## 2 Materials and Methods
### Multivariate Statistical Analysis  
The stated objective of the multivariate analysis performed was to determine the correct number of clusters and their composition in order to be able to explain the multivariate dynamics of swelling, entrapping, and the in vitro release capacity of the extract from the analyzed microsphere samples. In the multivariate PCA sequence, only the principal components (PC)--PC1, PC2, and PC3--were taken into account. These components describe more than 85% of the total variance in DOE. The graphics in 2D and 3D were produced from all the main components, PC1-PC17. The objective of this analysis was to compare and delimit the components to determine which presented the best results in terms of entrapping, in vitro release, the swelling index, and roughness. In Figure S16c, it can be seen that the samples prepared by method 2 showed the highest values in the entrapping efficiency, roughness, and in vitro release tests, and the lowest values in the swelling index. To achieve a clearer separation of the components, the linear discriminant analysis (LDA) test was applied. The coordinates of each variable in the PCA were used as input data for this test. From the graphs obtained (Figure S17), a clearer spacing of the components was observed, but since this method did not have statistical significance either, it was not possible to correctly generate sample clusters. With this in mind, the MANOVA test with a statistical significance threshold of \(p\) = 0.05 was applied, with the option of generating the multiple-sample comparison. Through the MANOVA analysis of each sample, the correct number and composition of the clusters were generated, and this can be seen in Tables 7 and 8. Finally, obtaining six clusters, the clustering dendrogram is presented in Figure S18 and the clustering heatmap chart is presented in Figure S19. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**RGrel (\%)** & \multicolumn{6}{c}{**Allosteric Sigmoidal**} \\ \hline Best-Fit Values & m1\_Ch01 & m1\_Ch02 & m1\_Ch04 & m2\_Ch01 & m2\_Ch02 & m2\_Ch04 \\ \hline Vmax & 1004 & 1044 & 1123 & 2109 & 2142 & 2196 \\ H & 1.53 & 1.498 & 1.463 & 1.565 & 1.62 & 1.675 \\ Khalf & 2.199 & 2.466 & 2.745 & 3.57 & 3.592 & 3.637 \\ Kprime & 3.339 & 3.866 & 4.38 & 7.326 & 7.941 & 8.696 \\ Std. Error & & & & & & \\ Vmax & 14.35 & 15.04 & 17.75 & 29.1 & 24.17 & 23.93 \\ H & 0.054 & 0.05007 & 0.05055 & 0.04587 & 0.0401 & 0.04114 \\ Khalf & 0.06922 & 0.07554 & 0.08973 & 0.0899 & 0.07224 & 0.06891 \\ Kprime & 0.1532 & 0.1722 & 0.2044 & 0.3476 & 0.3353 & 0.3841 \\ Goodness of Fit & & & & & & \\ Degrees of Freedom & 33 & 33 & 33 & 33 & 33 & 33 \\ R-square & 0.9947 & 0.9954 & 0.9951 & 0.9969 & 0.9979 & 0.998 \\ Adjusted & & & & & & \\ R-square & 0.9944 & 0.9951 & 0.9948 & 0.9967 & 0.9978 & 0.9979 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Results of the non-linear regression of the extract in vitro release time series for all samples. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**MANOVA** & \multirow{2}{*}{**m1\_Ch01**} & \multirow{2}{*}{**m1\_Ch02**} & \multirow{2}{*}{**m1\_Ch04**} & \multirow{2}{*}{**m2\_Ch01**} & \multirow{2}{*}{**m2\_Ch02**} & \multirow{2}{*}{**m2\_Ch04**} \\ \(p\)**-Values** & & & & & & \\ \hline m1\_Ch01 & 1 & \textless{}0.0001 & \textless{}0.0001 & \textless{}0.0001 & \textless{}0.0001 & \textless{}0.0001 \\ m1\_Ch02 & \textless{}0.0001 & 1 & \textless{}0.0001 & \textless{}0.0001 & \textless{}0.0001 & \textless{}0.0001 \\ m1\_Ch04 & \textless{}0.0001 & \textless{}0.0001 & 1 & \textless{}0.0001 & \textless{}0.0001 & \textless{}0.0001 \\ m2\_Ch01 & \textless{}0.0001 & \textless{}0.0001 & \textless{}0.0001 & 1 & \textless{}0.0001 & \textless{}0.0001 \\ m2\_Ch02 & \textless{}0.0001 & \textless{}0.0001 & \textless{}0.0001 & \textless{}0.0001 & 1 & \textless{}0.0001 \\ m2\_Ch04 & \textless{}0.0001 & \textless{}0.0001 & \textless{}0.0001 & \textless{}0.0001 & \textless{}0.0001 & \textless{}0.0001 \\ \hline \hline \end{tabular}
\end{table}