## 2 Materials and Methods
### Phantom Studies  
As physical phantom tests represent the gold standard for validating image processing methods before designing patients studies [20], the first part of the experimental work was carried out based on physical liver phantoms to allow estimation of accuracy, correlation, and agreement of the estimated absorbed dose distributions (ADD) derived from the SPECT and PET images. The phantom used was designed and built specifically for this research work. The complete phantom is shown in Figure 1. It is a cylinder made of transparent 5 mm thick acrylic, with a 300 mm base diameter and a 250 mm height. The outer body of the phantom can be filled with water to better model photon attenuation in the human body. This cylindrical outer body can accommodate in its core a cylindrical body of 150 mm base diameter by 100 mm height. The inner cylinder has a capacity of 1600 mL, simulating the volume of an adult liver. This inner cylinder contains five small fixed cylinders inside to simulate the PTV; two peripheral cylinders with 15 mL volume each (i.e., cylinders 1 and 3 in Figure 1, with 15 mm base diameter and 85 mm height), two other peripheral cylinders with 8 mL volume each (i.e., cylinders 2 and 4 in Figure 1, with 13 mm base diameter and 6 mm height), and a larger central cylinder with a volume of 108 mL (i.e., cylinder 5 in Figure 1, with 50 mm base diameter and 55 mm height) in the center of the inner cylinder.  
This experimental work used MAA and MS microparticles. These, after mixing with a liquid (such as water), are deposited by gravity on the bottom of the container. Thus, the microparticles were mixed with agarose at a ratio of 3 parts of agarose to 100 parts of water. This way, the agarose powder mixed with water and heated at around 90 \({}^{\circ}\)C takes the form of a gel when cooled to 30 \({}^{\circ}\)C in less than 10 min, forming an immobilizing net with the microparticles in suspension. This is the white (milky) matter that can be seen inside the inner cylinder in the main photograph of Figure 1.  
Two phantom experiences were thus created, one using the \({}^{99m}\)Tc-MAA and the other using the \({}^{90}\)Y-MS. First, a 150 mL mixture of agarose with 261 MBq of \({}^{99m}\)Tc-MAA at the time of the SPECT-MAA image acquisition was made. This mixture was used to fill the five internal cylinders simulating the PTV. The remaining volume of the phantom (liver) was filled with agarose without \({}^{99m}\)Tc-MAA. The outer body of the phantom was filled with water. SPECT-MAA and CT images of the phantom were acquired after the mixture solidified at room temperature. Subsequently, after being thoroughly cleaned and all of the agarose residues were removed, the acrylic phantom was loaded identically into the five inner cylinders with a gel-type agarose mixture with \({}^{90}\)Y-MS. These cylinders were loaded with 3184 MBq of \({}^{90}\)Y-MS at the time of the PET-MS image acquisition. PET-MS and CT images of the phantom were acquired after the mixture solidified at room temperature. Both \({}^{99m}\)Tc-MAA and \({}^{90}\)Y-MS activity concentrations in the cylinders simulated the PTV of about 1.63 MBq/mL and 20 MBq/mL, respectively. These values are approximately three times higher than the average PTV for all our patients.  
Figure 1: The main photograph shows the entire phantom on its stand with the inner insert containing several PTV cylinders. This inner insert is shown completely empty in the upper-right corner, where the five PTV cylinders are indicated by numbers 1 to 5. The entire phantom shown in the main photograph was captured with the inner insert in place filled with agarose.