## 2 Materials and Methods
### Experimental Design  
A 3 rd degree (3\({}^{3}\)) Central Composite Design (CCD) was used to mathematically optimise the extraction process of _S. officialis_ and the response surface methodology (RSM). The main functions describing the extraction process were as follows: (i) antioxidant activity (AA-DPPH) assessed by the DPPH method; (ii) antioxidant activity (AA-FRAP) assessed by the FRAP method; and (iii) total polyphenol content (TPC-FC) assessed by the Folin-Ciocalteu method. The experimental plan and contour drawings for the three studied functions were made using the computer program Statistica 13.3 PL software 7 (StatSoft, Krakow, Poland), followed by an analysis of the plotted drawings. The coefficients of the regression equations (a\({}_{1}\)-a\({}_{9}\)) for the standardised input quantities, the (1) raw material content, (2) ethanol concentration, and (3) extraction time, were determined by the least squares method using matrix calculus (Tables 1 and 2). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Raw Material Content**} & \multicolumn{2}{c}{**Ethanol Concentration**} & \multicolumn{2}{c}{**Extraction Time**} \\ \cline{2-5}
**No.** & \multicolumn{2}{c}{**[g/100 mL of Solvent]**} & \multicolumn{2}{c}{**[\% v/v]**} & \multicolumn{2}{c}{**[min]**} \\ \cline{2-5}
**Experiment** & \(\mathbf{X_{R}}\)1 ** & \(\mathbf{X_{K}}\)2  & \(\mathbf{X_{R}}\)1  & \(\mathbf{X_{K}}\)2  & \(\mathbf{X_{R}}\)1  & \(\mathbf{X_{K}}\)2  \\ \hline
1 & 7.5 & 1 & 20 & \(-1\) & 1 & \(-1\) \\  
5 & 0 \\
27 & 4 & \(-1\) & 60 & 1 & 10 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Factors of the designed experiment. \begin{table}
\begin{tabular}{c c c} \hline \hline  & **AA-DPPH** & **AA-FRAP** & **TPC-F-C** \\ \hline a\({}_{0}\) & \(-4.06156\) & 17.03347 & 0.271311 \\ a\({}_{1}\) & 2.05059 & \(-7.53573\) & \(-0.149816\) \\ a\({}_{2}\) & \(-0.12100\) & 0.75467 & 0.026748 \\ a\({}_{3}\) & 0.22767 & 0.53006 & 0.044819 \\ a\({}_{4}\) & \(-0.00326\) & \(-0.00631\) & \(-0.000440\) \\ a\({}_{5}\) & 0.32177 & 0.40814 & 0.065938 \\ a\({}_{6}\) & \(-0.05023\) & \(-0.06310\) & \(-0.009276\) \\ a\({}_{7}\) & 0.00028 & \(-0.01391\) & \(-0.001604\) \\ a\({}_{8}\) & 0.04664 & 0.09204 & 0.014078 \\ a\({}_{9}\) & 0.00665 & 0.01223 & 0.000189 \\ R\({}^{2}\) & 0.872 & 0.709 & 0.860 \\ AdjR\({}^{2}\) & 0.804 & 0.555 & 0.786 \\ SD & 0.75 & 2.4 & 0.14 \\ PRESS & 15.4 & 150 & 0.529 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The most relevant response surface methodology (RSM) optimisation statistics. Table 1 shows the experiment planning matrix, with coded values (\(X_{K}\)) calculated using the following Formula (1):  
\[X_{Ki}=\frac{2\cdot X_{i}-X_{max}-X_{min}}{X_{max}-X_{min}} \tag{1}\]  
where  
\(X_{i}\)--the raw material content, ethanol concentration and extraction time, respectively;  
\(X_{max}\), \(X_{min}\)--the minimum and maximum values in the optimisation area. In addition, an approximating function was determined for the three tested functions describing the extraction process, after which their adequacy was checked using an ANOVA test. The correlation coefficients (R\({}^{2}\)), the adjusted R-square (AdjR\({}^{2}\)), the standard deviation (SD), and the PRESS value were also calculated. The effect of the normalised independent factors (\(X_{i}\), \(X_{j}\)) of the extraction process of \(S\). _officialis_ on the value of the response function (\(Y_{i}\)) is presented by means of a second-degree algebraic polynomial (2):  
\[Y_{i}=a_{0}+\sum_{i=1}^{3}a_{i1}\cdot X_{i}+\sum_{i=1}^{3}a_{i2}\cdot X_{i}^{2 }+\sum_{i=1}^{2}\sum_{j=i+1}^{3}a_{ij}\cdot X_{i}\cdot X_{j} \tag{2}\]  
where  
\(X_{i}\)--the raw material content, ethanol concentration, and extraction time, respectively;  
\(a_{i}\)--regression coefficients;  
\(Y_{i}\)--the relevant dependent variable, i.e., the antioxidant activity (AA-DPPH and AA-FRAP) and total polyphenol content (TPC-FC).