## 4 Extraction Procedure via Experimental Design  
Finding the appropriate conditions for the best recovery of the APIs from the cream substrate was investigated using experimental design methodology and the \"Design Expert 11\" software. The study was carried out in order to determine the extraction solvents (methanol or acetonitrile), their participation ratios, as well as the time the samples should be subjected to ultrasonication and freezing. Thus, a D-optimal model was utilized, where the combination data of a mixture (A: acetonitrile B: methanol) was crossed with two factors, which were C: sonication and D: freezing time (Table 6). Based on the experimental design methodology, 18 proposed combinations were applied, aiming at the optimal % recovery of the 5 APIs from the substrate (10 mg cream). For their quantification, a standard (STD) solution with the same concentration (FERA = 5 \(\upmu\)g\(\cdot\)mL\({}^{-1}\), RSV = 10 \(\upmu\)g\(\cdot\)mL\({}^{-1}\), QR = 10 \(\upmu\)g\(\cdot\)mL\({}^{-1}\), RTN = 3 \(\upmu\)g\(\cdot\)mL\({}^{-1}\), a-TOC = 20 \(\upmu\)g\(\cdot\)mL\({}^{-1}\)), a blank (cream substrate), and a blank (diluent) were also analyzed (Figure 2). According to the results, only in three of the five responses were the understudy experimental factors significant (\(p\)-value \(<\) 0.05) and affecting the recoveries (% RTN, % RSV, and % FERA). The order of the combined (_mixture \(\times\) 2 factors_) model was cubic \(\times\) mean (\(p\) < 0.0001) for RTN, quadratic \(\times\) mean (\(p\) = 0.0001) for RSV, and mean \(\times\) linear (\(p\) = 0.0001) for FERA. Subsequently, the significance of the examined factors (A, B, C, D) was assessed by the analysis of variance (ANOVA). Based on the results, it was found that factor D was not significant in any of these cases. Indeed, freezing is used to ensure the purity of the samples (precipitation of fatty impurities due to very low temperature) and is likely not affecting the recoveries of the substances. The overall evaluation of the models is determined by the values of specific statistical parameters presented in Table 7. The F value expresses the ratio of the systematic dispersion to the random unsystematic. When F is \(>\)1, it means that the factor has a significant effect, and the modeling is satisfactory. It is desirable that \"R-squared\" and \"adjusted R-squared\" have similar values, as this ideal case means that 100% of the observed variance can be explained by the model. Finally, the term \"adequate precision\" expresses the signal-to-noise ratio, and values \(>\) 4 are preferred. The good predictive ability of the proposed models can also be revealed by the plots of the residuals against the predicted values, where there are no outliers (Supplementary Figure S6). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Mixture** & & **Units** & **Limits** \\ \hline A & MeOH & mL & 0–10 \\ B & ACN & mL & 0–10 \\ \hline \multicolumn{4}{c}{Process} \\ C & sonic & min & 20–80 min \\ D & freezing & min & 30–80 min \\ \hline \multicolumn{4}{c}{Responses} \\ R1 & RTN & \% Recovery \\ R2 & A-TOC & \% Recovery \\ R3 & RSV & \% Recovery \\ R4 & FERA & \% Recovery \\ R5 & QR & \% Recovery \\ \hline \hline \end{tabular}
\end{table} 
Table 6: D-optimal experimental design considered limits and selected responses. In order to find the best solution of the model, two approaches were applied:  
(A) Graphical: It is based on the superimposition of the isometric curves in a single diagram (overlay plot) and then graphical localization of the region of optimal solutions (Supplementary Figure S7). Based on the diagram, it was found that the shaded region, characterized as the region of optimal solutions, satisfies the objectives set. (B) Mathematical: It is based on the construction of mathematical optimization relations such as the _desirability function_ whose value must tend to 1.0 (Figure 4). The model suggested solvent composition was 5 mL MeOH and 5 mL ACN; the ideal sonication time was 80 min, and the freezing time was 30 min (desirability = 0.865).