# Investigation into the Uniformization of Proton Beams for FLASH Therapy
## 4 Comparison of Uniformization Results  
Without altering the central trajectory of the beam in the aforementioned transport line, particle cross-sections at the target were selectively intercepted using a collimator with a radius of 50 mm. The resulting beam spot cross-sections are shown in Figure 9, with particle loss rates of 0.004%, 0.02%, and 1.4%, respectively. Overall, the results indicate that the use of nonlinear magnets in the design of the proton therapy device's transport line significantly enhances beam uniformity, yielding positive effects for various applications. To assess the uniformity of the beam spot, in Figure 10, a 20 \(\times\) 20 grid was applied to a cross-sectional area of 100 mm \(\times\) 100 mm, totaling 400 individual grids with each grid measuring 5 mm \(\times\) 5 mm. Particle counts within each corresponding grid were then recorded and analyzed. From left to right, the figures correspond to the selection of horizontal and vertical grids from the 5 th to the 16 th. Specifically, the middle picture selects horizontal grids ranging from the 5 th to the 16 th and the vertical grids range from the 3 rd to the 17 th. The horizontal grids span from the 4 th to the 17 th and the vertical grids range from the 6 th to the 16 th. The utilization rates for particles are 78.78%, 84.38%, and 82.5%, respectively. The coordinates of each grid are determined by the indices \(i,j\), where \(i\) and \(j\) represent the  
Figure 8: Shows the transverse beam profiles at the locations of Nonlinear Magnet Group 1 (**left**) and Nonlinear Magnet Group 2 (**right**). At these two positions, the transverse size ratios are respectively approximated 4.8 and 5.5.  
horizontal and vertical directions, and the particle count in each grid is denoted as \(N_{ij}\), \(i,j=1\dots n\). The formula for calculating non-uniformity is defined as [30]:  
\[u=\pm\frac{n}{i,j}\frac{\left|\overline{N_{ij}-\overline{N_{ij}}}\right|}{ \overline{N_{ij}}} \tag{8}\]  
Through calculations, the obtained results for non-uniformity are \(\pm\)10.09%, \(\pm\)13.23%, and \(\pm\)12.3% for the methods using octupole magnet, multipole magnet components, and step-like field. The comparison results are shown in Table 4. The results indicate that the method of using multipole magnet components achieves the highest beam utilization, while the method using only the octupole magnet yields the best non-uniformity result, but the difference is very small. Considering the need for flatness in clinical settings to achieve the FLASH treatment effect, the design with the octupole magnet-only approach is deemed more optimal. Due to the impact of beam energy dispersion, the beam spot cross-section generated by the octupole magnet exhibits an asymmetric distribution, to reduce the non-uniformity of the beam spot, enhance beam uniformization, and improve the flatness and symmetry of the beam scanned in the horizontal direction, a sext-pole magnet with the same size gradient of \(-\)8.62/m\({}^{2}\) is added after the first octupole magnet in the horizontal direction. Additionally, the gradient component of the second octupole magnet is optimized and reduced to \(-\)1199/m\({}^{3}\). The results at the target location are shown in Figure 11, the same area as the grid drawn above, with a 20 \(\times\) 20 grid applied to a cross-sectional area of 100 mm \(\times\) 100 mm, totaling 400 individual grids with each grid measuring 5 mm \(\times\) 5 mm. Figure 10: The corresponding two-dimensional normalized grid particle count heatmaps. From (**a**–**c**), the sequentially inserted components are the octupole magnet, multipole magnet components, and step-like field. Figure 9: Compare the cross-sectional results of the beam spot at the target location. From (**a**–**c**), the sequentially inserted components are the octupole magnet, multipole magnet components, and step-like field. Selecting i values from the 7 th to the 14 th grid and j values from the 6 th to the 14 th grid, the particle utilization rate is 99.23%.