## 2 Materials and Methods
### Experimental Design and Statistical Analysis  
The extraction conditions for freeze-dried horseradish were established according to a Box-Behnken experimental design (Table 1), using Design-Expert 13 software (free trial version, Stat-Ease Inc., Minneapolis, MN, USA). A total of 12 experiments were conducted, complemented by 5 replicates to assess reliability, resulting in a Box-Behnken design (BBD) consisting of 17 runs for each solvent (Table 2). The same experimental design was applied to two different solvents (water and methanol-water 1:1, \(v/v\)) for the comparative evaluation of extraction efficiency depending on the solvent. \begin{table}
\begin{tabular}{c c c c} \hline \multirow{2}{*}{**Coded Values**} & \multicolumn{3}{c}{**Independent Variables**} \\ \cline{2-4}  & **A—Time (min)** & **B—Temperature (\({}^{\circ}\)C)** & **C—Potency (W)** \\ \hline \(-1\) & 10 & 30 & 80 \\ \hline \(0\) & 20 & 40 & 140 \\ \hline \(+1\) & 30 & 50 & 200 \\ \hline \end{tabular}
\end{table} 
Table 1: Coded vs. actual values of exposure parameters. \begin{table}
\begin{tabular}{c c c c} \hline \multirow{2}{*}{**Experiment**} & **A—Time** & **B—Temperature** & **C—Potency** \\  & **(min)** & **(\({}^{\circ}\)C)** & **(W)** \\ \hline \(1\) & 10 & 40 & 200 \\ \hline \(2\) & 20 & 50 & 80 \\ \hline \(3\) & 20 & 40 & 140 \\ \hline \(4\) & 20 & 30 & 80 \\ \hline \(5\) & 10 & 40 & 80 \\ \hline \(6\) & 20 & 50 & 200 \\ \hline \(7\) & 30 & 50 & 140 \\ \hline \(8\) & 20 & 40 & 140 \\ \hline \(9\) & 20 & 40 & 140 \\ \hline \(10\) & 20 & 40 & 140 \\ \hline \(11\) & 30 & 30 & 140 \\ \hline \(12\) & 10 & 50 & 140 \\ \hline \(13\) & 30 & 40 & 80 \\ \hline \(14\) & 20 & 40 & 140 \\ \hline \(15\) & 10 & 30 & 140 \\ \hline \(16\) & 20 & 30 & 200 \\ \hline \(17\) & 30 & 40 & 200 \\ \hline \end{tabular}
\end{table} 
Table 2: Box–Behnken Design (BBD) matrix for ultrasonic extraction conditions of bioactive compounds with potential antiallergic properties from freeze-dried horseradish powder. To analyze the experimental data, a quadratic polynomial model was fitted using multiple regression analysis within the framework of the Box-Behnken design. The response surface was described by the following second-order Equation (2):  
\[\mathrm{Y=b_{0}+\sum_{i=1}^{n}(b_{i}x_{i})+\sum_{i=1}^{n}\left(b_{ii}x_{ii}^{2} \right)+\sum_{ij=1}^{n}\left(b_{ij}x_{i}x_{j}\right) \tag{2}\]  
where Y denotes the predicted response (TPC, TFC, and DPPH), \(X_{i}\) refers to the coded levels of the independent variables (time, temperature and potency), \(b_{0}\) is the intercept, \(b_{i}\) corresponds to the linear coefficients, \(b_{ii}\) represents the quadratic terms, and \(b_{ij}\) accounts for the interaction effects between variables. The statistical differences in the content of antioxidant compounds, polyphenols, and flavonoids in horseradish extracts were evaluated by analysis of variance (ANOVA) to determine the influence of ultrasonic parameters in the extraction process. To improve the extraction process, a numerical optimization function was used, a statistical method frequently applied in Response Surface Methodology (RSM), with the goal of identifying the ideal experimental conditions that would lead to maximum efficiency. In this regard, three essential factors were considered: antioxidant capacity, total polyphenol content, and flavonoid content. The experimental data were analyzed using response surface methodology (RSM) with two types of regression models: the two-factor interaction (2 FI) model, which includes linear terms and interaction terms between pairs of variables, and the quadratic model, which also incorporates squared terms to account for curvature in the response. The quality assessment and model accuracy validation were performed based on statistical indicators: the coefficient of determination (R\({}^{2}\)), the adjusted coefficient of determination (adjusted R\({}^{2}\)), and the lack of fit test. The significance level considered was \(p<0.05\).