## 2 Materials and Methods  
### Absorbed Dose Calculation  
MIRD-5 Phantom Geant4 MC Simulations  
According to Pamphlet 5 of the Medical Internal Radiation Dose (MIRD), a mathematical hermaphrodite adult phantom (Figure 1) was used in this study, which consisted of entire anatomical organs [16], with male and female reproductive organs (testicles) and (ovaries and uterus). Female breasts were also incorporated into the phantom by adopting the breasts from [17]. The MIRD-5 phantom is made up of three different types of tissues with densities of bone 1.4862 g/cm\({}^{3}\), lung 0.2958 g/cm\({}^{3}\), and soft tissue 0.9869 g/cm\({}^{3}\).  
The Geant4 version 10. 6 toolkit was used in this study [18; 19] as an advanced example human phantom. Within the liver, a single tumour was created with the same shape as the MIRD-5 mathematical liver, positioned in the centre and variable in mass (Figure 2). Equation (1) was used to calculate the tumour involvement \((TI)(\%)\).  
\[\textit{Tumour Involvement}\ (TI)(\%)=\frac{\textit{Mass of tumour}}{\textit{Mass of liver}}\times 100\% \tag{1}\]  
We used tumour shape as the shape of the mathematical MIRD-5 liver in this study. Due to the geometrical boundaries which were allowed, only spheres with a radius of up to 4.3 cm could be confined within the liver, resulting in a tumour mass of 333 g and a maximum tumour involvement of only 18.2% [13].  
Figure 1: The MIRD-5 adult hermaphrodite human phantom as constructed and visualized using Geant4.  
Figure 2: Antero-superior view of the tumour model for (**a**) 10, (**b**) 30, (**c**) 50, and (**d**) 70% tumour involvement (TI).  
The electromagnetic interactions of photons and electrons were modelled using the low-energy electromagnetic package [20], which was based on the Livermore Evaluated Data Libraries. The generation threshold for secondary particles was set at one millimetre. The decay of \({}^{159}\)Gd and its distribution in the tumour, normal liver, lungs, and other organs were modelled using the Geant4 radioactive decay and general particle source components. The \({}^{159}\)Gd radionuclide point sources within each organ were evenly distributed, with activity uptake based on the T/N and LS, having emissions that are randomized in their direction. Normal liver, tumour, lungs, and other organs were set at sensitive volumes. The simulation's result was defined as mean energy (MeV) transferred into each volume. Several patient scenarios, including varying TI (10, 30, 50, and 70%), LS (0, 5, 10, 15, and 20%), and T/N (1, 2.5, 5, 7.5, and 10) were simulated. The Geant4 MC package was used to simulate the setup with \(10^{7}\) histories. The simulation was run thrice for each parameter combination to achieve a less than 1% standard deviation. To obtain a tumour dose of 120 Gy, the tumour, normal liver, lungs, and other organ doses acquired via simulation were multiplied by the same factor that provides a tumour dose of 120 Gy. When a recommended dose of 120 Gy [15] was delivered to the tumour, the absorbed dose (Gy) to normal liver, lungs, and other organs was calculated by converting the mean energy (MeV) transferred within the organs to joules (J) and dividing it by the mass of the organ.