## 3 Results and Discussion
### Response Surface Methodology (RSM) Analysis for the TSPs and FLAs from Pigmented Corn Powder
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Response** & **Source** & **SS** & **DF** & **MS** & **F-Value** & **\(p\)-Value** & **Lack of Fit** (\(p\)-Value)** & **R-Square** & **R-Adjust** \\ \hline TSP & \(X_{ET}\) & 1.19 & 1 & 1.19 & 0.98 & 0.3286 \({}^{++}\) & 0.4120 \({}^{++}\) & 0.8260 & 0.788 \\  & \(X_{ET}\)\({}^{2}\) & 16.13 & 1 & 16.13 & 13.25 & 0.0009 \({}^{+}\) & & & \\  & \(X_{PC}\) & 7.43 & 1 & 7.43 & 6.10 & 0.019 \({}^{++}\) & & & \\  & \(X_{PC}\)\({}^{2}\) & 32.19 & 1 & 32.19 & 26.43 & \textless{}0.0001 \({}^{+}\) & & & \\  & \(X_{UP}\) & 0.36 & 1 & 0.36 & 0.29 & 0.5903 \({}^{++}\) & & & \\  & \(X_{UP}\)\({}^{2}\) & 24.18 & 1 & 24.18 & 19.85 & \textless{}0.0001 \({}^{+}\) & & & \\  & \(X_{ET}\)\({}^{*}X_{PC}\) & 3.32 & 1 & 3.32 & 2.73 & 0.1080 \({}^{++}\) & & & \\  & \(X_{ET}\)\({}^{*}X_{PC}\)\({}^{2}\) & 0.006 & 1 & 0.006 & 0.005 & 0.9414 \({}^{++}\) & & & \\  & \(X_{ET}\)\({}^{*}X_{PC}\) & 24.18 & 1 & 24.18 & 19.85 & \textless{}0.0001 \({}^{+}\) & & & \\  & \(X_{ET}\)\({}^{*}X_{UP}\) & 9.18 & 1 & 9.18 & 7.53 & 0.0098 \({}^{+}\) & & & \\  & \(X_{ET}\)\({}^{*}X_{UP}\) & 112.19 & 1 & 112.19 & 92.12 & \textless{}0.0001 \({}^{+}\) & & & \\  & \(X_{PC}\)\({}^{*}X_{UP}\) & 12.18 & 1 & 12.18 & 10.08 & 0.0034 \({}^{+}\) & & & \\  & Error & 38.97 & 32 & 1.21 & & & & & \\ FLA & \(X_{ET}\) & 29.89 & 1 & 29.89 & 66.32 & \textless{}0.0001 \({}^{+}\) & 0.0551 \({}^{++}\) & 0.9648 & 0.9540 \\  & \(X_{ET}\)\({}^{2}\) & 13.26 & 1 & 13.26 & 29.42 & \textless{}0.0001 \({}^{+}\) & & & \\  & \(X_{PC}\) & 61.84 & 1 & 61.84 & 137.21 & \textless{}0.0001 \({}^{+}\) & & & \\  & \(X_{PC}\)\({}^{2}\) & 1.80 & 1 & 1.80 & 4.01 & 0.0536 \({}^{+}\) & & & \\  & \(X_{UP}\) & 39.47 & 1 & 39.47 & 87.57 & \textless{}0.0001 \({}^{+}\) & & & \\  & \(X_{UP}\)\({}^{2}\) & 280.82 & 1 & 280.82 & 623.07 & \textless{}0.0001 \({}^{+}\) & & & \\  & \(X_{ET}\)\({}^{*}X_{PC}\) & 9.14 & 1 & 9.14 & 20.28 & \textless{}0.0001 \({}^{+}\) & & & \\  & \(X_{ET}\)\({}^{*}X_{PC}\)\({}^{2}\) & 17.48 & 1 & 17.48 & 38.80 & \textless{}0.0001 \({}^{+}\) & & & \\  & \(X_{ET}\)\({}^{*}X_{UP}\) & 1.05 & 1 & 1.05 & 2.33 & 0.1359 \({}^{++}\) & & & \\  & \(X_{ET}\)\({}^{2*}X_{PC}\) & 16.38 & 1 & 16.38 & 36.36 & \textless{}0.0001 \({}^{+}\) & & & \\  & \(X_{ET}\)\({}^{*}X_{UP}\) & 13.24 & 1 & 13.24 & 29.29 & \textless{}0.0001 \({}^{+}\) & & & \\  & \(X_{PC}\)\({}^{*}X_{UP}\) & 12.17 & 1 & 12.17 & 27.01 & & & & \\ Error & 14.42 & 32 & 0.45 & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: ANOVA for the quadratic model using ultrasound-assisted extraction conditions for native Mexican pigmented corn kernel powder. used; nonetheless, their relative error was less than 10% (Table 1), similar to errors reported by other authors during the optimization of UAE using BBD and RSM [48; 49]. \[ \begin{split}\text{Total soluble phenols (mg GAE/g) = 272.82 - 81.39X_{ET}+10.58X_{ET}{}^{2}-3.97~{}X_{UP}+0.01X_{PC}{}^{2}+\\ 4.76X_{ET}{}^{*}X_{PC}-0.56X_{ET}{}^{2*}X_{PC}+0.82X_{ET}{}^{*}X_{UP}-0. 10X_{ET}{}^{2*}X_{UP}-0.10X_{PC}{}^{*}X_{UP}\end{split}  \tag{3}\]  
\[ \begin{split}\text{Total flavonoids (mg CE/g) = 419.79 - 3.07X_{ET}+0.81X_{ET}{}^{2}-8.89X_{UP}+0.05X_{UP}{}^{2}+2.81X_{ET}{}^{*}X_{PC}- \\ 4.12X_{ET}{}^{2*}X_{PC}-0.002X_{ET}{}^{2*}X_{UP}-0.10X_{PC}{}^{*}X_{UP} \end{split}  \tag{4}\]  
The three-dimensional response surface plots display the significant interaction effects (\(p\) < 0.01) of UAE parameters on the TSP (Figure 2a,c,e) and FLA (Figure 2b,d,f) contents in pigmented corn powder. Nonetheless, from these 3D plots, it is possible to visually identify the region that presented the highest UAE of TSP and FLA contents [50]. The results suggest that TSP and FLA extraction from native Mexican pigmented corn kernel powder under UAE occurred under all extraction conditions, regardless of the extraction time, pulse cycle, or ultrasound power. Nonetheless, the elliptical shape of the plots indicates a significant interaction of the independent variables [45].