## 2 Materials and Methods
### Experimental Design  
In this work, we focused on optimizing the ultrasound-assisted extraction conditions. The experimental design applied in this study was developed based on the Box-Behnken Design of response surface methodology (RSM) using Design-Expert(r) 13 software. Three factors (solid-to-liquid ratio (\(X_{i}\)), extraction temperature (\(X_{j}\)), and extraction time (\(X_{k}\))) at three levels (\(-1\), \(0\), \(+1\)) (shown in Table 1 and Figure 1) were selected to control for their linear, interactive, and quadratic effects on the following dependent variables: yield, total phenolic content (TPC), and total flavonoid content (TFC). The fifteen-test series is summarized in Table 2, including three replicates at the center point to determine the pure error. A second-order polynomial equation was executed for each response to characterize the relationships between the independent variables (extraction factors) and the response. Thus, the general formula that was used in the response surface analysis is given by Equation (1):  
\[\mathrm{Y}=\beta_{0}+\sum\
olimits_{i=1}^{k}\beta_{i}X_{i}+\sum\
olimits_{i=1 }^{k}\beta_{ii}X_{i}^{2}+\sum\
olimits_{i=1}^{k}\sum\
olimits_{j=i=1}^{k-1} \beta_{ij}X_{i}X_{j} \tag{1}\]  
where Y: the dependent variable, \(\beta_{0}\): constant regression coefficient, \(\beta_{i}\): linear coefficient, \(\beta_{ii}\): quadratic coefficient, \(\beta_{ij}\): interaction coefficient, and \(X_{i}\)\(\rightarrow\)\(X_{j}\): independent variables.  
Indeed, assuming that the regression coefficients of the factors do not significantly affect the response when its \(p\)-value is significantly greater than 0.05 (\(p>0.05\)), removing them from the second-order polynomial equation is possible.  
This second-order polynomial equation of each response allows the 3D visualization of multiple response surfaces, including the relationship between the obtained response  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Factor**} & \multicolumn{3}{c}{**Response Value**} \\ \cline{2-6}
**Run** & **Solid-to-Liquid** & **Temperature** & **Time** & **Yield** & **TPC** & **TFC** \\  & **Ratio (g/mL)** & (\({}^{\circ}\)C) & **(min)** & **(\%)** & **(ug GAE/mg DE)** & **(ug QE/mg DE)** \\ \hline
1 & \(-1\) (0.5:10) & \(-1\) (35) & 0 (20) & 11.09 & 27.22 & 3.77 \\
2 & \(+1\) (1.5:10) & \(-1\) (35) & 0 (20) & 7.50 & 32.96 & 4.38 \\
3 & \(-1\) (0.5:10) & \(+1\) (45) & 0 (20) & 13.50 & 32.07 & 5.36 \\
4 & \(+1\) (1.5:10) & \(+1\) (45) & 0 (20) & 8.14 & 29.26 & 4.26 \\
5 & \(-1\) (0.5:10) & 0 (40) & \(-1\) (10) & 12.01 & 33.21 & 3.81 \\
6 & \(+1\) (1.5:10) & 0 (40) & \(-1\) (10) & 8.28 & 29.25 & 2.23 \\
7 & \(-1\) (0.5:10) & 0 (40) & \(+1\) (30) & 13.80 & 33.71 & 4.94 \\
8 & \(+1\) (1.5:10) & 0 (40) & \(+1\) (30) & 7.95 & 39.78 & 5.13 \\
9 & 0 (1.0:10) & \(-1\) (35) & \(-1\) (10) & 8.57 & 31.38 & 5.14 \\
10 & 0 (1.0:10) & \(+1\) (45) & \(-1\) (10) & 10.55 & 19.89 & 2.56 \\
11 & 0 (1.0:10) & \(-1\) (35) & \(+1\) (30) & 11.10 & 25.79 & 4.59 \\
12 & 0 (1.0:10) & \(+1\) (45) & \(+1\) (30) & 11.47 & 34.77 & 7.94 \\
13 & 0 (1.0:10) & 0 (40) & 0 (20) & 9.46 & 30.77 & 5.21 \\
14 & 0 (1.0:10) & 0 (40) & 0 (20) & 10.01 & 31.58 & 4.94 \\
15 & 0 (1.0:10) & 0 (40) & 0 (20) & 10.99 & 30.26 & 4.77 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Box-Behnken Design for coded and uncoded factors and experimental response values.  
Figure 1: Graphical depictions summarize the experimental methodology for optimizing phenolic compound extraction, utilizing the Box-Behnken Design of the response surface methodology and the different phytochemical and antioxidant analyses evaluated.  
and two evaluated independent variables. In contrast, the third independent variable is maintained at its zero level.