Chapter 3: Experimental design application and interpretation in pharmaceutical technology – Computer-Aided Applications in Pharmaceutical Technology

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Experimental design application and interpretation in pharmaceutical technology

Jelena Djuris, Svetlana Ibric and Zorica Djuric,     Department of Pharmaceutical Technology and Cosmetology, Faculty of Pharmacy, University of Belgrade

Abstract:

This chapter provides a basic theoretical background on experimental design application and interpretation. Techniques described include screening designs, full and fractional factorial designs, Plackett–Burman design, D-optimal designs, response surface methodology, central composite designs, Box–Behnken design, and mixture designs, etc. The reader will be introduced to the experimental domains covered by specific design, making it easier to select the one appropriate for the problem. After theoretical introduction, a number of illustrative examples of design of experiments application in the field of pharmaceutical technology are presented.

Key words

design of experiments

screening designs

full and fractional factorial designs

response surface methodology

mixture designs

3.1 Introduction

Experimental design is a concept used to organize, conduct, and interpret results of experiments in an efficient way, making sure that as much useful information as possible is obtained by performing a small number of trials. Instead of using the trial-and-error approach, where independent variables (ingredients of a formulation, processing parameters, etc.) are varied by chance, pharmaceutical scientists are nowadays urged to apply experimental design in their product/process development to demonstrate their knowledge of it. This knowledge of product/process is defined by so-called design space, a multidimensional combination and interaction of input variables (e.g. material attributes) and process parameters, which have been demonstrated to provide assurance of quality (ICH Q8R2, 2009). Definition of a design space is done by applying concepts of experimental design (design of experiments, DoE). DoE was first used as a tool mainly by academic researchers, whereas development of pharmaceuticals in the industry was done mostly on an empirical basis or by relying on previous experience. With the introduction of user-friendly software tools, and encouraged by regulatory guidelines and advices, DoE is surely finding its way to becoming an everyday tool in the pharmaceutical industry.

Proper organization of experiments is a foundation of every thoughtful research. The number of experiments conducted is not always a direct measurement of the amount of information gained about the problem being studied. If multiple independent factors are being varied in an unorganized manner, then it is impossible to determine what affected the outcome. If factors are varied in an unreasonable range, optimization strategies can become difficult to manage. These, among others, are some of problems that can easily be anticipated by DoE.

There are many purposes of DoE application: screening studies to determine the most influential factors affecting the product/process being studied; full or fractional designs to quantify factorial effects; in-depth response surface studies particularly useful for optimization; mixture designs, etc. The selection of a particular experimental design depends upon the nature of the problem being studied and the desired level of information to be gained.

It is proposed that, in the case of pharmaceutical product development, screening designs are used at the beginning of the experimental procedure for investigation of a large numbers of factors, with the aim of revealing the most important ones. Optimization is used for finding a factor combination corresponding to an optimal response profile, and robustness testing is used as a last test before the release of a product, to ensure that it stays within the specifications (Eriksson et al., 1998).

3.2 Theory

The main goal of any experimental study is to find the relationships between independent variables (factors) and dependent variables (results, outcomes) within the experimental framework. Even though it sounds easy to accomplish, this task can be cumbersome when it is not organized correctly. In the field of pharmaceutical technology, independent variables are usually formulation factors (ingredients amount, materials attributes, etc.) or processing parameters, whereas dependent variables are product properties or parameters indicating process performance. Experimental design is, in general, used to simultaneously study the effects of multiple independent variables (factors) on response variable(s); therefore, it is a multivariate analysis technique. DoE requires definition of levels (values) of analyzed factors and often this phase uses knowledge from previous experience about the problem being studied.

In the simplest screening experimental design, a relatively large number of factors can be studied in a small number of experiments. In this way, the most influencial factors are identified and further examined in more detail using full factorial or response surface designs. The screening design usually varies the factors on two levels and only a few of all possible combinations of different factors on different levels are used. Response surface design enables optimization of the most influential factors. In this design, factors are varied on at least three levels, and many more combinations of factors on different levels are used (in comparison to screening designs). A mixture design is used for studies where examined factors are mixture related, such as in the amounts of formulation ingredients. There is a constraint that the total sum of ingredient masses must remain the same and the factors represent a fraction of the given ingredients in the formulation.

The reader is advised to consult relevant textbooks in the field of pharmaceutical experimental design for further explanations of experimental design concepts (Montgomery, 1997; Lewis et al., 1999; Armstrong, 2006).

3.2.1 Screening designs

Screening designs are used to identify the most influential factors from those that potentially have an effect on studied responses. A huge number of factors, f, can be screened by varying them on two levels in a relatively small number of experiments N ≥ f + 1. Typical two-level screening designs are fractional factorial or Plackett–Burman designs (Montgomery, 1997; Lewis et al., 1999). When the number of factors, f, is small, then full factorial design can also be used for screening purposes (Dejaegher and Heyden, 2011). Screening designs allow simultaneous investigation of both qualitative (discrete) and quantitative (continuous) factors, which makes them extremely useful for preliminary formulation development.

When f factors are varied on two levels, all possible combinations of these variations make up the two-level full factorial design. The number of experiments, N, in this design is Lf = 2f. Note that designs are usually denoted as ef, meaning that in a 23 design, 3 factors (f) are varied on 2 levels (e) (as represented in Table 3.1). Note that factor levels are in coded values, which enables them to be compared. The lower factor level is denoted as − 1, and 1 stands for the upper factor level.

Table 3.1

23 full factorial design

When the experiments are organized and conducted according to an experimental design, the results are used to calculate factor effects, which demonstrate to what extent certain factors influence the output (i.e. studied dependent variable).

Factor effects are used to build the regression model:

[3.1]

where y is the response (dependent variable), β0 the intercept, and βi the regression coefficients (regression coefficients stand for factor effects).

Full factorial designs allow identification of factor interactions. Independent variables, that is, factors can interact meaning that the influence of one factor on the response is different at different levels of another factor(s). Interactions are calculated from the columns of contrast coefficients (Table 3.2).

Table 3.2

Determination of factor interactions

Fractional factorial designs are denoted as 2f-v, where 2−v (1/2v) represents the fraction of experiments from the full factorial designs that are omitted (v = 1, 2, 3 …). An example of fractional factorial design for 4 factors, 24-1 design, is represented in Table 3.3. By fractioning the combinations of factors levels, some of the information is lost.

Table 3.3

24-1 fractional factorial design

Fractional factorial design does not indicate potential factor interactions and if it is highly fractioned, some factors effects are estimated together (factors are confounded).

A special type of screening design, the Plackett–Burman design (1946), allows estimation of factor effects for f = N – 1 factors, where N is the number of experiments with a multiple of 4. These designs are especially useful for preliminary investigations of huge numbers of potentially influential factors, as represented in Table 3.4 for a 27-4 design.

Table 3.4

Plackett–Burman design for seven factors

Other special kinds of screening designs are asymmetrical, supersaturated, or mixed-level designs. D-optimal design can also be adapted for screening purposes (Dejaegher and Heyden, 2011).

When fractional factorial designs are applied, there is always a possibility that a significant factor effect is not detected, due to all possible factor level combinations not being investigated.

3.2.2 Response surface designs

Response surface designs are used to analyze effects of the most significant factors that are recognized by screening studies (or are known from the previous experience), where the number of these factors is usually 2 or 3. Factors are varied on at least three levels. The main goal of response surface designs is usually optimization. Note that qualitative (discrete) factors cannot be used in these designs. Response surface designs are accompanied by visual representation of the factors’ influence on the studied response. Therefore, it is possible to display the influence of the two factors on the studied response in a graphically comprehensible manner. For more than two factors, only fractions of the entire response surface are visualized (Dejaegher and Heyden, 2011).

Response surface designs can be symmetrical or asymmetrical (Montgomery, 1997). Symmetrical designs cover the symmetrical experimental domain (Dejaegher and Heyden, 2011). Some of the most often used symmetrical designs are three-level full factorial, central composite, Box–Behnken design (BBD), etc. Three-level full factorial design for three factors is represented in Figure 3.1.

Figure 3.1 Three-level full factorial design for 3 factors, 33 design with 27 experiments

In order to determine the experimental error, the central point is often replicated several (3–5) times.

Central composite design (CCD) is composed of a two-level full factorial design (2f experiments), a star design (2f experiments), and a center point, therefore requiring N = 2f + 2f + 1 experiments to examine the f factors (Montgomery, 1997) (Table 3.5). The points of the full factorial design are situated at factor levels ‒1 and + 1, those of the star design at the factor levels 0, ‒α and + α, and the center point at factor level 0. Depending on the value of α, two types of designs exist, a face-centered CCD (FCCD) with |α| = 1, and a circumscribed CCD (CCCD) with |α| > 1. Therefore, in the case of FCCD and CCCD, factors are varied on three or five levels, respectively.

Table 3.5

CCD for three factors

A BBD is described for a minimum of three factors and contains N = (2f(f – 1)) + c0 experiments, of which c0 is the number of center points (Box and Behnken, 1960). The BBD is the most common alternative to the CCD (Vining and Kowalski, 2010). BBDs are second-order designs based on three-level incomplete factorial designs (Ferreira et al., 2007). It can be seen, from Table 3.6, that the first 4 experiments (i.e. the first experimental block) is a full 22 design for factors A and B, whereas factor C is constantly at the level0. The second experimental block is a full 22 design for factors A and C (factor B is at level 0), whereas the third experimental block is a full 2f design for factors B and C (factor A is at level 0). Therefore, BBD can be presented in a simplified manner (Table 3.7). When there are 5 or more factors, Box and Behnken recommended using all possible 23 designs, holding the other factors constant (Vining and Kowalski, 2010). One of the main advantages of BBD is that it does not contain combinations for which all factors are simultaneously at their highest or lowest levels, meaning that experiments performed under extreme conditions (for which unsatisfactory results might occur) are avoided (Ferreira et al., 2007).

Table 3.6

BBD for three factors (center point is replicated three times)

Table 3.7

BBD for three factors (simplified)

A Doehlert (uniform shell) design has equal distances between all neighboring experiments (Doehlert, 1970). In this design, factors are varied at different numbers of levels, in the same design. This enables the researcher to select the number of levels for each factor, depending on its nature and desired experimental domain. The Doehlert design describes a spherical experimental domain and stresses uniformity in space filling. For two variables, the design consists of one central point and six points forming a regular hexagon, and therefore is situated on a circle (Ferreira et al., 2004) (Table 3.8). Doehlert designs are efficient in the mapping of experimental domains: adjoining hexagons can fill a space completely and efficiently, since the hexagons fill space without overlapping (Massart et al., 2003). In this design, one variable is varied on five levels, whereas the other is varied on three levels (Table 3.8). Generally, it is preferable to choose the variable with the stronger effect as the factor with five levels, in order to obtain most information from the system (Ferreira et al., 2004).

Table 3.8

Doehlert matrix for two variables

A comparison between the BBD and other response surface designs (central composite, Doehlert matrix, and three-level full factorial design) has demonstrated that the BBD and Doehlert matrix are slightly more efficient than the CCD, but much more efficient than the three-level full factorial designs, where the efficiency of one experimental design is defined as the number of coefficients in the estimated model divided by the number of experiments (Ferreira et al., 2007).

Table 3.9

Doehlert matrix for three variables

Asymmetrical designs are used for investigation in the asymmetrical experimental domain. Typical examples are D-optimal designs. These (asymmetrical) designs can also be adapted for investigation of the symmetrical experimental domain, which is not the case for application of symmetrical designs for the asymmetrical domain. D-optimal designs are computer-generated designs tailor-made for each problem, allowing great flexibility in the specifications of each problem and are particularly useful when it is necessary to constrain a region and no classical design exists. D-optimal design is an efficient tool in experimental design, making it possible to detect the best subset of experiments from a set of candidate points. Starting from an initial set, several subsets with different type and number of experiments are selected. When analyzing the quality criteria (i.e. determinant of the information matrix, inflation factors) of each subset of different size, it is possible to find a good compromise between the quality of information obtained and the number of experiments to be performed (Frank and Todeschini, 1994). D-optimal designs are used for irregular experimental regions, multi-level qualitative factors in screening, optimization designs with qualitative factors, when the desired number of runs is smaller then required by a classical design, model updating, inclusions of already performed experiments, combined designs with process, and mixture factors in the same experimental plan (Eriksson et al., 2008). In D-optimal designs, N experiments forming the D-optimal design are selected from the candidate points, forming a grid over the asymmetrical domain. These experiments are the best subset of experiments selected from a candidate set (Eriksson et al., 2008). The term ‘best’ refers to the selection of experimental runs according to a given criterion. The criterion most often used is that the selected design should maximize the determinant of the matrix X’X for a given regression model. This is the reason why these designs are referred to as D (from ‘D’ in determinant) (Eriksson et al., 2008). More detailed information on the construction of D-optimal designs is provided in Lewis et al. (1999) and Eriksson et al. (2008).

In all of the above described response surface designs, the regression model is defined as:

[3.2]

where y is the response, β0 the intercept, βi the main coefficients, βij the two-factor interaction coefficients, and βii the quadratic coefficients. Usually, higher-order interactions (higher then two-factor interactions) are ignored and non-significant model terms are eliminated after statistical analysis. More details on regression analysis are provided in the following sections.

3.2.3 Mixture designs

Mixture designs are used to study mixture variables such as excipients in a formulation. All mixture components are examined in one design. The characteristic feature of a mixture is that the sum of all its components adds up to 100%, meaning that the mixture factors (components) cannot be manipulated completely independently of one another (Eriksson et al., 1998). In comparison to other (unconstrained) experimental designs, mixture designs cannot be viewed as squares, cubes, or hypercubes. Furthermore, data analysis is more complicated, since mixture factors are correlated. In the case of a three-component mixture, available designs are represented in Table 3.10 and Figure 3.2.

Table 3.10

Factor levels in mixture designs

Figure 3.2 Experimental points for the mixture design

Simplex lattice mixture designs can be defined with three (experiments 1–3 in Table 3.10) or six experiments (experiments 1–6 in Table 3.10). If experiment 7 is included, then it is a simplex lattice-centroid design and if all ten experiments are considered, then it is an augmented simplex lattice–centroid mixture design.

The three most commonly used mixture designs support linear, quadratic, and special cubic models (Figure 3.3). The linear design is taken from the axial designs, whereas quadratic and special cubic designs are derived from simplex centroid designs. The design supporting a linear model is useful when the experimental objective is screening or robustness testing, whereas the designs supporting quadratic or special cubic models are relevant for optimization purposes (Eriksson et al., 1998).

Figure 3.3 Three most commonly used mixture designs for three-component mixtures supporting linear (left), quadratic (center), and special cubic (right) models. Solid dots represent compulsory experiments, whereas open circles are optional and useful extra experiments. The three open circles positioned at the overall centroid correspond to replicated experiments (adapted from Eriksson et al., 1998)

Generally, mixture designs are of K – 1 dimensionality, where K is the number of factors (mixture components). The mixture regions of two-, three-, and four-component mixtures are line, triangle, and tetrahedron, respectively.

Previously described mixture designs are regular, since there are no bounds on the proportion of mixture components (other than the total sum of 100%). However, there are often certain limitations to mixture components, making it necessary to define some constraints. When constraints are defined (e.g. all three mixture components must be present, and weight ratio of one of the components should not exceed a certain percentage, etc.), experimental points are not part of the triangle represented in Figure 3.2 (in the case of a three-component mixture). Domains of different shape within the triangle (tetrahedron, etc.) are then selected. In this case, regular mixture designs no longer apply, and irregularity in experimental design is best handled with D-optimal design (Eriksson et al., 1998). D-optimal design maps the largest possible experimental design for selected model (linear, quadratic, or special cubic). It is therefore necessary to carefully define the purpose of experimental study (screening, optimization, or robustness testing) prior to selection of adequate mixture design.

Once D-optimal search algorithms for dealing with very constrained mixtures were improved (DuMouchel and Jones, 1994), it was possible to create efficient statistical experimental designs handling both mixture and process factors simultaneously (Eriksson et al., 1998).

3.2.4 Data analysis

There are different ways to determine the effects of different factors investigated in an experimental design. In the case of a simple regression model of screening design:

[3.3]

regression coefficients βi can be determined by solving a system of equations. It is therefore necessary that the number of experiments performed is equal to or exceeds the number of factors being investigated. The regression model can also be written in the form of vector components:

[3.4]

where response vector y is an N × 1 matrix, model X is an N × t matrix (t is the number of terms included in the model), β is the t × 1 vector of regression coefficients, and ε is the N × 1 error vector. Regression coefficient b is usually calculated using least squares regression:

[3.5]

where XT is the transposed matrix of X.

Also, the effect of each factor x on each response y is estimated as:

[3.6]

where ∑ y(+ 1) and ∑ y(− 1) represent the sums of the responses, where factor x is at the (+ 1) and (− 1) levels, respectively, and N is the number of design experiments.

Because effects estimate the change in response when changing the factor levels from − 1 to + 1, and coefficients between levels 0 and + 1, both are related as follows:

[3.7]

In order to determine the significance of the calculated factor effect, graphical methods and statistical interpretations are used. Graphically, normal probability or half-normal probability plots are drawn (Montgomery, 1997). On these plots, the unimportant effects are found on a straight line through zero, while the important effects deviate from this line (Dejaegher and Heyden, 2011). Statistical interpretations are usually based on t-test statistics, where the obtained t value or the effect Ex value is compared to critical limit values tcntical and Excrttical. All effects greater than these critical values (in absolute terms) are then considered significant:

[3.8]

where (SE)e is the standard error of an effect. The critical t value, tcritical, depends on the number of degrees of freedom associated with (SE)e and on the significance level, usually α = 0.05. The standard error of an effect is usually estimated from the variance of replicated experiments, but there are also other methods (Dejaegher and Heyden, 2011).

In the case of response surface designs, the relationship between factors and responses is modeled by a polynomial model, usually second-order polynomial. Interpretation of the model effects is similar to previously described screening design interpretation. Graphically, the model is visualized by two-dimensional contour plots or three-dimensional response surface plots. These plots become more complicated when the number of factors exceeds two. The fit of the model to the data can be evaluated statistically applying either Analysis of Variance (ANOVA), a residual analysis, or an external validation using a test set (Montgomery, 1997). Also, the previously described procedure for determination of significant factor effects can be applied and non-significant factors are then eliminated from the model.

Interpretation of mixture design models is similar to response surface designs. But, since mixture factors (components) are dependent on each other (the main constraint that the sum of all components is 100% is always present), application of multiple regression models requires data parameterization in order to alleviate the impact of the mixture constraint. Chemometric techniques, such as partial least squares regression (PLS), described in more detail in Chapter 4) do not assume mathematically independent factors and are, therefore, directly applicable to mixture data analysis (Kettaneh-Wold, 1992).

3.3 Examples

In the 1980s, the use of experimental design, especially the factorial design, was generalized in the development of solid dosage forms, and appropriate statistical analysis allowed determination of critical process parameters (CPP), the comparison between materials and improvement, or optimization of formulations. In 1999, Lewis suggested mathematical modeling and pointed out the statistical background needed by pharmaceutical scientists. The recent regulations from the key federal agencies, to apply quality-by-design (QbD), have pursued researchers in the pharmaceutical industry to employ experimental design during drug product development.

The following examples present several case studies (among many of them), presented in the pharmaceutical literature, from screening studies, through analysis of factor effects, to the optimization of formulation and/ or pharmaceutical processes.

Example 1

DoE was applied to evaluate influence of the granulation processing parameters on the granule properties and dissolution characteristics of a modified release drug (Ring et al., 2011). This work accentuated that understanding the relationship between high shear wet granulation processing parameters and the characteristics of intermediate and final products is crucial in the ability to apply QbD and process analytical technologies (PAT) to secondary pharmaceutical processes. The objective of the work was to map the knowledge domain for a high shear granulation/tableting process, by analyzing the relationship between critical granulation processing parameters and critical quality attributes (CQA) of the intermediate and final products. The following critical controlled parameters (CCP) were investigated: impeller speed, wetting rate, granulation time, and jacket temperature, with additional control of granule particle size by two different milling techniques. A Taguchi L-9 orthogonal design methodology was chosen for the study, and in addition to determining the dissolution of the resulting tablets after 2 and 4 h, a comprehensive range of granule and tablet characteristics were monitored. Four factors were varied on 3 levels in 36 experiments, which were further split into 2 separate 18 experimental runs by using 2 different milling techniques. The three-level Taguchi designs require relatively few data points and assume that all two-way interactions are negligible. The consequence is that some of the unexplained variance can be due to the effect of interactions that are not negligible, thus increasing the possible sources of error. The overall data analysis was performed by the ANOVA test and p-values were used to relate CQA of the system to CPP It was demonstrated that with respect to granule properties, granulation time followed by wetting rate and jacket temperature have an important influence on the product/ intermediate quality. Statistically significant CPPs were identified for granule hardness, granule density, and granule particle size. These granule properties were also identified as contributing to the dissolution release characteristics. Dissolution modeling and prediction was achieved within the DoE structure.

Example 2

Asymmetrical factorial design was used for screening of high shear mixer melt granulation process variables using an asymmetrical factorial design (Voinovich et al., 1999). The factors under investigation were binder grade, mixer load, presence of the deflector (all analyzed at 2 levels), binder concentration, impeller speed, massing time, type of impeller blades (these 4 at 3 levels), and jacket temperature (considered at 4 levels). Two granule characteristics were analyzed: the geometric mean diameter and the percentage of particles finer than 315 pm. The factorial arrangement 233441//25 was used, where 25 represents the number of runs. Asymmetrical factorial design allowed reduction in the number of runs from 2592 to 25. In addition, this technique permitted the selection of the factor levels, which have the major 'weight' on the 2 granule characteristics under study. Two additional trials were performed to attest the screening validity. The weight of each factor level was estimated by means of the least squares method.

Example 3

The Plackett–Burman design was used to study the effects of 11 different factors on stabilization of multicomponent protective liposomal formulations (Loukas, 2001). These formulations contain riboflavin in either free form or complexed with cyclodextrin as a model drug, sensitive to photochemical degradation, as well as various light absorbers and antioxidants incorporated into the lipid bilayer and/or the aqueous phase of liposomes. The following factors were varied on 2 levels: riboflavin complexation (free vitamin or γ-cyclodextrin complex), presence or absence of light absorbers oil red O, oxybenzone, dioxybenzone, sulisobenzone, and the antioxidant β-carotene. The multilamelar liposomes were prepared either by the dehydration-rehydration technique or by the disruption of lipid film method containing cholesterol in low or high concentrations, 1,2-distearoyl-sn-glycero-3-phosphocholine (DSPC) as an alternative lipid, and sonicated through a bath or probe sonication for a low or higher period of time. All these variations comprise the 11 factors that directly affect the physical stability of liposomes, as well as the chemical stability of the entrapped vitamin. To perform a full factorial design for the examination of 11 factors, at 2 levels for each, it would be necessary to prepare 211 = 2048 liposomal formulations. The Plackett–Burman design allows reduction in the number of experiments from 2048 to 12, for 11 factors studied, where each is at its 2 levels. As described in the introduction, the Plackett–Burman design is a 2-level fractional factorial design. The effect of each factor in the presented study can be determined as:

[3.9]

The authors have highlighted that the Plackett–Burman design is especially useful for a multivariate system with many factors that are potentially influential on system properties. Once these influential factors are recognized (by Plackett–Burman design) and their number significantly reduced, other forms of experimental design, such as full or fractional factorial design, are used. Plackett–Burman designs cannot be used for detection of factors interactions. It was found that the presence of light absorber oil red O demonstrates the most significant effect on liposome stabilization, followed by the complexation with the γ-cyclodextrin form of the vitamin, the preparation method (dried reconstituted vesicles - DRV or multilamellar vesicle - MLV), sonication type, and molar ratio of cholesterol as significant factors.

Example 4

The Plackett–Burman design was used to model the effect of Polyox–Carbopol blends on drug release (El-Malah and Nazzal, 2006). The aim of the study was to screen the effect of 7 factors – Polyox molecular weight and amount, Carbopol, lactose, sodium chloride, citric acid, and compression force on theophylline release from hydrophilic matrices. A Plackett–Burman experimental design of 12 experiments was performed to investigate the effects of 7 factors. A polynomial model was generated for the response, cumulative amount of theophylline released after 12 h, and validates using ANOVA and residual analysis. Results showed that only the amounts of Polyox and Carbopol polymers have significant effects on theophylline release. Regular Plackett–Burman design for 7 factors requires 8 experiments. In the presented study, 12 experimental runs were performed, where additional runs allow derivation of regression equations, since some of information from experiments can be used for error estimation. Note that if there are 8 experiments, there are 8 degrees of freedom - one for each of the factors and one for the intercept of the equation.

Example 5

Experimental design was used to optimize drug release from a silicone elastomer matrix and to investigate transdermal drug delivery (Snorradottir et al., 2011). Diclofenac was the model drug selected to study release properties from medical silicone elastomer matrix, including a combination of 4 permeation enhancers as additives and allowing for constraints in the properties of the material. The D-optimal design included 6 factors and 5 responses describing material properties and release of the drug. Limitations were set for enhancer and drug concentrations, based on previous knowledge, in order to retain elastometric properties of matrix formulations. The total concentration of the excipient and drug was limited to 15% (w/w ratio) of the silicone matrix; the maximum drug content was set to 5% and the minimal drug content to 0.5%, as drug release was one of the most important responses of the system. With these constraints, the region of experimental investigation becomes an irregular polyhedron and the D-optimal design was therefore used. The first experimental object was screening, in order to investigate the main and interaction effects, based on 29 experiments. All excipients were found to have significant effects on diclofenac release and were therefore included in the optimization, which also allowed the possible contribution of quadratic terms to the model and was based on 38 experiments. Generally, the enhancers reduced tensile strength and increased drug release, thus it was necessary to optimize these 2 responses. Screening and optimization of release and material properties resulted in the production of 2 optimized silicone membranes, which were tested for transdermal delivery. The results confirmed the validity of the model for the optimized membranes that were used for further testing for transdermal drug delivery through heat-separated human skin.

Example 6

Statistical experimental design was applied to evaluate the influence of some process and formulation variables and possible interactions among such variables, on didanosine release from directly-compressed matrix tablets based on blends of 2 insoluble polymers, Eudragit RS-PM and Ethocel 100, with the final goal of drug release optimization (Sanchez-Lafuente et al., 2002). Four independent variables were considered: compression force used for tableting, ratio between the polymers, their particle size, and drug content. The considered responses were the percentage of drug released at 3 predetermined times, the dissolution efficiency at 6 h, and the time to dissolve 10% of the drug. These responses were selected, since the percentage of drug released after certain time points is considered the key parameter for any in vitro/in vivo correlation process. The preliminary screening step, carried out by means of a 12-run asymmetric screening matrix according to a D-optimal design strategy, allowed evaluation of the effects of different levels of each variable. Different levels of each independent variable on the considered responses were studied: compression force and granulometric fractions of polymers were varied on 2 levels, drug content was varied on 3 levels, and the ratio between the polymers was varied on 5 levels. Starting from an asymmetric screening design 223151//18, D-optimal design strategy was applied and a 12-run asymmetric design was generated. The drug content and the polymers ratio had the most important effect on drug release, which, moreover, was favored by greater polymers particle size; to the contrary, the compression force did not have a significant effect. The Doehlert design was then applied for a response-surface study, in order to study in-depth the effects of the most important variables. In general, the Doehlert design requires k2+ k+n experiments, where k is the number of factors and n the number of central points. Replicates at the central level of the variables are performed in order to validate the model by means of an estimate of the experimental variance. In this study, drug content was varied on 3 levels and the polymer ratio was varied on 5 levels. Response surfaces were generated and factors interactions were investigated. The desirability function was used to simultaneously optimize the 5 considered responses, each having a different target. This procedure allowed selection, in the studied experimental domain, of the best formulation conditions to optimize drug release rate. The experimental values obtained from the optimized formulation highly agreed with the predicted values.

Example 7

A modified co-acervation or ionotropic gelation method was used to produce gatifloxacin-loaded submicroscopic nanoreservoir systems (Motwani et al., 2008). It was optimized using DoE by employing a 3-factor, 3-level Box–Behnken statistical design. Independent variables studied were the amount of the bioadhesive polymers: chitosan, sodium alginate, and the amount of drug in the formulation. The dependent variables were the particle size, zeta potential, encapsulation efficiency, and burst release. Response surface plots were drawn, statistical validity of the polynomials was established, and optimized formulations were selected by feasibility and grid search. An example of the response surface plot, showing effect of chitosan and sodium alginate concentration on encapsulation efficiency, is displayed in Figure 3.4.Objective function for the presented study was selected as maximizing the percentage encapsulation efficiency, while minimizing the particle size and percentage burst release. BBD was used to statistically optimize the formulation parameters and evaluate the main effects, interaction effects, and quadratic effects of the formulation ingredients on the percentage encapsulation efficiency of mucoadhesive nano-reservoir systems. A 3-factor, 3-level design was used to explore the quadratic response surfaces and for constructing second-order polynomial models. This cubic design is characterized by a set of experimental points (runs) lying at the midpoint of each edge of a multidimensional cube and center point replicates (n = 3), whereas the ‘missing corners’ help the experimenter to avoid the combined factor extremes. A design matrix comprising of 15 experimental runs was constructed and used for generation of regression equation accounting for linear and quadratic factor effects, as well as for factor interactions. The Box–Behnken experimental design facilitated the optimization of a muco-adhesive nano-particulate carrier systems for prolonged ocular delivery of the drug.

Figure 3.4 Response surface plot showing effect of chitosan and sodium alginate concentration on encapsulation efficiency (reprinted from Motwani et al., 2008; with permission of Elsevier)

3.4 References

Armstrong, A.N. Pharmaceutical Experimental Design and Interpretation, 2nd edition. Boca Raton, FL: CRC Press, Taylor & Francis Group; 2006.

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