# Modern Techniques for Flavonoid Extraction--To Optimize or Not to Optimize?
## 5 Statistical and Mathematical Modeling Techniques for Optimization Flavonoid Extraction Parameters  
Numerous variables, such as solvent concentration and type, extraction time, extraction temperature, solvent pH, solid-to-liquid phase ratio, etc., affect the extraction efficiency in terms of the quantity of extracted chemicals and their biological activity [171]. Therefore, all these variables and their interactions must be taken into account to obtain the highest extraction efficiency. In order to determine the best extraction settings, statistical and mathematical modeling techniques are utilized. More literature data have recently emerged that highlight the significance of using mathematical modeling tools to optimize the extraction process [12; 15; 172; 173]. As a result, multivariate analysis approaches (such as response surface methodology) are rapidly replacing and/or being compared to one-at-atime methodologies. In the one-factor-at-a-time method, the value of one factor is changed while the values of the other factors remain unchanged. Regardless of which optimization method is chosen, the optimization process follows the same steps as shown in Figure 9.  
Experimental design-based response surface methodology (RSM) consists of mathematical and statistical methods that can be used to model and explore processes where a significant number of factors affect the final outcome [174]. To find the output-input interactions, it is essential to understand how the variables interact during the process [175]. The RSM approach ensures the selection of the appropriate process conditions by mathematically modeling the relationships between variables based on observable data [176]. According to [PERSON] et al. [17] RMS application include four stages: (i) variables selection, (ii) experimental design, (iii) development of the model for description of the data and (iv) determination of the optimal process conditions (Figure 10.). In order to establish the best settings for these variables that produce the greatest response, it is crucial to fit a mathematical model equation. This will allow a rough prediction of the link between the response and the independent variables [177]. The most important step presents statistical validation of the developed RSM model and if the model is not adequate, the irrelevant variables must be removed and the experimental runs must be redone [178]. After carefully choosing the variables that have a significant impact on the replies, RSM should be conducted [179]. Screening studies such as factorial designs can be used for this. The linear effects of the variables on the output are estimated using these first order designs [180]. These designs do not estimate curvature. The curvature-interaction of the variables is estimated via second-order designs such central composite and its variations (the rotatable CCD, the spherical CCD, the small composite design, and the face-centered cube), Box-Behnken, and Doelhert designs, and is then presented  
Figure 9: Flow chart for process optimization. as a quadratic equation [181]. The application of RMS modeling for optimization of the flavonoids extraction process is given in Table 4. It is observed that different types of experiment designs and different extraction methods were used for different flavonoids sources but the general conclusion of all listed researches is that the design of the experiment coupled with RSM modeling reduces the number of experiments performed to achieve the maximum extraction yield.