Statistical Sequential Experimentation: Preliminary Mixed Factorial Design, I-Optimal Mixture Design Then Finally Novel Design Space Expansion for Optimization of Tazarotene Cubosomes

Figures & data

Table 1 Mixture Components Studied in the I-Optimal Mixture Design with Their Constraints Expressed in Different Values

Mixture Components	Actual Values (mg)		Real values (Proportions)		L-Pseudo Values
	Upper Limit	Lower Limit	Upper Limit	Lower Limit	Upper Limit	Lower Limit
A: Glyceryl monooleate	420 (510)	300	0.568 (0.689)	0.405	1	0
B: Pluronic F68	240 (330)	120	0.324 (0.446)	0.162	1	0
C: Polyvinyl alcohol	320	200 (110)	0.432	0.720 (0.149)	1	0
Measured Responses (Yx)	Targeted Constraints
Y1: Entrapment Efficiency (%) (EE)	Maximize
Y2: Particle Size (nm) (PS)	Minimize
Y3: Drug percent released after 24h (%) (Q24)	Maximize
Y4: Absolute Zeta Potential (AZP)	Maximize

Notes: Altered limits for the expanded mixture design are shown in parentheses. Studied responses and their targeted constraints are also shown.

Table 2 Composition and Characterization of Cubosomes Prepared According to Different Design Points of I-Optimal Mixture Design (C1-C10) and Extra Points Added in Expanded Design Space (C11–C14)

Std	A: GMO*	B: PL-F68*	C: PVA*	EE **	PS**	Q24h**	PDI**	ZP**
C1	420	120	200	88.5 ± 0.1	207.9±11.5	58.8 ± 1.0	0.36 ± 0.05	–40.1±0.3
C2	360	180	200	82.0 ± 1.4	256.8 ±5.3	57.0 ± 2.7	0.42 ± 0.05	–33.5±1.5
C3	360	180	200	80.2 ±1.1	240.9 ±12.9	55.6 ± 1.2	0.40 ± 0.04	–38.1±0.4
C4	300	240	200	85.3 ± 0.8	317.2 ± 51.7	62.5 ± 0.9	0.47 ± 0.01	–33.5±1.7
C5	340	160	240	79.5 ± 1.4	245.1 ± 6.1	60.9 ± 7.2	0.50 ± 0.04	–36.9±1.7
C6	340	160	240	81.1 ± 0.4	251.5 ± 2.7	60.8 ± 4.1	0.38 ± 0.03	–31.9±0.6
C7	360	120	260	87.2 ± 0.6	203.6 ± 15.7	56.3 ± 2.9	0.29 ± 0.02	–35.5±1.3
C8	360	120	260	89.6 ± 0.4	202.9 ± 12.5	54.1 ± 6.9	0.30 ± 0.03	–50.7±0.6
C9	300	180	260	86.3 ± 2.0	258.8 ± 41.0	56.7 ± 2.8	0.43 ± 0.05	–41.4±0.2
C10	300	120	320	87.3 ± 1.0	204.8 ± 1.9	51.9 ± 2.8	0.41 ± 0.01	–35.1±4.5
C11	510	120	110	83.7 ± 2.6	209.7 ± 12.5	42.5 ± 2.6	0.43 ± 0.05	–36.5±0.9
C12	405	225	110	81.7 ± 0.9	236.8 ± 1.7	58.1 ± 6.2	0.47 ± 0.02	–46.5±1.8
C13	300	330	110	82.1 ± 1.1	273.0 ± 11.2	68.8 ± 6.6	0.47 ± 0.04	–36.8±0.6
C14	370	190	180	76.6 ± 0.4	229.9 ± 4.4	61.0 ± 6.9	0.50 ± 0.01	–37.3±0.5

Notes: *Studied mixture components amounts: A: Glyceryl monooleate (GMO) (mg); B: Pluronic acid F68 (PL-F68) (mg); C: Polyvinyl alcohol (PVA) (mg). **Measured Responses. Data are duplicate measurements of each trial (mean ±SD).

Abbreviations: EE, Entrapment efficiency; PS, Particle size; PDI, Polydispersity index; Q24h, drug percent released after 24 hours; ZP, zeta potential.

Figure 1 I-optimal mixture design (IOMD) space - defined by the constraints 300≤A≤420, 120≤B≤240 and 200≤C≤320 - and its DPs (C1–C11) together with extra added DPs (C11–C14) to give the expanded mixture design (EMD) space defined by the constraints 300≤A≤510, 120≤B≤330 and 110≤C≤320. Where; A = Glyceryl monooleate amount (mg), B = Pluronic F68 amount (mg), and C = Polyvinyl alcohol amount (mg). Design points (DPs): O single trial, ● replicated trial, ◊ selected optimized formulation from IOMD (IOMD-OF), Δ selected optimized formulation from EMD (EMD-OF).

Figure 2 Release profiles of TAZA from trials (C1 – C14) compared with TAZA release from suspension.

Figure 3 Contour plots showing the changes in (A) Entrapment efficiency, (B) Particle size, (C) Drug percent released after 24h (Q24h) and (D) desirability over the I-optimal mixture design space. Arrows represent regions of maximum: (C) Q24h and (D) desirability, ✖ represents the selected optimized formulation based on this design (IOMD-OF).

Table 3 Coefficient Estimates for Different Model Terms – Appearing in the Final Equations from the Expanded Mixture Design – for Each Response and Their Significance, Together with Models’ Types and Evaluation

Terms	EE**		PS^−1.62**		Q24h^2.24**
	Coefficient	p-value	Coefficient	p-value	Coefficient	p-value
A*	83.4	0.0146	0.00017	< 0.0001	4591.5	< 0.0001
B*	82.2	0.0146	0.00011	< 0.0001	13,080.2	< 0.0001
C*	87.1	0.0146	0.00018	< 0.0001	6880.6	< 0.0001
AC	11.7	0.1160	….	….	11,383.8	0.0026
BC	….	….	−0.00021	< 0.0001	….	….
ABC	−202.0	0.0002	….	….	….	….
Model type	Reduced special cubic		Reduced quadratic		Reduced quadratic
Model evaluation
RMSE	1.76		8.04E-06		781.29
C.V. %	2.1		5.52		8.81
R^2	0.8528		0.9427		0.8777
Adjusted R^2	0.7873		0.9255		0.841
Prediction R^2	0.7342		0.9225		0.8145

Notes: *Studied mixture components amounts: A: Glyceryl monooleate (GMO) (mg); B: Pluronic acid F68 (PL-F68) (mg); C: Polyvinyl alcohol (PVA) (mg). **Measured Responses. p-values < 0.05 indicating significant coefficients are shown in bold.

Abbreviations: EE, entrapment efficiency; PS, particle size; Q24h, drug percent released after 24 hours; ZP, zeta potential; RMSE, relative mean square error; CV%, coefficient of variation percent.

Figure 4 Box-Cox plot for data transformation showing ln residual sum of squares (ln residual SS) without transformation and after recommended transformation for the responses: (A) Particle size, (B) Drug percent released after 24h.

Figure 5 Contour plots and Piepel’s trace plots showing the changes in Entrapment efficiency (A and B), Particle size (C and D), Drug percent released after 24h (E and F) over the expanded mixture design space.

Figure 6 Transmission electron microscope morphological examination for stained expanded mixture design optimized formulation showing its cubic shape.