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Contents
1 Introduction
2 An Overview of Bayesian Computation
5
1
3 Basic Bayesian Model Checking
25
4 Quantitative Decision-Making, a CMC Application to Analytical Method
41
Equivalence
5 Bayesian Dissolution Testing
57
6 A Non-Normal Bayesian Model for the Estimation and Comparison of
Immunogenicity Screening Assay Cut-Points
85
7 Application of Bayesian Hierarchical Models to Experimental Design
119
8 Bayesian Prediction for Staged Testing Procedures
135
9 A Bayesian Approach to Multivariate Conditional Regression Surrogate
Modeling with Application to Real Time Release Testing
1
10 Bayesian Approach for Demonstrating Analytical Similarity
177
11 Bayesian Evaluation and Monitoring of Process Comparability
195
vii
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viii
Contents
12 Bayesian Alternatives to Traditional Methods for Estimating Product
Shelf-Life and Internal Release Limits
225
13 Application of Bayesian Methods for Specification Setting
257
14 Calculating Statistical Tolerance Intervals Using SAS
275
15 A Bayesian Application in Process Monitoring – Establishing Limits for
Dosage Units in Early Phase Process Control
299
Bibliography
321
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1
Introduction
CONTENTS
1.1
.
1.2
.
1.3
.
Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1
Motivation
Regulatory Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Objectives and Organization of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
2
3
Nonclinical biostatistics can be broadly defined as statistical methods applied to areas in
the pharmaceutical/biotechnology industries that do not include clinical studies (Zhang and
Su, 2016). These areas include preclinical discovery, nonclinical development, translational
science, safety, pharmacology, and chemistry, manufacturing and controls (CMC).
CMC studies and data are focused on the development, characterization, and control of
biopharmaceutical products, processes, and analytical methods. They represent an integral
part of the drug development life-cycle and are required for market approval. New Drug
Application (NDA) and Biologics License Application (BLA) submissions include many
lengthy sections devoted to CMC aspects. Indeed, the quality, safety, and efficacy of the
final drug product are ultimately defined through CMC results and the statistical tools used
to evaluate them.
CMC data describe the manufacturing process and associated control measures, as well
as detail the capability of measurement systems to adequately characterize the critical quality attributes of the drug substance and drug product. For example, CMC scientists must
propose and justify lot release and shelf-life specifications for the quality attributes of future manufactured lots. They must demonstrate the robustness, precision, and accuracy of
analytical methods used to measure the quality attributes as well as describe and justify
process control measures. They must also demonstrate the long-term chemical and biological stability of the molecule throughout the supply-chain and proposed shelf-life. And
they must provide evidence of comparability of the drug throughout its various stages of
the process and product development. When changes are made to analytical methods or
manufacturing processes, comparability or bridging studies must be performed to demonstrate the continuity of the measurement system and quality attributes of the drug. For
biosimilar development, CMC scientists must demonstrate analytical similarity between
the proposed and reference products. Analytical similarity is foundational for providing
the totality-of-evidence of biosimilarity to regulatory agencies. Comparative analytical data
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2
Introduction
influence decisions about the type and amount of animal and clinical data needed to support
a demonstration of biosimilarity.
As a candidate molecule advances to late-stage development, CMC studies become more
complex and undergo greater scrutiny by health authorities. Moreover, the sponsor is expected to achieve greater understanding of how CMC aspects impact the quality, safety,
and efficacy of the drug. In this way, CMC studies are ideally suited for the application of
Bayesian methods. Knowledge regarding the product, process, and measurement systems
accumulates and improves as molecules move from early to late-phase development. For
example, early screening studies of critical manufacturing process parameters evolve into
process characterization and optimization studies in late phase. Novick et al. (2021) and
Faya et al. (2022) also illustrate the application of Bayesian methods to a “continuous”
learning approach to method validation. Moreover, it is common for biopharmaceutical
manufacturers to adopt “first-to-try” or “platform” manufacturing processes and analytical methods, particularly for proteins of similar structure (e.g., monoclonal antibodies).
Knowledge from such platform processes and methods can naturally be quantified across
molecules using Bayesian methods. As such, high-quality and carefully collected prior information can be available from the current project as well as from similar molecules. There
are also benefits to using Bayesian statistics that extend beyond using prior knowledge.
These include the probabilistic quantification of uncertainty, the clarity and coherence of
posterior-based inference (Robert, 2007), and the ability to analyze complex models directly
instead of relying on approximations, as can be common in frequentist methods.
1.2
Regulatory Considerations
Regulatory reviewers of new drug and biologic applications insist that CMC studies for
new molecular entities and biosimilars meet the highest of standards. The current guidance documents governing CMC practice (from Food and Drug Administration (FDA),
European Medicines Agency (EMA), International Council for Harmonisation (ICH), and
United States Pharmacopeia (USP)) are silent on the use of Bayesian methods, not even
mentioning them as alternatives to traditional approaches. This is curious given the enormous emphasis on risk control in pharmaceutical development and quality metrics (see ICH
Q8 (2005) and ICH Q9 (2005)). In contrast, as far back as 1998, ICH E9 (1998), which
governs statistical principles for clinical trials, noted that “the use of Bayesian and other
approaches may be considered when the reasons for their use are clear and when resulting
conclusions are sufficiently robust.”
Demonstrating equivalence of biosimilars is similarly bereft of Bayesian guidance. This
is despite the natural suitability of Bayesian approaches for such studies. Bayesian posterior
and predictive distributions can be used to make probabilistic (i.e., risk-based) conclusions
regarding the similarity of the physiochemical and functional properties of two products.
However, in the most recent FDA draft guidance (2019) for the development of therapeutic
protein biosimilars, only frequentist equivalence testing is proposed, with no mention of the
potential for Bayesian methods.
In a recent cover article of the American Institute of Chemical Engineers Journal,
Tabora et al. (2019) called for chemical engineers to adopt Bayesian-based approaches,
which quantify risk more effectively and meet the FDA’s vision for robust pharmaceutical
manufacturing. They argued that Bayesian methods are exceptionally effective at characterizing variability from limited data and anticipate a much wider adoption across the
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Objectives and Organization
of the Book
3
pharmaceutical industry. However, the authors also noted that regulatory agencies need to
encourage the use of such probabilistic risk-based methods.
At a broader level, the FDA’s vision for Pharmaceutical Quality for the 21st Century
– A Risk Based Approach (FDA, 2004) has been around for nearly 20 years. One of the
key pillars of the initiative was the concept of “Quality by Design” (QbD). QbD involves
developing a thorough understanding of the product and process along with a knowledge
of the risks involved in manufacturing the product. Extensive research has been published
by the industry (see for example, Peterson (2008), Peterson and Yahyah (2009), Peterson
and Lief (2010), Rozet et al. (2013)) on how Bayesian methods are ideally suited for QbD
applications. Yet, in the years since the launch of the FDA QbD initiative, no regulatory
document has been revised to reflect this fact.
We can also learn from the “least burdensome” provisions of the Center for Devices
and Radiological Health (CDRH), which are linked to the FDA Modernization Act of 1997.
Although written from the perspective of medical device clinical trials, the spirit of the
provisions is completely relevant to other spheres such as CMC. Principles such as using
only necessary (minimum required) information, acceptance of alternative approaches, the
efficient use of resources, the use of alternative sources of data, leveraging existing data,
and considering the “most efficient means” of obtaining scientific evidence are important
in CMC, and in many ways, can be realized through Bayesian statistics. The current draft
guidance documents for ICH Q2(R2) (2022) and ICH Q14 (2022) published in 2022 make
encouraging steps toward a “least burdensome” principle applied to analytical method validation. For example, the draft Q2(R2) explicitly allows for the use of pre-existing knowledge
in addition to, or in lieu of, a formal validation study to provide evidence of the validated
state of a method. The draft Q14 notes that prior knowledge can come from internal or
external sources and that existing platform procedures can be leveraged to evaluate the
attributes of a specific molecule without conducting additional method development. In addition, Q2(R2) introduces a predictive approach to method performance assessment, combining the evaluation of accuracy and precision into a measure of total error. Despite the
fact that Bayesian methods are naturally suited for the new validation strategies offered in
the draft Q2(R2) and Q14, there still remains an inherent frequentist method bias, with a
focus on confidence interval-based inference and no mention of Bayesian techniques.
In sum, adopting Bayesian methods in CMC has the potential to enhance decisionmaking and improve the quality, safety, and efficacy of biopharmaceutical products. Modernizing CMC regulatory guidance will open new paths for CMC scientists, engineers, and
statisticians to properly and fruitfully apply Bayesian principles throughout the product
development and manufacturing life-cycle. The clinical space has already paved the way
with guidance documents and examples of industry-regulatory collaborative efforts that the
nonclinical community can learn from and leverage. The need for initiative, clarity, and
collaboration from the CMC regulatory community on both the industry and government
sides is high. For further discussion on some of the reasons for the slower uptake of Bayesian
methods in the CMC space, see Faya and Berry (2022).
1.3
Objectives and Organization of the Book
The following chapters offer examples of Bayesian methods motivated by real case studies in traditional CMC areas. The contributing authors include current and former
CMC statisticians with extensive experience in the field as well as academics who have
collaborated in CMC research projects. In selecting the authors and topics for this book,
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4
Introduction
our goal was to provide as diverse a set of perspectives as possible. Many of the contributors to this book are members of the DIA-ASA Nonclinical Bayesian Working Group. While
some case studies are presented with simulated data (for the sake of data anonymity), the
Bayesian methodologies proposed in each chapter are those that have been used by the
authors to support CMC decision-making. In addition, a variety of software applications
and packages are used to demonstrate the implementation of Bayesian inference throughout
the book. We view one of the main bottlenecks to the adoption of Bayesian methods in the
CMC space as the lack of available easy-to-understand resources for implementing Bayesian
techniques practically (see industry survey results in Faya et al. (2021)). Our hope is that
the numerous practical examples and reproducible code will alleviate this bottleneck.
Recent advances in computational tools have reduced the barriers to the adoption of
Bayesian methods for many practitioners. However, the need for simulation methods to
derive inference for most problems means that there is a learning curve for those unfamiliar with computational techniques. Chapters 2 and 3 of this book provide an overview of
Bayesian computational strategies and model checking. While the content in these chapters
is not a pre-requisite for understanding the subsequent case studies, the concepts, theoretical
foundations, and mathematical derivations offer readers a sound foundation for simulationbased Bayesian modeling. Of particular value are the “quick-kill” and “due diligence” lists
in Chapter 3, which provide practical checks for diagnosing convergence of posterior distributions.
One of the primary aims of this book is the reproducibility of the statistical methods
presented. As such, each chapter contains code chunks and details integrated within the
body of the text. While some chapters may be reproduced by beginners, others are more
advanced and require a sound understanding of Bayesian ideas and methods. Our hope is
that regardless of the level of Bayesian training, CMC statisticians, scientists, and engineers
can find scenarios that will motivate, challenge, and equip them to use Bayesian inference
in their work. We also hope that this book will provide material to fill a gap in teaching
biostatistics, a field that extends beyond the design and analysis of clinical trials.
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