## 2 Materials and Methods
### Response Design and Optimization of MAE Variables  
The operational parameters were optimized through response surface methodology (RSM) combined with a face-centered central composite design (FCCD) to further assess the key factors influencing the MAE for PCL production. In FCCD, the axial points are set at \(\pm\)1 from the center along each axis, placing them on the faces of the factorial cube, which is why it is called \"face-centered\". This is an adjustment from standard central composite  
Figure 1: Solvent-based microwave-assisted extraction (MAE).  
design (CCD), where the axial points are positioned at \(\pm\alpha\) along each axis, with \(\alpha\) typically greater than 1 to achieve rotatability [15].  
Based on single-factor tests, the extraction time (X\({}_{1}\): 5-25 min), microwave power (X\({}_{2}\): 70-350 W), and solid-to-solvent ratios (X\({}_{3}\): 1:60-1:300 g/mL) were used as the MAE independent variables (Table 1). The yields of PCL (Y\({}_{1}\)), TPC (Y\({}_{2}\)), TPC (Y\({}_{3}\)), and 50% inhibition concentration on DPPH (IC\({}_{50}\)) (Y\({}_{4}\)) were selected as a dependent variable.  
The MAE experimental setting was designed, and the data that were gathered were examined using Minitab software (trial version 18, Minitab Inc., State College, PA, USA) to construct models based on response surface methodology (RSM). All responses were collected under varying conditions based on the 20-run experimental design, with each experiment conducted in triplicate. A second-order polynomial model was applied to fit the response data from the MAE experiments. The polynomial regression was given as Equation (2).  
\[\mathrm{Y}=\beta_{0}+\sum_{i=1}^{3}\beta_{i}\mathrm{X}_{i}+\sum_{i=1}^{3}\beta _{ii}\mathrm{X}_{i}^{2}+\sum_{i=1}^{2}\sum_{i=i+1}^{3}\beta_{ii}\mathrm{X}_{i} \mathrm{X}_{i}+\epsilon \tag{2}\]  
where Y represents the response variables; X\({}_{1}\), X\({}_{2}\), and X\({}_{3}\) represents the MAE independent variables; \(\beta_{0}\) is the intercept; \(\beta_{1}\) represents the linear coefficients; \(\beta_{ii}\) represents the quadratic coefficients; \(\beta_{ij}\) represents the interaction coefficients; and \(\epsilon\) is the random error.  
The model analysis and optimization of MAE conditions for the responses (yield, TPC, TFC, and IC\({}_{50}\)) were conducted using analysis of variance (ANOVA) with a 95% confidence interval. The regression coefficient (R\({}^{2}\)) and the lack-of-fitness statistic were used to test the adequacy of the model fitting. Based on the data from the FCCD models, MAE conditions were optimized. The optimum MAE conditions led to the maximum yield, TPC, TFC, and minimum IC\({}_{50}\). The interaction effects of each MAE parameter on the response level were depicted in a 3D surface plot. The interaction effects of each parameter on the response data were illustrated through 3D surface plots constructed using Design Expert V12 Trial Software (Stat-Ease Inc., Minneapolis, MN, USA). By performing extraction three times with the optimal MAE setting, the data model validity for the MAE optimized results was confirmed. Finally, the predicted values and the actual results were compared in order to determine the validity of the MAE model.