1. A Bayesian Approach to the ICH Q8 Definition of Design Space
2008 Graybill Conference, Fort Collins, Co.
June 11th-13th, 2008
John J. Peterson
Senior Director, Research Statistics Unit
GlaxoSmithKline Pharmaceuticals
Graphically accessible
Historical information (hard or soft) can be used

2. ICH Q8 Definition of Design Space
The ICH Q8 FDA Guidance for Industry defines "Design Space" as:
"The multidimensional combination and interaction of input variables
(e.g. material attributes) and process parameters that have been
demonstrated to provide assurance of quality.“
Further more….
“Working within the Design Space is not considered as a change. Movement out of the Design Space is considered to be a change and would normally initiate a post regulatory approval change process. Design Space is proposed by the applicant and is subject to regulatory assessment and approval”.

3. ICH Q8 Definition of Design Space
The ICH Q8 FDA Guidance for Industry defines "Design Space" as:
"The multidimensional combination and interaction of input variables
(e.g. material attributes) and process parameters that have been
demonstrated to provide assurance of quality.“

4. ICH Q8 Definition of Design Space
The ICH Q8 FDA Guidance for Industry defines "Design Space" as:
"The multidimensional combination and interaction of input variables
(e.g. material attributes) and process parameters that have been
demonstrated to provide assurance of quality.“
Three key concepts:
1. Measurement For example: controllable factors, input material attributes, in-process measurements, quality response measurements.

5. ICH Q8 Definition of Design Space
The ICH Q8 FDA Guidance for Industry defines "Design Space" as:
"The multidimensional combination and interaction of input variables
(e.g. material attributes) and process parameters that have been
demonstrated to provide assurance of quality.“
Three key concepts:
1. Measurement For example: controllable factors, input material attributes, in-process measurements, quality response measurements.
2. Prediction - Models to relate the measurements to the relevant quality responses. These need to be compared to specifications for quality.
- Need to be able to predict means AND variances of responses.

6. ICH Q8 Definition of Design Space
The ICH Q8 FDA Guidance for Industry defines "Design Space" as:
"The multidimensional combination and interaction of input variables
(e.g. material attributes) and process parameters that have been
demonstrated to provide assurance of quality.“
Three key concepts:
1. Measurement For example: controllable factors, input material attributes, in-process measurements, quality response measurements.
2. Prediction - Models to relate the predictive measurements to the quality responses. These need to be compared to specifications for quality.
- Need to be able to predict means AND variances of quality responses.
3. Reliability To quantify “How much assurance?”
The QbD-oriented guidance (PAT, ICH Q8, Q9, Q10, etc) is inundated with the words “risk” and “risk-based”.)
See presentation by H. Gregg Claycamp (CDER), “Room for Probability in ICH Q9”

7. Measurements In-process measurements (Z1, Z2, …) heat transfer, NIR
Input material measurements (W1, W2, …)
A Generic Process (or Unit Operation)
In-process measurements
(Z1, Z2, …) heat transfer, NIR
Control Factors/Parameters (F1, F2, …)
responses (Y1, Y2,…)
Responses have specification limits which define Quality: { AiL < Yi < AiU }
Vector of predictive variables, x = (f,w,z)

8. Prediction Models x = (x1,…xk) vector of predictive variables
The Standard Multivariate Regression Model
The Seemingly Unrelated
Regressions model
Other models?
e.g. Nonlinear, PLS, Wavelets, etc.
x = (x1,…xk) vector of predictive variables
r = no. of response types.
Fitted models give us predicted responses, i.e but we need to know the variances of the also to assess risk.

9. Reliability Model
How much assurance do we have of meeting specifications?
Consider
If we knew we could define a Design Space as
for some reliability level R.
From a Bayesian perspective one could consider the posterior expectation: to obtain the Bayesian Design Space:

10. Aside….what is a posterior predictive distribution?
A posterior predictive distribution is used to compute
If is the pdf for Y , then g(y | x, data) is the posterior predictive pdf with
where is the posterior distribution of b and S. So

11. In most situations, Markov Chain Monte Carlo techniques will be used to compute
MCMC
,……………....,

12. Design Space
From a Bayesian perspective one could consider the posterior expectation:
Computationally, is straightforward to compute using MCMC.
Experiments with multiple batches, split plots, missing data, noise variables and even heavy-tailed residual distributions can be handled with MCMC.
In theory it is also possible to handle latent variable models that may be needed for “functional data” from in-process measurements (e.g. Bayesian PLS, Wavelets, etc.)
The classical multiple response surface approaches found in Design Expert, JMP, Statistica, etc. fall short of providing a good reliability models!

13. ICH Q8 Annex: Design space can be determined from the common region of successful operating ranges for multiple CQA’s. The relations of two CQA’s, i.e., friability and dissolution, to two parameters are shown in Figures 2a and 2b. Figure 2c shows the overlap of these regions and the maximum ranges of the potential design space.
What do these contours represent? Mean
response surfaces?
Taken from the ICH Q8 Annex.
(August 2007)
This overlay plot does not quantify “How much assurance?”!

14. Taken from PQLI* Design Space
by Lepore and Spavins (J. of Pharm. Innovation, 2008)
*PQLI = Pharmaceutical Quality Lifecycle Implementation
What do these contours represent? Mean
response surfaces?
(The paper does not say.)
This overlay plot does not quantify “How much assurance?”!

15. Overlapping Means vs. Bayesian Reliability Approach to Design Space:
An Example – due to Greg Stockdale, GSK.
Example: An intermediate stage of a multi-stage route of manufacture for an Active Pharmaceutical Ingredient (API).
Measurements:
Four controllable quality factors (x’s) were used in a designed experiment.
(x1=‘catalyst’, x2= ‘temperature’, x3=‘pressure’, x4=‘run time’.)
A (face centered) Central Composite Design (CCD) was employed. (It was a Full Factorial (30 runs), with no aliasing.)
Four quality-related response variables, Y ’s, were measured. (These were three side products and purity measure for the final API.) Y1= ‘Starting material Isomer’, Y2=‘Product Isomer’, Y3=‘Impurity #1 Level’, Y4=‘Overall Purity measure’
Quality Specification limits: Y1<=0.15%, Y2<=2%, Y3<=3.5%, Y4>=95%. Multidimensional Acceptance region,

16. Overlapping Means vs. Bayesian Reliability Approach to Design Space:
An Example – due to Greg Stockdale, GSK.
Model Terms
Response x1 x2 x3 x4
x11
x22
x33
x44
x12
x13
x14
x23
x24
x34
SM Isomer D
Prod Isomer
Impurity
Purity
Prediction Models:
Temperature = x1 Pressure = x2 Catalyst Amount = x3 Reaction time = x4

17. An Overlapping means approach to Design Space for an Active Pharmaceutical Ingredient (API)
The Design Space is the “Sweet Spot” Highlighted in Yellow below
The so-called “sweet spot”
highlighted in yellow

18. Why is the “sweet spot” not so sweet?
If the mean of Y at a point x is less than an upper bound, u, then all that guarantees is that
(For there is no guarantee that
Suppose If Y1 and Y2 were independent, then all that is guaranteed is that For k independent Yi’s the situation becomes:
If Y1 and Y2 are positively correlated then it may be easier to find x-points to make large. Likewise, if Y1 and Y2 are negatively correlated (for each x) then it may be more difficult. Note: Corr(Y3, Y4 | x) is about -0.8 for the API experiment.

19. Why is a multivariate reliability approach needed?
(Accounting for correlation among the responses…a simple example)
Suppose we have a process with four key responses, Y1, Y2, Y3, Y4
For simplicity, let’s assume that
Let
Consider If S = I , then
But if then
and if

20. Overlapping Means vs. Bayesian Reliability Approach to Design Space:
An Example
Overlapping Mean Response Surface Approach –
Can be computed using SAS/JMP, Design Expert, Minitab, etc.
The “sweet spot” region is determined by the overlapping mean response surfaces that are all simultaneously within their specification limits.
However the overlapping mean response approach: (i) Does not take into account the model parameter uncertainty (ii) Does not provide a measure of assurance to say “How likely it is that future responses will meet their specifications.” (iii) Does not take into account the correlation structure of the multivariate distribution of future responses.
Thus the overlapping means approach does not address the question begged by the ICH Q8 definition of Design Space…namely, “How much assurance do we have of meeting process quality specifications?”

21. Overlapping Means vs. Bayesian Reliability Approach to Design Space:
An Example
A Bayesian Reliability Response Surface Approach –
A posterior predictive approach: (i) Takes into account the model parameter uncertainty (ii) Provides a measure of assurance to say “How likely it is that future responses will meet their specifications.” (iii) Takes into account the correlation structure of the multivariate distribution of future responses.
Thus the Posterior predictive approach addresses the question begged by the ICH Q8 definition of Design Space…namely, “How much assurance do we have of meeting process quality specifications?”
Both graphical and tabular approaches can be used with the posterior predictive approach to better understand the resulting Design Space.

22. A Bayesian Reliability Approach to Design Space:
API Example
A posterior Predictive Response Surface Approach –
How likely is it that a future multivariate response will meet specifications for a factor configuration in the sweet spot?
Consider the posterior predictive probability p(x)=Pr (Y is in A | x, data).
Here, Y is assumed to have a multivariate normal distribution. A is the multidimensional acceptance region. The standard noninformative prior for b and S is used, , where r = 4, the number of response types.
Pr (Y is in A | x, data) is computed using Gibbs Sampling, one of the Markov Chain Monte Carlo (MCMC) simulation methods.
The largest probability of meeting specifications is only about 0.75.
- This is corresponds to the best p(x) value within the yellow “sweet spot” of overlapping mean response surfaces. The worst p(x) value in the “sweet spot” is only 0.23 !

23. Optimal Reaction Conditions
Design Space Table of Computed Reliabilities1 for the API (sorted by joint probability2)
Note that the largest probability of meeting specifications is only about 0.75
Temp
Pressure
Catalyst
Rxntime
Joint
Prob SM
Isomer
Prod
Impurity
Purity 35 60 6 3
0.752 1
0.9985
0.8435
0.79
32.5 7
0.743
0.9995
0.7875
0.8295
37.5
0.7375
0.7855
0.8255
6.5
0.737
0.9975
0.821
0.7845 30
7.5
0.7335
0.7775
0.8175
0.725
0.7485
0.845
0.7225
0.77
0.812
0.7195
0.9955
0.864
0.7415
0.717
0.999
0.8075
0.759
0.716
0.734
0.859
5.5
0.7145
0.993
0.8065
0.7565
0.712
0.731
0.8555
Optimal Reaction Conditions
[1] This is only a small portion of the Monte Carlo output.
[2] values were computed using SAS IML
Marginal Probabilities

24. Overlapping Mean Contours from analysis of each response individually.
This x-point (in the yellow sweet spot)
has only a probability of
But this x-point (in the yellow sweet spot)
has a probability of only 0.23 !
Posterior Predicted Reliability with
Temp=20 to 70, Catalyst=2 to 12, Pressure=60, Rxntime=3.0
Rxntime
Pressure 70
0.7
0.6 60
= Design Space
0.5
Contour plot
of p(x) equal to
Prob (Y is in A
given x & data).
The region inside the
red ellipse is the
design space. 50
0.4
x2=
Temp
0.3 40
0.2 30
0.1
0.0 20 2 4 6 8 10 12
x1=
Catalyst

25. Greg Stockdale

26. A Question for the Audience
Question…. “How large should R be to calibrate the Design Space:
Note: “Based upon some historical precedents (e.g. three out of three successful manufacturing validation runs), some deductions about a value of R can be made”
“Three consecutive successful batches has become the
de facto industry practice, although this number is
not specified in the FDA guidance documents.”
Schneider, Huhn, and Cini, (2006), PAT Insider Magazine, April issue.

27. “How large should R be to calibrate the Design Space?”
Suppose that Z ~ Bernoulli(p) and that p has a beta prior, Beta(a,b).
Consider the likelihood based on 3Bernoulli trials,
Consider
Let Then p has a beta posterior distribution Beta(s+a, 3-s+b).
The posterior predictive distribution of new Z is beta-binomial and
For a uniform prior on [0,1], i.e. Beta(1,1), and s=3 (out of 3 trials) we get:
So is R=0.8 a reasonable value with which to calibrate a Design Space?

28. “How large should R be to calibrate the Design Space?”
If R=0.8 is not large enough (e.g. we want R=0.95, say), but some manufacturing processes have been approved, based upon 3 out of 3 successfully manufactured batches, what does this mean?
Clearly, some prior information must have been utilized.
Consider a beta prior Beta(a,b) with a=16, b=1. Then for 3 out of 3 successes:
What does a beta prior distribution with a=16, b=1 look like?
The 5th percentile is 0.83

29. “How large should R be to calibrate the Design Space?”
Based on historical precedent….
This deduction implies that either: R=0.8 is an acceptable de facto lower bound for calculating a Design Space. OR
Strong prior information should be allowed in the
calibration of a Design Space.

30. The Flexibility Offered by a Bayesian Approach to Design Space
A “pre-posterior analysis” can be performed to identify where additional information may be needed to improve design space calibration (by reducing model parameter uncertainty).
Noise variables are easily incorporated so that “robust parameter design” optimization can be done. (This is important for multistage processes.)
Can accommodate small amounts of missing data in a straightforward fashion.
The Bayesian approach can handle mixed-effect models in a straightforward manner. This has useful applications for split-plot designs and (random) batch effects.
The Bayesian approach can also be adapted to nonlinear mechanistic models.

31. Challenges to Constructing the ICH Q8 Design Space
Summary
The challenges….my opinion….
Getting clients to recognize the key elements of MPR: Measurements, Prediction model, Reliability Model,
particularly the importance of a reliability model to quantify “How much assurance?”
Computational issues. These can be solved with sufficient effort. The Bayesian approach provides a unifying paradigm.

32. From the ICH Q8 Annex: (my highlights in red)
An enhanced quality by design approach to product development would additionally include the following elements:
• A systematic evaluation, understanding and refining of the formulation and
manufacturing process, including:
Identifying, through e.g., prior knowledge, experimentation, and risk
assessment, the material attributes and process parameters that can
have an effect on product CQAs;
Determining the functional relationships that link material attributes
and process parameters to product CQAs.
• Using the enhanced process understanding in combination with quality risk
management to establish an appropriate control strategy which can, for
example, include a proposal for design space(s) and/or real-time release.
The Bayesian approach can address the concerns of ICH Q8 in a coherent, unifying manner.

33. Acknowledgements Gregory Stockdale Aili Cheng Tim Schofield
Paul McAllister
Michael Denham
Gillian Amphlett
Mohammad Yahyah
Kevin Lief
Val Fedorov
Darryl Downing

34. References
Claycamp, H. G. (2008), “Room for Probability in ICH Q9: Quality Risk Management“, presented at the Pharmaceutical Statistics 2008: Confronting Controversy conference., March 2008, Arlington, VA. (sponsored by the Institute of Validation Technology).
ICH Q8 (2006), “Guidance for Industry Q8 Pharmaceutical Development”.
ICH Q8 (2007), “Pharmaceutical Development Annex to Q8”
Miró-Quesada, G., del Castillo, E., and Peterson, J. J. (2004), “A Bayesian Approach to for Multiple Response Surface Optimization with Noise Variables”, Journal of Applied Statistics, 31,
Peterson, J. J. (2004), “A Posterior Approach to Multiple Response Surface Optimization, Journal of Quality Technology,
Peterson, J. J. (2007) “A Bayesian Approach to the ICH Q8 Definition of Design Space”. Proceedings of The American Statistical Association, Biopharmaceutical Section. (Also to appear in the Journal of Biopharmaceutical Statistics in fall 2008.)
Peterson, J. J. (2008). “A Bayesian Reliability Approach to Multiple Response Surface Optimization with Seemingly Unrelated Regressions Models”, Quality Technology and Quantitative Management, (to appear).
Stockdale, G. and Chen, A. (2008), “Finding Design Space and Reliable Operating Region using a Multivariate Bayesian Approach with Experimental Design”, Quality Technology and Quantitative Management, (to appear).
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