## 3 Results and Discussion  
### Experimental Design  
To optimize the UAE of the antioxidant capacity and the total phenolic content from aqueous ginger's extracts, a CCRD was used. Table 2 shows the CCRD matrix and the total phenolic content corresponding responses. Data show the effect of amplitude and temperature (X\({}_{1}\) and X\({}_{2}\)) on the antioxidant capacity and total phenolic content extracted from ginger. Responses for Y\({}_{1}\) (DPPH\(\bullet\)) and Y\({}_{2}\) (ABTS\(\bullet\)+) ranged from 13 to 157 mg Trolox/100 g. Lower values corresponded to ABTS method. For Y\({}_{3}\) (FRAP) and Y\({}_{4}\) (total phenolic content), values were from 93 to 168 mg Fe\({}^{2+}\)/100 g and from 8 to 17 mg GA/100 g, respectively. The CCRD allowed us to estimate the effects of the factors (amplitude and temperature) and their second order interactions (quadratic and cross-product effects) which are presented in Table 3. In general, amplitude (x\({}_{1}\)) presented a positive effect in all responses, except for Y\({}_{2}\) (ABTS**+**), permitting a higher extraction of bioactive compounds. It is well known that amplitude plays an important role in the intensification of the extraction due to its impact in cavitation [31]. On the other hand, temperature had a negative effect in all responses since it can damage the structure and properties of thermosensitive antioxidant and phenolic compounds promoting higher degradation rates [31; 32]. Regarding to the correlation coefficient values (Table 3), these were superior to 0.93, and Y\({}_{2}\) (ABTS**+**) showed the lowest R\({}^{2}\) (0.9178). Therefore, CCRD turns out to be useful to optimize the ultrasound-assisted aqueous extraction conditions of antioxidant compounds from ginger. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Variable Coded Values** & \multicolumn{4}{c}{**Responses**} \\ \hline \(\mathbf{x_{1}}\) & \(\mathbf{x_{2}}\) & \(\mathbf{Y_{1}}\) & \(\mathbf{Y_{2}}\) & \(\mathbf{Y_{3}}\) & \(\mathbf{Y_{4}}\) \\ \hline \(-1\) & \(-1\) & 154.62 & 20.63 & 149.30 & 14.33 \\ \(+1\) & \(-1\) & 128.21 & 13.28 & 109.77 & 11.74 \\ \(-1\) & \(+1\) & 141.15 & 18.88 & 140.70 & 13.62 \\ \(+1\) & \(+1\) & 125.91 & 13.10 & 108.20 & 10.38 \\ \(-\alpha\) & 0 & 138.48 & 17.95 & 136.23 & 12.94 \\ \(+\alpha\) & 0 & 109.65 & 12.23 & 93.18 & 8.18 \\ \(0\) & \(-\alpha\) & 157.15 & 22.53 & 168.80 & 17.11 \\ \(0\) & \(+\alpha\) & 139.80 & 16.59 & 130.92 & 13.54 \\ \(0\) & 0 & 133.03 & 13.58 & 138.92 & 12.40 \\ \(0\) & 0 & 133.14 & 14.24 & 139.65 & 12.80 \\ \(0\) & 0 & 134.06 & 14.40 & 136.77 & 12.07 \\ \(0\) & 0 & 132.05 & 13.64 & 137.35 & 11.99 \\ \(0\) & 0 & 134.61 & 14.07 & 136.70 & 12.62 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Experimental results of the central composite rotatable design (CCRD) for the optimization of the variables involved (x\({}_{1}\) and x\({}_{2}\)) in the ultrasound-assisted extraction of antioxidants from ginger for the four responses, Y\({}_{1}\), Y\({}_{2}\), Y\({}_{3}\), and Y\({}_{4}\). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter** & \multicolumn{4}{c}{**Response**} \\ \cline{2-5}  & \(\mathbf{Y_{1}}\) & \(\mathbf{Y_{2}}\) & \(\mathbf{Y_{3}}\) & \(\mathbf{Y_{4}}\) \\ \hline Intercept & & & & \\ \(\beta_{0}\) & \(-615\) & 230 & \(-3084\) & \(-239.6\) \\  Linear effect & & & & \\ \(\beta_{1}\) & 25.33 & \(-3.25\) & 83 & 6.71 \\ \(\beta_{2}\) & \(-Data obtained from CCDR were used to plot the response surfaces showed in Figure 1A-D. In this research, amplitude (X\({}_{1}\)) played a relevant role in the release of antioxidant compounds from ginger. Figure 1A shows DPPH\(\bullet\) antioxidant activity responses; it is observed that ginger extracts under amplitudes of 80 to 82.5% presented the highest values (>160 mg Trolox/100g). In the case of extracts measured through ABTS\(\bullet\)+ (Figure 1B), using an amplitude of 80%, the highest quantity of antioxidant components (>24 mg Trolox/100g) was obtained.