Experimental design application and interpretation in pharmaceutical technology
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.
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).
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).
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.
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).
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).
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.
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.
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).
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.
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.
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).
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).
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).
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).
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.
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.
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).
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:
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:
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:
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).
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.
Motwani, S.K., Chopra, S., Talegaonkar, S., Kohli, K., Ahmad, F.J., Khar, R. Chitosan-sodium alginate nanoparticles as submicroscopic reservoirs for ocular delivery: formulation, optimization and in vitro characterization. Eur. J. Pharm. Biopharm.. 2008; 68:513–525.
Ring, D.T., Oliveira, J.C.O., Crean, A. Evaluation of the influence of granulation processing parameters on the granule properties and dissolution characteristics of a modified release drug'. Adv. Powder Technol.. 2011; 22(2):245–252.
Sanchez-Lafuente, C., Furlanetto, S., Fernandez-Arevalo, M., Alvarez-Fuentes, J., Rabasco, A.M., et al. Didanosine extended-release matrix tablets: optimization of formulation variables using statistical experimental design. Int. J. Pharm.. 2002; 237:107–118.
Snorradottir, B.S., Gudnason, P.I., Thorsteinsson, F., Masson, M. Experimental design for optimizing drug release from silicone elastomer matrix and investigation of transdermal drug delivery. Eur. J. Pharm. Sci.. 2011; 42:559–567.
Voinovich, D., Campisi, B., Moneghini, M., Vincenzi, C., Phan-Tan-Lu, R. Screening of high shear mixer melt granulation process variables using an asymmetrical factorial design. Int. J. Pharm.. 1999; 190:73–81.