## 2 Materials and Methods
### Sample Size Justification  
The majority of KBP studies in the literature utilise historical datasets of previous clinical plans and therefore substantial training datasets can be curated with low effort. In contrast, the approach in this study required Pareto navigation on each training patient to define the ground truth dataset. Autonomous generation of the Pareto plans for a three PG navigation took 31 h (125 plans each taking 15 min), with approximately five minutes of operator time required for navigation. Whilst Pareto plans could be generated for 3 patients concurrently on a single application server, the time required to generate the ground truth dataset was non-trivial. The size of the training dataset therefore had to balance the competing demands of model accuracy and practicality. \begin{table}
\begin{tabular}{l l l l l l l} \hline \hline
**ROI** & **Fall Off** & **High Dose** & **Low Dose** & **Dose Gradient** & **Goal** & **Weighting** \\
**Name** & **Type** & **Level (Gy)** & **Level (Gy)** & **(\% Dose)** & **Number** & **Factor** \\ \hline PTV48 & Falloff & 57.0 & 40.8 & 50\% & 3 & 23.6 \\ PTV48 & Intra PTV & 54.0 & 52.8 & 50\% & 6 & 1.47 \\  & Falloff & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Priority 3—Trade-off Goals (Dose Fall Off). Dose Gradient represents the percentage of the overall treatment prescription. \begin{table}
\begin{tabular}{l l l l l} \hline \hline
**ROI Name** & **Dose Parameter** & \begin{tabular}{l} **Target** \\ **(Gy or \% Volume)** \\ \end{tabular} & \begin{tabular}{l} **Goal** \\ **Number** \\ \end{tabular} & \begin{tabular}{l} **Weighting** \\ **Factor** \\ \end{tabular} \\ \hline \hline \begin{tabular}{l} Rectum \\ Bladder \\ \end{tabular} & \(\text{D}_{\text{mean}}\) (Gy) & 5.0 & 1 & 21.3 \\ \begin{tabular}{l} Rectum \\ \end{tabular} & \(\text{D}_{\text{mean}}\) (Gy) & 5.0 & 2 & 6.86 \\ \begin{tabular}{l} Rectum \\ \end{tabular} & \(\text{D}_{\text{max}}\) (Gy) & 60.0 & 4 & 195 \\ \begin{tabular}{l} Bladder \\ \end{tabular} & \(\text{D}_{\text{max}}\) (Gy) & 54.0 & 5 & 0.880 \\ \
\begin{tabular}{l} Bowel \\ \end{tabular} & \(\text{V}_{\text{36.0 Gy}}\) & 0.0 & 7 & 0.762 \\
\begin{tabular}{l} Bowel \\ \end{tabular} & \(\text{V}_{\text{45.6 Gy}}\) & 0.0 & 7 & 0.762 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Priority 3—Trade-off Goals (Standard). Target represents dose in Gy (i.e., \(\text{D}_{\text{mean}}\) and \(\text{D}_{\text{max}}\)) or percentage of ROI volume (i.e., \(\text{V}_{\text{36.0 Gy}}\) and \(\text{V}_{\text{45.6 Gy}}\)). Targets are automatically adjusted during optimisation (via the PBAIO algorithms) to ensure PGs are minimised. Therefore initial values have negligible influence on the final plan, but may decrease planning time if correctly defined. [PERSON] et al. [40] presented evidence on the sample size requirements for KBP in prostate cancer. For DVH curve prediction using principle components and linear regression, 75 and 20 samples were required to minimise modelling errors for bladder and rectum DVHs, respectively. For objective function weight prediction, 150 patients were required for a k-nearest neighbour clustering methodology before a statistically insignificant difference from the benchmark was observed. However, only 10 were required for a logistic regression model. The large difference in dataset size requirements is because regression can exploit underlying distributions of the data (e.g., linear or logistic relations), whereas clustering cannot as it is a non-parametric approach. In our study, it is not objective function weights or DVH curves that were to be predicted, but rather patient specific weighting factors for an already validated AP solution. In this regard we hypothesised that the underlying variance of the data had already been substantially reduced through utilisation of a PBAIO framework. Therefore we considered that sample size requirements would therefore align with the lower of those proposed by [PERSON] et al. (i.e., 10 \(<\) n \(<\) 20).