## 6 Results: Simulations of Extended Scenarios
### Tracking a Tracer Particle on a Spiral Path  
The parameters that describe the spiral simulation were Spiral [0.1 mm, 0.8 m s\({}^{-1}\)] for a simulation time of 1.0 s with the tracer PointTracer [1000 uCi, 300 um, \({}^{68}\)Ga]. The location data with time can be seen in Figure 11. The combination of a path with low speed and a  
Figure 10: The distribution of cutpoints in the \(X\) dimension for the oscillatory motion [100 mm, 62.5 Hz] using a PointTracer [1100 uCi, 300 um, \({}^{68}\)Ga] of the sample of LORs used to find two tracer locations on different parts of the oscillatory motion. Corresponding to locations given at times on Figure 9b, (**a**) is a sample over a turning point at 0.064 s, and (**b**) is a sample of linear motion at 0.056 s. Also given are the values of the “tracked location”, which is derived from the simulated LORs, and the “true location”, which is the ground truth input to the simulation.  
tracer particle with high radiolabelled activity was used to ensure that as many locations as possible were present on the parts of the spiral path with smaller radii. The spiral gets larger in radius as time increases and allows for the determination of the smallest structure that can be tracked on a Lagrangian path with the HR++ scanner in terms of the minimally tracked radius.  
In the first 0.2 s of Figure 11d, the residuals are the largest, showing an inability to track the spiral path of the smallest radii considered, which correspond to the higher rates of acceleration experienced by the tracer.  
To analyse the spiral tracking results in more depth, the uncertainty in the path \(\Delta(R)\) was calculated with Equation (1), where the residual \(\delta R\) is defined as the difference between samples of locations from the true and tracked paths. Figure 12 shows the results as a function of radius and magnitude of acceleration. In Figure 12a, the uncertainty in the path tends to decrease with increasing radius for a tracer moving along the spiral path. For radii larger than 2.0 mm, the uncertainty tends to an asymptotic value of around 0.25 mm, which is similar in magnitude to the radius of the tracer particle. For radii smaller than 2.0 mm, the uncertainty is considerably larger than the size of the tracer, suggesting that, within this spiral, the PEPT measurement is unable to reliably track structures smaller than 2.0 mm with a Lagrangian path. The uncertainty in the path as a function of acceleration is shown in Figure 12b, where the acceleration tends to increase with a decreasing radius. For locations on the spiral path with acceleration values less than approximately 400 m s\({}^{-2}\), the uncertainty in the path was smaller than the radius of the tracer particle.  
Figure 11: The location data of the simulation of Spiral [0.1 mm, 0.8 m s\({}^{-1}\)] with a PointTracer [1000 \(\mu\)Ci, 300 \(\mu\)m, \({}^{68}\)Ga] in the (**a**) \(X\), (**b**) \(Y\), and (**c**) \(Z\) dimensions with time \(t\). Shown is the “tracked path”, the location data derived from the simulated LORs, and the “true path”, which is the ground truth input to the simulation. Note the different scale in subplot (**c**). The tracked path residuals \(\delta R\) are shown in (**d**).