A Better Strategy
Figure 4

First, we must quantitatively understand the core relationship illustrated in Figure 3. This is similar to how the commonly
used strategy begins. The first step is to run experiments for a small number of points along the range of the core response
relationship, in this case pH and the quality attribute (QA) in the expected target setting of the other process parameters.
Figure 4 shows this relationship. Knowing the nature of the core relationship, we can create the experimental design space
in the region of interest.
Figure 5

In this case, the region of interest is where the derivative is changing, i.e., the "knee," where the curve is bending over.
When the knee of the curve has been found, a designed experiment is set up in the region of interest to explore the effects
and interactions of the factors of interest, as illustrated in Figure 4. The workhorse central composite design is centered
at the knee. From a relatively small number of experiments we can gain a good quantitative calibration of the region of interest
and can establish a reasonable design space (Figure 5).
We are now ready to establish the acceptable region of process operation. It has been found that a logit function models the
relationship between pH and the QA very well. Knowing the nonlinear model form, we can determine the optimal design points
and the most precise estimate of the acceptance region using the following equation:
in which A is the asymptotic maximum, b is the point of inflection where the quality attribute (QA) is equal to A/2, and d is the steepness of the curve close to the inflection point.
The optimal design points to fit a logit model are known based on point of inflection and range. These are shown in Figure
6 for two different situations.
Figure 6

Different products have different points of inflection. Working from these points, a sensible design is found that takes into
account the features of the nonlinear model and provides minimum error of prediction in the region of interest.
Generalize Across Products
Knowledge of the general relationship or model can be used to learn more efficiently with new products of the same type. Using
historical data and data from some small studies, we determined that many products share the same rate of decline (parameter
d in the logit model) but differ by the point of inflection (half point b = (A/2)). Identifying the location for different products is equivalent to sliding the curve along the pH axis.
Knowing that the general model holds for most new products allows the quick identification of an optimal experimental design
scheme for new products after running only a few preliminary experimental points. Thus, the knowledge allows us to obtain
new knowledge more quickly, thus reducing development time.
Automate the Design and Analysis
After we have the design procedure worked out, the speed can be accelerated further by automating and validating design and
analysis software. An intranetbased tool is being developed to produce an optimal design series. The tool performs analysis
with standard polynomial and nonlinear model fits. This feature will allow those not skilled in the details of the method
to make effective designs and analyses output.
A focused and riskbased control scheme also is being developed using the knowledge gained and will be used to develop the
design space.
