bayesian approach to specification of design space · 2010-05-18 · quality target product-profile...

inferencefrom Blue Reference

Bayesian Approach to Specification of Design Space

Paul van Eikeren and Corey Dow-Hygelund

userR! 2009 (July 8-10, 2009)

©2009 Blue Reference, Inc. All rights reserved.


Quality by Design: Pharmaceutical Quality Vision

2


inferencefrom Blue ReferencePharmaceutical Quality Vision

3

Develop a harmonized pharmaceutical quality system applicable across the

lifecycle of the product emphasizing an integrated approach to quality

risk management and science.


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.”

4


inferencefrom Blue ReferenceQuality by Design Workflow

5


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

6

parameters

processdrug

product

attributes

CQA_1

CQA_2

CQA_3

...

inputs


inferencefrom Blue ReferenceQ8(R1) Example Design Space

7

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?



Risk-based approach to Design Space

8


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

9


inferencefrom Blue ReferenceIssue with Conventional Design Space

10

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?


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

11



Illustrative Example: Process for Coating Polymer

12


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

13


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)

14


inferencefrom Blue ReferenceMCMC Bayesian Regression

15

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



16

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


inferencefrom Blue ReferencePosterior Distributions

17

65 66 67 68 69 70

64

66

68

70

72

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


inferencefrom Blue ReferenceKnowledge Space

18

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


inferencefrom Blue Reference95% Credibility Intervals

19

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

13


inferencefrom Blue ReferenceMean Response Plots

20

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


inferencefrom Blue ReferenceResponse Probability Distributions

Conversion > 68% 38 < Viscosity < 42

21

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


inferencefrom Blue ReferenceJoint Probability Distribution

22

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


inferencefrom Blue ReferenceOptimization: Pareto Frontier

23

64 66 68 70

30

35

40

45

y1

y2


inferencefrom Blue ReferencePareto/Joint Probability Overlay

24

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


inferencefrom Blue Reference50% Credibility Design Space

25

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



take-home message

26


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.”

27


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.

28



www.InferenceforQbD.com

For additional information contact:

Paul van Eikeren, Ph.D.

[email protected]

29

bayesian approach to specification of design space · 2010-05-18 · quality target product-profile...

Documents