## 6 Results: Simulations of Extended Scenarios
### Tracking a Particle in Oscillatory Motion  
Two simulations were run using a PointTracer [1100 \(\upmu\)Ci, 300 \(\upmu\)m, \({}^{68}\)Ga] with Oscillatory [100 mm, \(f\in[1.25,62.5]\) Hz] in the \(X\) direction. This particle was chosen to represent the type of particle used in published examples of PEPT measurements of multiphase flow and the type of tracer particle that is routinely fabricated for experiments at PEPT Cape Town with the HR++ scanner [13].  
Two cases of oscillatory motion for extreme values of velocity have been highlighted in Figure 9, with Figure 9a,c representing the low-speed (0.5 m s\({}^{-1}\)) path and Figure 9b,d showing the high-speed (25.0 m s\({}^{-1}\)) path. For the low-speed path (Figure 9a,c), the location data of the tracked path tend to be close in value to those of the true path, as illustrated by small values of the residuals. Near the turning points, the absolute values of the residuals tend to be larger but are mostly within an acceptable range of \(\pm\)300 \(\upmu\)m defined by the size  
Figure 9: Simulated oscillatory motion in the \(X\) dimension of the NRW-100 PointTracer [1100 \(\upmu\)Ci, 400 \(\upmu\)m, \({}^{68}\)Ga] against time and corresponding path residuals \(\delta X\) for speeds of 0.5 m s\({}^{-1}\) (**a,c**) and 25.0 m s\({}^{-1}\) (**b,d**). The time series location data plots show the “true path”, which is the ground truth input to the simulation, and the “tracked path”, which is derived from the simulated LORs.  
of the tracer. This range is referred to as acceptable because it is expected that positrons from the tracer generally annihilate within or close to the material of the tracer particle when placed in dense media such as water and solid particle mixtures [25].  
Looking at the high-speed path (Figure 9b,d), there is a significant difference as compared to the low-speed case in the tracking fidelity of the oscillatory motion. On the straight portions of the trajectory, the tracer particle experiences a constant velocity, and the residuals are low along this path. However, near the turning points, the residuals are considerably larger. During constant linear motion and within a given time window, an average of multiple tracer positions will tend to be a good measure of the true position of the tracer particle at the middle of the time window. In contrast, during linear acceleration, the averaging process will not cancel out those tracer positions that were taken before or after the middle of the time window. This will lead to large uncertainty in the tracer location whenever a tracer is undergoing acceleration during any given time window. Calculation of an average residual over several cycles of the oscillatory motion would tend to a small value, close to zero, but would misrepresent the overall distribution of residuals over each cycle, which are significantly larger near the turning points.  
Therefore, the large deviation in \(X\) arising at the turning points of the oscillatory motion in Figure 9 is due to the non-symmetric distribution of the LORs in the samples around the turning points. This asymmetry is caused by the infinite acceleration about the turning points, which shifts the mean of the distribution of cutpoints used in each location to a point before/after the turning point.  
Examples of the distributions of cutpoints used in samples both at and in between the turning points of the motion are given in Figure 10 to illustrate this behaviour. The distribution of cutpoints in the sample near the turning point (see Figure 10a) is narrower and tends to have more weight to the right (in this case) of the true value of the location. The tracked location occurs 3.89 mm away from the true location. A smaller deviation of the tracked location from the true location of 0.30 mm is visible in Figure 10b for the location on the linear part of the motion, with cutpoints distributed symmetrically and around the true location.