Statistical Considerations in Design Space Development (Part III of III) - Pharmaceutical Technology

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Statistical Considerations in Design Space Development (Part III of III)
In the final article of a three-part series, the authors discuss how to present a design space and evaluate its graphical representation.
 Sep 2, 2010 Pharmaceutical Technology Volume 34, Issue 9, pp. 52-61

 Figure 9: Response surface contour plot of assay and degradate 1 for Factors A and B. All figures are courtesy S.Altan et al.
A contour plot is a two-dimensional graph of two factors and the fitted model for the response**. A contour plot is defined by vertical and horizontal axes that represent factors from the DoE. The lines on the contour plot connect points in the factor plane that have the same value on the response producing a surface similar to a topographic map. The contour lines show peaks and valleys for the response over the region studied in the DoE. When there are more than two factors in the experiment, the contour plots can be made for several levels of the other factors. (Figure 9* shows the contour plots for the models presented in Table III). In Figure 9, the red points included on these plots are the experimental design points; note that the axes extend from -1.5 to +1.5 coded units although the experimental space is from -1.41 to +1.41 coded units.

 Figure 10: Three-dimensional surface plot for assay and degradate 1 for Factors A and B.
Another useful display of the design space is the three-dimensional surface plot (see Figure 10). Figure 10 shows a three-dimensional plot of Factors A and B, the assay response surface on the left, and Degradate 1 response surface on the right; note that the contour plot is projected at the bottom of the graphic. Note that three-dimensional plots are ideal for showing the process shape, however, contour plots are more useful for determining or displaying acceptable operating ranges for process parameters.

When there is more than one quality characteristic in the design space, the use of overlay plots is helpful. Question 21 and Figure 8 in Part II of this article series provide an example of an overlay plot (2).

Multiple response optimization techniques can also be used to construct a design space for multiple independent or nearly independent responses. Each response is modeled separately and the predictions from the models are used to create an index (called a desirability function) that indicates whether the responses are within their required bounds. This index is formed by creating functions for each response that indicate whether the response should be maximized, minimized, or be near a target value. The individual response functions are combined into an overall index usually using the geometric mean of the individual response functions.

 Figure 11: Desirability contours for assay and degradate 1 by factors A and B—multiple response optimization.
Figure 11 is a desirability contour plot in which both the assay and degradate 1 best meet their specifications. If the desirability index is near one, then both responses are well within their requirements. If the desirability is near or equal to zero, one or both responses are outside of their requirements. The most desirable simultaneous regions for these responses are the upper right and lower left. The desirability index, however, combines the responses into a single number that may hide some of the information that could be gained from looking at each response separately.

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