## 3 Results
### Evaluation of Robust Optimization Algorithm with Various Cases  
Further validation of our robust strategy compared plan robustness in the nominal and robust plans. Four case-sets of applicator displacements can be divided into two categories. Two cases were a challenge to fit with the EMBRACE protocol guidance, but the other two were easier because of the shape of targets. There was not an enormous difference in total time between the two nominal and robust plans, whereas the dose-delivered results of EQD2 of all cases were affected differently as shown in Figure 8.  
It shows the comparison of the EQD2 and the standard deviation values of each target and OAR using both strategies. Regarding quantitative results, this proposed robust optimization could reduce the standard deviation of the target and OAR compared to the nominal plan in most of the cases, whereas the standard deviation of a few organs was not promising. In addition, the proposed robust optimization also reduced EQD2 in most of the cases. Although the standard deviation in a few organs was not sufficiently minimized, the resulting robustness plans from our strategy might be worth considering according to EMBRACE protocol guidance.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{5}{c}{**EQD2 and Standard Deviation of Dose Distributions**} \\ \hline
**Target/OAR** & **EMBRACE Recommendation**\({}^{*}\) & \begin{tabular}{c} **Nominal Plan EQD2** \\ **(Gy)(\(\sigma_{BT}\))** \\ \end{tabular} & \
\begin{tabular}{c} **Inverse Plan EQD2** \\ **(Gy)(\(\sigma_{BT}\))** \\ \end{tabular} &
\begin{tabular}{c} **Robust Plan EQD2** \\ **(Gy)(\(\sigma_{BT}\))** \\ \end{tabular} \\ \hline \multirow{4}{*}{HRCTV} & \(\alpha/\beta=10\) Gy & - & - & - \\  & D98 \(>75\)\({}^{*}\) & 88.59 (\(\pm\)1.0177) & 86.71 (\(\pm\)0.9393) & 84.84 (\(\pm\)0.9085) \\ - & D90 \(>90\)\({}^{*}\) & 92.15 (\(\pm\)1.8840) & 92.63 (\(\pm\)0.10983) & 91.56 (\(\pm\)1.0341) \\ ICRTV & D98 \(>60\)\({}^{*}\) & 74.96 (\(\pm\)0.6424) & 73.29 (\(\pm\)0.6063) & 72.24 (\(\pm\)0.5883) \\ - & D90 \(>66\)\({}^{*}\) & 82.18 (\(\pm\)0.7418) & 80.66 (\(\pm\)0.6927) & 78.83 (\(\pm\)0.6737) \\ \hline \multirow{4}{*}{Rectum} & \(\alpha/\beta=3\) Gy & - & - & - \\  & D2\({}_{cc}<75\)\({}^{*}\) & 55.29 (\(\pm\)0.4927) & 54.94 (\(\pm\)0.4620) & 54.09 (\(\pm\)0.4052) \\ Sigmoid & D2\({}_{cc}<75\)\({}^{*}\) & 68.20 (\(\pm\)1.2491) & 69.13 (\(\pm\)1.2718) & 67.25 (\(\pm\)1.1199) \\ Bladder & D2\({}_{cc}<90\)\({}^{*}\) & 84.98 (\(\pm\)2.1211) & 81.25 (\(\pm\)1.9764) & 81.35 (\(\pm\)2.1459) \\ Bowel & D2\({}_{cc}<60\)\({}^{*}\) & 51.06 (\(\pm\)0.1864) & 51.41 (\(\pm\)0.2270) & 51.58 (\(\pm\)0.2474) \\ \hline \hline \end{tabular}  
* Hard constraints which can be used in the treatment plan optimization.  
\end{table}
Table 1: The dose constraints in accordance with EMBRACE recommendation and the total EQD2 (EBRT + BT) of each target and OAR for nominal, inverse and robust plan strategy.  
For the evaluation of the effects of irregularly shaped targets on robust optimization outcomes, the degree of asymmetry for HRCTV was measured. The measurements of asymmetric ratios were obtained from the distance difference between the maximum and minimum distances from the activated dwell positions inside the HRCTV to the surface of the HRCTV.  
Figure 9 gives a comparison of the measured asymmetric ratios along the given dwell positions inside the HRCTV. When the asymmetric ratio is close to the value of 1, the target shape has symmetry. The results of cases that have lower asymmetric ratio variations along the given dwell positions had a greater standard deviation reduction than the results from the cases of the higher asymmetric ratio as compared with the optimized results as shown in Figure 8.  
Figure 8: Comparison of the total EQD2 (EBRT + BT) of each target and OAR for both the nominal and robust plan for all cases.  
Figure 9: The asymmetric ratio variations along the given dwell positions located inside the HRCTV in all cases.