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How to implement Bayesian statistics to Make Lifecycle
Strategy a Reality that Serves Quality
Steven Novick, Katherine Giacoletti, Tara Scherder, and Bruno Boulanger
[email protected]
Quality by Design overview
Product Profile
• Quality Target Product Profile (QTPP)
• Determine critical quality attributes (CQAs) and
Specifications
CQA’s
Risk Assessments
• Perform risk assessment
Design Space
• Develop a design space
Control Strategy
Continual
Improvement
• Design and implement a control strategy: SPC
• Manage product lifecycle, including continual
improvement: Transfer
Prove the objectives will be met surely in the future
2011 FDA Guidance Process Validation
Stage1
Process Design
the commercial
process is defined
based on knowledge
gained through
development and
scale-up activities
Stage 2
Process
Performance
Qualification
the process design is
evaluated and assessed to
determine if the process is
capable of reproducible
commercial manufacturing
SCIENTIFIC EVIDENCE
ACROSS LIFECYCLE
Stage 3
Continued
Process
Verification
ongoing assurance is
gained during routine
production that the
process remains in
a state of control
Excerpt from Guidance
• “…high degree of assurance on the
performance of the manufacturing process that
will consistently produce….”
• “ ….collection and evaluation of data … which
establishes scientific evidence that a process is
capable of consistently delivering quality
product….”
• “ … the assurance should be obtained from
objective information and data from laboratory,
pilot batches….”
Excerpt from Guidance
• “During the process qualification (PQ) stage of
process validation, the process design is
evaluated to determine if it is capable of
reproducible commercial manufacture…”
What is “capable”?
Capability is defined as the ability of
a process to meet specification
Bayesian principle
“PRIOR DISTRIBUTION”
from previous studies, expert
opinion, literature,…
Available
Data
+
Observed
Data
“LIKELIHOOD”
data coming from the
experiment
=
Total
Data
“POSTERIOR DISTRIBUTION” for parameters
combination of information collected before the experiment
and what comes from the experimental data
Bayesian principle
• Uncertainty is described in terms of probability :
P(θ>5.5)=0.401
Bayesian principle
PRIOR distribution
distribution
BATCH data
+
POSTERIOR

P(potency in Specs)= P(quality)
Bayes directly tests hypotheses:
 P(performance|data)
Frequentist method is indirect
 P(data|performance)
Bayesian principle
• PPQ batches are produced to collect evidence of
the quality of the process
Frequentist analysis:
• Point estimate and confidence intervals as summaries of process
(mean and sd)
 What do PPQ batches tell us about the process?
Bayesian analysis:
• Before the PPQ: a priori opinion on the process
 How should those PPQ batches change our opinion about the
process?
 How should those PPQ batches provide assurance about future
batches?
• Motivations for adopting Bayesian approach:
Natural and coherent way of thinking about learning and risk
How to make predictions
Monte-Carlo Simulations
Bayesian Predictions
• “new observations” ỹ ~ F(m,s)
• (m,s) are [erroneously] fixed
• “new observations” ỹ ~ F(m,s)
• (m,s) ~ p(m0,s0| data)
Bayesian Predictive Distribution
The Bayesian theory provides a definition of the
Predictive Distribution of a new observation given past
data.
𝑝 𝑦|𝜇, 𝜎 2 , 𝑑𝑎𝑡𝑎 × 𝑝 𝜇, 𝜎 2 |𝑑𝑎𝑡𝑎 𝑑𝜇𝑑𝜎 2
𝑝 𝑦 𝑑𝑎𝑡𝑎 =
𝜇,𝜎 2
Integrate over parameter distribution
Model
Joint posterior
Integral typically
computed by Monte
Carlo methods
𝑝 𝑦|𝜇, 𝜎 2 , 𝑑𝑎𝑡𝑎 × 𝑝 𝜎 2 |𝑑𝑎𝑡𝑎 × 𝑝 𝜇|𝜎 2 , 𝑑𝑎𝑡𝑎 𝑑𝜇𝑑𝜎 2
=
𝜇,𝜎 2
Model
Marginal
Conditional
Comparison Frequentist vs Bayesian
• When NON-informative priors are envisioned
Posteriors and HPD (~quantiles) are the same as the Frequentist results
Are non-informative priors defensible in Stage 2?
There are defensible priors
• Once decision is made to go through PPQ, there is belief it will work.
• Translate those scientific evidence and data based into priors
• Priors contain the whole uncertainty about this belief.
 This is the prior elicitation process.
• Classical statistics ignores prior available information.
Stage 2 and Bayesian Method
Prior
Distribution
P
-∞
+∞
X
X
X
X
PPQ batches
X
X
X
X
Predictive
Distribution
Based on a point estimate of µ, σ
Frequentist
Based on a distribution of µ and σ
Bayesian
Stage 2 and Bayesian Method
Prior
Distribution
P
-∞
+∞
X
X
X
X
X
PPQ batches
X
X
X
Predictive
Distribution
Based on a point estimate of µ , σ
Frequentist
Based on a distribution of µ and σ
Bayesian
Probability being in specifications
vs Tolerance intervals
• Use the Predictive distribution to compute the prob. to be within specs
𝑈
𝑃 𝐿<𝑌<𝑈 =
𝑝 𝑦 𝑑𝑎𝑡𝑎 𝑑𝑦
𝐿
Predictive
posterior
X
X
X
X
Bayesian method
directly calculates risk
[---------------------]
Tolerance Interval
Frequentist tolerance interval indirectly assesses risk
Number of Batches
• Number of batches required to guarantee 95% of success in
PPQ, i.e. that 96% of future results will be within specifications.
• Classical Stats requires more
than 10 batches
• Bayesian statistics using prior
(defensible) information only
requires 4 batches.
Why?
• The Posterior of performance
parameters is more precise.
Other Benefits of Bayesian Approach
• Capability is defined as the ability of a process to meet specification,
that is, the probability of meeting specification
Bayesian provides a true prediction of future performance
• Handles complicated hierarchy/ sampling plan
Between batch, sample within batch, within sample variation can
be incorporated
Unbalanced sampling
• Joint prediction of multiple CQAs is possible
• Uncertainty of parameters included, thus improving prediction and
reducing risk
• Not affected by non-centering within specification range
• Systems approach to unit operations (simultaneous prediction)
Stage 1 - Design Space and Predictions
• In Stage 1 the objective is to identify the Design Space
• DoE are performed to understand the relationships
between the CPP and the CQA
• The known or assumed control/uncertainty on CPPs can
be integrated into Predictions
𝑝 𝑦|𝜇, 𝜎 2 , 𝑋, 𝑑𝑎𝑡𝑎 × 𝑝 𝑋 × 𝑝 𝜇, 𝜎 2 |𝑑𝑎𝑡𝑎 𝑑𝑋 𝑑𝜇𝑑𝜎 2
𝑝 𝑦 𝑑𝑎𝑡𝑎) =
𝜇,𝜎 2 𝑋
• The set of CPP (X) that guarantee results are in
specifications is called the Design Space.
An example: Spray-drying process
• Spray-drying is intended to create a powder with small and controlled
particle size for pulmonary delivery of a drug substance
• Several Critical Process Parameters (CPP) have an influence on
several Critical Quality Attributes (CQA)
 Inlet temperature
 Spray flow-rate
 Feed rate
(other process parameters are kept constant)
• Specifications on CQA defined as minimal
satisfactory quality
 Yield > 80%
 Moisture < 1%
 Inhalable fraction > 60%
…
Focusing only on the mean
(average) can put us at risk!
Average depth of river is 3 feet.
The Flaw of Averages:
Why We Underestimate Risk in the Face of Uncertainty
by Dr. Sam Savage
From John Peterson, 2012
Spray-drying process
• Risk-based design space: predicted P(CQAs∈ l)
Feed.Rate
Feed.Rate
Spray.Flow.Rate
•
In the Design Space, there is 45% of chance to observe
each CQA within specification, jointly
Inlet.Temperature
Inlet.Temperature
Inlet.Temperature
~ 45% probability to jointly observe CQAs within specification
 100-45% = 55% of risk not to observe the CQAs within
specification (jointly) !
Spray-drying process
• Validation
Experiments have been repeated 3 times independently at optimal
condition, i.e.
Inlet Temperature: 123.75°C
Spray Flow Rate: 1744 L/h
Feed Rate: 4.69 ml/min
Jointly, 2 out of the 3 runs within specification
Spray-drying process
• Post-analysis (« How they are statistically distributed »)
Marginal predictive densities of the CQAs
Inhalable fraction is predicted to
be widely distributed
Predictive uncertainty =
data uncertainty + model
uncertainty
Model Uncertainty can be reduced
with an appropriate DoE
Stage 3
An example: Vaccine compounding
Buffer
Drug Substance 1
Drug Product
Titration
Titration
Drug Substance 2
Decision:
Proportion of
DS1 / DS2/ DS … / Buffer
(% / % / … / %)
Estimated concentrations
Estimated
concentration
Release
Discard
Overall view of the dilution problem
• Optimize the assay format, and the concentration of DS such that
it will result in a drug product looking like…
Each black line is the predicted behavior of one individual realization of DP
LSL
At mix
After filtration
At release
At shelf-life
99% guarantee
meeting LSL at
shelf-life…
Control strategy
• Control strategy is defined based on the (simulated) outcome of
the process profile at strategic intermediate testing (red)
• The prediction interval (b-expectation tolerance interval) can be
used as control limits.
Control strategy
• Raise appropriate out-of-control, alert, and reject at release
30
100 (pg/ml)
Relative Error
20
10
0
-10
-20
-30
30
10
0
-10
-20
-30
30
LSL
Concentration
Relative Error
20
1000 (pg/ml)
It allows to control the risks and
keep the quality constant over
time.
You maintain your initial claim and
monitor it with appropriate levels
of risk.
Release Routine
Conclusion
• Bayesian statistics provide a natural answer to all Stages of
process or method development
• Bayesian statistics provide predictive distribution to permit
prediction-based decision
• Prediction are key to Design Space
• Prediction are key to PPQ
• Bayesian statistics make multivariate modeling easy and allows to
compute joint probability of success
• Bayesian statistics are easy to compute today with languages
such as SAS, BUGS, JAGS, or STAN.
Thank you!