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1.1 Introduction

Design of experiments (DoE) or also referred to as the experimental designs are the systematic tools used for decades by the technology-driven industries for rationally optimizing the product and process performance and robustness during manufacturing. As any scientific experiment tends to involve a series of factors which encounter high variability in the end results, thus use of systematic tools has now become a routine practice [1]. DoE particularly emphasizes on rational planning and execution on the basis of a planned set of experiments and utilizes the mathematical and statistical principles to suggest an optimal solution. Experimental designs are very helpful in establishing the relationship between the input and output variables and furnishes the end results in the form of output and reduces the variability [2]. In another way, the experimental designs are very efficient for their predictive ability to detect and plug the error and to reduce the experimental variability to attain batch-to-batch consistency. However, the experimental designs are unable to reduce the impact due to the uncontrollable or noise factors [3, 4].

1.2 Early History and Evolution of Experimental Designs

The concept of experimental design was first envisioned by Sir Ronald A. Fisher in 1920s with his work pioneering to deal with agricultural models implementing the statistical methods. After Fisher, Rothamsted also put forward stepping stones for the application of systematic mathematical models in the agricultural experiments [3]. Later, George E.P. Box and K.B. Wilson in 1951 worked on the response surface mapping (RSM) technique for implementation in the manufacturing sector to improve the product and process robustness. Additionally, Kiefer and Wolfowitz in 1959 worked on multifactorial optimization techniques using factorial designs, while Genichi Taguchi in 1987 proposed the concept of Taguchi experimental designs for robustness testing and optimization [2].

1.3 Experimental Design Principles

Every experimental design relies on three key principles such as replication, randomization, and blocking (or error control), which should be carefully adhered for meaningful execution of the experimental conditions and rational prediction of the optimum response variables [1]. The details regarding the aforesaid principles of experimental designs are provided below.

1.3.1 Replication

It is ideal to use a good number of replicates on any point of an experimental design for reducing the experimental error, if any. There is no fixed rule for selection of the number of replicates, but it is better to use at least three or more replicates for any design.

1.3.2 Randomization

It is ideal to randomize all the experimental runs and perform them in a random manner. This helps in avoiding the experimental bias which may be encountered owing to performing the experiments in a serial manner. Use of randomization helps in reducing the effect of extraneous factors that may present during experimental practice.

1.3.3 Blocking or Error Control

Blocking is used to reduce or eliminate the variability transmitted from nuisance factors, which may influence the experimental response. Block represents a set of relatively homogeneous experimental conditions and helps in classifying the experiments performed in identical set of conditions.

1.4 General Considerations in Performing DoE

The application of DoE into actual practice involves a multistep process, thus requires defining the experimental objectives, identification of the critical factors and responses, mathematical modeling and statistical validity analysis, and selection of the optimum solutions [5,6,7]. Figure 1.1 illustrates the typical five-step approach used for implementation of DoE in an experimental setup.

Fig. 1.1
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Six-step approach used for implementation of the DoE in pharma product development

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1.4.1 Defining the Problem

Before execution of an experimental design, the study objectives are defined to prepare a list of specific problems that are to be addressed by the experiment. It is also important to keep all the objectives of the experiment in mind before execution of an experimental design.

1.4.2 Establishment of Cause-and-Effect Relationship

The cause-and-effect relationship is established by drawing rational relationship between the input and output variables based on the prior knowledge, experience, and expertise on the similar type of experiments. The cause-and-effect diagram is drawn, which is also known as fishbone diagram by considering the various causes of variability such as men, materials, machine, methods, milieu, and management.

1.4.3 Selection of Critical Factors by Screening

When a system or process is new, it is usually important to identify the vital factors which have the most influence on the response(s) of interest. Often there are a lot of factors involved in any pharmaceutical product or process, and the experimenters do not know much about the system. Hence, it is better to perform screening in the beginning of the experiment in order to eliminate any unwanted factors. Usually, prioritization exercise is carried out on the basis of prior knowledge and experience. Also, screening experimental designs are used for identifying suitable factors. Figure 1.2 pictorially depicts a list of commonly used screening experimental designs.

Fig. 1.2
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Examples of various experimental designs used for factor screening study, (a) fractional factorial design, (b) Taguchi design, (c) Plackett-Burman design

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1.4.4 Selection of the Critical Response Variable(s)

The selection of critical response variables is carried out on the basis of product-specific and patient-centric attributes. While selecting the response variable(s), the experimenter must consider useful information related to the impact of input factors on the safety and efficacy of the product. It is ideal to access the likely impact of the factors on the response and measures to mitigate the same.

1.4.5 Experimental Design-Based Factor Optimization

Use of response surface mapping (RSM) designs is very helpful for optimizing the factor effect on the related responses. Figure 1.3 pictorially depicts a list of commonly used screening experimental designs. With the help of a RSM design, a set of experiments are performed containing rational combination of the factors. The obtained data is subjected to analysis by fitting with various mathematical models such as linear, quadratic, cubic, and quartic models. A suitable model was chosen based on the validity of the model for statistical parameters like correlation coefficient, predicted error, and lack of fit analysis.

Fig. 1.3
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Examples of various experimental designs used for factor screening study, (a) full factorial design, (b) central composite design, (c) Box-Behnken design, (d) optimal design, (e) mixture design

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1.4.6 Selection of Optimum Solution and Model Validation

The selection of optimum solution is carried out on the basis of varied tools like graphical optimization (by brute-force method, overlay plots) and numerical optimization (by desirability and objective function) and artificial neural networks (ANN) (Fig. 1.4). The optimum solution is usually demarcated in the design space which is the narrower region within the knowledge space, as shown in Fig. 1.5.

Fig. 1.4
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Various optimum search techniques used for experimental design-based optimization

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Fig. 1.5
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Various types of spaces encountered during experimental design optimization

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1.5 Experimental Design Selection and Types

The approach of experimental design selection depends on the nature of experiment, number of factors, and flexibility of conducting the number of experiments. Design selection also depends on the selection of empirical model to describe statistical cause-and-effect relationship. Experimental designs are basically classified into two types such as the screening designs and the response surface designs [1, 8, 9]. The details regarding various experimental designs have been discussed below.

1.5.1 Screening Designs

Screening designs are an efficient way to identify the significant main effects of the factors. The term “screening” refers to an experimental plan that is intended to find the few significant factors from a list of many potential ones. Alternatively, a screening design is primarily used to identify significant main effects only, while the interaction effects are biased due to confounding nature of the factors. Fractional factorial design, Taguchi design, and Plackett–Burman design are primarily used for the purpose of factor screening.

1.5.1.1 Fractional Factorial Design

FFDs are recommended when the number of factors in an experiment ranges between 3 and 7. Such designs require two levels (−1 and +1) for preparing a matrix with combination of the factors. FFDs are expressed by Xk−p, where “X” indicates the number of levels and “k” indicates the number of factors and p describes the fractionation used.

1.5.1.2 Taguchi Design

Such design relies on the use of orthogonal arrays which provide a set of well balanced (minimum) experiments and serve as objective functions for optimization. Taguchi design starts with minimum of three factors and produces four experimental runs at two levels without requisite of any center point runs.

1.5.1.3 Plackett–Burman Design

Plackett–Burman design (PBD) is used for screening study for a minimum of 11 factors and produces 12 experimental runs. The number of runs (n) produced by the design is produced in the multiple of 4, 8, 12, 24. Hence, PBD is useful in screening factors in an experiment with large number of factors.

1.5.2 Response Surface Designs

Response surface designs or optimization design are used for optimizing the critical factors identified by factor screening. Such design employs nonlinear models like quadratic, cubic, and quartic models. Select instances of the response surface designs include full factorial design, central composite design, Box–Behnken design, optimal design, and mixture designs, which have been discussed below.

1.5.2.1 Full Factorial Design

These designs are represented by Xk, where X indicates number of factors and k indicates number of levels. A full factorial design may also be called a fully crossed design. A full factorial design generates experimental runs based on the factorial points and generates a linear polynomial model. Moreover, center points can also be placed in a full factorial design to increasing its power for better predictability. Such design is also capable of identifying the main effect as well as interaction effects.

1.5.2.2 Central Composite Design

Central composite design (CCD) is an effective statistical design which provides information exclusively on the effect of experiment variables. It is widely used for response surface optimization of the experiments, which employs second-order (quadratic) model for the response variable without using a complete three-level factorial experiment. It is employed when factorial designs detect the presence of curvature in the data, thus requires augmentation from an erstwhile linear design to the quadratic response surface design.

1.5.2.3 Box–Behnken Design

Box–Behnken design (BBD) is an independent quadratic design in that it does not contain an embedded factorial or fractional factorial design. In this design the treatment combinations are at the midpoints of edges of the process space and at the center. These designs are rotatable (or near rotatable) and require three levels of each factor. The designs have limited capability for orthogonal blocking compared to the central composite designs.

1.5.2.4 Optimal Designs

The optimality of a design depends on the statistical model and is assessed with respect to a statistical criterion, which is related to the variance-matrix of the estimator. Specifying an appropriate model and specifying a suitable criterion function both require understanding of statistical theory and practical knowledge with designing experiments. Further, optimal designs are of different types such as D-optimal, I-optimal, and A-optimal. These designs utilize three levels for each of the selected factors and are most commonly used for factor optimization study.

1.5.2.5 Mixture Designs

In a mixture experiment, the independent factors are proportions of different components of a blend. In another way, the fact that the proportions of the different factors must sum to 100% complicates the design as well as the analysis of mixture experiments. Mixture designs can be of different types such as simplex lattice designs, simplex centroid designs, and optimal designs. Among these variants of mixture designs, optimal designs are most commonly used for optimization of factors. Further, optimal designs are of different types such as D-optimal, I-optimal, and A-optimal. These designs utilize three levels for each of the selected factors and are most commonly used for factor optimization study.

1.6 Applications of DoE in Drug Product Development

For the development of pharmaceutical drug products, the use of systematic approaches for product development has now become a routine practice. Federal regulatory authorities are bringing new regulations for efficient, effective, and cost-effective product development techniques for ultimately improving the cost of drug products to make them affordable without compromising their quality, safety, and efficacy [10, 11]. DoE, a multivariate approach to product development, has invariably proved its worth for producing robust drug products. Hence, applicability of DoE is possible at all stages of product development, which also helps in ultimately saving time, efforts, and resources. A variety of pharmaceutical products containing multitude of excipients for diverse functional applications for oral and non-oral drug delivery have been developed.

1.6.1 Oral Drug Delivery Systems

A variety of oral drug delivery systems have been developed using the systematic DoE approach. Such delivery systems include tablets, capsules, powders, granules, pellets, microspheres, etc., while liquid dosage forms include dry syrups, emulsions, suspensions, etc. Select instances of the literature reports available on DoE optimization of oral drug delivery systems have been described in Table 1.1.

Table 1.1 Examples of the DoE-based optimization of oral drug delivery systems
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1.6.2 Non-oral Drug Delivery Systems

A variety of non-oral drug delivery systems have been developed using the systematic DoE approach, which include dosage forms such as parenteral, topical, transdermal, ocular, otic, and inhalational preparations. Select instances of the literature reports on DoE optimization of non-oral drug delivery systems have been provided in Table 1.2.

Table 1.2 Examples of the DoE-based optimization of non-oral drug delivery systems
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1.7 Conclusion and Future Perspectives

In a holistic development practice, product and process understanding are the twin keystones of DoE approach. The more the formulator knows about the system, the better he can define it, and the higher precision he can monitor it with. As with any other coherent scientific methodology, DoE also requires holistic envisioning of the formulation development characteristics. Notwithstanding the enormous utility of this DoE-based philosophy in developing optimal solutions, it leads research mindsets to evolve the “out-of-box” strategies too. Besides, the current regulations laid by ICH through Q8 guidance highlight the pharmaceutical development using Quality by Design (QbD) principles. QbD particularly emphasizes on the application of DoE tools for systematic product development practice. Hence, the regulatory importance of DoE has augmented manifold in attaining efficient pharmaceutical development.