Response Surface Designs Part 1 — Types and Properties - Response surface designs are used during method optimization to determine optimal conditions for the factors that have the most influence

Response Surface Designs Part 1 — Types and Properties

Response surface designs are used during method optimization to determine optimal conditions for the factors that have the most influence on the response(s) of interest. Those factors might have been identified in a preceding screening, for example, from a screening design. The design results allow the determination of the optimal experimental conditions by modelling the response(s) as a function of the examined factors. In this column, the types and properties of regularly applied response surface..

May 1, 2009
By: Bieke Dejaegher, Yvan Vander Heyden
LCGC Europe
Volume 22, Issue 5

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Experimental designs are used in method development and robustness testing and have been discussed in an earlier article.¹ An experimental design is an experimental set-up that allows the simultaneous examination of a predefined number of factors in a predefined number of experiments. Method development is often divided into a screening and an optimization step. During the first step, many factors, potentially affecting the method, are screened to determine the most important factors, which are then further optimized.¹

During screening, so-called screening designs are applied. These designs allow evaluating the effects of a relatively high number of factors in a relatively small number of experiments and were already discussed thoroughly in previous columns.^2,3

In the optimization step, either sequential simplex approaches or response surface designs are applied to further examine those factors found most important from the screening step.¹ Usually, two or three important factors are then optimized further. The response surface design results allow modelling the response(s) as a function of the factors to determine the optimal experimental conditions. When applying these designs, it is either assumed that the optimum is situated in the examined experimental domain, defined by the chosen factor levels, or the maximally feasible domain is examined.

Both symmetrical and asymmetrical response surface designs exist.¹ A symmetrical design forms a symmetrical figure when its experiments are plotted as a function of the factor levels, whereas an asymmetrical design does not when considering an asymmetrical domain of the factors. An asymmetrical design is a design applicable in an asymmetrical domain. Symmetrical designs do not fit well in such a domain.¹ Symmetrical designs are thus used to examine a symmetrical experimental domain, while asymmetrical designs are usually applied in cases where an asymmetrical area needs to be explored.⁴ For example, when optimizing the factors pH and percentage organic modifier in a chromatographic mobile phase, the domain with suitable retention for all compounds can be irregular. In such cases, it is recommended to perform an asymmetrical design to cover the experimental domain well.

Figure 1: Three-level full factorial design for two factors (N = 9).

In this column, the types and properties of the response surface designs, most currently applied in the context of method optimization, are discussed. The analysis of the design results and the simultaneous optimization of multiple responses, which is often required during optimization with response surface designs, will be discussed in a future column.

Symmetrical Response Surface Designs

Figure 2: Three-level full factorial design for three factors (N = 27).

Within the symmetrical response surface designs, we consider the three-level full factorial, central composite, Doehlert and Box-Behnken designs as most frequently applied.

Figure 3: Central composite design for two factors (N = 9).

A three-level full factorial design contains all possible combinations between the f factors and their L = 3 levels (–1, 0, +1). Thus, N = L^f = 3^f experiments are required to examine f factors in this design. For two and three factors, 9 and 27 experiments, respectively, need to be performed. In Figure 1, the experiments from a three-level full factorial design for two factors are both tabulated and plotted. In Figure 2, a three-level full factorial design for three factors is shown. In both figures, the red experiment represents the centre point. It is often replicated (represented by etc.) to estimate the experimental error in a later data handling.

Figure 4: Central composite design for three factors (N = 15).

Central composite designs (CCD) are the most frequently applied response surface designs. They consist of a two-level full factorial design (2^f experiments), a star design (2f experiments) and a centre point. As a consequence the CCD require N = 2^f + 2f + 1 experiments to examine f factors. The experiments of the full factorial design are situated at levels –1 and +1, those of the star design at levels –α or +α for one factor and the centre point at levels 0 (see Figures 3 and 4). Depending on the α value, two common types of CCD are distinguished. A face-centred CCD (FCCD) with |α| = 1 examines all factors at three levels (–1, 0, +1), while a circumscribed CCD (CCCD) has |α| > 1 and evaluates five levels for each factor (–α, –1, 0, +1, +α). To obtain a so-called rotatable CCCD, the extreme levels of the star design (–α, +α) should fulfil the requirement; |α| = (2^f)^¼. Therefore, |α| is equal to 1.41 and 1.68 for two and three factors, respectively. For two and three factors, a CCD requires 9 and 15 experiments, respectively. In Figures 3 and 4, a CCCD for two and three factors, respectively, is presented. In both figures, the blue experiments represent the full factorial design, the green represent the star design, while the red again represents the centre point.

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