Bayesian Adaptive Design for Immunotherapy and Targeted Therapy 9811981744, 9789811981746

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Haitao Pan Ying Yuan

Bayesian Adaptive Design for Immunotherapy and Targeted Therapy

Bayesian Adaptive Design for Immunotherapy and Targeted Therapy

Haitao Pan · Ying Yuan

Bayesian Adaptive Design for Immunotherapy and Targeted Therapy

Haitao Pan Department of Biostatistics St. Jude Children’s Research Hospital Memphis, TN, USA

Ying Yuan Department of Biostatistics The University of Texas MD Anderson Cancer Center Houston, TX, USA

ISBN 978-981-19-8174-6 ISBN 978-981-19-8176-0 (eBook) https://doi.org/10.1007/978-981-19-8176-0 © Springer Nature Singapore Pte Ltd. 2023 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

To my wife Chen and daughter Candice. Haitao Pan To my wife Suyu and daughter Selina. Ying Yuan

Preface

The advent of targeted therapies and immunotherapies has changed the treatment of cancer and offers new highly effective treatment options to patients with advanced stage or relapsed disease. Due to the unique characteristics and new mechanism of action of these novel therapies, the traditional clinical trial design paradigm, developed mainly based on cytotoxic chemotherapies, has been increasingly inefficient or dysfunctional for developing targeted therapies and immunotherapies. The objective of this book is to fill this gap and provide a comprehensive review of novel adaptive trial designs for targeted therapies and immunotherapies. This book covers a wide range of novel statistical designs for various clinical settings, including early phase dose-escalation study, proof-of-concept trials, and confirmatory studies with registrational purposes. Chapters 1–2 introduce novel designs for phase I clinical trials, in particular on how to handle late-onset toxicity. Chapters 3–5 cover both model-based and model-assisted Bayesian adaptive phase I-II designs to identify the optimal biological dose to accommodate the fact that the efficacy of targeted therapies and immunotherapies does not necessarily increase with the dose. Chapters 6–7 focus on single-arm and randomized phase II clinical trials to address complicated endpoints in targeted therapies and immunotherapies. Chapters 8–9 discuss some novel design options for basket and platform trials to expediate the development of targeted therapies and immunotherapies. This book may serve as a textbook for a graduate-level course and a reference for statisticians who are working in research medical hospital or pharmaceutical companies. After reading this book, readers are expected to understand modern adaptive clinical trial designs and statistical methodology and be able to design novel targeted therapy and immunotherapy clinical trials on their own. We would like to express our sincere thanks and gratitude to our colleagues at both institutes for their enormous encouragement and support. We are extremely grateful to all of our students, postdocs, and colleagues who developed the novel methodology and software in this book. In particular, Haitao Pan would like to thank Tomi Mori, Ph.D., for insightful discussion and strong support, as well as Jianrong Wu, Ph.D., Hongyu Miao, Ph.D., Arzu Onar-Thomas, Ph.D., Cheng Cheng, Ph.D., Stan Pounds, Ph.D., Yimei Li, Ph.D., Yisheng Li, Ph.D., Rongji Mu, Ph.D., Chen Li, vii

viii

Preface

Ph.D., Xiaomeng Yuan, Ph.D., Fang Wang, M.S., and Chia-Wei Hsu, M.S., for their productive collaboration especially during the pandemic era. Ying Yuan would like to thank Beibei Guo, Ph.D., Ruitao Lin, Ph.D., Peter Thall, Ph.D., Suyu Liu, Ph.D., Guosheng Yin, Ph.D., Yong Zang, Ph.D., Fangrong Yan, Ph.D., Yiyi Chu, Ph.D., Heng Zhou, Ph.D., Yanhong Zhou, Ph.D., Yujie Zhao, Ph.D., Liyun Jiang, Ph.D., and Rongji Mu, Ph.D. Special thanks goes to Vani Shanker, Ph.D., for proofreading. Finally, we thank our families, whose ongoing love and support made all of this possible. Memphis, USA Houston, USA

Haitao Pan Ying Yuan

Contents

Part I

Phase I Trials

1 Introduction to Phase I Dose-Finding Clinical Trials . . . . . . . . . . . . . . . 1.1 Phase I Dose-Finding Trials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Continual Reassessment Method (CRM) . . . . . . . . . . . . . . . . 1.1.2 Modified Toxicity Probability (mTPI) Design . . . . . . . . . . . . 1.1.3 Keyboard Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.4 Bayesian Optimal Interval (BOIN) Design . . . . . . . . . . . . . . . 1.1.5 Deviation from the Planned Design . . . . . . . . . . . . . . . . . . . . . 1.1.6 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Challenges of Immunotherapies, Targeted Therapies and New Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11 13

2 Phase I Designs for Late-Onset Toxicity . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Model-Based Phase I Designs for the Late-Onset Toxicity . . . . . . . . 2.1.1 Time-to-Event CRM (TITE-CRM) . . . . . . . . . . . . . . . . . . . . . 2.1.2 Fractional CRM (fCRM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Data Augmentation CRM (DA-CRM) . . . . . . . . . . . . . . . . . . . 2.2 Model-Assisted Phase I Design for the Late-Onset Toxicity . . . . . . . 2.2.1 Time-to-Event BOIN (TITE-BOIN) . . . . . . . . . . . . . . . . . . . . 2.3 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15 15 15 22 25 31 33 40 41

Part II

3 3 4 5 7 9 10 11

Phase I/II Trials

3 Optimal Biological Dose and Phase I/II Trials . . . . . . . . . . . . . . . . . . . . . 3.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Elements of a Phase I/II Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47 47 49 51 52

ix

x

Contents

4 Model-Based Designs for Identification of Optimal Biological Dose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 EffTox Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Logistic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 A Bayesian Phase I/II Trial Design for Immunotherapy . . . . . . . . . . 4.4 Nonparametric Isotonic Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53 53 56 59 64 69

5 Model-Assisted Designs for Identifying the Optimal Biological Dose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 BOIN12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 U-BOIN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71 71 79 86 88

Part III Phase II Trials 6 Single Arm Phase II Clinical Trial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Simon’s Two-Stage Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Limitations of Simon’s Two-Stage Design . . . . . . . . . . . . . . . . . . . . . 6.3 Approaches Based on Posterior and Predictive Probabilities . . . . . . 6.4 Bayesian Optimal Phase II Design (BOP2) . . . . . . . . . . . . . . . . . . . . . 6.5 Time-to-Event Bayesian Optimal Phase II (TOP) Trial Design . . . . 6.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91 91 94 95 100 112 117 117

7 Randomized Phase II Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Frequentist Randomized Two-Arm Phase II Designs . . . . . . . . . . . . . 7.3 Bayesian Randomized Two-Arm Phase II Design . . . . . . . . . . . . . . . 7.3.1 Randomized BOP2 Design with Binary, Ordinal, and Co-primary Endpoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Randomized BOP2 Design with the Survival Endpoint . . . . 7.3.3 Two-Stage Screened Selection Design (SSD) . . . . . . . . . . . . . 7.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

119 119 122 125 125 133 136 145 145

Part IV Master-Protocol Trials 8 Introduction to Basket Trials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Basket Design Based on the Binomial-Test . . . . . . . . . . . . . . . . . . . . . 8.2.1 Two-Stage Basket Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 Optimal Two-Stage Basket Design . . . . . . . . . . . . . . . . . . . . . . 8.3 Bayesian Hierarchical Model (BHM)-Based Basket Design . . . . . . . 8.3.1 Bayesian Hierarchical Model (BHM) . . . . . . . . . . . . . . . . . . . 8.3.2 Calibrated Bayesian Hierarchical Model (CBHM) . . . . . . . .

149 149 151 152 158 163 164 166

Contents

8.3.3 Optimal Bayesian Hierarchical Model (OBHM) . . . . . . . . . . 8.4 Basket Trials with the Multisource Exchangeability Model (MEM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Bayesian Basket Trial Design with Predictive Sample Size Determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 Single-Drug Non-randomized and Multiple-Individual-Drug Randomized Basket Trials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7 Extensions to Randomized and Confirmatory Basket Designs . . . . . 8.7.1 Randomized Basket Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7.2 Confirmatory Basket Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Platform Trials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Introduction to Platform Trials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Adaptive Platform Design with the Binary Endpoint . . . . . . . . . . . . . 9.2.1 Controlled Multi-arm Platform Design . . . . . . . . . . . . . . . . . . 9.2.2 A Bayesian Drug Combination Platform Trial Design . . . . . 9.3 Adaptive Platform Design with the Survival Endpoint . . . . . . . . . . . . 9.3.1 Adaptive Phase I/II Platform Design for the Immunotherapy Drugs . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2 Adaptive Phase II Platform Design with the Survival Endpoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.3 A Rolling-Arms Platform Design . . . . . . . . . . . . . . . . . . . . . . . 9.3.4 A Bayesian Platform Trial Design with Borrowing from Historical Control Data . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 More Discussions on Adding New Arms to Ongoing Clinical Trials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.1 Randomization Procedures and Interim Analyses When Adding New Treatment Arms . . . . . . . . . . . . . . . . . . . . 9.4.2 Impact on Error Rates of Multi-arm One-Stage Platform Trials When Adding New Treatment Arms . . . . . . 9.4.3 Statistical Considerations of Phase III Platform Trials from a Single Institutional Perspective . . . . . . . . . . . . . . . . . . 9.4.4 Planning a 2+2-Experimental Arm Trial that Controls the PWER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

168 176 185 189 190 191 192 197 198 201 201 203 204 206 211 211 219 225 231 236 238 243 248 258 259 260

Part I

Phase I Trials

Chapter 1

Introduction to Phase I Dose-Finding Clinical Trials

1.1 Phase I Dose-Finding Trials Phase I clinical trial designs aim to identify the maximum tolerated dose (MTD) of a new drug, which is defined as the dose with a dose-limiting toxicity (DLT) probability that is closest to the target probability. Traditionally, phase I dose-finding designs can generally be classified as algorithm-based and model-based. Algorithm-based designs use simple, prespecified rules to govern dose escalation and de-escalation. Examples include the 3 + 3 design, the “rolling-six” design (Skolnik et al., 2008), the biased-coin design (Wei, 1978), and its variations. Despite the 3 + 3 design poorly identifying the MTD, it is the most common phase I trial design used in practice, mainly because of its transparency and simplicity. Model-based designs have been proposed that improve upon the performance of algorithm-based designs. The most well-known model-based design is the continual reassessment method (CRM) (O’Quigley et al., 1990). The CRM begins with a prior dose-toxicity curve and continuously updates this curve based on the accruing toxicity outcomes from patients in the trial. Each new patient or cohort of patients are assigned to the dose that corresponds to the estimated DLT probability closest to the prespecified target, where the estimated DLT probabilities are derived from the updated dose-toxicity curve. Various extensions of the CRM have been proposed, including dose escalation with overdose control (EWOC) (Babb et al., 1998), timeto-event CRM (Cheung & Chappell, 2000), Bayesian model averaging CRM (Yin & Yuan, 2009), Bayesian data-augmentation CRM (Liu et al., 2013), partial order CRM (Wages et al., 2011), and bivariate CRM (Braun et al., 2002). Cheung (2011) in his book titled “Dose finding by the continual reassessment method” provides a comprehensive review of the CRM and related methods. Compared with algorithmbased designs, model-based designs typically have superior operating characteristics. However, because model-based designs require repeated model fitting and estimation, many practitioners view them as conceptually and computationally complex,

© Springer Nature Singapore Pte Ltd. 2023 H. Pan and Y. Yuan, Bayesian Adaptive Design for Immunotherapy and Targeted Therapy, https://doi.org/10.1007/978-981-19-8176-0_1

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1 Introduction to Phase I Dose-Finding Clinical Trials

as if the decisions are coming from a “black box.” This perception has likely limited the use of model-based designs, such as the CRM, in practice. Recently, a new class of designs, known as model-assisted designs (Yuan et al., 2019), have been proposed to combine the simplicity of algorithm-based designs with the superior performance of model-based designs. Model-assisted designs use a model for efficient decision making like model-based designs, whereas their dose escalation and de-escalation rules can be tabulated before the onset of a trial as with algorithm-based designs. Unlike the model-based design, such as the CRM, which assumes a dose-toxicity curve across all doses, the model-assisted design often models only local data (ie, the data observed at the current dose), typically using a binomial model, which renders it possible to enumerate the dose escalation and de-escalation rules before the trial begins. Examples of model-assisted designs include the modified toxicity probability interval (mTPI) design (Ji et al., 2010) and its variation mTPI-2 (Guo et al., 2017), Bayesian optimal interval (BOIN) design (Liu & Yuan 2015), Keyboard design (Yan et al., 2017), BOIN combination design and phase I/II design (Zhou et al., 2021), and Keyboard combination design (Pan et al., 2020). Recently, Mu et al. (2019) proposed a generalized BOIN design that handles toxicity grades, binary or continuous toxicity endpoints, under a unified framework (Yuan et al., 2022). In this chapter, we briefly introduce some popular phase I designs, including the CRM, mTPI, BOIN, and Keyboard designs. The following notations will be used throughout this chapter. We use d1 , . . . , d J to denote the J prespecified doses of the new drug that is under investigation in the trial, p j to denote the DLT probability that corresponds to d j , and φ to denote the target DLT probability for the MTD. We use n j to denote the number of patients who have been assigned to d j , and y j to denote the number of DLTs observed at d j , j = 1, . . . , J . Therefore, at a particular time point during the trial, the observed data are D = {D j , j = 1, . . . , J }, where we call D j = (n j , y j ) the “local” data observed at dose level j.

1.1.1 Continual Reassessment Method (CRM) The CRM (O’Quigley et al., 1990) is a model-based dose-finding approach that assumes a parametric model for the dose-toxicity curve. As information accrues during the trial, the dose-toxicity curve is reevaluated by updating the estimates of the unknown model parameters and the corresponding DLT probability at each investigational dose. The current estimates for the DLT probabilities are used to determine the dose allocation for the next patient or cohort of patients. One commonly used model for the CRM is the power model (also known as the empiric model) that assumes ex p(α)

p j (α) = a j

, f or j = 1, . . . , J,

(1.1)

1.1 Phase I Dose-Finding Trials

5

where α is the unknown parameter and 0 < a1 < · · · < a J < 1 are prior guesses for the DLT probability at each dose. The a = {a j , j = 1, . . . , J } is called the “skeleton” of the CRM in literature. Under the power model in (1.1), the likelihood function for α is L(α|D) =

J 

ex p(α) y j

{a j

ex p(α) n j −y j

} {1 − a j

}

,

(1.2)

j=1

and then the posterior distribution of α is given by f (α|D) = 

L(α|D) f (α) L(α|D) f (α)dα

(1.3)

and thus, the posterior mean estimate for p j is calculated as  pˆ j =

ex p(α)

aj



L(α|D) f (α) dα, L(α|D) f (α)dα

(1.4)

where f (α) denotes the prior distribution for α, e.g., N (0, 2). Upon updating the posterior mean estimate of the DLT probability at each dose, the next patient or cohort of patients is assigned to the dose with an estimated DLT probability closest to the target φ. That is, the next patient or cohort of patients is assigned to dose level j ∗ such that j ∗ = argmin | pˆ j − φ| j∈(1,...,J )

(1.5)

The trial continues in this manner until the prespecified sample size is exhausted. At that point, the MTD is selected as the dose with an estimated DLT probability closest to the target φ. In practice, typically dose escalation and de-escalation are restricted to one level at a time, and a safety stopping rule is included such that the trial is terminated if Pr( p1 > φ|D) > 0.9 (ie, the lowest dose d1 has more than 90% chance of being above the MTD).

1.1.2 Modified Toxicity Probability (mTPI) Design The mTPI design (Ji et al., 2010) requires the investigator to prespecify three clinically indifference intervals, the underdosing interval (0, δ1 ), the proper dosing interval (δ1 , δ2 ), and the overdosing interval (δ2 , 1). For example, given a target rate of φ= 0.20, the three intervals may be elicited as (0, 0.15), (0.15, 0.25), and (0.25,1), respectively.

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1 Introduction to Phase I Dose-Finding Clinical Trials

The mTPI design assumes the following Beta-Binomial model. y j |n j , p j ∼ Binom(n j , p j ) p j ∼ Beta(1, 1) ≡ Unif(0, 1),

(1.6)

thus, the posterior distribution arises as p j |D j ∼ Beta(y j + 1, n j − y j + 1), for j = 1, . . . , J.

(1.7)

Unlike the CRM, which models the toxicity across doses using the power model in (1.1), the mTPI models the toxicity locally only at the current dose d j . To determine the next dose, based on D j , the mTPI design uses the unit probability mass (UPM) corresponding to each of the three intervals, which are defined as UPM1 = Pr( p j ∈ (0, δ1 )|D j )/δ1 , UPM2 = Pr( p j ∈ (δ1 , δ2 )|D j )/(δ2 − δ1 ), UPM3 = Pr( p j ∈ (δ2 , 1)|D j )/(1 − δ2 ),

(1.8)

That is, the UPM is the posterior probability that p j lies in the corresponding interval divided by the length of that interval. Suppose j is the current dose level. The mTPI design determines the next dose as follows (see Fig. 1.1a): • If UPM1 = max{UPM1 , UPM2 , UPM3 }, escalate the dose to level j + 1. • If UPM2 = max{UPM1 , UPM2 , UPM3 }, stay at the current dose level j. • If UPM3 = max{UPM1 , UPM2 , UPM3 }, deescalate the dose to level j − 1. Because the three UPMs can be determined for all possible outcomes D j = (n j , y j ), the dose escalation and de-escalation rules can be tabulated before the trial starts, which makes the mTPI design easy to implement in practice. The trial continues until the prespecified sample size is exhausted. At that point, the MTD is selected based on isotonic estimates of the p j that are calculated using the pooled adjacent violators algorithm. As the decision of dose escalation and de-escalation is based only on the local data at the current dose, it is possible that the dose transition oscillates between a safe dose and the next higher dose that is toxic. To avoid that issue, similar to the CRM design, the mTPI design includes a dose exclusion/safety stopping rule: if Pr( p j > φ|n j , y j ) > 0.95, dose level j and higher are excluded from the trial. If the lowest dose is excluded, the trial is stopped for safety. One deficiency of using the UPM to guide dose escalation is that it lacks clear interpretation and leads to a high risk of overdosing patients. To exemplify the problem, consider a trial with a target toxicity rate of 0.20 and underdosing, proper dosing, and overdosing intervals of (0, 0.17), (0.17, 0.23), and (0.23, 1), respectively. Suppose at a certain stage of the trial, the observed data indicate that the posterior probabilities of the underdosing interval, proper dosing interval, and overdosing interval are 0.01, 0.09, and 0.9, respectively. That is, there is a 90% chance that the current dose is

1.1 Phase I Dose-Finding Trials

7

overdosing patients and only a 9% chance that the current dose is properly dosing patients. Despite such dominant evidence of overdosing, the mTPI design stays at the same dose for treating the next patient or patient cohort, since the UPM that corresponds to the proper dosing interval is the largest. In particular, the UPM that corresponds to the proper dosing interval is 0.09/(0.23 − 0.17) = 1.5, whereas the UPM that corresponds to the overdosing interval is 0.9/(1 − 0.23) = 1.17. Due to this safety concern, mTPI is not recommended for practical use.

1.1.3 Keyboard Design The Keyboard design (Yan et al., 2017) resolves the overdosing issue of the mTPI by defining a series of equal-width dosing intervals (or keys) that correspond to the potential locations of the true DLT probability of a particular dose and using the interval (or key) with the highest posterior probability to guide dose escalation and de-escalation; see Fig. 1.1b. Specifically, the Keyboard design starts by specifying a proper dosing interval I∗ = (δ1 , δ2 ), referred to as the target key, and then populates this interval toward both sides of the target key, forming a series of keys of equalwidth that span the range of 0 to 1. For example, given the proper dosing interval or target key of (0.25, 0.35), on its left side, we can form 2 keys of width 0.1, ie, (0.15, 0.25) and (0.05, 0.15), and on its right side, we can form 6 keys of width 0.1, ie, (0.35, 0.45), (0.45, 0.55), (0.55, 0.65), (0.65, 0.75), (0.75, 0.85), and (0.85, 0.95). We denote the resulting intervals/keys as I1 , . . . , I K . As all keys have equal width and must be within [0, 1]; some DLT probability values at the two ends (e.g., < 0.05 or > 0.95 in the example) may not be covered by keys because they are not long enough to form a key. As explained in Yan et al. (2017), ignoring these “residual” DLT probabilities at the two ends does not pose any issue for decision making of dose escalation and de-escalation. To make the decision of dose escalation and de-escalation, given the observed data D j = (n j , y j ) at the current dose level j, the Keyboard design identifies the interval Imax that has the largest posterior probability, ie, Imax = {Ik : argmax {Pr( p j ∈ Ik | D j )}}, k∈{1,...,K }

(1.9)

which can easily be evaluated based on p j ’s posterior distribution given by Eq. (1.7) assuming that p j follows a Beta-Binomial model (1.6). Imax represents the interval where the true value of p j is most likely located, referred to as the “strongest” key. Graphically, the strongest key is the one with the largest area under the posterior distribution curve of p j (see Fig. 1.1b). If the strongest key is on the left (or right) side of the target key, that means that the observed data suggest that the current dose is most likely underdosing (or overdosing), and thus, dose escalation (or de-escalation) is needed. If the strongest key is the target key, the observed data support that the current dose is most likely to be in the proper dosing interval, and thus, it is desirable to retain the current dose for treating the next patient. In contrast, the UPM used by

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1 Introduction to Phase I Dose-Finding Clinical Trials (a)

(b)

(c)

Fig. 1.1 Illustration of (a) the modified toxicity probability interval (mTPI) design, (b) the keyboard design, and (c) the Bayesian optimal interval (BOIN) design. The curves are the posterior distributions of p j . To determine the next dose, the mTPI design compares the values of the 3 unit probability masses (UPMs), whereas the keyboard design compares the location of the strongest key with respect to the target key. Bayesian optimal interval compares the observed dose-limiting toxicity (DLT) rate at the current dose with the prespecified dose escalation boundary λe and deescalation boundary λd

the mTPI design does not have such an intuitive interpretation and tends to distort the evidence for overdosing, as described previously. Suppose j is the current dose level. The Keyboard design determines the next dose as follows: • If the strongest key is on the left side of the target key, then escalate the dose to level j + 1. • If the strongest key is the target key, then stay the current dose level j. • If the strongest key is on the right side of the target key, then de-escalate the dose to level j − 1. The trial continues until the prespecified sample size is exhausted, and the MTD is selected based on isotonic estimates of p j as described previously. During conducting the trial, the Keyboard design imposes the dose exclusion/early stopping rule such that if Pr( p j > φ|n j , y j ) > 0.95 and n j ≥ 3, dose level j and higher are eliminated from the trial, and the trial is terminated if the lowest dose is eliminated, where Pr( p j > φ|n j , y j ) > 0.95 is evaluated based on the posterior distribution (1.7). Similar to the mTPI design, the dose escalation and de-escalation rules of the Keyboard design can be tabulated before the trial begins, making it easy to implement in practice. As the location of the strongest key approximately indicates the mode of the posterior distribution of p j , the Keyboard design can be approximately viewed as a posterior-mode-based Bayesian dose-finding method. This makes the Keyboard design a new method different from the UPM-based mTPI design, despite some structural similarities between the two designs (e.g., partitioning the toxicity probability into intervals and the dose escalation and de-escalation rules can be pretabulated). Pan et al. (2020) showed that the Keyboard design is optimal under the 0–1 loss, long-memory coherent, and can be extended to drug-combination trials.

1.1 Phase I Dose-Finding Trials

9

1.1.4 Bayesian Optimal Interval (BOIN) Design Compared with the mTPI and Keyboard designs, the BOIN design (Liu & Yuan, 2015) is more straightforward and transparent. The dose escalation and de-escalation in the BOIN design is determined simply by comparing the observed DLT rate at the current dose with a pair of fixed dose escalation and de-escalation boundaries. Specifically, let pˆ j = y j /n j denote the observed DLT rate at the current dose, and λe and λd denote the predetermined dose escalation and de-escalation boundaries. Suppose j is the current dose level. The BOIN design determines the next dose as follows (see Fig. 1.1c): • If pˆ j ≤ λe , then escalate the dose to level j + 1. • If pˆ j > λd , then de-escalate the dose to level j − 1. • Otherwise (i.e., λe < pˆ j ≤ λd ), stay at the current dose level j. The trial continues until the prespecified sample size is exhausted. At that point, select the MTD based on the isotonic estimates of DLT probabilities as described previously. During the trial conduct, the BOIN design imposes a dose elimination (or overdose control) rule as follows: if Pr( p j > φ|n j , y j ) > 0.95 and n j ≥ 3, dose level j and higher are eliminated from the trial, and the trial is terminated if the lowest dose is eliminated, where Pr( p j > φ|n j , y j ) > 0.95 is evaluated based on the posterior distribution in (1.7). To determine the dose escalation and de-escalation boundaries (λe , λd ), the BOIN design requires the investigator(s) to specify φ1 , which is the highest DLT probability that is deemed to be underdosing such that dose escalation is required, and φ2 , which is the lowest DLT probability that is deemed to be overdosing such that dose deescalation is required. Liu & Yuan (2015) provided general guidance to specify φ1 and φ2 and recommended default values of φ1 = 0.6φ and φ2 = 1.4φ for general use. When needed, the values of φ1 and φ2 can be calibrated to achieve a particular requirement of the trial at hand. For example, if more conservative dose escalation is required, setting φ2 = 1.2φ may be appropriate. Given φ1 and φ2 and assuming a noninformative prior (ie, a priori the current dose is equally likely to be below, equal to, or above the MTD), the optimal escalation and de-escalation boundaries (λe , λd ) that minimize the decision error of dose escalation and de-escalation arises as     1−φ 1 log 1−φ log 1−φ 1−φ2  , λd =    λe = (1.10) φ(1−φ1 ) φ2 (1−φ) log φ1 (1−φ) log φ(1−φ2 ) Table 1.1 provides the dose escalation and de-escalation boundaries (λe , λd ) for commonly used target DLT rate φ using the recommended default values φ1 = 0.6φ and φ2 = 1.4φ. For example, given the target DLT rate φ = 0.25, the corresponding escalation boundary λe = 0.197 and the de-escalation boundary λd = 0.298. That is, escalate the dose if the observed DLT rate at the current dose pˆ j ≤ 0.197 and de-escalate the dose if pˆ j > 0.298. It has been shown that λe and λd are the bound-

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1 Introduction to Phase I Dose-Finding Clinical Trials

Table 1.1 Values of λe and λd under the BOIN design for different target toxicity rates Interval boundaries Target toxicity rate φ 0.15 0.2 0.25 0.3 0.35 0.4 λe λd

0.118 0.179

0.157 0.238

0.197 0.298

0.236 0.358

0.276 0.419

0.316 0.479

aries corresponding to the Bayes factors, and thus, the resulting BOIN design is optimal with desirable finite-sample and large-sample properties, i.e., long-memory coherence and consistency (Liu & Yuan, 2015). One interesting note is that the decision rule of the BOIN (with the noninformative prior) has an appearance of the classical frequentist design and only involves the observed DLT rate. This is common in Bayesian statistics. Many well-established Bayesian methods (e.g., estimation for normal linear regression models) result in the same estimators as for the frequentist approach when noninformative priors are used. Actually, the BOIN can also be derived as a frequentist design, and its decision rule is equivalent to using the likelihood ratio test to determine dose escalation/deescalation (Liu & Yuan, 2015), providing another way to prove its optimality. Having both Bayesian and frequentist interpretations is a strength of the BOIN, making it appealing to wider audiences. In contrast, the mTPI and Keyboard designs only have a Bayesian interpretation and require specifying priors and calculating posterior distributions.

1.1.5 Deviation from the Planned Design Cohort size is one important component of phase I clinical trials and is typically prespecified in the trial protocol. However, during trial implementation, the cohort size often deviates from the planned one, which shifts the schedule of the interims. Park et al. (2021) reviewed 45 phase 1 dose-escalation trials published between January 2017 and May 2018 in 3 peer-reviewed journals (Journal of Clinical Oncology, Clinical Cancer Research, and Cancer), combined with a simulation study to systematically investigate the association of cohort size deviation with the operating characteristics of the trials. The results showed that when cohort size deviation was random, it had little association with the performance of novel model-based and model-assisted designs (mean reduction in the percentage of correct selection of the maximum tolerated dose was 0.87% point for the CRM and 0.84% points for the BOIN). When the cohort size deviation was informative and made based on the observed data on toxicity (e.g., if dose-limiting toxicity was observed, the size of the next or current cohort was reduced or expanded), the variation of the design performance increased. The range of the change in the percentage of correct selection was 3.7–1.3% points for the CRM and 4.5–0% points for the BOIN. Thus, the authors made conclusions that when novel phase I clinical trial designs are used, some cohort size deviation is acceptable and has little association with the performance of the designs.

1.2 Challenges of Immunotherapies, Targeted Therapies and New Methods

11

1.1.6 Software The software for implementing the CRM can be freely available at the MD Anderson software download website https://biostatistics.mdanderson.org/SoftwareDownload/ SingleSoftware/Index/81. The R code for implementing the mTPI design is available at https://biostatistics.mdanderson.org/SoftwareDownload/SingleSoftware/Index/72. The software for the BOIN design can be available in three ways, including a standalone graphical user interface-based Windows desktop program freely available from the MD Anderson software download website https://biostatistics.mdanderson.org/ SoftwareDownload/SingleSoftware/Index/99, an R Shiny online app freely available at www.trialdesign.org, and an R package BOIN available from the R CRAN. The gBOIN can be implemented by an R package UnifiedDoseFinding. The Keyboard design can be implemented using the Shiny online app freely available at www.trialdesign.org or an R package Keyboard from the R CRAN.

1.2 Challenges of Immunotherapies, Targeted Therapies and New Methods The paradigm for phase I clinical trial design, like the above-introduced designs, was initially established in the era of cytotoxic chemotherapies, for which toxicities were often acute and ascertainable in the first cycle of therapy. Over the past decade, noncytotoxic therapies such as molecularly targeted therapies and immunotherapies have entered the clinic. The problem of late-onset toxicity is particularly common and important in the era of targeted therapy and immunotherapy. A recent study reported that in 36 clinical trials of molecularly targeted agents, more than half of the grade 3 or 4 toxicity events occurred after the first treatment cycle (PostelVinay et al., 2011). Immune-related toxicity is often of late-onset. For instance, endocrinopathies have been observed between post-treatment weeks 12 and 24 (Weber et al., 2015; June et al., 2017). Late-onset toxicity is also associated with conventional radiochemotherapy, which may occur several months after treatment. However, most existing designs, like the above-introduced methods in the previous section, require the DLT evaluation of all the previously enrolled cohorts to be completed before they can make a treatment assignment for the next cohort. We refer to this type of designs as complete-data designs. To account for late-onset toxicity, it is imperative to use a relatively long toxicity assessment window (e.g., over multiple treatment cycles) to define the dose-limiting toxicity (DLT) such that all DLTs relevant to the dose escalation and MTD determination are captured. This, however, causes a major logistic difficulty when conducting phase I trials. For example, if the DLT takes up to 8 weeks to evaluate and the accrual rate is 1 patient/week, on average 5 new patients will be accrued while waiting to evaluate the previous 3 patients’ outcomes. The question is: how can new patients receive timely treatment when the previous patients’ outcomes are pending? The

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1 Introduction to Phase I Dose-Finding Clinical Trials

same difficulty arises with rapid accrual. Suppose that the DLT of a new agent can be assessed in the first 28-day cycle; if the accrual rate is 8 patients/28 d, then on average 5 new patients will accrue while waiting to evaluate the previous 3 patients’ outcomes, and we must determine how to provide them with timely treatment. This logistic difficulty persists throughout the trial and cripples most existing novel adaptive designs, such as the continuous reassessment method (CRM) and mTPI, etc. To make a real-time decision of dose assignment, these designs require that the DLT is quickly ascertainable, such that by the time of enrolling the next new cohort of patients or, patients previously enrolled have completed their DLT assessment. If some of the enrolled patients’ DLT data are pending, these designs have difficulty informing a real-time decision of dose assignment for the new patients. One possible approach to circumvent this difficulty is to suspend accrual after each cohort and wait until the DLT data for the already accrued patients have cleared before enrolling the next new cohort. This approach of repeatedly interrupting accrual, however, is highly undesirable and often infeasible in practice. It delays treatment for new patients and slows down the trial. Several phase I designs have been proposed to allow for continuous accrual and real-time dose assignment for new patients when some previous patients’ DLT data are still pending due to late-onset toxicity or rapid accrual. A number of model-based designs and algorithm-based designs have been proposed to address this issue. Cheung and Chappell (2000) proposed the time-to-event continual reassessment method (TITE-CRM), a model-based design, where the likelihood of each patient is weighted by his/her follow-up proportion. Taking a different perspective, Yuan and Yin (2011) and Liu and others (2013) treated the pending DLT data as missing data problem, and proposed the expectation-maximization algorithm and Bayesian data augmentation method to facilitate real-time decision making. These model-based designs (called DA-CRM) yield excellent operating characteristics, but their use in practice have been limited because they are often perceived by practitioners as difficult to understand, due to the black-box-style of decision making, and complicated to implement, because of the requirement of repeated model fitting and estimation. Thus, in practice, the rolling six (R6) design (Skolnik et al., 2008) and TITE-BION design (Yuan et al., 2018) are often used. The rolling six (R6) design is a modification of the 3 + 3 design that allows for continuous accrual of up to 6 patients when some of the patients’ DLT data are pending. Specifically, given that 3–6 patients have been treated at the current dose, the R6 enumerates all possible outcomes shown in Fig. 1.2, (i.e., DLT/no DLT/pending) from these patients and provides the corresponding decision rule of dose assignment for the new patients. For example, among 3 patients treated, if 1 has DLT, 1 has no DLT, and 1 has a pending outcome, the R6 assigns the next cohort to the same dose. The main advantage of the R6 is its transparency and simplicity. Implementation of the R6 does not require complicated model fitting and estimation. Users only count the number of patients with DLTs, the number of patients without DLTs, and the number of patients with pending outcomes, and then use the decision table to determine the dose assignment for the next new cohort. However, as an algorithmbased design, the R6 inherits the drawbacks of the 3 + 3 design such as low accuracy

References

13

Fig. 1.2 Decision rules of rolling six (R6) design

for MTD identification, treating a large proportion of patients at low (potentially subtherapeutic) doses, and inability to target a specific DLT rate for the MTD (Zhao, 2011). Subsequently, in Sect. 2.1 model-based TITE-CRM, DA-CRM, and fractional CRM (fCRM) designs will be introduced. In Sect. 2.2, we will introduce a modelassisted design, TITE-BOIN. Software to implement these methods will be described.

References Babb, J., Rogatko, A., & Zacks, S. (1998). Cancer phase I clinical trials: Efficient dose escalation with overdose control. Statistics in Medicine, 17(10), 1103–1120. Braun, T. M. (2002). The bivariate continual reassessment method: Extending the CRM to phase I trials of two competing outcomes. Controlled Clinical Trials, 23(3), 240–256. Cheung, Y. K., & Chappell, R. (2000). Sequential designs for phase I clinical trials with late-onset toxicities. Biometrics, 56(4), 1177–1182. Cheung, Y. K. (2011). Dose finding by the continual reassessment method. CRC Press.

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Guo, W., Wang, S. J., Yang, S., Lynn, H., & Ji, Y. (2017). A Bayesian interval dose-finding design addressing Ockham’s razor: mTPI-2. Contemporary Clinical Trials, 58, 23–33. Ji, Y., Liu, P., Li, Y., & Nebiyou Bekele, B. (2010). A modified toxicity probability interval method for dose-finding trials. Clinical Trials, 7(6), 653–663. June, C. H., Warshauer, J. T., & Bluestone, J. A. (2017). Is autoimmunity the Achilles’ heel of cancer immunotherapy. Nature Medicine, 23, 540–547. Liu, S., Yin, G., & Yuan, Y. (2013). Bayesian data augmentation dose finding with continual reassessment method and delayed toxicity. The Annals of Applied Statistics, 7(4), 1837. Liu, S., & Yuan, Y. (2015). Bayesian optimal interval designs for phase I clinical trials. Journal of the Royal Statistical Society: Series C: Applied Statistics, 507–523. Mu, R., Yuan, Y., Xu, J., Mandrekar, S. J., & Yin, J. (2019). gBOIN: A unified model-assisted phase I trial design accounting for toxicity grades, and binary or continuous end points. Journal of the Royal Statistical Society: Series C (Applied Statistics), 68(2), 289–308. O’Quigley, J., Pepe, M., & Fisher, L. (1990). Continual reassessment method: A practical design for phase 1 clinical trials in cancer. Biometrics, 33–48. Skolnik, J. M., Barrett, J. S., Jayaraman, B., Patel, D., & Adamson, P. C. (2008). Shortening the timeline of pediatric phase I trials: The rolling six design. Journal of Clinical Oncology, 26(2), 190–195. Pan, H., Lin, R., Zhou, Y., & Yuan, Y. (2020). Keyboard design for phase I drug-combination trials. Contemporary Clinical Trials, 92, 105972. Park, M., Liu, S., Yap, T. A., & Yuan, Y. (2021). Evaluation of deviation from planned cohort size and operating characteristics of phase 1 trials. JAMA Network Open, 4(2), e2037563–e2037563. Postel-Vinay, S., Gomez-Roca, C., Molife, L. R., Anghan, B., Levy, A., Judson, I., & Paoletti, X. (2011). Phase I trials of molecularly targeted agents: Should we pay more attention to late toxicities. Journal of Clinical Oncology, 29(13), 1728–1735. Wei, L. J. (1978). The adaptive biased coin design for sequential experiments. The Annals of Statistics, 6(1), 92–100. Wages, N. A., Conaway, M. R., & O’Quigley, J. (2011). Continual reassessment method for partial ordering. Biometrics, 67(4), 1555–1563. Weber, J. S., Yang, J. C., Atkins, M. B., & Disis, M. L. (2015). Toxicities of immunotherapy for the practitioner. Journal of Clinical Oncology, 33, 2092–2099. Yan, F., Mandrekar, S. J., & Yuan, Y. (2017). Keyboard: A novel Bayesian toxicity probability interval design for phase I clinical trials. Clinical Cancer Research, 23(15), 3994–4003. Yin, G., & Yuan, Y. (2009). Bayesian model averaging continual reassessment method in phase I clinical trials. Journal of the American Statistical Association, 104(487), 954–968. Yuan, Y., Lin, R., Li, D., Nie, L., & Warren, K. E. (2018). Time-to-event Bayesian optimal interval design to accelerate phase I trials. Clinical Cancer Research, 24(20), 4921–4930. Yuan, Y., Lin, R., & Lee, J. J. (2022). Model-assisted Bayesian designs for dose finding and optimization. Boca Raton, FL: Chapman and Hall/CRC. Yuan, Y., Lee, J. J., & Hilsenbeck, S. G. (2019). Model-assisted designs for early-phase clinical trials: Simplicity meets superiority. JCO Precision Oncology, 3, 1–12. Yuan, Y., & Yin, G. (2011). Robust EM continual reassessment method in oncology dose finding. Journal of the American Statistical Association, 106(495), 818–831. Zhao, L., Lee, J., Mody, R., & Braun, T. M. (2011). The superiority of the time-to-event continual reassessment method to the rolling six design in pediatric oncology phase I trials. Clinical Trials, 8(4), 361–369. Zhou, Y., Lin, R., Kuo, Y. W., Lee, J. J., & Yuan, Y. (2021). BOIN suite: A software platform to design and implement novel early-phase clinical trials. JCO Clinical Cancer Informatics, 5, 91–101.

Chapter 2

Phase I Designs for Late-Onset Toxicity

2.1 Model-Based Phase I Designs for the Late-Onset Toxicity 2.1.1 Time-to-Event CRM (TITE-CRM) As introduced in Chap. 1, the original CRM design is a model-based approach for dose escalation decision-making based on an underlying dose-toxicity relationship. This model uses all available data from previously enrolled participants to determine the next participants dose and subsequently the maximum tolerated dose (MTD). The CRM requires all participants currently on the trial to be followed up for the entire observation window before their data can be used to estimate the next participants dose. This is not always feasible in practice, especially if the observation period is very long, such as in molecularly targeted therapies and immunotherapies trials. The time-to-event continual reassessment method (TITE-CRM) (Cheung et al., 2000) is a modification of the original CRM developed to address the issue of late-onset toxicities. In addition to those participants who complete follow-up or experience a toxicity, it accounts for participants who have not been followed up completely. Data are weighted according to how much information each participant provides. Participants can be continually recruited, thus reducing the overall duration of the trial, and all information is used to assign new participants to the best dose. The resulting weighted dose toxicity model incorporates both fully and partially observed participants. A comprehensive practical guidance for designing, setting up, and running TITE-CRM trials, from the grant application to dissemination with two phase I radiotherapy oncology trials as examples can be found (Werkhoven et al., 2020). When designing a trial using the TITE-CRM, the following parameters are needed: • A maximum sample size of N participants to be recruited; • A target toxicity level, φ, denoting an acceptable probability of dose-limiting toxicity (DLT); © Springer Nature Singapore Pte Ltd. 2023 H. Pan and Y. Yuan, Bayesian Adaptive Design for Immunotherapy and Targeted Therapy, https://doi.org/10.1007/978-981-19-8176-0_2

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2 Phase I Designs for Late-Onset Toxicity

• J dose levels to be explored, labeled d1 , . . . , d J ; • A DLT observation time period of length T, also called the DLT window; • A prior guess of the DLT probability at each dose, also called the skeleton α = {α1 , . . . , α J }; • A functional form for the dose-toxicity curve, for example, the power function ex p(α) , j = 1, . . . , J ; and p j = F(d j , α) = d j • A prior distribution for the model parameter(s) of the dose-toxicity curve, for example, a normal distribution for the parameter α of the power function, p(α) = N (0, 2). At the start of the trial, dose labels d1 , . . . , d J are calculated by solving F(d j , E(α)) = α j where E(α) is the prior mean of α. This choice of the dose labels guarantees that at the start of the trial the model fits the probabilities given by the prior estimates α. In other words, these re-scaled values assigned to d j s are not the actual clinical dose but instead a numeric value determined to promote good fit of the mode during the study. After each participant is recruited, these estimates are updated to the posterior estimates. The crucial aspect of the TITE-CRM is that it uses the actual amount of follow-up, u i , collected on subject i, as well as an indicator as to whether the subject experienced DLT (yi = 1) or is still being followed without DLT (yi = 0). The likelihood used by the TITE-CRM is a weighted binomial likelihood L n (α) =

n 

F(d j , α) yi {1 − w(u i )F(d j , α)1−yi }

(2.1)

i=1

in which n is the number of up-to-now enrolled subjects, and 0 ≤ w(u i ) ≤ 1 is a weight given to subject i based upon how long they have been followed. Theoretically, the weight function w(u i ) should reflect the expected distribution of DLT times, which is often unknown and difficult to estimate. A simple solution, which is shown to work well in many settings, is to assume DLTs occur uniformly over the DLT window interval [0, T ], so that w(u i ) = u i /T . To be more specific, this weight function is chosen to account for the proportion of the observation period that each currently enrolled patient has been observed. It is linear in the follow-up time u i until the end of the observation window: ⎧ ⎪ ⎨0 u i < 0 (2.2) w(u i ) = uTi 0 ≤ u i ≤ T ⎪ ⎩ 1 T ≤t Other weight functions could be used if DLTs are more likely to occur early or late during follow-up; the functional form of w(u i ) could also be adaptively changed during the study. An example of a weight function that takes into account an acute DLT period and a late-onset DLT period is given below (Werkhoven et al., 2020).

2.1 Model-Based Phase I Designs for the Late-Onset Toxicity

17

It gives half the weight to the acute DLT period and half to the remainder of the observation time window:

w(u i ) =

⎧ 0 ⎪ ⎪ ⎪ ⎨ ui

2tlate

1 ⎪ + ⎪ ⎪ ⎩2 1

u i −tacute 2(tlate −tacute )

ui < 0 0 ≤ u i < tacute tacute ≤ t < tlate tlate ≤ t

(2.3)

for instance, tacute = 30 days, and tlate = 120 days. For each dose j, the plug-in estimate of the toxicity probability is then calculated using αˆ n , the posterior mean of α based on the enrolled n subjects, as follows: exp(αˆ n )

pˆ j = F(d j , αˆ n ) = d j

.

(2.4)

The MTD is defined as the dose level j ∗ such that pˆ j ∗ is maximized but remains below the target toxicity level, φ, or is the closest to the φ, depending on the definition used. When a dose decision is reached, the current best guess for the MTD is calculated based on all data accrued so far. The trial continues to recruit participants until one of the stopping rules is satisfied (for example, the maximum sample size is reached) and the MTD is declared. By using a weighted likelihood, subjects can be enrolled continuously throughout the trial without recruitment pauses, thereby shortening the duration of the trial relative to the CRM, as well as allowing every patient to be enrolled as soon as they are eligible. A comparison of the TITE-CRM and the rolling six design (Zhao et al., 2011) (a rule-based design allowing 6 at a dose level) concluded that the TITE-CRM was superior as it treated all available participants, identified the MTD more accurately, and did not increase the probability of exposing participants to toxic doses. Similarly the TITE-CRM has been compared with the 3 + 3 design and was found to be superior in its performance (Normolle et al., 2006). A comprehensive comparison of the TITECRM and the TITE-BOIN was conducted (Yuan et al., 2018), in which the details will be introduced in the next section. Example for time-to-event continual reassessment method Motivating trial: A Phase I trial at St. Jude Children’s Research Hospital (St. Jude) seeks to determine the MTD for intravenous liposomal irinotecan (Onivyde), given together with a fixed dose of talazoparib, for children with recurrent solid malignancies. Talazoparib was administered at a fixed dose of 600 mcg/m2 , given orally twice on day 1, the second dose given approximately 12 h after the first dose, and once daily on days 2–6 of each cycle (21 d). Onivyde was administered intravenously over 60 min on Days 1 and 8, and on Day 1, the onivyde dose will immediately follow the oral talazoparib dose of each treatment cycle. A DLT is defined as one that occurs during the first two cycles (42 d from the therapy is initiated). The study will examine five dose levels of Onivyde: 70, 90, 120, 180, and 220 mg. The trial will enroll 24

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2 Phase I Designs for Late-Onset Toxicity

patients with an estimated accrual rate of 10 children per year. The investigators have defined the MTD as the dose with a DLT probability of no more than 0.25. The a priori DLT probabilities (skeleton) for the five doses are, in dose order, 0.08, 0.16, 0.25, 0.35, and 0.46, which will be introduced below by using R code to show how to elicit this skeleton. The first child is assigned to 90 mg, as the lowest dose of 70 mg is included in the trial only as a fall-back dose should deescalation be required early in the trial. We used the power model described earlier to model the DLT rate of each dose and assumed uniformly distributed DLT times for our weight function. Software: There are two generic packages in R CRAN that can be used to implement the TITE-CRM designs, the dfcrm package via functions of titecrm for conducting and titesim for simulating trials, and the trialr package via stan_crm function. Additionally, a SAS tool to design and manage TITE-CRM trials was also developed and the SAS program can be accessed via https://sph.umich.edu/ccb/titeresources.html. We will use the dfcrm package to implement the TITE-CRM design for the above motivating trial. The dfcrm package implementations. From the motivating example, we can set the following parameters: > PI target library (dfcrm) > target delta mtd0 # i n i t i a l DLT rates with indifference intervals [0.20 , 0.30]. > prior prior [1] 0.08397349 0.15674102 0.25000000 0.35450043 0.46034311

That is, the skeleton is estimated to be α = {α1 , · · · , α5 } = {0.08, 0.16, 0.25, 0.35, 0.46}. We then use the following code to conduct 2000 simulations for the above setting. The parameters in the following code include the following: n=24 refers to sample size of the trial, x0=2 indicates the starting dose (in our motivating trial, the starting dose is dose level 2 of 90 mg for the Onivyde), nsim refers to the number of simulations, obswin is the observation window with respect to which the MTD is defined, model="empiric" means that the empiric or power model is used, accrual="poisson" means that the Poisson process is used to simulate patient arrivals, scheme="linear indicates that the weight function is linear/uniform, and seed refers to the random number generator for replicating the simulation results. restrict=TURE applies the two restrictions during the trials to avoid (1) skipping doses in escalation and (2) escalation immediately after a toxic outcome (i.e., incoherent escalation). The final parameter rate is the patient arrival rate

2.1 Model-Based Phase I Designs for the Late-Onset Toxicity

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which is the expected number of arrivals per observation window. Since the patient accrual rate is 10 patients per year, that is, given the 42 d of the DLT evaluation 10 × 42 ≈ 1.15 patients; therefore, we set window, there would be approximately 365 the rate=1.15 in the following code. > library (dfcrm) > foo