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

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PharmTech Europe

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

Q17: When developing a design space, one tends to increase batch size from small laboratory batches to medium and large production batches. Because production size batches are more expensive, how can one minimize the resources used to develop the final design space?

Table II2: Advantages and challenges in the use of laboratory-scale models.
A: A significant amount of resources are required to establish a detailed design space using large-scale batches. Using a laboratory scale model is beneficial to understanding the chemistry and effect of other factors that are scale-independent on the desired responses. Most of the effort to define ranges and evaluate factors should be done at the laboratory scale(s). One strategy is to reduce the number of factors and the number of batches as the batch size increases while maintaining the most important factors. Using laboratory scale models has certain advantages and challenges (see Table II2 ).

One challenge when using a laboratory-scale model is not being able to simulate the impact of scale-dependent factors. Some factors may be known to be dependent on batch size. In other cases, mechanistic knowledge may enable a scale-independent factor to be identified rather than perhaps a more obvious scale-dependent factor. Increasing the batch size from laboratory scale to a medium size may identify other scale-dependent factors that need to be studied at production scale.

There are a few options for addressing scale-up:
  • Identify the least desirable, yet acceptable, combination(s) of the factors that have been studied in the laboratory. The least desirable combination for each critical quality attribute (CQA) is determined by scientific knowledge and information obtained from laboratory data that is likely to give the least desired response. In addition, identify the operating target combination obtained from the laboratory scale model. Perform the worst-case combination(s), the operating target combination, and preferably the best-case combination(s) in duplicate at the large scale. If the combined experimental runs provide results that pass the specifications, then the ranges from the large scale can be used to establish the ranges of the design space for the factors.
  • Identify the scale-dependent factors and their potential ranges based on laboratory and medium-size batches. Perform a very small designed experiment with these factors (at two levels) with the other factors at their worst combination(s) so that any potential interaction of the scale-dependent and scale-independent factors can be studied at large scale.
  • Generate scale-dependent data on intermediate scales using approach (b) above and perform approach (a) on the actual production scale. This approach might be useful when material cost is expensive.
  • Augment a design space over time using information from batches that are made at large scale. The data should include deviation data and data from other small designed studies that may have been performed to gain more knowledge about the process in real time.
  • Use mechanistic models that can provide a good understanding of scale-dependent rates. These models allow one to change factor settings in large scale to ensure the desired response. Typical examples of this change include heat transfers and mass transfers in chemical reactions. Equipment and scale effects can be studied using rate kinetics studies in the laboratory or by performing in silico experiments on a mechanistic model. These can provide an optimum combination for large scale production.

Q18: Are there any other statistical principles to consider when designing the experiment that might reduce the number of batches required or improve identification and/or estimation of important factors and their effects?

A: There are several statistical principles that can be used to ensure that the important factors are identified and their effects are estimated without ambiguity, as outlined below.

  • Randomization refers to running experiments in such a way that the order in which the experimental runs are performed do not induce systematic bias in the results. To accomplish randomization, each run should be treated as though it is the first; that is, all factors should be reset before each run. Any divergence from complete randomness for convenience in running the experiment introduces a potential change into the analysis approach. Divergence from traditional randomization can often be economically desirable or physically necessary, but should be discussed with a statistician before running the experiment. For example:
  • Splitting the batch into sub-batches. In a DOE, there could be factors applied to the parent batch and other factors applied to the sub-batches. An example would be to manufacture tablet batches with different combinations of factors (e.g., mixing time, lubricant level, mixing speed) and then split the core tablets from each batch into sub-batches for coating, which would have its own set of factors (e.g., spray rate, pan speed). These types of designs are called split-plots and are analyzed in a different way than a factorial experiment because the error structure is more complicated.
  • Performing an experiment in a convenient manner. It may be convenient to make all of the high-temperature batches first and the low-temperature batches afterward. However, this arrangement can introduce a split-plot structure and other biases into the experiment and lead to a misleading conclusion.
  • Not planning the order of analytical (or other) testing. It is important to ensure that measurement sources of variability are not confounded with the effects being studied. Consider the following examples demonstrating the impact of analytical variability on the samples from the experimental phase. Suppose that six batches were manufactured in the experiment and sent to the analytical department for dissolution testing. Assume that six analytical tests are performed with six tablets from the same batch and all tested in the same analytical run. The batch results are now confounded with the analytical run. A better approach would be to perform six analytical runs with six tablets per run but within each run, use one tablet from each batch. This approach will separate the batch-to-batch variability from the analytical variability, thereby improving the comparison between batches. Analytical runs could also be confounded with factor effects. If samples generated from all of the high levels of a factor (e.g., temperature) in an experiment are tested in one analytical run and the low levels in another analytical run, it is possible that a significant difference between the high and low levels is not due to a temperature effect but rather may be due to day-to-day variation in the analytical method (i.e., the factor temperature is confounded with analytical days). Instead, one might randomly assign the samples from the experimental study to the different analytical runs.
  • Prior to running the experiment, the sampling strategy determines where and how many samples should be obtained to maximize information. Collecting samples throughout the execution of a given experimental run provides significant information that can be used to understand mechanisms across the run. Preliminary experiments are useful for developing the sampling strategy. This exercise also helps to identify what type of analytical methods would be beneficial for obtaining accurate and precise information about the process.
  • Replication refers to running one or more independent repeats of the same experimental condition or setting to provide reliable estimates and/or conclusions from the data. Replication should not be confused with repeated measurements of the same sample or measurements of multiple samples taken at the same experimental condition. To have adequate ability or power to distinguish factor effects and estimate with good precision the coefficients of the statistical model, it is essential to have sufficient replication of the design points in the study. Most of the time, the center design point of the statistical study is replicated. However, there may be situations in which replication of the extreme design points provide precision information across the statistical design. In general, the number and location of replicates depend on the size of the effects to be detected, the desired statistical power, and the range of the factors in the study.
  • Statistical thinking refers to recognizing that all work occurs in a series of interconnected processes. Before running a designed experiment, the process and potential sources of variability should be discussed, understood, and controlled as much as possible. Diligence in this manner will aid in assuring more accurate and precise data. The scientist should be well trained and knowledgeable about the process flow and the equipment. In this case, performing a preliminary experiment(s) can be helpful. Learning how to operate the equipment while conducting the experiment adds a confounding factor to the experiment, which may adversely affect the conclusions. In addition, because the analysis usually depends on analytical results, the methods should have adequate accuracy and precision before performing an experiment.

Analysis of DoE

This section provides an overview and interpretation of the analysis of DoE data beginning with the simple single response and single-factor case progressing to the single response with multiple-factor case, and concludes with multiple responses and multiple factors. Analysis of variance (ANOVA) along with partial least squares (PLS) analyses are discussed.

Q19: How do I analyze the data and define the design space for one response and one factor?

Figure 2: Categorical Factor X and continuous product attribute Y.
A: Analyzing an experiment to study the effect of a single material attribute or a process factor, X, on a single response, Y, varies based on whether X and Y are categorical or continuous and whether the relationship is described by a well-defined mechanistic model or an empirical model. Typically, a design space has more than one CQA and more than one factor, but this example is useful to understand more complex situations. Figure 22 shows an example where X is categorical. Figure 3 shows an example where X is continuous and the relationship between X and Y is described by a mechanistic model.

When X is categorical as shown in Figure 2, statistical intervals on the response are constructed for different values of the categorical factor. Figure 2 demonstrates a box–plot display of the response for each of the X values (A, B, C, D). Box plots summarize the data for each X value, showing the spread and center of the response at the value of X. The design space would include all the X values where the statistical limits on Y provide acceptable quality.

Figure 3: Arrhenius relationship.
When X is continuous, functional relationships are fitted between X and Y as shown in Figure 3. When the relationship between X and Y is not well-defined, or is too complicated, empirical models may be used to approximate the underlying relationship between X and Y. First-order and second order polynomial models are frequently used and have been found to be very useful in approximating the relationship between X and Y over limited ranges of X.

Figure 4: Empirical model of degradate (Y) on Factor B (Degradate = 0.72 + 0.57 * Factor B).
Figure 4 illustrates an example of an empirical model of Y on X. A linear regression equation is used to model how the level of a degradate changes with varying levels of Factor B. If the specification for degradate is not more than (NMT) 1.00%, then –1.00 to 0.49 may be defined as a design space for Factor B. This is shown as the green region in Figure 4. In this region, the predicted values from the regression equation meet specification. It should be noted that there is some uncertainty associated with the just defined area. Evaluation of the design space is discussed in Question 25, which will be covered in Part III of this series.

Q20: Once I have data from the designed experiment, how do I define a design space for one response and multiple factors?

A: The design space can be built using a statistical model that results from the DoE that has been executed. The particular model employed will vary depending on the design and will be subject to the appropriate caveats. For example, a screening design may produce a model with linear terms and may assume no curvature. An interaction design may assume no quadratic effects. An optimization experiment is an approximation to a more complicated response function. As described in Questions 8–18, a risk-based approach is used in determining the DoE structure and determines the analysis.

The modeling step of the analysis is often an iterative process where the statistically significant model terms that contribute to explaining the response are determined. The final fitted model can be used to generate interaction profiles and contour plots to help visualize and understand the effect of the factors on the response. An interaction profile shows how the response changes as one factor changes at given levels of another factor. A contour plot is a two-dimensional graph of two factors and the fitted response. Vertical and horizontal axes of the contour plots represent factors from the DoE while the lines on the contour plot connect points in the factor plane that have the same response value (y), thereby producing a surface similar to a topographic map. The contour lines show peaks and valleys for the quality characteristic over the region studied in the DoE. When there are more than two factors in the experiment, contour plots can be made for several levels of the other factors (see Figures 5 and 6).

Figure 5: Assay interaction profile depicting the interaction effect between Factors A and B on assay. The subgraph on the left shows how the assay (y-axis) changes from the low to high level of A (x-axis). The red line indicates the assay values for the low level of B (–1.41 coded units) and the blue line indicates assay values for the high level of B (1.41 coded units). The subgraph on the right shows levels of B on the x-axis and the low and high levels of A coded as red and blue lines. This subgraph provides insight into how assay is related to level of factor A and that this relationship is dependent on the level of factor B.
The same fitted equation,

Assay = 96.9+3.2* A-1.7* B+3.0* A* B* - 2.2 A2

Figure 6: Assay contour plot shows the expected assay responses for different combinations of Factors A and B. Red points are the experimental design points; note that although the axes extend from –1.5 to +1.5 coded units, the experimental factors only spanned -1.41 to +1.41 coded units. Color is used to indicate the value of assay with red representing lower values and blue indicating higher values.
that is used to draw Figures 5 and 6 can then be used to determine the factor ranges that would produce acceptable results; results that meet specifications. If there were no variability, Figure 7 offers an example of the region (in yellow) for Factors A and B where the assay values are predicted to be at least 95%. The levels of factors A and B are presented in coded/standardized form with -1 representing the low level of the factor and 1 representing the high level of the factor in the study. This coding allows the factors to be viewed on a common scale that is not dependent on units of measure. Because this DoE is a response surface design, the experiment includes star points that extend from the center of the design to the -1.41 and +1.41 levels.

Similar to Question 19, there can be variability and some degree of uncertainty associated with the edges of a design space region (see Figure 7). The use of a probability region to define a design space to protect against the uncertainty of meeting specification at the edges of the region described in Figure 7 is discussed in Question 26, which will appear in Part III of this series.


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