## 2 Materials and Methods
### Statistical Analysis and Model Training and Simulation  
The patient background of the collected data was tabulated. Groups were classified into those in which PIHF occurred and those in which it did not, and the age and propofol dose were compared using \(t\)-tests. The significance level was set at 0.05.  
Of the analyzed cases, 80% were randomly assigned to the training dataset, with 20% allocated to the test dataset. The model creation data were further randomly divided into a training set (80%) and a test set (20%). A fully connected neural network model was developed using the training and validation data to predict the propofol induction dose. The input variables included the age, sex, height, weight, American Society of Anesthesiologists physical status (ASA-PS) scale, systolic blood pressure (SBP), diastolic blood pressure (DBP), and heart rate (HR) before anesthesia induction, as well as the induction dose of fentanyl. The output variable was the induction dose of propofol. The model comprised three layers: input, intermediate, and output layers, with 9-7-1 units, respectively (Figure 2). The Adam optimizer was used for model training. The hyperparameters--batch size, learning rate, and number of epochs--were manually tuned to achieve optimal performance. The proposed model is called \"model N\".  
The performance of both the neural network model and the previously reported prediction equation (\"model P\") was evaluated using the RMSE and MAE calculated based on the test data. To estimate the 95% confidence intervals (CIs) for these metrics, a bootstrap resampling method was employed. Specifically, the bootstrap procedure involved generating 1000 random samples with replacement from the test dataset, maintaining the same sample size as the original test data. For each resampled dataset, the RMSE and MAE were recalculated. The 95% CIs were then derived by determining the 2.5 th and 97.5 th percentiles of the bootstrapped distribution of each metric. The prediction equation for model P was defined as follows: propofol dose (mg) = [2.374 \(-\) 0.0113 \(\times\) age (years) \(-\) 0.0788 (if ASA-PS 3 or 4) + 0.057 (if female) + 0.1087 \(\times\) fentanyl dose (ug/kg)] \(\times\) body weight (kg) [12].  
According to the study that proposed model P, it is reported that if the residual error difference between the doses predicted by model P and the actual doses is large, PIHF is likely to occur. Therefore, a chi-square test was used to compare the frequency of PIHF between the groups with large and small residual errors, divided into doses of \(\geq\)5 mg and \(<\)5 mg. The significance level was set at 0.05.  
Furthermore, utilizing the developed model, simulations were conducted to predict the changes in the propofol induction dose based on age. In each case, the predicted value of the propofol dose was calculated by model N using a hypothetical age range of 10 years, from 20 to 90 years old. After calculating the propofol dose per body weight at each age, we simulated the change in the dose for each 10 years of aging. Specifically, we simulated the changes in the doses with aging, starting at ages 20 and 30 and continuing at every 10 years until ages 80 and 90.  
Figure 2: Structure of the neural network. The input layer consists of nine clinical parameters: age, sex, height, weight, ASA-PS status, systolic blood pressure (SBP), diastolic blood pressure (DBP), heart rate (HR) before induction, and induction dose of fentanyl. The intermediate layer represents a fully connected hidden layer. The output layer provides the predicted propofol induction dose (mg).  
All the statistical analyses were conducted using Python (version 3.9) and the scikit-learn library (version 1.3.0), while the model construction and training were performed using PyTorch (version 2.0.1).