Expected Design Space: a Bayesian perspective based on modeling, prediction and multicriteria decisi

1
Expected Design Space a Bayesian perspective
based on modeling, prediction and multi-criteria
decision method

Pierre Lebrun ULg, BelgiumBruno Boulanger UCB
Pharma, SA, BelgiumPhilippe Lambert Ulg,
BelgiumBenjamin Debrus ULg, BelgiumPhilippe
Hubert ULg, Belgium
Friday , October 23
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Overview

• The process
• Liquid chromatography
• Multivariate regression correlated responses
• Definition
• Design Space
• Objective functions
• Bayesian model
• Introduction
• Priors
• Predictions
• MCDM acceptance limits and desirability
• Results
• Conclusions

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Example of application

• - A chromatographic method is to be optimized
  using DOE and response surface models
• - P3 peaks to be separated in the shortest time

Gradient time (min.)
pH
pH 2.6 6.3 10
Gradient (min.) 10 20
30
(N x 3P)
Design Space set of conditions (pH, Gradient,)
in the domain, such that separation and short
run are guaranteed for the future
These 9 responses are correlated
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Overview

• The process
• Liquid chromatography
• Multivariate regression correlated responses
• Definition
• Design Space
• Objective functions
• Bayesian model
• Introduction
• Priors
• Predictions
• MCDM acceptance limits and desirability
• Results
• Conclusions

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ICH Q8 (may 2006) definition

• ? The Design Space is the set of conditions
  giving solution within Acceptance Limits
• the established range of process parameters and
  formulation attributes that have been
  demonstrated to provide assurance of quality.
• Working within is not considered as a change in
  the analytical method.
• n.b. If the Design Space is large w.r.t. control
  parameters or conditions, the solution is
  considered as robust

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Proposal definition of Design Space

• When the process is known
• Design Space (DS)
• domain of factors
• set of combinations of factors
• the responses obtained for the
  condition
• the set of acceptance limits (e.g.
  resolutiongt1.2 )
• the quality level (e.g. P( resolutiongt1.2) gt
  0.8)

However - in development validation, the
process is unknown, its performances are
estimated with uncertainty - purpose predict
the space that will likely in the future provide
most outputs within acceptance limits
Peterson, J. Qual. Tech, 36, 2, 2004
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Proposal definition of design space

• When the process is unknown
• Expected Design Space (DS)

• The predictive probability of achieving the
  acceptance limits is larger than , the
  quality level
• Given the process parameters
• The DS is located using predictions from models
  estimated during development validation
  experiments

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Objective functions
Specific problem Criteria / Objective functions

• - Sum, product, min/max of the responses
• - Discontinuity
• - Non linearity

Ex
? i.e. DS is the set of conditions, such that the
probability that objectives will be
simultaneously (jointly) within the Acceptance
Limits is higher than
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Overview

• The process
• Liquid chromatography
• Multivariate regression correlated responses
• Definition
• Design Space
• Objective functions
• Bayesian model
• Introduction
• Priors
• Predictions
• MCDM acceptance limits and desirability
• Results
• Conclusions

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Bayesian model

• Multivariate multiple linear regression model

• The joint posterior distribution of the
  parameters is obtained as follow

• and are assumed independent a priori,
  therefore

Go to results !
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Bayesian model
Posterior distribution of the parameters
Non informative
priors
(Box and Tiao, 1973)
Informative priors
reflects the certainty put in , the
prior correlation matrix
Elements of
Meff takes into account the correlation between
the responses (Sattertwhaite, 1946)
Meff lt M
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Prediction

• Plausible values of one prediction ,
  conditional to the available information
  predictive posterior distribution

• A draw from the joint posterior of parameters
• A draw from the Normal (model) conditionally to
  the posterior of parameters

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MCDM acceptance limits and desirability

• From the predictive distribution
• of responses
• of objective functions (criteria to optimize)
• Use of desirability functions for Multi-Criteria
  Decision Making (MCDM)

Ex Desirability functions of Le Bailly and
Govaerts Each criteria, conditional to x, is
transformed gt Domain 0,1, using the CDF of
the Normal distribution, F
D(O) is the global Desirability Index
Weights allow flexibility and balanced decision
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MCDM acceptance limits and desirability

• Joint predictive distribution of two objectives
  at a given x0

Brown Arithmetic mean Red Geometric
mean Green Harmonic mean Blue Acceptance
limits
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MCDM acceptance limits and desirability

• Desirability to acceptance limits
• Acceptance limits can be viewed as a special case
  of classical desirability-based MCDM

bz 0 az
Acceptance limit
? Step desirability functions
az
az
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MCDM acceptance limits and desirability

• When bz comes close to 0

bz sd(crz) az
bz sd(crz)/2 az
bz sd(crz)/5 az
bz sd(crz)/10 az
bz 0 az
When bz is 0, weights have no importance any
longer
-Using acceptance limits has the advantage to
have clear limits expressed in the criteria
space -Using classical desirability functions
allows trade-off between objectives ? this is
the experimenter choice
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Overview

• The process
• Liquid chromatography
• Multivariate regression correlated responses
• Definition
• Design Space
• Objective functions
• Bayesian model
• Introduction
• Priors
• Predictions
• MCDM acceptance limits and desirability
• Results
• Conclusions

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Results

• Informative prior

Gradient time fixed at 20 min.
Reflects the within peak correlation
03 Mean minimal resolution
Objective function is non-linear
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Results - MCDM - uncertainty

• From the joint predictive distribution of
  objective, using acceptance limits

? correlation is taken into account ?
uncertainty

• Ex - separation gt 0, resolution gt 1.2, Run lt 7
  min.
• - p gt 0.8

(Predictive) probability map that the three
objectives are achieved
X0
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Validation
Mean predicted Real
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Considerations

• DS can be subject to subjectivity
• Model choice (consider Bayesian model averaging)
• Choice of prior distribution
• Choice of responses or criteria used in MCDM
  (separation, resolution or both ?)
• What is not subjective
• Data and parameters (model) uncertainty
• Good statistical models are prerequisite
• Good data as well
• Design of experiments
• Control of non modelled but influential factors
  (to reduce noise)
• If it is assumed the responses do not follow the
  same regression equation, an alternative is to
  use seemingly unrelated regression (SUR)
• In some cases, constraints apply on the responses
  or criteria. They can be included
• using truncated distributions (e.g. Geweke, 1991)
• via rejection sampling in MCMC, if constraints
  are complex

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Conclusions

• Design Space must be defined on prediction of
  future results given past experiments
• The uncertainties in the model should be taken
  into account in predictions
• Bayesian multivariate multiple regression is
  powerful and flexible to model correlated
  responses and to manage uncertainty
• The Design Space is straightforward to obtain
  with Bayesian models
• The joint predictive posterior distribution of
  objective functions allows the development of
  Multi-Criteria Decision Methods (MCDM)
• About expected future performance
• under uncertainty
• taking into account dependencies between criteria
• Bayesian models in chromatography can take
  advantage from the long history of the domain,
  e.g. to set up informative priors
• Acceptance limits is a special case of
  desirability-based MCDM

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• Thank you !