## 3 Results and Discussion
### Response Surface Methodology (RSM) Analysis for the TSPs and FLAs from Pigmented Corn Powder  
An RSM analysis determined the optimal UAE conditions for TSPs and FLAs from native Mexican pigmented corn kernel powder. The statistically significant influence of the independent factors (\(X_{ET}\), \(X_{PC}\), and \(X_{UP}\)) on the UAE of the TSPs and FLAs was verified by the analysis of variance (ANOVA, \(p\) < 0.01), as listed in Table 2. Regarding TSPs, the ANOVA showed that most of the experimental factors and their interactions were significant (\(p\) < 0.01), except for \(X_{PC}\), \(X_{UP}\), \(X_{ET}\)*\(X_{PC}\), and \(X_{ET}\)*\(X_{PC}\)2, and similar trends were observed for FLAs (except for \(X_{PC}\)2 and \(X_{ET}\)*\(X_{UP}\); \(p\) > 0.01). [PERSON] et al. [41] reported that some factors (time) and their interactions (solvent*time) were not significant. Similar trends were observed during the UAE of phenolic compounds from _Mycriaria cauliflora_ fruit; the extraction temperature, sonication amplitude, pulse cycle, and their interactions were not significant (\(p\) > 0.01) [42]. Table 3 shows the \(\beta\)-coefficients of the fitted regression models for the TSP and FLA contents of UAE of native Mexican pigmented corn kernel powder. Most of the \(\beta\)-coefficients were significant (\(p<0.01\)) for TSP, except \(X_{PC}\), \(X_{PC}\)\({}^{2}\), and \(X_{ET}\)\({}^{2*}X_{PC}\), and similar trends were observed for FLA, where \(X_{PC}\)\({}^{2}\) and \(X_{ET}\)\({}^{2*}X_{UP}\) were nonsignificant (\(p>0.01\)). In both cases, the \(X_{PC}\) and \(X_{ET}\)\({}^{*}X_{PC}\) exhibited a positive effect during extraction. The pulse cycle has been reported to influence bubble formation during cavitation, improving/affecting the UAE [34]. From this information, a second-order polynomial equation was obtained for each response variable, excluding coefficients that were not significant (\(p>0.01\)) aimed to improve the prediction ability of the model [38]. These mathematical models can predict the effect of the evaluated factors on the TSP and FLA contents (Equations (3) and (4)). Moreover, TSPs and FLAs showed an advisable relation with the predicted value and the experimental data (R\({}^{2}\) = 0.8260 and R\({}^{2}\) = 0.9648) for the estimated model. The nonsignificant lack of fit (\(p>0.01\)) values imply good prediction of the model, indicating the effectiveness of the model [43], but also the R\({}^{2}\) is relevant to verify the model's suitability [44]. According to [PERSON] et al. [42], a low value of R\({}^{2}\) during the UAE of phenolic compounds from a vegetable matrix can be associated with the type of bioactive compound extracted, considering its polarity and size, as demonstrated during the UAE of phenolic compounds from _Myricaria cauliflora_ fruit (R\({}^{2}\) of 0.706). Several authors have reported R\({}^{2}\) values ranging from 0.96 to 0.99 when applying UAE methods for corn-based materials and adequate adjustment of the experimental data to the models (lack of fit, \(p>0.05\) or \(p>0.01\)) [14; 15; 16]. Moreover, it has also been reported that an R\({}^{2}\) of 0.78 to 0.85 in polynomial equations obtained from UAE processes exhibited adequate adjustment between the experimental and predicted data [45; 46; 47]. In this study, all experimental conditions exhibited similar values of TSP and FLA contents between experimental and predicted data when UAE was