## 5 Results: Model Validation
### Validation with Location Data from a Stationary Tracer  
The experiments with the tracer particle at different stationary positions within the FOV of the HR++ scanner, as shown in Figure 3, were also simulated using the ButtonTracer [60 \(\upmu\)Ci, \({}^{22}\)Na] in the HR++ GATE model. Figure 5 shows examples of the LOR distributions  
Figure 4: Coincidence rate as a function of activity \(A\) for both the simulation and experiment using a ButtonTracer [20 \(\upmu\)Ci, \({}^{68}\)Ga] placed at position Stationary [\(-1.60\) mm, \(0.49\) mm, \(-1.50\) mm]. Both fits to the experimental and simulated coincidence data had a reduced \(\chi^{2}\) of approximately 28,000.  
from experiments and simulations at Position O (see Figure 3). There were two missing buckets during the experiments, and they were removed from the simulation. Both plots show the sites of the missing buckets as areas where no events were detected. Due to the stochastic nature of the nuclear interactions involved in PET, the simulation will never reproduce an identical combination of LORs as found in the experiments.  
The experimental and simulated LOR data from a stationary tracer particle were processed using PEPT-ML to produce a series of tracer locations as shown in Figure 6. Gaussian functions were fitted to the distributions of locations to find a mean location with a standard uncertainty value in order to compare the simulated and experimental values, the full results of which can be found in Appendix C, Table 11. In order to compare the deviations from the ground truth between the experiments and simulations, Equation (1) was applied to the location data with time, where \(\delta P\) is the difference in position as calculated between a location \(i\) on the tracked path (\(P_{track,i}\);\(t_{track,i}\)) and the ground truth path at the time value taken from the tracked path (\(P_{true,i}\);\(t_{track,i}\)), and \(N\) is the number of locations. The parameter \(\Delta(P)\) will be referred to as the average path uncertainty.  
\[\Delta(P)=\sqrt{\frac{\sum_{i=1}^{N}(\delta P)^{2}}{N}} \tag{1}\]  
Figure 6 shows agreement between locations derived from the HR++ model and the experiments, as the fitted Gaussian distributions have similar profiles. This point is reinforced by a comparison of the mean and standard uncertainty values from the fits as given in Table 2 where the intervals overlap, and thus the experiment and simulation results agree within experimental uncertainty. A similar analysis for the other FOV positions is given in Appendix C, Table 11.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Location** & \(\mathbf{X\pm\Delta(X)}\) **(mm)** & \(\mathbf{Y\pm\Delta(Y)}\) **(mm)** & \(\mathbf{Z\pm\Delta(Z)}\) **(mm)** \\ \hline
1 (Exp) & \(326.50\pm 0.78\) & \(2.30\pm 0.51\) & \(-2.31\pm 0.29\) \\
1 (Sim) & \(326.66\pm 0.81\) & \(2.20\pm 0.48\) & \(-2.21\pm 0.22\) \\ \end{tabular}
\end{table}
Table 2: The mean location and path uncertainty from Gaussian fits to the distribution of locations from PEPT experiments (“Exp”) and GATE simulations (“Sim”) at Position 1 in the FOV in each of the \(X\), \(Y\), and \(Z\) dimensions. The fits correspond to the data shown in Figure 6.  
Figure 5: The distribution of 300 LORs projected in 2D in the \(XY\) plane from a tracer particle at Position O (see Figure 3) in the HR++ PET scanner as derived from (**a**) PEPT experiments and (**b**) GATE simulations.