Design of Experiments

Design of experiments (DoE) is a statistical optimisation approach. A series of predefined experiments are conducted, with the aim of identifying the effect of each individual (main) factor and the effect of synergistic and antagonistic interactions between the factors. This enables polynomial modelling of a response surface for the defined experimental space and prediction of the global optimum.

Full factorial designs (FFD) are used to screen the factors that are assumed to have a significant effect on the response, by determining the coefficient for each main and interaction term. Each factor is assigned a discrete level and experiments are conducted at all possible combinations of these levels across all factors. In addition, experiments are conducted in replicate at the centre-point conditions to give a measure of repeatability. The number of experiments required, N, is given by [Eq (9)], where n = number of levels, k = number of factors and m = number of

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centre-point replicates. Hence, for a 2-level 3-factor FFD with upper and lower levels +1 and -1 respectively, the experimental space is defined by a cube with experiments at each vertex (Figure 10).³⁹

𝑁 = 𝑛^𝑘+ 𝑚 (9)

Figure 10. A 2-level 3-factor FFD with 3 centre-point replicates. The design requires 11 experiments to determine the coefficients, b, of the main and interaction terms.

2^k factorial experiments = red circles, centre-point experiments = orange circles.

When processes have long reaction times and/or involve high value reagents it is important to keep the number of experiments required to a minimum. The coefficients of the main and interaction terms can be determined by conducting an initial FFD screening. When a factor is found to have no significant effect on the response, it can be excluded from any subsequent optimisation studies, thus reducing the overall number of experiments required. However, the number of experiments for the initial FFD increases exponentially with an increase in the number of factors. Hence, when the number of factors is high, the use of an FFD can be impractical.

In these cases, fractional factorial designs can be used, which confound main effects and/or interaction terms to reduce the number of experiments. This approach utilises the sparsity-of-effects principle, which states that a system is dominated by main effects and low-order interactions. Therefore, confounding

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high-order interactions with main effects will have a negligible effect on model accuracy. An important property of the design is the resolution, which defines the ability to separate main effects and low-order interactions. The extent of confounding for resolution III, IV and V designs are summarised in Table 2. The number of experiments required, and expression notation, for a fractional factorial design is given by [Eq (10)], where n = number of levels, k = number of factors and p = number of generators (design settings).^{40, 41}

𝑁 = 𝑛^𝑘−𝑝 (10)

Table 2. Extent of confounding for different resolution fractional factorial designs.

Resolution Extent of Confounding

III Main effects confounded with two-factor interactions.

IV Main effects not confounded with two-factor interactions;

two-factor interactions confounded with two-factor interactions.

V

Main effects not confounded with ≤ three-factor interactions;

two-factor interaction not confounded with two-factor interactions; three-factor interactions confounded with

two-factor interactions.

For example, a 2^4-1 fractional factorial design was used to optimise the reaction variables for the synthesis of active pharmaceutical ingredient (API) AZD0530 1.18 (Scheme 5).⁴² The variables studied were: equivalents of base (A), equivalents of water (B), temperature (C) and equivalents of alcohol 1.17 (D). The design was constructed using the generator I = ABCD, which implies D = ABC. A 3-factor FFD was produced for A, B and C, which was used to construct the levels of D based on D = ABC. This resulted in a resolution IV design, as indicated by the number of letters in the generator. Hence, each main effect was confounded with a three-factor interaction (A = BCD, B = ACD, C = ABD, D = ABC) and each two-factor interaction was confounded with another two-factor interaction (AB = CD, AC = BD, AD = BC). This enabled the main effects of the four factors to be studied in half the number of experiments compared to a 2⁴ FFD.

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Exp A B C D = ABC 1 +1 +1 +1 +1 2 -1 +1 +1 -1 3 +1 -1 +1 -1 4 -1 -1 +1 +1 5 +1 +1 -1 -1 6 -1 +1 -1 +1 7 +1 -1 -1 +1 8 -1 -1 -1 -1

9 0 0 0 0

10 0 0 0 0

Scheme 5. A 2-level fractional factorial design (resolution IV) with 2 centre-points used to optimise the reaction variables for the synthesis of AZD0530 1.18.

In some cases, the response surface may have a significant degree of curvature.

Therefore, accurate model predictions cannot be made using a factorial design, which assumes a linear relationship between each factor, X, and the response, Y. The inclusion of centre-point replicates provides a useful indication for the presence of curvature, as a bad fit would be observed if the relationship was non-linear. If curvature is detected, an optimisation design is required to characterise the square terms. 2-level full factorial designs can readily be extended to central composite optimisation designs by the inclusion of axial points, which are defined by [(±α, 0, 0), (0, ±α, 0), (0, 0, ±α)] for a 3-factor optimisation. The face centred composite (CCF, α = 1, Figure 11) design is best suited for the optimisation of chemical reactions, as the parameter limits are usually dictated by the experimental limits of the equipment, making α > 1 infeasible. The number of experiments required for a CCF design is given by [Eq (11)].³⁹ An alternative to central composite optimisation designs is the Box-Behnken, which is discussed in more detail in Chapter 3.

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𝑁 = 2^𝑘+ 2𝑘 + 𝑚 (11)

Figure 11. A 3-factor CCF design with 3 centre-points. The design requires 17 experiments to determine the coefficients, b, of the main, interaction and square terms. 2^k factorial experiments = red circles, centre-point experiments = orange circles, axial experiments = blue circles, optimum = ☆.