In the latest version "JMP 17" at the moment, as an option of "Prediction Profile" "Design Space Profile" function has been added.
This function "When modeling the relationship between Y (response) and X (factor), find the rectangular region of the factor that is within the specification limits of the response" It can be done, but what was the purpose of this function in the first place? What exactly is a design space? I will explain including questions such as.
"Design space" is a term used in drug development, and this is a function that is conscious of this. However, even for those who are not involved in drug development, we believe that the utility value is high.
What is design space
In drug development, the concept of QbD (Quality by Design) is advocated. In order to ensure the quality of pharmaceuticals, the idea is to properly understand and respond to risks at the design stage. It is described as
design space :
A multidimensional combination and interaction of input variables (e.g., raw material properties) and process parameters proven to ensure quality. Operating within this design space is not considered a change.
In addition, the guidelines show an example of design space presentation as shown below.
Source: ICH-Q8 Guidelines for Drug Product Development
In the granulation process, a response surface model is applied to the factors that affect the dissolution rate of tablets (the amount of additives, the amount of water), and the contour line for the predicted value of the dissolution rate is displayed. The area where the dissolution rate is 80% or more is a curved line, and a rectangular area within this area is cut out as a design space.
Use case
Consider the following example with two responses and two factors.
Response (Y)
Dissolution (%) : Desirable dissolution rate of 80% or more
Uniformity of Content (RSD%) : content uniformity (relative standard error %), preferably 4% or less
Factor (X)
PSD d90(μm): 90% point of particle size distribution (10-30)
Disintegrant(%): Percentage of disintegrant (1 to 3)
Perform an experiment based on the design of experiments, fit a response surface model to each Y, Areas that meet the Y specification range (white areas) are indicated by contour profiles.
The design space will be set based on the management strategy and the practical aspects. Since there is an advantage that operating within the space is not regarded as a change, we would like to set the space as wide as possible.
If I were to cut out the marginal areas, the following green areas would look good.
Note, however, that the contours obtained here are plotted against the predicted values produced by the estimated model. Forecasts have forecast errors. Therefore, there is a way of thinking that it is dangerous to take out the last-minute area, and I myself have been asked about this point many times by customers.
As a countermeasure, there are methods such as obtaining a confidence interval for the mean (or prediction) and constructing a region based on the lower and upper limits, but let's introduce the "design space profile" here.
Use of “Design Space Profile”
A design space profile first specifies the LSL (lower specification limit) and USL (upper specification limit) of the response. You can then set the lower and upper limits of the factors graphically or enter them in the input boxes and consider the output fraction within specification and volume fraction.
Percent within specification : the probability that the response is within the specification limits given the limit settings for the current factor
volume ratio : The relative proportion of the area ("PSD d90" 13 to 30, and "Disintegrant" to 2.65 to 3, the area narrows to about 14.4% of the first factor area)
This example shows that there is a 90.9% chance that Y is within the specification range.
I want to raise the probability a little more, so let's set the lower part of Disintegrant to 2.8 and narrow the range. The graph on the left shows how the ratio within the specification changes when the specification limit is set for each factor.
Since the area is narrowed, the volume ratio is as small as 8%, but the ratio within the specification has increased to 95%. Contour lines of this area are shown below.
It's not a borderline area, but it looks like a safe area, a so-called conservative area, considering the prediction error. Using the "design space profile" like this conservative design space can be constructed.
How do you determine the probability of being within specifications?
The in-specification percentage is determined by Monte Carlo simulation. Therefore, it can be obtained even when modeling with a complicated model such as a neural network.
Main steps of simulation
- Generate a large number of uniformly distributed random numbers (10,000 by default) in the domain of the factor. Of these, consider the specification limit points of the set factor (bottom left figure: group of points).
- For each green point, plug the value into the resulting response surface equation and add the standard deviation of the error of the model (RMSE) as random error to get the predicted value of Y.
- Count the number of predicted values of Y that fall within the specification limits and calculate the percentage of all points. In the lower right figure, the green points are within the Y specification limits and the red points are not.
In JMP, there is an option called "Simulator" in the prediction profile, which generates random numbers within the range of the determined factor and calculates the percentage of Y that does not fall within the specification limits (non-conformance rate). On the other hand, the design space profile is the opposite, I'm looking for a percentage that fits.
It's okay to have many factors
So far, we have dealt with two-factor examples for ease of explanation, but of course Cases with three or more factors can also be handled. In the figure below, in addition to the previous two factors ("PSD D90" and "Disintegrant"), the specific surface area ("Specific Surface Area") is also added as a factor, and it is a design space profile when a response surface model is fitted with three factors. This function is useful because it is difficult to visualize a rectangular area with 2D contour lines for more than 3 factors.
In the case of three factors, the predicted value of Y is obtained by generating uniform random numbers in the three-dimensional factor domain as shown in the lower left figure.
by Naohiro Masukawa (JMP Japan)
