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A Box–Wilson central composite design, commonly termed as CCD, is frequently utilized to build a second‐order polynomial for the response variables (CQAs) in RSM without using a complete full factorial design of experiments. There are two main varieties of CCD, namely, face‐centered CCD and rotatable CCD. Due to its simplicity, regions of interest, and operability, the face‐centered CCD was chosen in the present study. As per the face‐centered CCD, the optimal composition and the experimental conditions to prepare the topical ophthalmic emulsions were fixed (Table 2.7). Although the goal of ZP is fixed at maximum (Table 2.7) by considering the previous observation that a stable cationic nanosized emulsion should contain the ZP value that ranges from +25 to +45 mV (Tamilvanan et al. 2010), the stable stability of emulsion is not increasing with the increase of ZP's values. With the help of polynomial regression equation, the factor–response relationship was examined for the response function (Yi) using the generalized response surface model [as given in Eq. (2.2)]. In Eq. (2.2), the terms X₁, X₂, and X₃ indicate the three different factors (CPPs, independent variables) such as amounts of castor oil, chitosan, and poloxamer, respectively. While the term a₀ represents intercept (a constant), the linear (first‐order), quadratic (second‐order), and interactive polynomial coefficients are given as a₁, a₂, and a₃ (in general, a_i), a₁₁, a₂₂, and a₃₃ (in general, a_ii), and a₁₂, a₁₃, and a₂₃ (in general, a_ij), respectively.

(2.2)

The estimated model equations (both coded and actual), regression coefficients, R², adjusted R², regression (P‐value and F‐value), and standard deviation related to the effect of the three CPPs (independent variables) are presented in Table 2.8. Although the actual model equation contains the levels specified in the original units for each CPP, this equation should not be used to determine the relative impact of each CPP (factor). Therefore, the coded model equation is used for identifying the relative impact of the CPPs by comparing the regression coefficient values of each factor. Nevertheless, a positive value in the regression equation represents an effect that favors optimization due to synergistic effect, while a negative value indicates an inverse relationship or antagonistic effect between the factor and the responses (Woitiski et al. 2009). It should be mentioned that nonsignificant (p < 0.005) linear terms (main CCD effect) were included in the final reduced model if quadratic or interaction terms containing these variables were found to be significant (p < 0.05). In the present study, the response surface analysis demonstrated that the second‐order polynomial used for MPS has a higher coefficient of determination (R² = 0.8186) compared to ZP value (R² = 0.6525) and PDI value (R² = 0.7188). The obtained coefficient of determination showed that >82% of the response variation of the MPS, PDI, and ZP values could be described by response surface models as the function of the CPPs. It was observed that the lack of fit had no indication of significant (p < 0.05) for the final reduced model, therefore providing the satisfactory fitness of the RSM to the significant independent variables (factors) effect. From Table 2.8, it was observed that two independent variables (A and B) exhibited a positive effect on the response of ZP (R₃). Both MPS (R₁) and PDI (R₂) showed a positive effect for all of the tested three independent variables (A, B, and C). The interaction coefficients with more than one factor, or higher order terms in the regression equation, represent the interaction between terms or the quadratic relationship, respectively, which suggest a nonlinear relationship between factors and responses (Motwani et al. 2008). Therefore, a different degree of response than it is originally predicted by regression equation may be expected from the independent variables as they are varied at different levels or more than one factor or variables is changed simultaneously (Woitiski et al. 2009). In the present face‐centered CCD modeling, all the responses (MPS, PDI, and ZP or R₁, R₂, and R₃) were affected by the interaction of independent variables and hence displaying a quadratic relationship. The interaction effects between B and C was favorable for response R₃ (showing a positive regression coefficient value of 2.06) and non‐favorable for R₁ and R₂ (showing negative regression coefficient values). The interaction effects between A and B was favorable for response R₁ (showing a positive regression coefficient value of 13.75) and non‐favorable for R₂ and R₃ (showing negative regression coefficient values). However, it was observed that the interaction between A and C had an inverse effect for all three responses variables studied. Noticeably, quadratic effects (A², B², and C²) were also seen between all three independent variables and all of the studied three response variables (MPS, PDI, and ZP or R₁, R₂, and R₃). The highest and positive quadratic effect for all three independent variables was noticed for the MPS response variable R₁ (showing a positive regression coefficient value of 15.35). Similarly, the highest and negative quadratic effect was also noted for the MPS response variable R₁ (showing a negative regression coefficient −118.30).

TABLE 2.8. Estimated Coded and Actual Model Equations Along with ANOVA of Regression Coefficients for Face‐Centered Central Composite Design (CCD) Effects (Main or Linear, Quadratic and Interaction) Against the Critical Quality Attributes (CQAs, Dependent Variables) to Determine the Best Fitted Quadratic Equation

CQAs	Model Equations	R 2	Adjusted R2
Mean particle size (R1)	Coded: R1= 569.86 + 65.20 × A + 25.21 × B + 43.03 × C + 13.75 × AB − 55.18 × AC − 10.38 × BC + 15.35 × A2 − 118.30 × B2 − 91.00 × C2	0.8186	0.6553
Actual: PS = −5992.56500 + 663.59545 × a + 88.29447 × b + 120.25931 × c + 4.58333 × a × b − 8.82800 × a × c − 0.138333 × b × c + 61.41818 × a2 − 3.28598 × b2 − 0.582371 × c2
Polydispersity index (R2)	Coded: R2 = 0.6038 + 0.0325 × A + 0.0234 × B + 0.0960 × C − 0.0106 × AB − 0.0649 × AC − 0.0494 × BC + 0.0180 × A2 + 0.0075 × B2 + 0.1455 × C2	0.7188	0.4658
Actual: PDI = −9.27017 + 0.800295 × a + 0.061847 × b +0.194161 × c −0.003542 × a × b − 0.010380 × a × c − 0.000658 × b × c + 0.071818 × a2 + 0.000207 × b2 − 0.000931 × c2
Zeta potential (R3)	Coded: R3 = 28.89 + 0.0500 × A + 1.32 × B − 2.76 × C − 2.21 × AB − 0.9125 × AC + 2.06 × BC − 0.8136 × A2 − 1.56 × B2 − 4.26 × C2	0.6525	0.3397
Actual: ZP = −180.64250 + 31.48864 × a − 0.037576 × b + 4.44347 × c − 0.737500 × a × b − 0.146000 × a × c + 0.027500 × b × c − 3.25455 × a2 − 0.043434 × b2 − 0.027287 × c2
CCD effects	Variables	R 1	R 2	R 3
F‐value	P‐value	F‐value	P‐value	F‐value	P‐value
Linear term	A	7.61	0.0202	1.09	0.3219	0.0013	0.9714
B	1.14	0.3111	0.5629	0.4704	0.9391	0.3554
C	3.32	0.0986	9.47	0.0117	4.11	0.0702
Interaction term	AB	0.2709	0.6141	0.0928	0.7668	2.11	0.1769
AC	4.36	0.0633	3.46	0.0924	0.3590	0.5624
BC	0.1542	0.7028	2.01	0.1872	1.83	0.2054
Quadratic term	A 2	0.1161	0.7403	0.0911	0.7689	0.0981	0.7605
B 2	6.89	0.0254	0.0157	0.9027	0.3624	0.5606
C 2	4.08	0.0711	5.99	0.0344	2.69	0.1317

Note: A/a: Castor oil, B/b: Chitosan, C/c: Poloxamer.

The coefficient significance of the quadratic polynomial models was evaluated by using Analysis of Variance (ANOVA). For any of the terms in the models, a large F‐value and a small p‐value indicated more significant effect on the respective response variables (Joglekar 1987). Table 2.8 shows the effect of independent variables on the variation of the physicochemical properties of topical ophthalmic emulsions. The variable that exhibited the largest and significant (p < 0.05) effect on the MPS of the emulsion for the linear term was castor oil amount. The other two variables (amounts of chitosan and poloxamer) showed insignificant effects. All the interaction terms showed insignificant effect on MPS. The quadratic terms of chitosan amount exhibited significant effect on MPS while other two terms showed insignificant effect.

The independent variables that most affect the ZP value of the emulsions for the linear term were poloxamer amount followed by the linear term of chitosan amount and castor oil amount but none of the linear terms has any significant effect. All the three interaction and quadratic effects have insignificant effect (p > 0.05) on the ZP value of the emulsions.

For the PDI, the linear and quadratic terms of poloxamer amount have the significant effect while all other linear, interaction, and quadratic terms have insignificant effect. Thus, it was indicated that in evaluating the response variation of PDI, it was important to consider the poloxamer amount.