## 4 Discussion  
Considering the results shown in Figure 5, two peaks can be observed from the PEMGRAPH measurements. In addition to these two peaks, a small tail peak was observed and vanished rather quickly. The main reason for the formation of these peaks was the proton interaction and scattering from the HDPE container used in the proton irradiation experiment; in particular, the wall effect from the HDPE container that mainly contains carbon (note that \({}^{1}\)H nuclei do not produce stable positron emitting radioisotopes). Upon the interaction of proton with matter, the protons lose energy, and hence lower energy protons should be found in the deeper parts of the water-gel phantom, and at shallow depths, there will be more higher energy protons; this satisfies the \({}^{11}\)C and \({}^{15}\)O threshold energies for their production. The threshold energies for the creation of \({}^{11}\)C, \({}^{15}\)O and \({}^{13}\)N were reported to be 20.61, 16.79 and 5.660 MeV, respectively [41]. As depth increases and in turn the proton energy decreases, the main production reaction is \({}^{16}\)O(p,2p2n)\({}^{13}\)N, which has a lower threshold energy compared to other positron emitting radioisotopes. The current finding using the present scheme is in good agreement with previously reported results [41]. The production of proton-induced radioisotopes was confirmed by the SA approach, which was applied to the dynamic data measured using the PEMGRAPH system. In Figure 6, it was confirmed that the two peaks formed at the entrance (i.e., lower depth) and at the end (i.e., higher depth) were mainly due to the \({}^{11}\)C radioisotopes. The \({}^{15}\)O peak was formed after the second \({}^{11}\)C peak, which is also in agreement with the knowledge of the  
Figure 8: Three-dimensional scatter visualization and the 1D profiles of \({}^{11}\)C, \({}^{15}\)O and \({}^{13}\)N radioisotopes, along with the simulated dose for the 80 MeV proton beam irradiating water-gel phantom. threshold energy for the production of these radioisotopes and subsequent energy loss of protons at higher depths. Interestingly, the peak of \({}^{13}\)N radioisotopes was formed after the \({}^{15}\)O peak, mainly due to its lower threshold energy of production; this can be seen in the 1D profile plot shown in Figure 6.  
Furthermore, the contribution of proton-induced radioisotopes was confirmed in the ROI study. In Figure 7, the main contributing radioisotopes in the entire edge and plateau regions are shown to be \({}^{11}\)C radioisotopes. Considering the Bragg region, the \({}^{13}\)N radioisotope makes the highest contribution in the Bragg region. The lowest contribution was found to be from \({}^{15}\)O radioisotopes. The SA analysis confirmed the activity in the distal fall-off region of proton-induced \({}^{13}\)N radioisotopes. From the 1D profiles in Figure 8, the offset between the simulated dose and extracted \({}^{13}\)N peaks is about 3 mm, which is in good agreement with our recent simulation study [42]. Our reported average offset value between the Bragg and the \({}^{13}\)N peak was found to be about 2 mm for 80 MeV of incident protons impinging on the water-gel phantom [42]. The present scheme could be applied to inhomogeneous phantoms and small animal experiments after standard irradiation. Further analysis suggests that there is a significant advantage of 3D visualization for proton range verification, especially for inhomogeneous targets (i.e., human organs/tissues). Considering the generated 3D spatial distribution of positron emitting radioisotopes shown in Figure 8, one can note that the present step-by-step scheme and developed numerical tools can accurately obtain such 3D visualizations. These 3D visualizations would be important when investigating non-uniform targets such as organs in the human body or animal subjects. As shown in Figure 8, \({}^{11}\)C radioisotopes formed two peaks at lower depth (near the entrance) and at higher depth (\(\sim\)33 mm).