*2.1. Materials*

Apigenin (purity 98%), chitosan medium molecular weight (190–310 kDa and degree of acetylation 85%), and polyethylene glycol (PEG) 400 were purchased from Baoji Guokang Bio-Technology Co., Ltd. Baoji, China. Sodium tripolyphosphate (TPP) and all other reagents and chemicals were of analytical grade in this study.

#### *2.2. Preparation of APG-Loaded PEGylated Chitosan Nanoparticles (PEGylated-CNPs)*

NPs were prepared by ionotropic gelation and were then homogenized and ultrasonicated [14,15]. For this, CS was dissolved in water that contained 1% *v/v* glacial acetic acid solution. Then, a weighed amount of APG was incorporated to the CS solution, and the added mixture has been homogenized at 12,000 rpm for 10 min. Throughout the process of homogenization, a TPP solution was added drop-wise to the mixture. Based on preliminary research, the CS-to-TPP ratio was determined as 4:1. Instantly, after homogenization, the dispersion was exposed to probe sonication (SONICS Vibra cell VC750, Newton, CT, USA) at 30% amplitude and pulsed at pulse 10 s for 20 min to generate nanoparticles. To keep the temperature increase under control, the dispersion was kept in an ice bath. NPs were coated

with PEG 400 immediately after they were formed. Several batches were prepared by using the Box–Behnken design (BBD) to optimize the formulation. The schematic representation for the preparation of PEGylated-CNPs is as follows (Scheme 1).

**Scheme 1.** The schematic representation for the preparation of PEGylated-CNPs.

#### *2.3. Experimental Design*

The BBD is a 3-factor 3-level design experiment that is recommended over others because it requires minimal experimental runs and provides the desired points under the cuboidal space; therefore, the probability of receiving unsuitable results reduced [16,17]. To examine the formulation variables influencing the studied, a three-factor, three-level BBD was applied, in which three formulation variables (amount of CS: TPP, amount of PEG 400, and sonication time) were differentiated at low (−1), middle (0), and high (+1), explored in Table 1. This design necessitates 15 runs with three replicated center points in order to obtain a more uniform estimation of the prediction variance across the overall experimental design. The responses considered in this studied were particle size (PS) (Y1), drug entrapment efficiency (DEE) (Y2), and zeta potential (ZP) (Y3). The BBD was developed by Design-Expert software (Version 12.0, Stat-Ease Inc., MN, USA), which created and analyzed fifteen experimental runs (Table 2). The ranges for each independent factor were preferred from the preliminary study, which can be seen in Table 1.

In order to identify the most desirable mathematical model by using F-tests, Design-Expert produced linear, two-factor interaction (2FI), and quadratic models. Among these, the quadratic model was considered as the best-fitting model with respect to all responses (Table 3). The predicted and adjusted R2 should be close with an approximate difference of 0.2 that must be achieved with a "reasonable agreement" [18]. Adequate precision is also one of the significant parameters in predicting the optimum response for a given variable. The chosen model was also subjected to a lack-of-fit test, and the lack of significance observed for this value in comparison to pure errors indicated that the independent variables and their responses had a significant correlation [19]. The polynomial equation obtained by experimental design is expressed mathematically as follows:

$$\mathbf{Y} = \mathbf{A}\_0 + \mathbf{A}\_1 \mathbf{X}\_1 + \mathbf{A}\_2 \mathbf{X}\_2 + \mathbf{A}\_3 \mathbf{X}\_3 + \mathbf{A}\_{12} \mathbf{X}\_1 \mathbf{X}\_3 + \mathbf{A}\_{13} \mathbf{X}\_1 \mathbf{X}\_3 + \mathbf{A}\_{23} \mathbf{X}\_2 \mathbf{X}\_3 + \mathbf{A}\_{11} \mathbf{X}\_1^2 + \mathbf{A}\_{22} \mathbf{X}\_2^2 + \mathbf{A}\_{33} \mathbf{X}\_3^2 \tag{1}$$

where Y is the dependent responses, Ao represents intercept, A1 to A33 represent the regression coefficients of Y, and X1 to X3 are presented as independent variables [20]. After producing polynomial equations directly related to dependent and independent variables, the optimization of minimum PS (Y1), maximum DEE (Y2), and low ZP (Y3) was performed using a desirability function to obtain the levels of X1, X2, and X3.

**Table 1.** Variables in BBD for formulation development of APG-loaded PEGylated-CNPs.


**Table 2.** Variables and observed responses in BBD.



**Table 3.** Summary of results of regression analysis for responses.

ND = not defined.
