## 3 Results
### Beam Spot Analysis  
Figure 5a,b are MAGAT-f and PAGAT polymer gels after 73.5 MeV proton irradiation, respectively. The readout of irradiated polymer gel was represented by the volumetric isotropic T\({}_{2}\)-weighted acquisition (VISTA) map to characterize the beam spot (Figure 5c,d).  
The central spot was chosen to measure the FWHM and standard deviation, \(\sigma\) (Table 4). Horizontal profiles of the central spot measured by MAGAT-f and PAGAT gels were analyzed (Figure 6). Single Gaussian fitting was performed to measure FWHM and standard deviation, as shown in Figure 6c,d. Goodness of fit were 0.9992 and 0.9948 for MAGAT-f and PAGAT, respectively.  
\begin{table}
\begin{tabular}{l l l l} \hline
**Dosimeters** & **Spot Width, \(\sigma\)/cm** & **Penumbra Width/cm** & **FWHM/cm** \\ \hline MAGAT-f & 0.58 & 0.65 & 1.37 \\ PAGAT & 0.50 & 0.57 & 1.17 \\ \hline \end{tabular}
\end{table}
Table 4: Spot width and penumbra width measured with polymer gel dosimeter (73.5 MeV).  
Figure 4: (**a**) Two crossed planes of the Bragg peak; (**b**) reconstructed 3D volume rendering of the Bragg peak; (**c**) transversal cut; and (**d**) longitudinal cut of the Bragg peak.  
The spot size of energy 73.5 MeV simulated with the MC method was 5.83 mm. The deviations between the gel measurement and MC simulation were \(-0.56\%\) and \(-14.23\%\) for MAGAT-f and PAGAT gels, respectively.  
Gaussian fitting results at different planes are shown in Figure 7. Detailed Gaussian fitting parameters are shown in Table 5. With the fitting parameters, the spot width and FWHM can be calculated for the pencil beam. Due to the 3D measurement specificity of polymer gel dosimeters, spot sizes at different depths were acquired (Table 6).  
Figure 5: (a) MAGAT-f gel and (**b**) PAGAT gel after 73.5 MeV proton irradiation. VISTA map of (**c**) MAGAT-f and (**d**) PAGAT gel dosimeter. (No. 1–4 in (**c**,**d**) demonstrate the spot spacing numbers measured by MAGAT-f and PAGAT gel dosimeter).  
Figure 6: Proton beam spot measured with (**a**) MAGAT-f gel (top left) and (**b**) PAGAT gel (top right); the lateral profile of (**c**) MAGAT-f gel and (**d**) PAGAT gel (73.5 MeV).  
\begin{table}
\begin{tabular}{l l l l l l} \hline \hline  & & & \multicolumn{3}{c}{**f(x) = a \(\times\) exp(\(-\)(x \(-\) b)/c)\({}^{2}\)**} \\ \hline
**Gel** & **Slice** & **a** & **b** & **c** & **R-Squared** \\ \hline \multirow{9}{*}{PAGAT} & 1 & 0.013 & 1.446 & 0.794 & 0.996 \\  & 2 & 0.015 & 1.455 & 0.8249 & 0.997 \\  & 3 & 0.016 & 1.463 & 0.882 &  
Figure 8 shows the linear correlation between FWHM and spot width \(\sigma\) at different depths. According to TG-224 [10], FWHM was equal to 2.35\(\sigma\) at a position. The slopes of linear fitting of PAGAT and MAGAT-f were 2.433 and 2.355, respectively. The discrepancies between the fitting slope and the theoretical value were 3.53% and 0.21% for PAGAT and MAGAT-f, respectively.  
\begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Depth/cm**} & \multicolumn{2}{c}{**FWHM/cm**} \\ \cline{2-3}  & **PAGAT** & **MAGAT-f** \\ \hline
1.244 & 1.319 & 1.346 \\
1.592 & 1.368 & 1.354 \\
1.940 & 1.470 & 1.347 \\
2.288 & 1.463 & 1.409 \\
2.636 & 1.501 & 1.441 \\
2.984 & 1.546 & 1.495 \\
3.332 & 1.550 & 1.518 \\
3.680 & 1.528 & 1.619 \\
4.028 & 1.497 & 1.678 \\ \hline \hline \end{tabular}
\end{table}
Table 6: FWHW at different depths measured with PAGAT and MAGAT-f for 73.5 MeV beam.  
Figure 7: (**a**) Dose distribution measured with PAGAT and (**b**) its Gaussian fitting results of different slices; (**c**) dose distribution measured with MAGAT-f and (**d**) its Gaussian fitting results of different slices.