## 3 Results and Discussion
### Optimization of the Extraction Process and Validation of the Models  
Optimizations based on numerical values were applied as suggested to optimize the UAE variables, resulting in the optimal response [29]. The optimal UAE conditions for extracting TSPs and FLAs from native Mexican pigmented corn kernel powder are shown in Figure 3. The best-optimized conditions for TSPs were a 3.15 min extraction time, a 1.58 s on/off pulse cycle, and 100% ultrasound power (Figure 3a); these new experimental conditions did not match any run in the original experimental trial. It must be noted that higher extraction times and pulse cycles and low ultrasound power produced lower extraction yields of TSPs [53]. On the other hand, an extraction time of 4.18 min, a 3 s on/off pulse cycle, and 80% ultrasound power were the best experimental conditions for FLAs (Figure 3b). These experimental conditions matched with the eight-standard order in the original experimental trial. [PERSON] and [PERSON] [49] reported that during the UAE of phenolic compounds from spinach roots, the optimized conditions obtained from response surface methodology can match some runs of the original experimental trial. In general, the UAE of FLAs required more extraction time than TSPs, and these results agree with those reported by [PERSON] et al. [53], who explained that FLAs needed a higher extraction time during UAE due to the electrosensitivity of these molecules.  
Figure 3: Optimization and desirability profile plots for total soluble phenol (**a**,**c**) and total flavonoid (**b**,**d**) contents from native Mexican pigmented corn kernel powder.  
Additionally, the UAE of TSPs and FLAs from native Mexican pigmented corn kernel powder exhibited desirability (D) values of 0.86 and 0.98 (Figure 3c,d), suggesting a good prediction performance. In the scale of desirability function, D values ranging from 1 to 0.8 are classified as \"very good\" [49]. Furthermore, the desirability function for optimizing the UAE of bioactive compounds from vegetable matrices has been widely used [49; 54]. On the other hand, these new experimental conditions produced predicted optimal responses of 35.07 mg GAE/g for TSPs and 27.51 mg CE/g for FLAs (Table 4).  
The predictive empirical models were validated through verification experiments conducted under optimal conditions. Table 4 shows the predicted and experimental values of TSPs and FLAs under optimal conditions by response surface methodology analysis. The experimental TSP (34.06 mg GAE/g) and FLA (27.04 mg CE/g) results were very close to the predicted values (35.07 mg GAE/g and 27.51 mg CE/g, respectively), demonstrating the adequacy and reliability of the fitted model for TSP and FLA responses.  
Additionally, the methanolic extracts obtained from the native Mexican pigmented corn kernel powder under optimal ultrasound-assisted extraction conditions for TSPs (extraction time of 3.158 min, ultrasound power 100%, and pulse cycle of 2 s on/off) and FLAs (extraction time of 3.188 min, ultrasound power 80%, and pulse cycle of 3 s on/off) exhibited antioxidant properties through DPPH, ABTS, and FRAP methods. Under the optimal conditions (Table 4), the antioxidant capacity of the pigmented corn kernel extract for both TSP and FLA extraction conditions was DPPH > FRAP > ABTS (\(p<0.01\)). Furthermore, no differences (\(p>0.01\)) were observed between TSP and FLA extraction conditions for DPPH (80.86 and 79.77 mmol TE/g). However, for the ABTS (43.78 and 31.31 mmol TE/g) and FRAP (59.43 and 54.65 mmol TE/g) methods, the extraction conditions for TSPs resulted in higher antioxidant properties (\(p<0.01\)). These findings could be related to the recovered phenols and how they neutralize radicals (donating electrons, chelating metals, or transferring electrons) [55]. Furthermore, these results indicate that phenolic compounds from native Mexican pigmented corn kernel powder could be utilized in various industrial applications.