Steven Novick, Katherine Giacoletti, Tara Scherder, and Bruno BoulangerSteven.Novick@arlenda.com

How to implement Bayesian statistics to Make Lifecycle Strategy a Reality that Serves Quality

Quality by Design overview

• Quality Target Product Profile (QTPP)

• Determine critical quality attributes (CQAs) and Specifications

• Perform risk assessment

• Develop a design space

• Design and implement a control strategy: SPC

• Manage product lifecycle, including continual improvement: Transfer

Prove the objectives will be met surely in the future

CQA’s

Product Profile

Risk Assessments

Design Space

Control Strategy

Continual Improvement

Stage1 Process Design

Stage 2Process

Performance Qualification

Stage 3Continued Process

Verification

the commercial process is defined based on knowledge

gained through development and scale-up activities

the process design is evaluated and assessed

to determine if the process is capable of

reproducible commercial manufacturing

ongoing assurance is gained during routine production that the

process remains in a state of control

SCIENTIFIC EVIDENCE ACROSS LIFECYCLE

2011 FDA Guidance Process Validation

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

TotalData

AvailableData

ObservedData =+

“LIKELIHOOD”data coming from the

experiment

“POSTERIOR DISTRIBUTION” for parameterscombination of information collected before the experiment

and what comes from the experimental data

“PRIOR DISTRIBUTION” from previous studies, expert

opinion, literature,…

• Uncertainty is described in terms of probability :

Bayesian principle

P(θ>5.5)=0.401

Bayesian principle

Bayes directly tests hypotheses: P(performance|data)

Frequentist method is indirect P(data|performance)

PRIOR distribution BATCH data POSTERIOR distribution

P(potency in Specs)= P(quality)

+

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• “new observations” ỹ ~ F( ,m s)• ( ,m s) are [erroneously] fixed

Bayesian Predictions• “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.

Joint posteriorModelIntegrate over parameter distribution

MarginalModel Conditional

𝑝 (~𝑦|𝑑𝑎𝑡𝑎 )=∬𝜇 ,𝜎 2

❑

𝑝 (~𝑦∨𝜇 ,𝜎2 ,𝑑𝑎𝑡𝑎 )×𝑝 (𝜇 ,𝜎 2∨𝑑𝑎𝑡𝑎 )𝑑𝜇𝑑𝜎2

¿∬𝜇 ,𝜎 2

❑

𝑝 (~𝑦∨𝜇 ,𝜎 2 ,𝑑𝑎𝑡𝑎 )×𝑝 (𝜎 2∨𝑑𝑎𝑡𝑎)×𝑝 (𝜇∨𝜎2 ,𝑑𝑎𝑡𝑎 )𝑑𝜇𝑑𝜎 2

Integral typically computed by Monte

Carlo methods

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

-∞ +∞

P

X X X X

X

X X X

Based on a point estimate of µ, σ Based on a distribution of µ and σ

PredictiveDistribution

Prior Distribution

PPQ batches

Frequentist Bayesian

Stage 2 and Bayesian Method

-∞ +∞

P

X X X X

X

X X X

Based on a point estimate of µ , σ Based on a distribution of µ and σ

PredictiveDistribution

Prior Distribution

PPQ batches

Frequentist Bayesian

Probability being in specificationsvs Tolerance intervals

• Use the Predictive distribution to compute the prob. to be within specs

X X X X

[---------------------] Tolerance Interval

Bayesian method directly calculates risk

Frequentist tolerance interval indirectly assesses risk

Predictive posterior

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.

Class

ical

Sta

tistic

s

Bayesian Statistics• 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

• The set of CPP (X) that guarantee results are in specifications is called the Design Space.

𝑝 (~𝑦|𝑑𝑎𝑡𝑎¿= ∫𝜇 ,𝜎 2

❑

∫𝑋

❑

𝑝 (~𝑦∨𝜇 ,𝜎2 ,𝑋 ,𝑑𝑎𝑡𝑎 )×𝑝 ( 𝑋 )×𝑝 (𝜇 ,𝜎2∨𝑑𝑎𝑡𝑎 )𝑑𝑋 𝑑𝜇𝑑𝜎2

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%- …

The Flaw of Averages: Why We Underestimate Risk in the Face of Uncertainty by Dr. Sam Savage

Focusing only on the mean (average) can put us at risk!

Average depth of river is 3 feet.

From John Peterson, 2012

Spray-drying process

Risk-based design space: predicted P(CQAs ∈ l)•

In the Design Space, there is 45% of chance to observe each CQA within specification, jointly

~ 45% probability to jointly observe CQAs within specification

100-45% = 55% of risk not to observe the CQAs within specification (jointly) !

Inlet.Temperature

Spr

ay.F

low

.Rat

e

Inlet.Temperature

Fee

d.R

ate

Inlet.Temperature

Fee

d.R

ate

Spray-drying process

• Validation- Experiments have been repeated 3 times independently at optimal

condition, i.e.Inlet Temperature: 123.75°CSpray Flow Rate: 1744 L/hFeed 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

Estimated concentrations

Titration

Decision:Proportion of

DS1 / DS2/ DS … / Buffer(% / % / … / %)

Drug Substance 1Buffer

Drug Product

Titration

Estimated concentration

Release Discard

Drug Substance 2

At mix After filtration At release At shelf-life

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

99% guaranteemeeting LSL atshelf-life…

LSL

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

-20

-10

0

10

20

30

Rela

tiv

e E

rror

100 (pg/m

l)

-30

-20

-10

0

10

20

30

Rela

tiv

e E

rror

1000 (pg/m

l)

-30

-20

-10

0

10

20

30

Rela

tiv

e E

rror

500 (pg/m

l)

Concentratio

n

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

series

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

LSL

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!