Bayesian Approach to Specification of Design Space · 2010-05-18 · Quality Target Product-Profile...
Transcript of Bayesian Approach to Specification of Design Space · 2010-05-18 · Quality Target Product-Profile...
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Bayesian Approach to Specification of Design Space
Paul van Eikeren and Corey Dow-Hygelund
userR! 2009 (July 8-10, 2009)
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Quality by Design: Pharmaceutical Quality Vision
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inferencefrom Blue ReferencePharmaceutical Quality Vision
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Develop a harmonized pharmaceutical quality system applicable across the
lifecycle of the product emphasizing an integrated approach to quality
risk management and science.
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inferencefrom Blue ReferencePharmaceutical Development
QbD Pharmaceutical Development Quality Target Product-Profile
Critical Quality Attributes (CQA)
Risk Assessment: link to Product CQA
Design Space
What is Design Space “The multidimensional combination and
interaction of input variables (e.g., material attributes) and process parameters that have been demonstrated to provide assurance of quality.”
“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 regulatory post-approval change process.”
“Design space is proposed by the applicant and is subject to regulatory assessment and approval.”
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inferencefrom Blue ReferenceElements of Design Space
Risk based approach to assurance of quality
Critical to quality attributes (CQA) are multivariate
Quantitative model links input attributes and process parameters to CQAs
Model quantifies risk of not meeting CQAs
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parameters
processdrug
product
attributes
CQA_1
CQA_2
CQA_3
...
inputs
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inferencefrom Blue ReferenceQ8(R1) Example Design Space
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Q1. How do we enable a risk-based approach to design space?
Q2. How do we best construct the design space?
Q3. How do we assess the reliability (assurance) of the design space?
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Risk-based approach to Design Space
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inferencefrom Blue ReferenceDesign Space Strategy
Multivariate Strategy:
multidimensional combination and interaction of material and process variables linked to responses
responses comprised of drug product critical-to-quality attributes (CQA)
drug products involve a multitude of specifications (response factors)
Overcome Limits of Conventional Design Space:
model: predictive response surface
model parameters: static variables
multiple responses: overlap of mean-response surfaces
Apply Bayesian Design Space:
model: posterior predictive probability function
model parameters: random variables
multiple responses: multivariate joint probability function
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inferencefrom Blue ReferenceIssue with Conventional Design Space
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J. Lepore and J. Spavins, “PQLI Design Space,” J. Pharmaceutical Innovation (2008) 3:79-87
Is the risk of failing to meet all four specifications the same throughout thedesign space?
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inferencefrom Blue ReferenceWhy Multivariate & Bayesian?
Multivariate because
Quality entails multiple responses (CQA defined by specifications)
Multiple responses are not independent
Correlation between responses can have a big effect on probability of meeting all specifications
Bayesian because conventional
fails to account for uncertainty in model parameters
fails to account for correlation structure in data
fails to provide a metric for “assurance of quality”
fails to enable use of prior information
fails to enable adaptive experimental design
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Illustrative Example: Process for Coating Polymer
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inferencefrom Blue ReferenceMultivariate Chemical Process
Chemical Process
Production of a coating polymer
Two critical to quality attributes (CQA)
Response 1: Percent conversion
Response 2: Polymer viscosity
Two critical process parameters
Parameter 1: reaction time
Parameter 2: reaction temperature
Product profile: target CQA
Percent conversion: > 68%
Viscosity: 40 ± 2 mPA-secs
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inferencefrom Blue ReferenceBayesian Response Surface Model
Experimental Design
Central composite design
8 setting combinations (4 corners and 4 faces of a square)
5 centerpoints
Models
Response surface models for conversion and viscosity
linear, cross-product and quadratic terms for each response
Bayesian Regression and Analysis in R (R-project)
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inferencefrom Blue ReferenceMCMC Bayesian Regression
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2000 6000 10000
1.0
1.2
1.4
1.6
1.8
Iterations
Trace of var1
1.0 1.2 1.4 1.6 1.8 2.0
01
23
45
6
N = 10000 Bandwidth = 0.011
Density of var1
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Model Coefficient DistributionRegression Coefficients: Conversion
iDistribution
Time
Temperature
I(Time 2̂)
I(Temperature 2̂)
Time:Temperature
-1 0 1 2
Regression Coefficients: Viscosity
iDistribution
Time
Temperature
I(Time 2̂)
I(Temperature 2̂)
Time:Temperature
-5 0 5
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inferencefrom Blue ReferencePosterior Distributions
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65 66 67 68 69 70
64
66
68
70
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Observed vs. Predicted: Conversion
Observed
Pre
dic
ted
32 34 36 38 40 42 44
20
30
40
50
60
Observed vs. Predicted: Viscosity
Observed
Pre
dic
ted
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inferencefrom Blue ReferenceKnowledge Space
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60 62 64 66 68 70 72
10
20
30
40
50
60
Conversion
Vis
co
sity
1
569
1138
1706
2274
2842
3410
3979
4547
5115
5684
6252
6820
7388
7956
8525
9093
Counts
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inferencefrom Blue Reference95% Credibility Intervals
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Conversion
Time
Tim
e
-1.0
-0.5
0.0
0.5
1.0
-1.0 -0.5 0.0 0.5 1.0
0.90
0.95
1.0
01.05
1.05
1.05
1.05
1.10
1.10
1.10
1.10
1.15
1.15
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
Viscosity
Time
Tim
e-1.0
-0.5
0.0
0.5
1.0
-1.0 -0.5 0.0 0.5 1.0
8.0
8.5
9.0
9.5
9.5
9.5
9.5
10.0
10.0
10.0
7
8
9
10
11
12
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inferencefrom Blue ReferenceMean Response Plots
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Conversion
Time
Tim
e
-1.0
-0.5
0.0
0.5
1.0
-1.0 -0.5 0.0 0.5 1.0
65
66
67
68
69
70
62
63
64
65
66
67
68
69
70
71
Viscosity
Time
Tim
e-1.0
-0.5
0.0
0.5
1.0
-1.0 -0.5 0.0 0.5 1.0
32
34
36
38
40
42 44
30
35
40
45
50
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inferencefrom Blue ReferenceResponse Probability Distributions
Conversion > 68% 38 < Viscosity < 42
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Conversion>68
Time
Te
mp
era
ture
-1.0
-0.5
0.0
0.5
1.0
-1.0 -0.5 0.0 0.5 1.0
0.1
0.20.30.40.50.60.70.80.9
1.0
0.2
0.4
0.6
0.8
1.0
Viscosity>38 & Viscosity<42
Time
Te
mp
era
ture
-1.0
-0.5
0.0
0.5
1.0
-1.0 -0.5 0.0 0.5 1.0
0.1
0.1
0.2 0.2
0.3
0.3
0.4
0.4
0.5
0.5
0.6
0.7
0.7
0.2
0.4
0.6
0.8
1.0
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inferencefrom Blue ReferenceJoint Probability Distribution
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Conversion>68 & Viscosity>38 & Viscosity<42
Time
Te
mp
era
ture
-1.0
-0.5
0.0
0.5
1.0
-1.0 -0.5 0.0 0.5 1.0
0.1
0.1
0.2
0.2
0.3
0.3
0.40.50.6
0.7
0.7
0.2
0.4
0.6
0.8
1.0
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inferencefrom Blue ReferenceOptimization: Pareto Frontier
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64 66 68 70
30
35
40
45
y1
y2
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inferencefrom Blue ReferencePareto/Joint Probability Overlay
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Conversion>68 & Viscosity>38 & Viscosity<42
Time
Te
mp
era
ture
-1.0
-0.5
0.0
0.5
1.0
-1.0 -0.5 0.0 0.5 1.0
0.1
0.1
0.2
0.2
0.3
0.3
0.40.50.6
0.7
0.7
0.2
0.4
0.6
0.8
1.0
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inferencefrom Blue Reference50% Credibility Design Space
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0.0
0.5
1.0-0.5
0.0
0.5
1.0
0.0
0.2
0.4
0.6
Time
Temperature
Probability
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
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take-home message
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©2009 Blue Reference, Inc. All rights reserved.
inferencefrom Blue ReferenceBenefits of Bayesian Approach
Provides estimates of uncertainty in model parameters.
Enables use of prior information leading to more efficient adaptive design and experimentation.
Provides a figure of merit (probability) for meeting product profile (specifications) in terms consistent and easy to understand by technical workers operating in a regulated environment.
Enables use of Pareto optimization in conjunction with risk minimization.
Enables a means to establish the reliability of the Design Space.
Provides a basis for selection of alternate process settings within the design space while ensuring “assurance of quality.”
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inferencefrom Blue ReferenceReferences
John J. Peterson (2004). “A posterior predictive approach to multiple response surface optimization.” Journal of Quality Technology, 36, 139-153.
John J. Peterson (2008). “A Bayesian approach to the ICH Q8 Definition of Design Space.” Journal of Biopharmaceutical Statistics, 18, 959-975.
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www.InferenceforQbD.com
For additional information contact:
Paul van Eikeren, Ph.D.
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