## 2 Materials and Methods
### Experimental Design and Statistical Analysis  
RSM was used to analyze the experimental data, enabling the optimization of the UAE-MAE process for lycopene recovery from tomato peels. The analysis was performed with StatSoft STATISTICA 12.0 software (Hamburg, Germany). A Central Composite Design (CCD) was employed within the RSM framework to determine the optimal conditions for the independent variables. Each experiment was conducted in triplicate. The independent variables examined were the solid/liquid ratio (g/mL, X\({}_{1}\)), microwave power (W, X\({}_{2}\)), and ultrasound power (W, X\({}_{3}\)), with each variable coded at three levels: \(-1\), 0, and 1. The dependent variables in the model included the lycopene content (mg/100 g of tomato peels) and extraction yield (EY, %). The responses from 16 experimental runs are presented in Table 1. The generalized second-order quadratic equation, which describes the relationship between the response and the independent variables, is as follows:  
\[Y=b_{0}+b_{1}X_{1}+b_{2}X_{2}+b_{3}X_{3}+b_{11}X_{1}^{2}+b_{22}X_{2}^{2}+b_{33} X_{3}^{2}+b_{12}X_{1}X_{2}+b_{13}X_{1}X_{3}+b_{23}X_{2}X_{3}, \tag{4}\]  
where Y is the dependent variable (lycopene content, EY); \(b_{0}\) is the intercept; \(b_{1}\), \(b_{2}\), and \(b_{3}\) are linear coefficients; \(b_{11}\), \(b_{22}\), and \(b_{33}\) are quadratic coefficients; and \(b_{12}\), \(b_{13}\), and \(b_{23}\) represent the interaction effects between variables. The independent variables \(X_{1}\), \(X_{2}\), and \(X_{3}\) correspond to the solid/liquid ratio, microwave power, and ultrasound power, respectively.  
To evaluate the significance of the effects of the independent variables on the lycopene content and EY, an analysis of variance (ANOVA) was performed, with a significance threshold set at \(p<0.05\). The independent variable with the lowest \(p\)-value was considered to have the most influence on the response, while variables with a \(p\)-value \(>0.05\) were excluded from the model, unless their quadratic or interaction effects were significant [26]. The model was then simplified by removing these non-significant factors, and the resulting reduced model was used for further data interpretation.  
A 95% confidence level was used for the error assessment. ANOVA was also used to determine the significance level for each response, while a lack-of-fit test was conducted to evaluate the adequacy of the mathematical model. The F-value was employed to determine how well the model fit the data, and the model's predictive accuracy was assessed using the R\({}^{2}\) value. To better visualize the relationships and interactions between the independent variables, three-dimensional surface plots were used. For enhanced clarity, only the interaction effects of the most significant variables on the dependent variables are shown.  
Additionally, one-way ANOVA was performed to compare differences between the samples, and [PERSON]'s post hoc test (\(\alpha=0.05\)) was applied to identify any statistically significant differences. All statistical analyses were carried out using StatSoft STATISTICA 12.0 software (Hamburg, Germany).