## 2 Materials and Methods  
### Representative Cases from AAPM TG-244  
This study was conducted on the prostate and head and neck clinical cases available on the AAPM website. CT image and structure sets of these two cases (in DICOM RTst format) were downloaded from [[https://www.aapm.org/pubs/MPPG/TPS/](https://www.aapm.org/pubs/MPPG/TPS/)]([https://www.aapm.org/pubs/MPPG/TPS/](https://www.aapm.org/pubs/MPPG/TPS/)) (accessed on the 15 January 2025), with the related proposal of sample objectives that can be used for optimization and prescription. Due to their complexity and clinical significance, these case studies are recommended for testing radiotherapy TPSs. Both have anatomical structures and critical organs partially overlapping PTVs, requiring a high dose gradient and precise delivery.  
In particular, for the PCa patient, the treatment plan includes a high-risk fossa and nodal region with two different dose levels, 68 Gy to the high-risk PTV and 56 Gy to the low-risk PTV, treated with the SIB technique. For the HNCa patient, the treatment plan includes high-risk head and neck treatment with three different dose levels, 70 Gy to the high-risk PTV, 63 Gy to the medium-risk PTV, and 56 Gy to the low-risk PTV using an SIB approach.  
The contoured OARs and PTVs are reported in Table 1 with the related plan objectives. Regarding the PCa case, a volume overlap of 8.9% and 17.6% was found between the total PTV and the rectum and the bladder, respectively, while, regarding the HNCa case, a volume overlap of 9.1%, 28.6%, and 14.0% was found between the total PTV and the larynx, mandible, and right parotid, respectively (as shown in Table 2). Notably, high overlap volumes between PTVs and OARs usually lead to difficulties and/or impossibility in achieving the OAR dose-volume constraints.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Site** & **PTV/OARs** & **Volume (cm\({}^{3}\))** & **Optimal Objectives** & **Mandatory Objectives** \\ \hline PCa & Prostate bed & 65.13 & V68 Gy \(\geq\) 99\% & \\  & PTV 68 & V68 Gy & V68 Gy \(\geq\) 95\% & \\  & & D0.03 cc & D0.03 cc \(<\) 71.5 Gy & \\  & PTV 56 & 256.15 & V56 Gy \(\geq\) 95\% & \\  & Bladder & & V65 Gy \(<\) 15\% & \\  & & & V40 Gy \(<\) 40\% & \\  & & 47.92 & V65 Gy \(\leq\) 5\% & \\  & & & D\({}_{\text{max}}\) (0 cc) \(\leq\) 68 Gy & D\({}_{\text{max}}\) (0 cc) \(\leq\) 70 Gy \\  & & & V40 Gy \(\leq\) 20\% & V40 Gy \(\leq\) 30\% \\ \hline HNCa & PTV 70 & V70 Gy & V70 Gy \(\geq\) 95\% & \\  & & D0.03 cc & D0.03 cc \(<\) 77 Gy & \\  & CTV 70 & V70 Gy & V70 Gy \(\geq\) 99\% & \\  & PTV 63 & V63 Gy & V63 Gy \(\geq\) 95\% & \\  & PTV 56 & V56 Gy & V56 Gy \(\geq\) 95\% & \\  & Spinal Cord & D0.03 cc & D0.03 cc \(<\) 48 Gy & \\  & Brainstem & D0.03 cc & D0.03 cc \(<\) 52 Gy & \\  & Right Inner Ear & D0.03 cc & D0.03 cc \(<\) 35 Gy & \\  & Left Inner Ear & D0.03 cc & D0.03 cc \(<\) 37 Gy & \\  & Lips & V30 Gy & \(<\)20\% & \\  & Right Parotid & Mean & \(<\)24 Gy & \\  & Mandible & V70 Gy & \(<\)10\% & \\  & Larynx & Mean & \(<\)45 Gy & \\ \hline \hline \end{tabular}
\end{table}
Table 1: List of the OARs and PTV volumes, the objectives suggested by the AAPM TG-244 [13], and mandatory objectives used in our center for the plan optimization for PCa and HNCa template cases.