## 2 Materials and Methods
### Experimental Design, Statistical Analysis & Process Optimization  
In this study, a Face-Centered Central Composite Design (FC-CCD) was applied for the effective study of 4 operational parameters of MAE at 3 levels (\(-\)1, 0, +1), and response surface methodology (RSM) was performed for data correlation. The independent parameters studied were extraction temperature (T) from 40 to 60 \({}^{\circ}\)C, duration (t) from 5 to 25 min, solvent-to-biomass ratio (R) from 20 to 90 mL\({}_{\text{solv}}\)/g\({}_{\text{biom}}\), and microwave power (P) from 300 to 800 watts. The examined responses were extraction yield and total phenolic (TPC), chlorophyll (CHL), and carotenoid (CAR) contents, selected carotenoid content (sel. CAR), and antioxidant activity (IC\({}_{50}\)). According to Table 2, the experimental design consisted of 8 axial points, 16 factorial points and 3 repetitions of the central point. The FC-CCD is considered one of the most popular designs of response surface methodology (RSM) and facilitates effect studies by avoiding a full-factorial design [33; 34].  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Time (min)** & **aq. Phosphoric Acid, 1\% \(v/v\) (\% \(v/v\))** & **Methanol (\% \(v/v\))** & **MTBE (\% \(v/v\))** \\ \hline
0 & 4 & 81 & 15 \\
15 & 4 & 66 & 30 \\
23 & 4 & 16 & 80 \\
27 & 4 & 16 & 80 \\
27.1 & 4 & 81 & 15 \\
35 & 4 & 81 & 15 \\ \hline \hline \end{tabular}
\end{table}
Table 1: The linear gradient adjusted for the RP-HPLC analysis.  
The data correlation of each response was expressed through the polynomial Equation (1). Response transformation, according to Equation (2), was also applied where considered mandatory.  
\[Y=b_{0}+\sum_{i=1}^{4}b_{i}\ \mathrm{X_{i}}+\sum_{i=1}^{4}b_{ii}\ \mathrm{X_{i}}^{2}+\sum_{i=1}^{3}\sum_{i=1}^{4}b_{ij}\ \mathrm{X_{i}}\ \mathrm{X_{j}}+\sum_{i=1}^{2}\sum_{j=2}^{3}\sum_{k=3}^{4}b_{ijk}\ \mathrm{X_{i}}\ \mathrm{X_{j}}\ \mathrm{X_{k}} \tag{1}\] \[\qquad\qquad+\sum_{i=1}^{3}\sum_{j=i+1}^{4}b_{1 ij}\ \mathrm{X_{i}}\ \mathrm{X_{j}}^{2}+\sum_{i=1}^{3}\sum_{j=i+1}^{4}b_{2 ij}\ \mathrm{X_{i}}^{2}\mathrm{X_{j}} \tag{2}\]  
where Y and Y\({}^{\prime}\) stand for the corresponding response and transformation, respectively, \(b_{0}\), \(b_{i}\), \(b_{ii}\), and \(b_{ij}\) stand for the constant, linear, quadratic and 2-factor interaction coefficients, respectively, \(b_{ijk}\), \(b_{1 ij}\), and \(b_{2 ij}\) stand for the cubic coefficients, and \(\mathrm{X_{i}}\), \(\mathrm{X_{j}}\) and \(\mathrm{X_{k}}\) stand for the examined independent variables.  
[PERSON]'s statistical test (F-test) was applied for the determination of the statistical significance with a 95% significance level. Finally, the experimental design, modeling, and statistical analysis of the experimental data were performed using the Design Expert(r) Version 13 software trial version (Stat-Ease Inc., Minneapolis, MN, USA).  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Run**} & **T** & **P** & **t** & **R** & **Yield** & **TPC** & **CHL** & **sel.** & **CAR** & **IC\({}_{\text{50}}\)** \\  & (\({}^{\circ}\)**C**) & **(watts)** & **(min)** & **(\(\text{mL}_{\text{solv}}\)/** & **(\% wtw)** & **(mg\({}_{\text{GA}}\)** & **(mg/** & **(mg/** & **(mg/** & **(mg\({}_{\text{extr}}\)**) \\  & & & &