## 4 Conclusions  
To develop healthy functional foods formulated with purple-colored bran, the quality and dough properties of flour-bran blends with different blending ratios and the quality and antioxidant activity of bread prepared with the blends were investigated in this study. Additionally, optimizing processing conditions for bread with the blend at 30% bran was conducted using response surface methodology. Increasing the blending ratio of bran resulted in increased water absorption and the contribution of damaged starch, and decreased gluten formation. With more than 20% bran, the weakening effect of gluten development was significantly noticeable. As the blending ratio of bran increased, the \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Response** & **Source** & **DF** & **Sum of Squares** & **Mean Square** & **F Value** & **R\({}^{2}\)** \\ \hline \multirow{3}{*}{\begin{tabular}{c} \multirow{2}{*}{\begin{tabular}{c} \multirow{2}{*}{\begin{tabular}{c} \multirow{2}{*}{\begin{tabular}{c} Model \\ \end{tabular} \\ height \\ \end{tabular} }} \\ \end{tabular} } & Model & 9 & 450.93 & 50.10 & 8.00 ** & 0.911 \\  & Residual & 7 & 43.82 & 6.26 & & \\  & Total & 16 & 494.75 & & & \\ \hline \multirow{3}{*}{\begin{tabular}{c} \multirow{2}{*}{\begin{tabular}{c} \multirow{2}{*}{\begin{tabular}{c} \multirow{2}{*}{\begin{tabular}{c} Model \\ \end{tabular} \\ volume \\ \end{tabular} }} \\ \end{tabular} } & Model & 9 & 36,822.12 & 4091.35 & 6.17 * & 0.888 \\  & Residual & 7 & 4642.61 & 663.23 & & \\  & Total & 16 & 41,464.73 & & & \\ \hline \multirow{3}{*}{\begin{tabular}{c} \multirow{2}{*}{\begin{tabular}{c} \multirow{2}{*}{\begin{tabular}{c} Model \\ \end{tabular} \\ firstrm \\ \end{tabular} }} \\ \multirow{3}{*}{\
\begin{tabular}{c} \multirow{2}{*}{
\begin{tabular}{c} Model \\ \end{tabular} \\ firstrm \\ \end{tabular} } } & Model & 9 & 758.65 & 84.29 & 8.03 ** & 0.912 \\  & Residual & 7 & 73.52 & 10.50 & & \\ \cline{1-1}  & Total & 16 & 832.17 & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Analysis of variance of the quadratic models for height, volume, and firmness of bread prepared with flour-bran blend. \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Optimum Condition** & \\ \hline Water (g) & 68.1 & 68.1 \\ Mixing time (min) & 4.0 & 4.0 \\ Fermentation time (min) & 67.3 & 67.3 \\ \hline Response & Predicted value & Experimental value \\ \hline Bread height (mm) & 64.6 & 64.7 \(\pm\) 1.5 \\ Bread volume (mL) & 513.2 & 530.5 \(\pm\) 9.0 \\ Bread firmness (N) & 8.8 & 8.2 \(\pm\) 0.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Predicted and experimental values of response at optimized conditions. -head-making performance of the flour-bran blends decreased. However, the total phenolic compound and total anthocyanin contents and the DPPH and ABTS radical scavenging activities as indicators of antioxidant activity significantly increased. The results were best fitted into a quadratic polynomial model with regression coefficient values close to or more than 0.900 for all responses based on RSM analysis. Optimized conditions were close to a center point: 68.1 g water, 4.0 min mixing time, and 67.3 min fermentation time. Overall, this study provides fundamental data for developing bread to promote health benefits through purple-colored wheat bran's excellent antioxidant activity. In particular, controlling the amount of water and mixing time could improve bread quality without requiring any additional ingredients. Conceptualization, M.K. and Y.M. ; methodology, Y.S. and Y.M. ; formal analysis, Y.S. and Y.M. ; investigation, M.K. ; data curation, Y.S. ; writing--original draft preparation, Y.S. and Y.M. ; writing--review and editing, M.K. ; supervision, M.K. ; project administration, Y.M. ; funding acquisition, M.K. All authors have read and agreed to the published version of the manuscript. This research was supported by the Cooperative Research Program for Agriculture Science and Technology Department (Project No. PJ014543012020) funded by the Rural Development Administration (Korea).