index PR computed from the phantom ADD became a reference for the later discussion on the \(\gamma\)-index PR results obtained from patients' data.  
The registration and resampling operations were performed by means of the open-source software 3D Slicer [32]. The ROI segmentation task was carried out in the open-source ITK-SNAP [33]. The commercial software MATLAB R2017b (The MathWorks) and  
Figure 2: Phantom ROI on a similar slice of CT and SPECT-MAA-based ADD. Cubic voxels of 2.21 mm length were used after resampling both images. The red circles represent the _ptv_5_cyl_backg_ mask (i.e., including four samples of the background near the PTV) or the _ptv_5_cyl_ mask (i.e., excluding the four background samples). The individual PTV cylinder masks (red circles) are numbered 1 through 5. The _liver_box_ and the _liver_cyl_ masks are represented by the blue square and the green circle, respectively.  
the open-source code Tools for NifTI and Analyze Image [34] were used for the computation of the voxelized ADD, voxelized \(\gamma\)-index distributions and PR values, PCC, and ICC values.