*3.1. Experimental Design*

The experimental matrix was composed of 30 experimental runs. Experiments were performed in triplicate and the tabulated results from each experimental run are described in Table 2. The maximum values achieved in the extraction process were 4.36 ± 0.04 mg GA/g seed and 0.51 ± 0.01% of bixin. These values were obtained at 2.5 min time for MAE, pH of 11, and a solvent-to-seed ratio of 6:1. The only variation between the extractions corresponded to the solvent concentration used at a 50% for the maximum extraction of polyphenol compounds and 96% for the maximum extraction of bixin.

ANOVA was used to evaluate the significance of the quadratic polynomial models. For each term in the models, a large *F*-value and a small *p*-value would imply a more significant effect on the respective response variable [24]. The ANOVA (Table 3) shows how factors such as treatment time, solvent concentration and the solvent-to-seed ratio had a statistically significant effect (*p* < 0.05) for the MAE.


**Table 2.** Experimental matrix and results obtained from the MAE.

Values are expressed as mean ± standard deviation (*n* = 3). MAE: microwave-assisted extraction, GA: gallic acid.



The response surface plots obtained from the polynomial equations are shown in Figure 2. All the models were subjected to an optimization process and the polynomial equations for the response variables are described as follows:

$$\begin{aligned} \text{Ln (Polyphenols)} &= 0.70 + 0.03 \times \text{X}\_2 - 0.26 \times \text{X}\_3 + 0.53 \times \text{X}\_4 + 0.26 \times \text{X}\_1 - 0.42 \times \text{X}\_2 \text{X}\_3 + 0.18 \times \text{X}\_1 \text{X}\_4\\ &= 0.18 \times \text{X}\_2 \text{X}\_4 - 0.41 \times \text{X}\_4^2 + 0.29 \times \text{X}\_1^2 \end{aligned} \tag{3}$$

$$\begin{aligned} \text{Ln (Bixin)} &= -0.59 - 1.12 \times \text{X}\_2 - 0.04 \times \text{X}\_3 + 0.76 \times \text{X}\_4 - 0.15 \times \text{X}\_1 + 0.04 \times \text{X}\_2 \text{X}\_4 \\ &- 0.01 \times \text{X}\_3 \text{X}\_4 + 0.06 \times \text{X}\_2^2 + 0.01 \times \text{X}\_4^2 + 0.10 \times \text{X}\_1^2 \end{aligned} \tag{4}$$

**Figure 2.** Response surface plots of solvent concentration (ethanol) and treatment conditions for: (**a**) polyphenol compounds and (**b**) bixin.

The response surfaces plots depict how the application of microwaves accelerates the process of extraction of the bioactive compounds. When a longer treatment time was applied the yield of bioactive compounds was larger. Once both dependent variables were optimized (polyphenol compounds and bixin), the estimated conditions for the MAE along with their relative error were as listed in Table 4. The absolute bias for the MAE process was high, and thus the experimental results obtained were greater than those predicted by the models.


**Table 4.** Predicted local maximum for the optimization of the MAE applying a Box–Behnken experimental design.

Values are expressed as mean ± standard deviation (*n* = 3).

The absolute error compares the results predicted by the optimized model to the experimental ones obtained under optimal conditions. It indicates that the bixin extraction models using MAE are more precise than those obtained for the extraction of polyphenol compounds.

#### *3.2. Effect of the MAE on the Antimicrobial and Antioxidant Activities of the Extract*

The antioxidant and antimicrobial properties of the extracts obtained at the optimal operation conditions of the MAE were compared to those obtained by leaching. Table 5 shows larger antimicrobial and antioxidant activities for the MAE than those obtained by leaching. This is explained by the higher content of polyphenol and bixin compounds present in MAE.


**Table 5.** Comparison of the antioxidant and antimicrobial activities of annatto extracts obtained by MAE and leaching.

Different superscript letters within row indicate significant differences (*p* < 0.05) according to LSD-Fisher; values are expressed as mean ± standard deviation (*n* = 3). ABTS: 2,2-azino-bis(3-ethylbenzothiazoline-6-sulphonic acid), FRAP: Ferric Reducing Antioxidant Power, and DPPH: 2,2-Diphenyl-1-picrylhydrazyl
