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

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Statistical Considerations in Design Space Development (Part II of III)
The authors discuss the statistical tools used in experimental planning and stategy and how to evaluate the resulting design space and its graphical representation.


Pharmaceutical Technology
Volume 34, Issue 8, pp. 52-60

Q10: What is a screening experiment?

A: A screening experiment provides broad but not deep information, and generally requires fewer experimental runs than an interaction or optimization experiment. In general, a screening DOE is more efficient than changing one factor at a time. There may be some interest in studying the interactions between factors although this is usually not the focus. Typically, performing a risk assessment to identify potential factors for a screening study is recommended.

Screening experiments are often used to identify which factors have the largest effect on the responses1 . These few significant factors can then be studied in a subsequent, more comprehensive DOE. Another use for screening experiments is to demonstrate that factors have no practical change in the responses across the range studied.

Designs used in these experiments generally include fractionated or highly fractionated factorial experiments with each factor at two levels (e.g., Plackett–Burman designs). Center points (i.e., each factor is set at the center of their range) are often added to measure replication variability and to perform an overall test of curvature over the experimental region.

Boundaries of the design space can be defined using a screening design; however, complete process understanding may not be achieved by studying main effects, thus requiring an interaction or optimization design. In some cases, the risk/benefit may not be great enough to proceed beyond a screening design.

Q11: What is an interaction experiment?

A: As underlying scientific phenomena become more complex, so too does the underlying mathematical model. The goal is to study interactions between factors along with the main effects. An interaction between two factors means that the effect that one factor has on the response depends on the level of the other factor. For example, the effect of mixing time on a response may depend on the mixing speed. Interaction designs generally consist of factorial or fractional-factorial experimental designs to assess factors at two levels along with a replicated center point. There are other designs that can evaluate interactions (e.g., unbalanced designs produced using D-Optimal algorithms).

The advantage of the factorial family of designs, including screening and interaction designs, is that additional points can be added to the design to obtain more information about the interactions in a sequential fashion. The analysis is fairly easy and can be performed by most statistical software packages. Some designs (e.g., Plackett–Burman) appear similar to factorial experiments but do not have the same properties and cannot be easily added on to in a sequential fashion.

Boundaries of the design space can be defined using an interaction design. Complete process understanding, however, may not be achieved by only studying main effects and interactions. In this case, an optimization design may be necessary.

Q12: What is an optimization experiment?

A: Again, as the underlying scientific phenomena become more complex so too does the underlying mathematical model. Optimization experiments are used to map a response surface to understand the intricacies of factors and their interactions, as well as nonlinearities and their combined effect on the responses of interest. These experiments can be used to find the combination of factors that predict the optimum response (a maximum or minimum), to find a region of possible factor combinations that predict acceptable results, or to predict process performance in the region studied. Optimization experiments are generally larger, requiring three or more levels to estimate quadratic or higher order terms. In addition, more factor levels may be required to capture the response more accurately, especially in cases where the factor ranges are large. Common designs used for optimization experiments are central composite, Box–Behnken, three-level factorials, or mixture designs. Analysis of designs used in optimization experiments can be more complicated than those used in screening experiments.

Q13: What are the differences among screening, interaction, and optimization experiments, and why choose one over the other?

A: The design type is selected based on experimental goals, number of factors, timing, and resources. Planning is a crucial part of experimental design. As discussed, screening designs are usually used to select a few significant factors out of many for more intense future study or to support a decision about robustness for the combination of factors under study. Interaction experiments usually include fewer factors than screening designs and result in more information about the factors and associated interactions. Optimization designs are used to obtain a complete picture of the factors effects plus the interactions plus any curvature or quadratic effect.

Factorial designs encourage sequential experimentation. For instance, one could start with a screening design, choose to create a new screening design with different factor ranges, or augment with additional runs an interaction design. If needed, augment an optimization design. If the timing of results will not permit this increased experimental time, then one larger experiment may be more appropriate.

Screening experiments should be performed before running an optimization experiment. It is possible that a response surface is not needed. If the goal is to show that acceptable results can be obtained over a defined experimental region, then a screening or interaction experiment may be all that is necessary. A common mistake is to run an optimization experiment too early in the process and find out that the ranges were not well chosen, thus resulting in wasted resources. One economical approach is to run a screening or interaction design, and if the results are reasonable, to then add points to the design to allow fitting a response surface.

There are pros and cons to performing studies in a sequential manner. Performing pre-experimental runs can provide a good start on the potential ranges to consider for the studies. Performing center points during screening could provide information not only on the replication error, but also on the presence of significant curvature in the experimental region. Although center points can identify curvature, they cannot by themselves, identify which factor(s) contribute to the curvature. If the center points do not provide evidence of curvature, the risk/benefit is unlikely to be great enough to proceed to an optimization experiment. However, if there is curvature, then the risk/benefit based on a priori knowledge and the magnitude/direction of the curvature should be used to determine whether to develop a response surface model. In general, if there is practically important curvature, either scientific knowledge can be used to help model the curvature, or an optimization experiment should be performed. The risk/benefit should be considered when two-level screening or interaction experiments with center points are used to define the design space. In any case, if there are unacceptable experimental results among the factorial points, additional runs may be required to confirm the model predictions of acceptable performance. If designs are done in stages, at least one point should be replicated (often the center points) at each stage. Such common points allow a comparison of the two sets of data to ensure that changes between stages are within experimental error.

Q14: If I conduct several small experiments, can they be analyzed together to produce a design space?

A: During the development process, a scientist may run several experimental designs. The information obtained from one design provides knowledge that helps determine the next design. Results from each design may be analyzed separately, or results may be combined and analyzed together. However, the results could be misleading if the analysis is not performed properly given the structure of the studies. In the planning stage of each experiment, the combined set of factor levels across the designs should be considered, so that when combined, a meaningful response surface model can be constructed. In many cases, a sequence of designs has the same factors (or a subset), but the factor levels have changed. Ideally, there should be common points from one design to the next to make sure that there is not a shift in the results, as well as common factors with similar levels between the experimental designs. In addition, if there are differences between the designs such as scale, these should be modeled appropriately during the analyses in order to obtain meaningful and accurate conclusions. Although it is generally advantageous in terms of power (i.e., the ability to detect an effect) and information to combine data from various experimental studies, additional planning of the analysis approach is required to guarantee success. When combining data from various experimental studies, a more complex analysis may be required. Also, caution should be used when extrapolating outside the range of the experimental space.

Q15: In a multiple step process, should I use a separate design for each step or a design that includes factors from multiple steps to see interactions between steps?

A: Most manufacturing processes have multiple steps (e.g., blend, mill, blend, compress or dissolve, sonicate). Each step may have several factors that affect responses either during that step or a subsequent step. If one factor in one step interacts with a factor from another step, it is better to include factors from multiple steps in the same design so that the between-step interaction can be evaluated. Suppose that temperature is a factor in one step, and there is a speed factor in the next step. If the effect of speed in the second step depends on the temperature in the previous step, then the design should include both steps. If, however, the effect in the second step does not depend on the level of each factor in the first step, then each step can have its own design. The existence of cross-step interactions should be considered during the risk assessment or the planning stage; cross-step-versus-single-step experimental designs may be beneficial. Another approach is to use a separate design for each step but to use the most important factors in a design across multiple steps. Sometimes, response at a step can be used as a factor in the subsequent step (see Question 16).

Q16: Can the experimental design used in a later step use a response, rather than factors, in an earlier step?

A: As discussed in Question 15, experiments can be performed at each step separately or they can incorporate several steps in the same experiment. If experiments are performed at each unit operation or step separately, there may be a response at an early stage that can be used as a factor in a later stage. Using the response from an earlier step as the control of a later response is desirable. This property could allow equipment interchangeability. The assumption is that the response in the earlier step used as the control contains all of the information required to predict the second response. At a minimum, the experiment performed in the subsequent step should have the response from the first step as a factor. For example:

  • In a roller-compaction formulation for tablets, ribbon thickness (and scientific understanding of ribbon density) from the compactor may depend on several factors in the roller-compaction process. If dissolution is an important response for the final tablets, the experimenter may want to control the ribbon thickness to ensure acceptable dissolution rather than controlling the factors in the compaction step that affect the ribbon thickness. Ribbons of different thickness (e.g., thin, middle, thick) should be generated and used as a factor in the design for the next stage. One should be able to demonstrate that, after removing the effect of ribbon thickness, the other factors used in making the ribbons no longer have an effect on the response at the later stage.
  • When developing an active pharmaceutical ingredient, if the focus is to understand impurity rejection at a subsequent step, then one could use the impurity at a previous step as a "factor" while making sure that the range is challenged. If one is not able to spike the input material for the current step with the impurity(ies) of interest, one would make a batch with the highest possible impurity level from the previous step as a level in the subsequent step.


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