![viii PREFACE
recurring concern in my previous CRM trials, and so is included as an extension of
the CRM (Chapter 11). All in all, while this is not intended to be a cookbook, the
inclusion of materials is based on their practical relevance.
This book does not cover dose finding in all possible clinical settings. In fact,
it has a singular focus on the simplest and the most common phase I trial setting,
where the study endpoint is defined as a binary outcome and the subjects are assumed
to come from a homogeneous population. I make no mention of the concerns with
multiple toxicity and the gradation of severe toxicities. The topic of individualized
dosing is omitted. While some basic ideas of dose finding using both efficacy and
toxicity are outlined in Chapter 13, the discussion is brief and does not do full justice
to this fast-growing area. All these are important topics in which I am intellectually
interested. Their omission, however, is mainly due to my limited practical experience
in dealing with these “nonstandard” situations in real dose finding studies; dealing
with these issues simply from a methodological and theoretical viewpoint does not
fit my intent of writing a practical book (although I think such a book is interesting
in its own right and hope someone more qualified than I will deliver it). I do have
a word or two to add from a methodological and theoretical viewpoint here, if not
already alluded to in the book’s final chapter (Section 14.4, to be precise). First,
a complete theoretical framework is crucial for these nonstandard methods to be
successfully translated into actual practice. In this book, I try to explicate possible
pathological behaviors (e.g., incoherence and rigidity) of some CRM modifications
for the simplest setting; it is reasonable to infer that these pathologies will multiply
for methods more complex than the CRM for the more complicated clinical settings.
Solid theoretical investigation will help us navigate the potential pitfalls. I also hope
the theoretical framework developed in this book for the simplest case will prove
useful when extended to the complicated settings. Second, and more specifically,
I think stochastic approximation offers partial solutions (albeit mostly theoretical)
to many of these nonstandard dose finding settings. This is why I close this book
with a chapter where I try to connect and compare the CRM with the rich stochastic
approximation literature.
The last points I just made give a hint about my methodological and theoretical
interests. I hope that this book will in some way simulate research in the CRM and
general dose finding methods, despite its practical nature. As I try to present the
CRM and the dose finding criteria at a rigorous level, and to cover the CRM literature
as comprehensively as possible, I also hope this book can serve as an introduction
for those interested in doing research in this area. I taught a course on sequential
experimentation at Columbia University from an early unpublished version of this
book. This final manuscript is, in turn, adapted from the course notes, and is suitable
for use in a course on sequential experimentation or clinical trials.
There are several statistics books on dose finding. The two most popular ones
are the edited volumes by Chevret [26] and by Ting [105]. Both give surveys of
dose finding methods and are good introductions to the dose finding literature. By
comparison, this book is a single-authored work on a specific dose finding method,
which I think is necessary if we are to get down to the nuts and bolts of the method.
By writing a book on the CRM, I do not imply that it is the best method out there.](https://image.slidesharecdn.com/1112001-250531235121-8e6f58df/85/Dose-Finding-By-The-Continual-Reassessment-Method-Chapman-Hall-Crc-Biostatistics-Series-1st-Edition-Ying-Kuen-Cheung-12-320.jpg)
![Chapter 1
Introduction
The clinical development of a new drug or a new treatment proceeds through three
phases of testing in human subjects. Phase I trials are small studies that evaluate
safety and identify a safe dose range of the treatment. Once a dose range is cho-
sen, its therapeutic efficacy will be examined in a phase II trial. Regimens that are
shown promising in phase II trials will be moved to multi-institutional phase III clini-
cal trials for randomized comparison to standard treatments. The ultimate goal of this
entire process is to translate promising discoveries in the laboratory into new medical
procedures that can be used in the general clinical settings. This division of clinical
trials, however, may give an oversimplified picture of the actual drug development
process. Often, several phase I-II trial sequels of a drug, possibly with minor vari-
ations in the treatment schedule and patient populations, are needed before a phase
III trial is warranted. This process is necessarily iterative rather than linear, as the
phase I-II-III paradigm appears to suggest. In addition, the taxonomy of trials is not
universal across disciplines, and may include finer divisions such as phase IA, IB,
IIA, and IIB. The recent trend to combine phases of trials, the so-called combined
phase I/II trials and seamless phase II/III trials, renders further refinement of the drug
development process.
This having been said, the phase I-II-III paradigm provides a conceptual frame-
work for in-depth study of statistical methods. The subject matter of this book is dose
finding using the continual reassessment method (CRM). The CRM [78] is among
the first model-based designs for phase I cancer trials in which toxicity is the primary
study endpoint. The role of toxicity in early-phase cancer trials had long been a sub-
ject for discussion in the medical literature [93, 85]. In particular, for cytotoxic drugs,
toxicity serves as evidence that the drug has reached a level that does harm not only
to the cancer cells but also to a patient’s normal organs. In other words, a therapeutic
dose is expected to cause a significant amount of severe but reversible toxicities in
the cancer patient population. Therefore, a primary goal of phase I cancer trials is to
identify the so-called maximum tolerated dose (MTD). For other disorders such as
acute stroke and HIV, identifying the MTD is also a primary objective of early-phase
safety studies (usually called phase IB trials). In addition, dose finding is important
in phase II proof-of-concept trials where the goal is to identify a dose range with
demonstrated biological activity. This objective is usually achieved through the esti-
mation of the minimum effective dose (MED) [106, 27]. From a statistical viewpoint,
3](https://image.slidesharecdn.com/1112001-250531235121-8e6f58df/85/Dose-Finding-By-The-Continual-Reassessment-Method-Chapman-Hall-Crc-Biostatistics-Series-1st-Edition-Ying-Kuen-Cheung-21-320.jpg)
![4 INTRODUCTION
the MTD in safety studies and the MED in efficacy studies can be formulated in an
analogous way. Therefore, this book is relevant to the design of phase I and II dose
finding trials. Under the modernized paradigm, the dose finding principles discussed
here also address the design issues in the combined phase I/II trials, in which both
the safety and the efficacy endpoints are considered as co-primary (cf. Section 13.3).
Another advantage of dividing the drug development process into phases is that
by doing so, we can set a clear and manageable benchmark to achieve in a particular
study. This entails a clearly defined set of study endpoints and an interpretable study
objective. Since clinical trials are conducted in human subjects, each benchmark is
to be reached within certain ethical constraints. In particular, in phase I dose finding
studies, randomization is not entirely appropriate because it may expose subjects to
excessively high and toxic doses without sufficiently testing the lower doses. (Some
would also argue randomization exposes subjects to low and inefficacious doses,
although this aspect is apparently not as alarming.)
We illustrate these points using a bortezomib dose finding trial [62]. Bortezomib
is a proteasome inhibitor with proven activity in lymphoma. In the trial, bortezomib
was given in combination with the standard chemotherapy as a first-line treatment
for patients with diffuse large B cell or mantle cell non-Hodgkin’s lymphoma. Each
patient would receive up to six 21-day cycles of the treatment combination. Table 1.1
describes the five dose schedules of bortezomib tested in the trial. The primary safety
concerns related to bortezomib were neuropathy, low platelet count, and symptomatic
non-neurologic or non-hematologic toxicity. Toxicity was graded according to the
National Cancer Institute Common Terminology Criteria for Adverse Events [71],
with grade 3 or higher defined as dose limiting. Generally, a grade 3 toxicity is se-
vere but can be resolved by symptomatic treatment, whereas a grade 4 toxicity is
irreversible; toxic death due to the treatment is invariably defined as grade 5. The
primary endpoint of each patient was the indicator of whether any dose-limiting tox-
icity (DLT) was experienced at any time during the six cycles. The objective of the
trial was to determine the MTD, defined as a dose associated with a 25% DLT rate.
Table 1.1 gives the number of patients and the number of DLTs per dose in the borte-
zomib trial. The data show strong evidence that the highest dose is adequately safe:
we pool the observations in dose levels 4 and 5 by assuming an increasing dose–
toxicity relationship; based on 1 DLT out of 16 patients, we obtain a 95% confidence
upper bound of 0.26 for the DLT probability.
Table 1.1 The bortezomib trial [62]: dose schedules of bortezomib, sample size (n), and the
number of DLT (z) at each dose
Level Dose and schedule within cycle n z
1 0.7 mg/m2 on day 1 of each cycle 0 0
2 0.7 mg/m2 on days 1 and 8 of each cycle 0 0
3 0.7 mg/m2 on days 1 and 4 of each cycle 4 0
4 1.0 mg/m2 on days 1 and 4 of each cycle 9 1
5 1.3 mg/m2 on days 1 and 4 of each cycle 7 0](https://image.slidesharecdn.com/1112001-250531235121-8e6f58df/85/Dose-Finding-By-The-Continual-Reassessment-Method-Chapman-Hall-Crc-Biostatistics-Series-1st-Edition-Ying-Kuen-Cheung-22-320.jpg)
![6 INTRODUCTION
This book is organized into three parts. Part I (Chapters 2–6) contains the back-
ground and introductory material of the CRM. Specifically, Chapter 2 provides the
clinical background, outlines the problem of dose finding in the context of several
real trial examples, and reviews the dose finding literature. Chapter 3 introduces the
basic approach of the CRM and presents its major modifications. The method will
be developed along with a description of an R package ‘dfcrm’. Chapter 4 presents
a unified framework for dose–toxicity models used in the CRM. Chapters 5 and 6,
respectively, discuss the theoretical and empirical properties of the CRM. The ob-
jective of Part I is for the readers to develop a basic understanding of the CRM and
be able to implement the method using a simple R code. Readers familiar with the
basic CRM methodology are also encouraged to review the materials, as they are
reorganized and presented in a unified framework in this book.
Part II (Chapters 7–10) details the calibration process of the CRM based on the
notation and the theory introduced in Part I. Chapter 7 introduces a system of design
parameters involved in the CRM, and classifies them into two categories: clinical
parameters and model parameters. The subsequent chapters then present fine-tuning
techniques of the model parameters: the initial guesses of the toxicity probabilities
(Chapter 8), the prior distribution of the model parameter (Chapter 9), and the initial
design of a two-stage CRM (Chapter 10). The objective of Part II is for the readers
to develop the ability to design a “good” CRM trial within a reasonable timeline.
Part III (Chapters 11–14) contains a variety of advanced topics related to the
CRM. Chapter 11 presents the TITE-CRM to deal with situations in which the toxic-
ity outcome is defined with respect to a nontrivial duration. Chapter 12 gives a critical
review of CRM using multiparameter models. Chapter 13 considers situations where
the CRM is an inappropriate design, and puts forward some alternatives. Chapter 14
connects the CRM and modern dose finding trials to the large literature of stochastic
approximation. The objective of Part III is to stimulate further research in the CRM
and general dose finding methodology.
The materials in this book are presented at a level that requires college algebra
and some basic calculus concepts. Sections marked with “†” in the table of contents
contain technical details that may be skipped without affecting the reading of the
other chapters. Exposition in the book will be supplemented by illustrations of the
usage of R functions in the ‘dfcrm’ package. While some basic knowledge of R will
enhance the reading experience, proficiency in R is not required. Interested readers
can find out more information about R from the Comprehensive R Archive Network
(CRAN) [83] at
http://www.r-project.org.](https://image.slidesharecdn.com/1112001-250531235121-8e6f58df/85/Dose-Finding-By-The-Continual-Reassessment-Method-Chapman-Hall-Crc-Biostatistics-Series-1st-Edition-Ying-Kuen-Cheung-24-320.jpg)
![8 DOSE FINDING IN CLINICAL TRIALS
denote the toxicity probability associated with dose level k for k = 1,...,K. The
MTD from a trade-off-for-efficacy perspective, denoted by γ, is defined as the largest
dose level with toxicity probability no greater than a prespecified threshold θ, that
is, γ ≡ max{k : pk ≤ θ}.
Both the surrogacy and the trade-off perspectives define the MTD with respect to
a target toxicity rate θ, and as such formulate dose finding as a percentile estimation
problem. However, from a statistical viewpoint, it is generally easier to estimate ν
than γ (if ν is well defined) by making use of monotonicity of the dose–toxicity
curve. Therefore, it is in some cases pragmatic to take ν as the operative objective of
a trial, even though the toxicity endpoint is generally not a surrogate for efficacy for
noncytotoxic drugs. In practice, it is important to discern the appropriate objective
for a given clinical setting, and the degree of tolerance in terms of the target θ.
Apparently, these decisions need to be made on a trial-by-trial basis.
Example 2.1 (acute ischemic stroke). A number of statins, when administered
early after stroke in animal models, have demonstrated neuroprotective effects
in a dose-dependent manner, with the greatest effects at the highest doses. The
NeuSTART (Neuroprotection with Statin Therapy for Acute Recovery Trial) drug
development program aimed to translate preclinical research and test the role of high-
dose statins in stroke patients. In a phase IB dose finding study under the NeuSTART
program [34], high-dose lovastatin was given to patients for 3 days after stroke fol-
lowed by a standard dose for 27 days. The primary safety concerns for giving high-
dose lovastatin included elevated liver enzyme; a toxicity was said to occur if the peak
enzyme levels at any posttreatment time points exceeded a prespecified threshold.
There were five test doses in the NeuSTART and the trial objective was to identify a
dose with toxicity rate closest to 10%, that is, θ = 0.10.
Example 2.2 (cancer prevention). Polyphenon E (Poly E) is a tea catechin extract
that is thought to block tumor promotion by inhibiting cell proliferation and inducing
cell cycle arrest and apoptosis. The multiple mechanisms of Poly E make it a good
candidate agent for chemoprevention. On the other hand, the agent has been shown
toxic and causing mortality in female beagle dogs after an overnight fast. Toxicity
generally involved the gastrointestinal (GI) system, producing vomiting and damage
to the lining of the GI tract, with hemorrhage and necrosis apparent at autopsy. A
Poly E trial was conducted in women with a history of hormone receptor-negative
breast cancer. Three different doses of Poly E were administered to subjects over
6 months. The study objective was to find the MTD defined as a dose that causes
25% DLT during the six-month period, where a DLT here meant any grade 2 or
higher toxicity that would persist for at least one week or requires stopping the study
drug. As a secondary objective, this study included biologic correlates such as tissue-
based biomarkers and mammography. Therefore, some subjects were randomized to
receive a placebo, to which the identified MTD would be compared.
Example 2.3 (early neuro-rehabilitation). Early rehabilitation was conjectured to](https://image.slidesharecdn.com/1112001-250531235121-8e6f58df/85/Dose-Finding-By-The-Continual-Reassessment-Method-Chapman-Hall-Crc-Biostatistics-Series-1st-Edition-Ying-Kuen-Cheung-26-320.jpg)
![10 DOSE FINDING IN CLINICAL TRIALS
Dose
Toxicity
probability
MTD
0
0.25
0.5
0.75
1
(a) Bortezomib trial
Dose
Toxicity
probability
MTD
0
0.25
0.5
0.75
1
(b) Poly E trial
Dose
Toxicity
probability
MTD
0
0.25
0.5
0.75
1
(c) ASCENT
Figure 2.1 Plausible shapes of dose–toxicity curve on a conceptual continuous dose range
in three studies and the corresponding MTD, defined as a dose associated with a 25% target
toxicity rate. Definition 2.1 is used to define MTD in (a) and (b), and Definition 2.2 in (c).
dose decreases. In contrast, because low-grade GI toxicities are not uncommon in
subjects with a history of cancer in the Poly E trial, there is a nonzero intercept
in Figure 2.1b. However, it is generally believed that the dose–toxicity curve will be
strictly increasing around the 25th percentile in both trials, and thus ν is well defined.
It is easy to verify that ν = γ when the dose–toxicity curve is a continuous and
strictly increasing function of dose. Thus, if a continuum of test doses is available
for the trial, it will be a practical choice to use ν as the operative objective, because
estimating ν is generally easier than estimating γ. On the other hand, most trials in
practice allow only a discrete number of test doses; in which case, Definition 2.1
may yield a slightly more aggressive recommendation than Definition 2.2, because
ν ≥ γ. Therefore, the choice between ν and γ depends on whether it is acceptable
to be (slightly) more aggressive than the target θ, given the clinical factors such as
the nature of treatment, the severity of the disease, and the seriousness of the study
endpoint.
As alluded to earlier, the CRM is motivated by applications in cancer trials with
a surrogacy view. Thus, the book will naturally focus on the estimation of ν using
the CRM. However, Chapter 13 will explore situations in which the CRM is not
applicable, and introduce an alternative approach that uses γ as trial objective. Also,
as the MED in efficacy trials may be defined analogously to ν, the CRM may be
applicable to dose finding in phase II efficacy trials. However, this book will focus
on the MTD finding by the CRM according to the method’s originally intended use;
the design strategy for the MED can be derived by analogy.
2.2 An Overview of Methodology
This section gives a brief overview of the development of the dose finding literature
since the late 1980s so as to put the CRM in a historical light. Limited by the scope
of this book, the review will be cursory. Interested readers can find additional topics
in the edited volume by Chevret [26] and the article by Le Tourneau et al. [59].](https://image.slidesharecdn.com/1112001-250531235121-8e6f58df/85/Dose-Finding-By-The-Continual-Reassessment-Method-Chapman-Hall-Crc-Biostatistics-Series-1st-Edition-Ying-Kuen-Cheung-28-320.jpg)
![AN OVERVIEW OF METHODOLOGY 11
The 3+3 algorithm. Traditionally, a 3+3 algorithm is used to dictate dose escalation
and to approach the eventual recommended dose. The method starts the trial at a
low dose (e.g., one-tenth of LD10 in mice) and escalates after every three to six
patients per dose; the recommended dose is defined as the largest dose with fewer
than two patients experiencing a predefined DLT during the first course of treatment.
Table 2.2 describes the dose escalation rules of this algorithm. In practice, there may
be slight variations from institution to institution. For example, in order to obtain
preliminary information about efficacy of the drug, it is common to treat additional
patients (usually 6 to 12) at the identified MTD.
Table 2.2 Escalation rules at a given dose with cumulative sample size n and total number z
of DLT in accordance with the 3+3 algorithm
n z Action
3 0 Escalate to the next higher dose
3 1 Treat three additional patients at the current dose
6 1 Escalate to the next higher dose
3 or 6 ≥ 2 Stop escalation and terminal triala
aThe MTD is estimated by the dose immediately below the terminating dose.
The 3+3 algorithm has historically been the most widely used phase I trial design.
Its main advantage is simplicity. Since the dose escalation rules can be tabulated
before a trial starts, the clinical investigators can make dose decisions during a trial
without help from a statistician.
However, the algorithm has two major shortcomings. First, due to a low starting
dose and the conservative escalation scheme, the 3+3 algorithm tends to treat many
patients at low and inefficacious doses. Since phase I cancer trials typically enroll
patients as subjects, as opposed to healthy volunteers, there is an intent to treat the
subjects at a therapeutic dose that is likely higher than the lowest test dose. As such,
the algorithm is discordant with the therapeutic intent of these trials [84, 55]. Second,
the 3+3 algorithm has no statistical justification. There is no intrinsic property in the
method to stop escalation at any given percentile, and thus the distribution of the
recommended MTD depends arbitrarily on the underlying dose–toxicity curve and
the number of test doses [97]. This deficiency is due to the fact that there is no
correspondence between the method and any quantitative definition of the MTD, as
the 3+3 algorithm does not involve an explicit choice of the target θ. As poor dose
selection in the early phase will likely be carried over to its subsequent developmental
phases, the use of the 3+3 algorithm will have lingering financial implications and
adverse scientific consequences. And thus, the simplicity of the method does not
justify its widespread application.
Stochastic approximation. Although discussions on phase I designs for cancer trials
can be traced back to the 1960s [93], a formal statistical formulation of the MTD
appeared at a much later time. Among the earliest discussions was Anbar [2], who
in 1984 proposed the use of stochastic approximation [86] in phase I trials. The](https://image.slidesharecdn.com/1112001-250531235121-8e6f58df/85/Dose-Finding-By-The-Continual-Reassessment-Method-Chapman-Hall-Crc-Biostatistics-Series-1st-Edition-Ying-Kuen-Cheung-29-320.jpg)
![12 DOSE FINDING IN CLINICAL TRIALS
procedure assigns dose sequentially by the recursion
xi+1 = xi −
1
ib
(Yi −θ) (2.1)
for some prespecified constant b > 0, where xi is the dose assigned to patient i, and
Yi is the toxicity indicator. The stochastic approximation is a nonparametric method
in that it does not assume any parametric structure on the dose–toxicity relationship.
Let π(x) = Pr(Yi = 1 | x) denote the probability of toxicity at dose x. Under very mild
assumptions of π(x), the dose sequence {xi} generated by (2.1) will converge with
probability 1 to a dose x∗ such that π(x∗) = θ.
However, the use of recursion (2.1) implicitly assumes a continuum of doses
is available for testing in the trial. This is not always feasible in practice. In many
situations such as the bortezomib trial, there may be no natural scale of dosage;
rather, “dose” is composed of drug dosage and treatment schedule. There are also
other difficulties from a statistical viewpoint. First, the stochastic approximation has
been shown to be inferior to model-based methods such as the maximum likelihood
recursion for binary data [112, 50]. Second, the choice of the constant b in (2.1) has
a large impact on the performance of the procedure. As a consequence, the stochastic
approximation has seldom been used in dose finding trials. We will return to this in
Chapter 14.
Up-and-down designs. Subsequent to Anbar’s work [2], Storer [98] considered the
up-and-down schemes originally described by Wetherill [110]. An example of the
up-and-down design, which Storer called design D, is to enroll patients in groups of
three; then, escalate dose for the next group if there is no toxicity in the most recent
group, deescalate if there is more than one toxicity in the group, and stay at the same
level if one of three patients has toxicity. By Markov chain representation, design
D can be shown to sample around a dose that causes toxicity with a probability
θ = 0.33. The group size and the decision rules in the up-and-down schemes can
apparently be modified to accommodate other target θ; for example, for a target
θ = 0.20, one will intuitively enroll patients in groups of five. Also, combinations of
schemes can be applied in stages. Storer, in particular, suggested using a group size
of one initially in the trial and switching to design D upon the first observed toxicity.
The idea here is to move the trial quickly through the low doses so as to avoid treating
many patients at low and inefficacious doses.
Durham et al. [32] proposed a randomized version of the up-and-down rule for
any target at or below the 50th percentile, that is, θ ≤ 0.50. The design deescalates
the dose for the next patient if the current patient has toxicity, and escalates according
to a biased coin with probability θ/(1−θ) if there is none. The method is thus called
a biased coin design.
In a trial that uses design D or the biased coin design for dose escalation, when
the enrollment is complete, we may naturally estimate the dose–toxicity curve using
logistic regression [98] or isotonic regression [99], and estimate the MTD by the
100θth percentile of the fitted curve. The consistency conditions for the estimation of
the MTD hold, because the design points {xi} “spread out” under this type of random
walk sampling plans. That is, technically, it can be shown that the number of patients](https://image.slidesharecdn.com/1112001-250531235121-8e6f58df/85/Dose-Finding-By-The-Continual-Reassessment-Method-Chapman-Hall-Crc-Biostatistics-Series-1st-Edition-Ying-Kuen-Cheung-30-320.jpg)
![AN OVERVIEW OF METHODOLOGY 13
treated at each dose will grow indefinitely as sample size increases. It can also be
shown by the properties of random walk that the asymptotic distribution of dose
allocation has a mode near the target MTD. However, from a design viewpoint, the
“memoryless” property of random walk may cause ethical difficulties: Since these
up-and-down rules make dose decisions based only on the most recent patient or
group of patients at the current dose, previously accrued data are ignored and a trial
will likely reescalate to a dose that appears toxic.
Model-based designs. In brief, a model-based design makes dose decisions based
on a dose–toxicity model, which is being updated repeatedly throughout a trial
as data are accrued. The continual reassessment method (CRM), proposed by
O’Quigley et al. [78] in 1990, is the first model-based design in the modern dose
finding literature. Several model-based designs proposed since 1990 share a similar
notion with the CRM. One example is the escalation with overdose control (EWOC)
design [4], which takes the continual reassessment notion but estimates the MTD
with respect to an asymmetric loss function that places heavier penalties on over-
dosing than underdosing; see (2.2) below. A list of model-based methods is given in
Section 2.3.
Most model-based methods take the myopic approach by which dose assignment
is optimized with respect to the next immediate patient without regard to the future
patients. For example, the EWOC at each step minimizes the Bayes risk with respect
to the loss function:
xi+1 = argmin
x
Ei
α(ν −x)+
+(1 −α)(x−ν)+
(2.2)
where x+ = max(x,0) and Ei(·) denotes expectation computed with respect to the
posterior distribution of the MTD ν, given the first i observations. As the EWOC
is intended to control overdose, the loss function (2.2) should be specified with a
value of α, that is, α 0.50. More recently, Bartroff and Lai [6] take a stochastic
optimization approach that minimizes the global risk and propose to choose the doses
{x1,x2,...,xN} sequentially so as to mininize
E
(
N
∑
i=1
α(ν −xi)+
+(1 −α)(xi −ν)+
)
,
where the expectation is taken with respect to the joint distribution of xis and ν.
Such sequential optimization is implemented by backward induction and requires dy-
namic programming which can be computationally intensive. However, this approach
presents a new direction for model-based design and warrants further research.
The model-based approach facilitates borrowing strength from information
across doses through the parametric assumptions on the dose–toxicity curve. This is
especially important in early-phase dose finding trials where sample sizes are small
and informational content is low [41]. In addition, since these methods allow starting
a trial at a dose higher than the lowest level, the in-trial allocation tends to concentrate
around the target dose.
Many model-based designs take a Bayesian approach. They update sequentially](https://image.slidesharecdn.com/1112001-250531235121-8e6f58df/85/Dose-Finding-By-The-Continual-Reassessment-Method-Chapman-Hall-Crc-Biostatistics-Series-1st-Edition-Ying-Kuen-Cheung-31-320.jpg)
![14 DOSE FINDING IN CLINICAL TRIALS
the uncertainty about the dose–toxicitycurve with respect to the posteriordistribution
given the interim data. This approach is by nature automated, so far as the posterior
computations can be efficiently programmed and reproduced. An advantage of such
automation is that a carefully calibrated dose–toxicity model can handle unplanned
contingencies in a manner that is coherent with the trial objective. In contrast, we
may imagine the predicament arising with the 3+3 rule in a trial where there are
two toxic outcomes among 7 patients at a dose; this contingency can be caused by
overaccrual due to administrative delays, and is not uncommon in practice.
Several practical difficulties may hinder the use of a model-based design. First,
there is skepticism among the clinicians because of the “blackbox” approach of these
designs. Second, there is the perception that the success of the method is sensitive
to the choice of the dose–toxicity model. Third, as a model-based design requires
specialized computations, the clinical team will need to interact regularly with the
study statistician for interim dose decisions. Such interaction may be perceived as
adding unnecessary burdens on both parties, in light of the fact that the standard 3+3
algorithm requires minimal statistical inputs.
This book attempts to address the second difficulty with a focus on the CRM.
By theoretical and empirical arguments, we will see that the method’s performance
does not depend on correctness of the model specification. This book also partially
addresses the third difficulty by illustrating how a CRM design can be calibrated and
implemented in practice. The ultimate goal is to alleviate the statistician’s burden
during the planning stage and the conduct of the trial. This endeavor is facilitated by
availability of software. This book focuses on the R package ‘dfcrm’ for the CRM
and its major variants. Software for some model-based designs is also available to
public access; see Table 2.3.
Table 2.3 Some software links for model-based approaches
Description Section Reference
Escalation with overdose control (EWOC)a 2.2 [4]
Late-onset toxicity monitor using predicted risksb 11.4.2 [8]
Modified CRM on a continuous dosage rangec 12.4.4 [81]
Phase I/II dose finding based on efficacy and toxicityb
13.3 [102]
ahttp://www.sph.emory.edu/BRI-WCI/ewoc.html
bhttp://biostatistics.mdanderson.org/softwaredownload
chttp://www.cancerbiostats.onc.jhmi.edu/software.cfm
Algorithm-based designs. Because of the above-mentioned difficulties with the
model-based designs, there seems to be a renewed interest in the algorithm-based
designs since the late 1990s. Generally, an algorithm-based design prescribes a set
of escalation rules for any given dose without regard to the outcomes at other doses.
As a result of the independence among observations across doses, the rules can be
tabulated and made accessible to the clinical investigators before a trial starts.
The 3+3 algorithm is the most prominent example of algorithm-based designs.
Efforts have recently been made to extend this traditional method so as to obtain](https://image.slidesharecdn.com/1112001-250531235121-8e6f58df/85/Dose-Finding-By-The-Continual-Reassessment-Method-Chapman-Hall-Crc-Biostatistics-Series-1st-Edition-Ying-Kuen-Cheung-32-320.jpg)
![BIBLIOGRAPHIC NOTES 15
well-defined statistical properties. In particular, Cheung [18] formulates dose finding
as a multiple testing problem and introduces a class of stepwise test procedures that
operate in a manner similar to the 3+3 algorithm. This approach has practical ap-
peal because clinicians are familiar with the 3+3 algorithm. Chapter 13 gives further
details of the stepwise procedures.
Ji et al. [49] propose a class of up-and-down designs that make dose decisions
based on the posterior toxicity probability intervals. Specifically, the parameter space
of the toxicity probability pk at dose k is partitioned into three sets: for some prespec-
ified constants K1,K2 0,
Θk,E = {pk −θ −K1σk}
Θk,S = {−K1σk ≤ pk −θ ≤ K2σk}
Θk,D = {pk −θ K2σk}
so that Θk,E ∪Θk,S ∪Θk,D = [0,1], where σk is the posterior standard deviation of pk.
Escalation from a current dose, say dose k, will take place if the posterior probability
of Θk,E is largest among the three sets. Similarly, deescalation occurs if Θk,D is the
most probable event according to the posterior distribution of pk. Otherwise, the next
dose will remain at level k. An important difference between this design and the
random walk up-and-down is that the posterior interval uses all observations accrued
to a dose to make a dose decision and avoids the memoryless problem of a random
walk design.
2.3 Bibliographic Notes
The practice and design of phase I trials are discussed in the medical community
by Schneiderman [93], Carbone et al. [13], Ratain et al. [85, 84], and Kurzrock and
Benjamin [55]. Korn [53] discusses the relevance of MTD in noncytotoxic targeted
cancer treatments. Discussion of dose finding strategies in the other disease areas
is relatively sporadic. Fisher et al. [38] present some phase I and II trial designs in
the context of acute stroke MRI trials, and make a case against the use of the 3+3
algorithm in the phase I safety studies. Cheung et al. [23] make analogous arguments
for early-phase trials in patients with amyotrophic lateral sclerosis. The dose finding
design of NeuSTART is reported in Elkind et al. [34] as a case study.
Robbins and Monro [86] introduce the first stochastic approximation method,
which has been studied extensively and has motivated a large number of subsequent
modifications. (See for example Sacks [92], Venter [107], Lai and Robbins [57, 58],
and Wu [112, 113].) In the more recent literature, Lai [56] gives a thorough review of
the advances of stochastic approximation. Cheung [19] draws a specific tie between
this area and modern dose finding methods.
Lin and Shih [64] study the operating characteristics of a class of A+B designs
that include the 3+3 algorithm as a special case. The theoretical properties of the
biased coin design are established by Durham and colleagues [29, 30, 31].
Several model-based designs have been proposed since the 1990s. These include
the Bayesian decision-theoretic design [111], the logistic dose-ranging strategy [70],](https://image.slidesharecdn.com/1112001-250531235121-8e6f58df/85/Dose-Finding-By-The-Continual-Reassessment-Method-Chapman-Hall-Crc-Biostatistics-Series-1st-Edition-Ying-Kuen-Cheung-33-320.jpg)
![16 DOSE FINDING IN CLINICAL TRIALS
and the Bayesian c-optimal design [44]. The CRM has generated a large literature
and will be reviewed in the next chapter. Chapter 12 will comment on two CRM-like
dose finding designs: the curve-free method [40] and the isotonic design [63]. For
the EWOC, Zacks et al. [117] prove that the method is Bayesian-feasible, Bayesian-
optimal, and consistent under the assumption that the specified dose–toxicity model
is correct. These Bayesian criteria are introduced in the previous work by Eichhorn
and Zacks [33].
2.4 Exercises and Further Results
Exercise 2.1. Discuss the MTD objective (surrogacy versus trade-off perspectives)
for cancer prevention in the context of the Poly E trial.
Exercise 2.2. Definition 2.1 formulates dose finding as estimating a percentile on a
dose–toxicity curve. Another possible alternative to define the MTD according to the
surrogacy perspective is
ν′
= argmin
k
|π−1
(pk)−π−1
(θ)|.
Show that |ν − ν′| ≤ 1, that is, the two definitions can differ by at most one dose
level. Discuss why ν′ may not be applicable for the bortezomib trial.
Exercise 2.3. By computer simulations, generate the outcomes of a trial using the
3+3 algorithm with K = 5 doses and true toxicity probabilities p1 = 0.02, p2 = 0.04,
p3 = 0.10, p4 = 0.25, and p5 = 0.50. Observe the recommended MTD. Repeat the
simulations 1000 times, and record the distribution of the recommended MTD in the
1000 simulated trials.](https://image.slidesharecdn.com/1112001-250531235121-8e6f58df/85/Dose-Finding-By-The-Continual-Reassessment-Method-Chapman-Hall-Crc-Biostatistics-Series-1st-Edition-Ying-Kuen-Cheung-34-320.jpg)
![18 THE CONTINUAL REASSESSMENT METHOD
and a one-parameter logistic function
F(x,β) =
exp(a0 +βx)
1 +exp(a0 +βx)
for −∞ x ∞
where the intercept a0 is a fixed constant. Another common dose–toxicity model is
the hyperbolic tangent function [78, 67]
F(x,β) =
tanhx+1
2
β
for −∞ x ∞.
To ensure an increasing dose–toxicity relationship, the parameter β in these models
is restricted to taking on positive values. The positivityconstraint could present some
difficulty in estimation, especially when the sample size is small. Hence, it is useful
to consider the following parameterization:
empiric: F(x,β) = xexp(β)
for 0 x 1 (3.2)
logistic: F(x,β) =
exp{a0 +exp(β)x}
1 +exp{a0 +exp(β)x}
for −∞ x ∞ (3.3)
hyperbolic tangent: F(x,β) =
tanhx+1
2
exp(β)
for −∞ x ∞ (3.4)
under which the parameter β is free to take on any real values while F(x,β) is strictly
increasing.
The original formulation of the CRM uses a Bayesian approach by which the
model parameter β is assumed random and follows a prior distribution G(β). We
will focus on the normal prior distribution, that is,
β ∼ N(β̂0,σ2
β )
where β̂0 and σ2
β are, respectively, the prior mean and variance.
3.2.3 Dose Labels
An important point about the CRM is that the numerical dose labels d1,...,dK are
not the actual doses administered, but rather are defined on a conceptual scale that
represents an ordering of the risks of toxicity. Consider the dose schedules used in
the bortezomib trial (Table 1.1). The first three levels prescribe bortezomib at a fixed
dose 0.7 mg/m2 with increasing frequency, whereas the next two levels apply the
same frequency with increasing bortezomib dose. While it is reasonable to assume
that the toxicity risk increases with each level, there is no natural unit for dose (e.g.,
mg/m2) in this application. Similarly, in the ASCENT trial (Example 2.3), “dose” is
a composite of timing and duration of physical therapy given. In these examples, it
is artificial to assume the dose–toxicity curve π(x) is well defined on a continuous](https://image.slidesharecdn.com/1112001-250531235121-8e6f58df/85/Dose-Finding-By-The-Continual-Reassessment-Method-Chapman-Hall-Crc-Biostatistics-Series-1st-Edition-Ying-Kuen-Cheung-36-320.jpg)
![ONE-STAGE BAYESIAN CRM 21
3.2.5 Normal Prior on β
Now, suppose that β has a normal prior distribution with mean β̂0. For the logistic
function (3.3), applying backward substitution (3.5) gives
dk =
logit(p0k)−a0
exp(β̂0)
where logit(p) = log{p/(1 − p)}. As a result, we have
F(dk,β) =
exp
h
a0 +exp(β −β̂0){logit(p0k)−a0}
i
1 +exp
h
a0 +exp(β −β̂0){logit(p0k)−a0}
i.
Since F(dk,β) depends on the parameter β only via β − β̂0, which is mean zero
normal, we may arbitrarily set β̂0 = 0 without affecting the computation. The logistic
model (3.3) is therefore invariant to the mean of a normal prior distribution. This
invariance property holds for the general class of dose–toxicity models described in
Chapter 4.
3.2.6 Implementation in R
The R package ‘dfcrm’ consists of functions for the implementation and the design
of the Bayesian CRM using the empiric (3.2) and logistic (3.3) models. In particular,
the function crm takes cumulative patient data and returns a dose for the next patient
according to the model-based estimate (3.7).
### Return the recommended dose level for patient 6
### based on data from five patients
library(dfcrm)
p0 - c(0.05,0.12,0.25,0.40,0.55) # initial guesses
theta - 0.25 # target toxicity rate
y - c(0, 0, 1, 0, 0) # toxicity indicators
lev - c(3, 5, 5, 3, 4) # dose levels
fooB - crm(p0,theta,y,lev,model=logistic,intcpt=3)
fooB$estimate # posterior mean of beta
[1] 0.2794614
fooB$mtd
[1] 4
fooB$doses # Dose labels
[1] -5.944439 -4.992430 -4.098612 -3.405465 -2.799329
The above R code illustrates the usage of crm for a trial with K = 5 test doses and
a target θ = 0.25; the function is applied to observations from the first 5 subjects who
receive dose levels 3, 5, 5, 3, and 4, where the third patient has a toxic outcome. The
one-parameter logistic function (3.3) is used when the argument model is specified](https://image.slidesharecdn.com/1112001-250531235121-8e6f58df/85/Dose-Finding-By-The-Continual-Reassessment-Method-Chapman-Hall-Crc-Biostatistics-Series-1st-Edition-Ying-Kuen-Cheung-39-320.jpg)
![TWO-STAGE CRM 23
toxic outcome is observed, the trial turns to the model-based CRM for dose assign-
ments. Because the 3+3 algorithm is familiar, there is an inclination to consider the
“group-of-three” initial design by which escalation takes place after every group of
three nontoxic outcomes. That is,
x1,0 = x2,0 = x3,0 = d1, x4,0 = x5,0 = x6,0 = d2, x7,0 = x8,0 = x9,0 = d3 ...
and so on. There is, however, no clear justification for using the group-of-three rule
apart from convention. In fact, this initial design is sometimes not in line with the
motivation of the CRM. We will study the calibration of the initial dose sequences in
Chapter 10.
3.3.2 Maximum Likelihood CRM
In response to the difficulty associated with the subjectivity of the Bayesian approach,
we may use maximum likelihood estimation in conjunction with the CRM [80]. The
idea is simple and analogous to the Bayesian CRM: with data observed in the first
i −1 patients, the dose for the next subject is computed as
xi = D̃1(Hi) = argmin
dk
|F(dk,β̃i−1)−θ| (3.11)
where β̃i−1 = argmaxβ Li−1(β) is the maximum likelihood estimate (mle) of β for
given Hi.
The R function crm also implements the maximum likelihood CRM through the
specification of the argument method:
### Compute the recommended dose level for patient 6
### using maximum likelihood CRM on the same data
### i.e., same values of p0, theta, y, lev
fooL - crm(p0,theta,y,lev,model=logistic,method=mle)
fooL$estimate # mle of beta
[1] 0.3142946
fooL$mtd
[1] 5
The function crm evaluates the maximum likelihood estimate of MTD (3.11) through
the specification of the method argument as “mle”. When method is not specified,
the Bayesian CRM is assumed.
Using (3.11) presupposes the existence of mle of β. For a one-parameter model
F, the mle β̃i−1 exists if and only if there is heterogeneity in the toxicity outcomes
among patients, that is, Yj = 0 and Yj′ = 1 for some j, j′ i. In the last R illustration,
there is one toxic outcome out of 5 patients, and thus β̃5 exists. In general, when one
plans to use the maximum likelihood CRM, it is necessary to consider a two-stage
design, which can be formed by replacing D1(Hi) with D̃1(Hi) in (3.10).
The posterior mean β̂i−1 and the mle β̃i−1 are generally different, thus leading](https://image.slidesharecdn.com/1112001-250531235121-8e6f58df/85/Dose-Finding-By-The-Continual-Reassessment-Method-Chapman-Hall-Crc-Biostatistics-Series-1st-Edition-Ying-Kuen-Cheung-41-320.jpg)
![24 THE CONTINUAL REASSESSMENT METHOD
to potentially different dose recommendations. For example, in the above R codes,
the Bayesian CRM chooses dose level 4 for the sixth patient, whereas the maximum
likelihood CRM chooses level 5. However, the posterior mean β̂5 = 0.279 and mle
β̃5 = 0.314 are only slightly different. Looking at the model-based estimates of the
toxicity probabilities reveals that both dose levels 4 and 5 are roughly equally apart
from the target θ = 0.25:
round(fooB$ptox,digits=2) # Posterior toxicity rates
[1] 0.01 0.03 0.08 0.18 0.33
round(fooL$ptox,digits=2) # Maximum likelihood estimates
[1] 0.01 0.02 0.07 0.16 0.30
The difference in the estimation of β will diminish as sample size increases; and
eventually, the choice of the estimation method per se will have minimal impact on
the dose assignments. However, the estimation method may have different implied
performance due to the fact that the maximum likelihood CRM always requires a
two-stage strategy whereas Bayesian CRM is typically used as a one-stage design.
Two-stage CRM
The two-stage CRM requires the specification of an initial design as part of the
design parameters in addition to those required in the one-stage CRM. Planning
and implementation of a two-stage CRM in a dose finding trial takes four steps:
1. Setting the clinical parameters:
• Target toxicity probability θ
• Number of test doses K
• Prior MTD ν0
• Sample size N
2. Calibrating the model parameters:
• Functional form of the dose–toxicity model F(·,β)
• Skeleton {p0k} and hence the dose labels dks via backward substitution†
• Prior distribution G(β) of β, if Bayesian CRM is used
3. Specifying an initial dose sequence: x1,0 ≤ x2,0 ≤ ... ≤ xN,0.
4. Execution: Treat patients initially according to {xi,0}. Upon the first observed
toxic outcome, treat the subsequent patient at the most recent model-based MTD
(3.7) or (3.11) as data accrue throughout the trial.
†Backward substitution for the maximum likelihood CRM can be carried out as
in (3.5) using an arbitrary initial value β̃0 without affecting the dose calculations.
More details will be given in Chapter 4.](https://image.slidesharecdn.com/1112001-250531235121-8e6f58df/85/Dose-Finding-By-The-Continual-Reassessment-Method-Chapman-Hall-Crc-Biostatistics-Series-1st-Edition-Ying-Kuen-Cheung-42-320.jpg)
![SIMULATING CRM TRIALS 25
3.4 Simulating CRM Trials
3.4.1 Numerical Illustrations
Computer simulation is a primary tool for evaluating the aggregate performance of
the method. To get a sense of how the CRM works, it is also useful to examine
individual simulated trials. Figure 3.1 shows the outcomes of simulated trials using
a one-stage and two-stage CRM for a trial with target θ = 0.25 and K = 5 under the
true dose–toxicity scenario
p1 = 0.02, p2 = 0.04, p3 = 0.10, p4 = 0.25, p5 = 0.50. (3.12)
Hence, dose level 4 is the true MTD. The dose assignments by the two methods in the
figure are quite different, although both select the correct MTD (dose level 4) based
on the 20 simulated subjects. The one-stage CRM, taking advantage of a high starting
dose, treats the majority of the subjects at the MTD but also seven patients at an
overdose. In contrast, the two-stage CRM takes a conventional escalation approach
initially and potentially treats many patients at low doses. As a result, the one-stage
CRM causes two toxic outcomes (patients 3 and 9) more than the two-stage CRM.
It is debatable whether the one-stage CRM is unsafe. On the one hand, the CRM has
been criticized on account of ethical concerns, as it is shown to cause more toxic
outcomes than the standard 3+3 method under some dose–toxicity scenarios [54].
On the other hand, overdosing is not necessarily a worse mistake than underdosing
when treating patients with severe diseases such as cancer. Rosa et al. [89] describe
a case study in which starting at a low dose (per the 3+3 algorithm) causes an ethical
dilemma when consenting a cancer patient. Overall, the risk–benefit trade-off should
be considered on a case-by-case basis. While the bortezomib trial in Chapter 1 started
at the third level, the NeuSTART (Example 2.1) exercised caution via a low starting
dose. In addition, the one-stage CRM in Figure 3.1 causes a 25% observed toxicity
rate (5 out of 20), which is on target. From a numerical viewpoint, the two-stage
design appears to be overconservative in this particular simulated trial.
3.4.2 Methods of Simulation
When simulating toxicity outcomes in a trial, each patient may be viewed to be car-
rying a latent toxicity tolerance ui that is uniformly distributed on the interval [0,1].
If the uniform variate is smaller than the true toxicity probabilityof the dose assigned
to the patient, the patient has a toxic outcome; otherwise, the patient does not have a
toxic outcome. That is,
Yi =
1 if ui ≤ π(xi)
0 otherwise
Table 3.2 displays the latent tolerance of the 20 simulated patients used in the
simulated trials in Figure 3.1. Consider, for example, patient 1 who receives dose
level 3 according to the one-stage CRM (left panel of the figure). He does not have a
toxic outcome because the tolerance u1 = .571 p3 = .10. Consequently, using the](https://image.slidesharecdn.com/1112001-250531235121-8e6f58df/85/Dose-Finding-By-The-Continual-Reassessment-Method-Chapman-Hall-Crc-Biostatistics-Series-1st-Edition-Ying-Kuen-Cheung-43-320.jpg)
![PRACTICAL MODIFICATIONS 27
one-stage and two-stage designs are treating the same patients, while the patients are
not necessarily treated at the same doses. The concept of toxicity tolerance is also
instrumental to the construction of a nonparametric optimal design. We will return to
this in Chapter 5.
The function crmsim in the ‘dfcrm’ package can be used to simulate multiple
CRM trials under a given dose–toxicity curve. The following R code runs 10 trials
using the one-stage Bayesian CRM specified in Figure 3.1 under the dose–toxicity
scenario (3.12):
### Generate 10 CRM trials
theta - 0.25
PI - c(0.02,0.04,0.10,0.25,0.50) # True MTD = 4
N - 20 # sample size
x0 - 3 # starting dose level
foo10 - crmsim(PI,p0,theta,N,x0,nsim=10,model=logistic,restrict=F)
simulation number: 1
simulation number: 2
simulation number: 3
simulation number: 4
simulation number: 5
simulation number: 6
simulation number: 7
simulation number: 8
simulation number: 9
simulation number: 10
foo10$MTD # Display the distribution of recommended MTD
[1] 0.0 0.0 0.1 0.8 0.1
In this illustration, dose level 4 is selected as the MTD in 8 of the 10 simulated trials.
3.5 Practical Modifications
3.5.1 Dose Escalation Restrictions
The one-stage CRM in Figure 3.1 assigns dose level 5 to the second patient after the
nontoxic outcome in the first patient who receives dose level 3. This escalation may
raise safety concerns because a high dose is tested without testing an intermediate
dose level. Several authors have noted the potential problem with dose skipping by
the CRM, and proposed the restricted CRM by imposing an escalation restriction:
The dose level for the next patient cannot be more than one level higher than that of
the current patient. Likewise, it is possible for the unrestricted CRM to skip doses in
deescalation; cf. patient 4 under the one-stage CRM in Figure 3.1. While skipping
doses in deescalation has not been perceived to be as problematic or unsafe, we may
apply a similar restriction that the dose level for the next patient cannot be more than
one level lower than that of the current patient. At any rate, these restrictions against
dose skipping will typically be applied, if ever, only to the first few patients because](https://image.slidesharecdn.com/1112001-250531235121-8e6f58df/85/Dose-Finding-By-The-Continual-Reassessment-Method-Chapman-Hall-Crc-Biostatistics-Series-1st-Edition-Ying-Kuen-Cheung-45-320.jpg)
![28 THE CONTINUAL REASSESSMENT METHOD
the change in the model-based estimates diminishes as the number of observations
increases (try plotting the sequence of β̂i in Table 3.2).
Another pathologyin escalation is illustratedin the two-stage CRM in Figure 3.1:
the dose for patient 13 is escalated from that of patient 12, who has a toxic outcome.
Such an escalation is called incoherent as it puts patient 13 at undue risk of toxicity
in light of the outcome in the previous patient [16]. To avoid the potential incoherent
moves by the CRM, we may apply the restrictions:
Coherence in escalation: If a toxicity is observed in the current patient, then
the dose level for the next patient cannot be higher than that of the current
patient.
Coherence in deescalation: If the current patient does not experience toxicity,
then the dose level for the next patient cannot be lower than that of the current
patient.
It is noteworthythat there is no incoherent move by the one-stage CRM in Figure 3.1.
That is, escalation occurs only after a nontoxic outcome, and deescalation after a
toxic outcome. In fact, the one-stage CRM will never induce an incoherent move—
even if no restriction is applied. This property indicates that the model-based CRM
is doing the right thing ethically. In contrast, an unrestricted two-stage CRM may
yield incoherent escalation as seen in Figure 3.1. It turns out that the problem is due
to poor calibration of the initial design. In this particular example, the group-of-three
escalation sequence is not an appropriate choice. This topic will be further discussed
in Chapters 5 and 10.
Figure 3.2 shows the outcomes of the restricted CRM trials using the same model
specifications as in Figure 3.1 and the same 20 patients as in Table 3.2. The dose
assignment patterns are similar to that of the unrestricted counterparts. In general,
the dose escalation restrictions have little impact on the CRM’s aggregate behaviors
in terms of the probability of correctly selecting the MTD and the average toxicity
number; try Exercise 3.4. Instead, these restrictions modify the pointwise properties
of the design: a pointwise property of a dose finding design concerns the behaviors
of individual outcome sequences (“points”). The pointwise properties of a design
should agree with principles deemed sensible to clinicians. Violations of any such
principles (e.g., making incoherent moves) will be perceived as worse than showing
poor statistical properties. After all, only a single outcome sequence is observed in
an actual trial. The aggregate operating characteristics are comparatively abstract to
clinicians.
In the rest of this book, unless specified otherwise, we assume that the restriction
of no dose skipping in escalation and the coherence restrictions are in effect. This
is the assumption used in the function crmsim if no value is given to the argument
restrict.
3.5.2 Group Accrual
The fully sequential CRM follows the current patient over an evaluation period,
called an observation window, before enrolling the next patient. A long observation](https://image.slidesharecdn.com/1112001-250531235121-8e6f58df/85/Dose-Finding-By-The-Continual-Reassessment-Method-Chapman-Hall-Crc-Biostatistics-Series-1st-Edition-Ying-Kuen-Cheung-46-320.jpg)
![PRACTICAL MODIFICATIONS 29
5 10 15 20
1
2
3
4
5
Patient number
Dose
level
x x x
x x
o
o ooo
oo o
ooooo oo
Restricted one−stage CRM
5 10 15 20
1
2
3
4
5
Patient number
Dose
level
x
x x
ooo
ooo
ooo
oo o
oo o
oo
Restricted two−stage CRM
Figure 3.2 Simulated trials using the one-stage and the two-stage Bayesian CRM with dose
escalation restrictions. Each point represents a patient, with “o” indicating no toxicity and “x”
indicating toxicity.
window (e.g., 6 months) will likely result in repeated interim accrual suspensions,
impose excessive administrative burdens, and cause long trial duration.
Goodman et al. [43] address this problem by assigning m 1 patients at a time
to each dose, and updating the model-base estimate (3.7) between every group of
m patients. Figure 3.3 shows the outcomes of the simulated trials by group accrual
CRM, using the models in Figure 3.1 and the same 20 patients in Table 3.2. In the
left panel of Figure 3.3, the model-based MTD estimate (3.7) is updated after every
m = 2 patients. The dose assignment pattern is similar to that of the fully sequential
CRM in Figure 3.1: half of the study subjects receive the MTD. The use of a larger
group size with m = 4, as shown in the right panel of the figure, gives the CRM fewer
occasions to adapt to the previous outcomes. This may partly explain the findings that
the accuracy of MTD estimation decreases as a large group size is used, when the
assumed model F is not a correct specification of the true dose–toxicity curve [43].
Therefore, although a large group size reduces the number of interim calculations
and shortens trial duration, it also undermines the advantage of the adaptiveness of
the CRM.
When the observation window is long in comparison to the recruitment period,
the reduction in trial duration by group accrual may not be adequate. Suppose, for
instance, each patient in the current group is to be followed for 6 months before the
next group is enrolled in Figure 3.3. Assuming instantaneous patient availability, a
group CRM with size 4 will take roughly 24 months to enroll 20 patients, whereas a
group size of 2 will take 54 months. We will return to consider clinical settings with
a long observation window due to late toxicities in Chapter 11, where we introduce
the time-to-event continual reassessment method (TITE-CRM) as an alternative to
the group accrual CRM.](https://image.slidesharecdn.com/1112001-250531235121-8e6f58df/85/Dose-Finding-By-The-Continual-Reassessment-Method-Chapman-Hall-Crc-Biostatistics-Series-1st-Edition-Ying-Kuen-Cheung-47-320.jpg)
![30 THE CONTINUAL REASSESSMENT METHOD
5 10 15 20
1
2
3
4
5
Patient number
Dose
level
x x xx
x
oo
oo
o oo o
ooooo oo
Group CRM (size 2)
5 10 15 20
1
2
3
4
5
Patient number
Dose
level
x xx x x
oooo
oooo
o
oooo
o o
Group CRM (size 4)
Figure 3.3 Simulated trials using the one-stage group accrual CRM. The MTD estimate is
updated after every two patients in the left panel, and every four patients in the right panel.
Dose escalation restrictions are applied. Each point represents a patient, with “o” indicating
no toxicity and “x” indicating toxicity.
3.5.3 Stopping and Extension Criteria
It is common to have provisions for early termination in clinical trials for economic
and ethical reasons. The standard 3+3 design for dose finding trials stops a trial when
two toxic outcomes are observed at a dose. In contrast, the original CRM works with
a fixed sample size N. From a practical viewpoint, neither the economic nor ethical
reason is sufficiently compelling for stopping a CRM trial early. Economically, the
sample size of a phase I trial is already small, and there is not much room to reduce
the sample size unless the early stopping rules are liberal. Ethically, the CRM is
expected to converge to the MTD as the trial accrues patients. Therefore, patients
enrolled towards the end of the trial will likely receive a good dose, and thus there
will be a reduced ethical imperative to stop the trial.
On the other hand, there may be good reasons for extending enrollment with
the CRM in some situations. Goodman et al. [43] consider continuing a CRM trial
beyond N until the recommended MTD has at least a certain number of patients
assigned to it. This is to avoid the case where only few patients are treated at the
recommended MTD. Consider the two-stage CRM in Figure 3.2 (right panel). Based
on the 20 observations shown in the figure, the recommended MTD is dose level 5,
which is one level higher than the dose given to the final patients, and has only had 5
patients with an observed toxicity rate of 40%. These are signs that the recommended
MTD is not adequately assessed. Figure 3.4 shows the outcomes of an extended
CRM trial with a minimum sample size criterion: the trial will go beyond 20 patients
until at least 9 patients have been treated at the recommended MTD. As a result
of this modification, the trial continues for another 7 patients. With a total of 27](https://image.slidesharecdn.com/1112001-250531235121-8e6f58df/85/Dose-Finding-By-The-Continual-Reassessment-Method-Chapman-Hall-Crc-Biostatistics-Series-1st-Edition-Ying-Kuen-Cheung-48-320.jpg)
![BIBLIOGRAPHIC NOTES 31
subjects, there is evidence that dose level 5 exceeds the acceptable toxicity level
(44% observed toxicity rate) and the final MTD is dose level 4.
0 5 10 15 20 25
1
2
3
4
5
Patient number
Dose
level
x
x x x x
x
ooo
ooo
ooo
oo o
oo o
oo
oo
oo
Restricted two−stage CRM with minimum MTD size 9
Figure 3.4 Simulated trials using the restricted two-stage CRM with a minimum sample size
criterion. Dose escalation restrictions are applied. Each point represents a patient, with “o”
indicating no toxicity and “x” indicating toxicity.
3.6 Bibliographic Notes
O’Quigley and Chevret [75] and Chevret [25] examine the operating characteristics
of the one-stage Bayesian CRM by simulation study. Moller [67] and O’Quigley and
Shen [80] study the two-stage CRM; the latter also propose the use of maximum
likelihood estimation. Several authors, including Faries [36], Korn et al. [54], and
Goodman et al. [43], note the potential problem with dose skipping by the CRM
and propose to modify the CRM to limit escalation by no more than one level at a
time. Faries [36] also suggests enforcing coherence in escalation by restriction; the
notion of coherence is subsequently formalized in Cheung [16]. Ahn [1] compares
various variants of the CRM by simulations. Early stopping of the CRM is considered
in Heyd and Carlin [45] and O’Quigley and Reiner [79]. A recent overview of the
CRM is given in Garrett-Mayer [39] and Iasonos et al. [48].
3.7 Exercises and Further Results
Exercise 3.1. Apply backward substitution (3.5) to the empiric function (3.2) with
β̂0 = 0, and show that
F(dk,β) = p
exp(β)
0k
for a given skeleton {p01,..., p0K}.](https://image.slidesharecdn.com/1112001-250531235121-8e6f58df/85/Dose-Finding-By-The-Continual-Reassessment-Method-Chapman-Hall-Crc-Biostatistics-Series-1st-Edition-Ying-Kuen-Cheung-49-320.jpg)
![36 ONE-PARAMETER DOSE–TOXICITY MODELS
by Jensen’s inequality. This example illustrates that parameterization via c(β) can be
an important consideration when the Bayesian CRM (3.7) is used.
When c(β) = exp(β), the ψ-representation (4.3) of a CRM model depends on
the parameter β only via β −β̂0. If the prior distribution of β is normal, the centered
variable β −β̂0 is always mean zero normal. Therefore, the posterior computations
will be identical regardless of the specified prior mean β̂0, and the Bayesian CRM is
invariant to the mean of a normal prior. In general, this invariance property holds for
Bayesian CRM when model (4.1) is used and the prior distribution of β constitutes a
location-scale family (Lehmann, 1983, page 20). In this book, we will focus on the
model class (4.1) with c(β) = exp(β) and a normal prior distributionfor β, so that the
model parameters of the Bayesian CRM are the function ψ(z), the skeleton {p0k},
and the variance σ2
β of the normal prior. Table 4.1 gives some simple examples of
ψ(z) and the corresponding CRM models. The calibration of {p0k} will be detailed
in Chapter 8, and specification of σ2
β in Chapter 9.
4.3 Model Assumptions
In this section, we state the regularity conditions on the one-parameter CRM model
Fk(β) assumed in this book. The role of these conditions in various theoretical results
will be discussed in the following chapters. In this chapter, we focus on the intuition
behind the conditions and their implications. A practical point: all these assumptions
are verifiable for any given Fk(β), and thus can serve as preliminary conditions to
remove certain models from consideration for use.
Condition 4.1. F(x,β) is strictly increasing in x for all β.
Condition 4.2. Fk(β) is monotone in β in the same direction for all k.
Conditions 4.1 and 4.2 are satisfied by many dose–toxicity functions. In particular,
the model class (4.1) satisfies these conditions when either h(dk) 0 for all k or 0
for all k. It is easy to verify that the logistic model with a0 = 0 in Table 3.1 does not
satisfy Condition 4.2 because h(dk) 0 for k ≤ 4 but h(d5) 0. Thus, this model
should not be considered for use. Graphically, Condition 4.2 implies that the family
of curves induced by F(x,β) do not cross each other; see Figure 4.1.
For the next condition, we first define gi j(β) = {1 −Fi(β)}/{1 −Fj(β)}.
Condition 4.3. The derivatives F′
k(β) and g′
i j(β) exist and g′
i j(β)F′
k (β) ≤ 0 for all
k and i j.
Condition 4.3 can be equivalently stated as
|F′
i (β)| ≥ |F′
j(β)|{1 −Fi(β)}|/{1 −Fj(β)}
for all i j, which puts a lower bound on |F′
i (β)| relative to |F′
j(β)|. Using the fact
that log{1 − Fk(β)} =
R β
−∞[−F′
k(φ)/{1 − Fk(φ)}]dφ when F′
k (φ) 0 for all k, one
can show that Condition 4.3 implies Condition 4.1 and hence is a stronger condition.](https://image.slidesharecdn.com/1112001-250531235121-8e6f58df/85/Dose-Finding-By-The-Continual-Reassessment-Method-Chapman-Hall-Crc-Biostatistics-Series-1st-Edition-Ying-Kuen-Cheung-54-320.jpg)
![manuscripts. The written history of Paris is there; and all workers
know the pretty, sculpture-ornamented room of Monsieur le Vayer,
the erudite, obliging Curator of these fine collections. Messieurs
Poète, Beaurepaire, Jacob, Jarach and Wilhem, in the Library;
Messieurs Pètre and Stirling in the History room are the wise and
welcoming hosts of this admirable Parisian Library.
All this Marais quarter, indeed, contains sumptuous mansions, not
one of which, alas! has been respected. All are given over to
business and manufacturing. The Lamoignon mansion is occupied by
glass-polishers and garden-seatmakers; the Albret mansion by a
bronze lamp-dealer; those of Tallard, Maulevrier, Sauvigny,
Brevannes, Epernon, c., are still standing, but in what a state! The
Rue des Nonnains-d'Hyères offers us its curious bass-relief, in
painted stone, representing a knife-grinder in eighteenth-century
costume. In 1748, a Madame de Pannelier kept a wit-office in this
same street; Lalande, Sautereau, Guichard, Leclerc de Merry used to
attend meetings there. They were held on Wednesdays, and were
preceded by an excellent dinner. The tradition has happily been
preserved in Paris.
In the Rue François-Miron, one sees a spacious, handsome mansion
with circular pediment, escutcheons and garlands. It is the Beauvais
mansion, built by Le Pautre in 1658.
To look at it now, old and in a dull street, one would hardly think
that the coaches of Louis XIV.—King Sun—had passed under the
dark vault of the entrance gate and that, from the top of the central
pavilion balcony, Queen Anne of Austria, in company with the Queen
of England, Cardinal Mazarin, Marshal de Turenne and other
illustrious nobles, had watched her son Louis XIV. and her daughter-
in-law, the new Queen Marie-Thérèse of Austria, go by as they
made, through Saint-Antoine's Gate, their solemn entry into Paris on
the 26th of August 1660![3]
On account of its picturesque aspect and the fine mansions it
contains, the Rue Geoffroy-l'Asnier is one of the most curious in
Paris. At No. 26 stands the Châlons-Luxembourg mansion, with its](https://image.slidesharecdn.com/1112001-250531235121-8e6f58df/85/Dose-Finding-By-The-Continual-Reassessment-Method-Chapman-Hall-Crc-Biostatistics-Series-1st-Edition-Ying-Kuen-Cheung-71-320.jpg)
Dose Finding By The Continual Reassessment Method Chapman Hall Crc Biostatistics Series 1st Edition Ying Kuen Cheung
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- 6. Dose Finding by the Continual Reassessment Method C9151_FM.indd 1 1/12/11 12:25 PM
- 7. Editor-in-Chief Shein-Chung Chow, Ph.D. Professor Department of Biostatistics and Bioinformatics Duke University School of Medicine Durham, North Carolina Series Editors Byron Jones Senior Director Statistical Research and Consulting Centre (IPC 193) Pfizer Global Research and Development Sandwich, Kent, U .K. Jen-pei Liu Professor Division of Biometry Department of Agronomy National Taiwan University Taipei, Taiwan Karl E. Peace Georgia Cancer Coalition Distinguished Cancer Scholar Senior Research Scientist and Professor of Biostatistics Jiann-Ping Hsu College of Public Health Georgia Southern University Statesboro, Georgia Bruce W. Turnbull Professor School of Operations Research and Industrial Engineering Cornell University Ithaca, New York C9151_FM.indd 2 1/12/11 12:25 PM
- 8. Adaptive Design Theory and Implementation Using SAS and R, Mark Chang Advances in Clinical Trial Biostatistics, Nancy L. Geller Applied Statistical Design for the Researcher, Daryl S. Paulson Basic Statistics and Pharmaceutical Statistical Applications, Second Edition, James E. De Muth Bayesian Adaptive Methods for Clinical Trials, Scott M. Berry, Bradley P. Carlin, J. Jack Lee, and Peter Muller Bayesian Methods for Measures of Agreement, Lyle D. Broemeling Bayesian Missing Data Problems: EM, Data Augmentation and Noniterative Computation, Ming T. Tan, Guo-Liang Tian, and Kai Wang Ng Bayesian Modeling in Bioinformatics, Dipak K. Dey, Samiran Ghosh, and Bani K. Mallick Causal Analysis in Biomedicine and Epidemiology: Based on Minimal Sufficient Causation, Mikel Aickin Clinical Trial Data Analysis using R, Ding-Geng (Din) Chen and Karl E. Peace Clinical Trial Methodology, Karl E. Peace and Ding-Geng (Din) Chen Computational Methods in Biomedical Research, Ravindra Khattree and Dayanand N. Naik Computational Pharmacokinetics, Anders Källén Data and Safety Monitoring Committees in Clinical Trials, Jay Herson Design and Analysis of Animal Studies in Pharmaceutical Development, Shein-Chung Chow and Jen-pei Liu Design and Analysis of Bioavailability and Bioequivalence Studies, Third Edition, Shein-Chung Chow and Jen-pei Liu Design and Analysis of Clinical Trials with Time- to-Event Endpoints, Karl E. Peace Difference Equations with Public Health Applications, Lemuel A. Moyé and Asha Seth Kapadia DNA Methylation Microarrays: Experimental Design and Statistical Analysis, Sun-Chong Wang and Arturas Petronis DNA Microarrays and Related Genomics Techniques: Design, Analysis, and Interpretation of Experiments, David B. Allsion, Grier P. Page, T. Mark Beasley, and Jode W. Edwards Dose Finding by the Continual Reassessment Method, Ying Kuen Cheung Elementary Bayesian Biostatistics, Lemuel A. Moyé Frailty Models in Survival Analysis, Andreas Wienke Handbook of Regression and Modeling: Applications for the Clinical and Pharmaceutical Industries, Daryl S. Paulson Measures of Interobserver Agreement and Reliability, Second Edition, Mohamed M. Shoukri Medical Biostatistics, Second Edition, A. Indrayan Meta-Analysis in Medicine and Health Policy, Dalene Generalized Linear Models: A Bayesian Perspective, Dipak K. Dey, Sujit K. Ghosh, and Bani K. Mallick Monte Carlo Simulation for the Pharmaceutical Industry: Concepts, Algorithms, and Case Studies, Mark Chang Multiple Testing Problems in Pharmaceutical Statistics, Alex Dmitrienko, Ajit C. Tamhane, and Frank Bretz Sample Size Calculations in Clinical Research, Second Edition, Shein-Chung Chow, Jun Shao, and Hansheng Wang Statistical Design and Analysis of Stability Studies, Shein-Chung Chow Statistical Methods for Clinical Trials, Mark X. Norleans Statistics in Drug Research: Methodologies and Recent Developments, Shein-Chung Chow and Jun Shao Statistics in the Pharmaceutical Industry, Third Edition, Ralph Buncher and Jia-Yeong Tsay Translational Medicine: Strategies and Statistical Methods, Dennis Cosmatos and Shein-Chung Chow C9151_FM.indd 3 1/12/11 12:25 PM
- 9. Ying Kuen Cheung Columbia University New York, New York, USA Dose Finding by the Continual Reassessment Method C9151_FM.indd 5 1/12/11 12:25 PM
- 10. Chapman & Hall/CRC Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2011 by Taylor and Francis Group, LLC Chapman & Hall/CRC is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number: 978-1-4200-9151-9 (Hardback) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information stor- age or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copy- right.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that pro- vides licenses and registration for a variety of users. For organizations that have been granted a pho- tocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com C9151_FM.indd 6 1/12/11 12:25 PM
- 11. Preface Despite its poor statistical properties, the 3+3 algorithm remains the most commonly used dose finding method in phase I clinical trials today. However, as clinicians begin to realize the important role of dose finding in the drug development process, there is an increasing openness to “novel” methods proposed in the past two decades. In particular, the continual reassessment method (CRM) and its variations have drawn much attention in the medical community. To ride on this momentum and overcome the status quo in the phase I practice, it is critical for us (statisticians) to be able to design a trial using the CRM in a timely and reproducible manner. This is the impetus to writing a detailed exposition on the calibration of the CRM for applied statisticians who need to deal with dose finding in phase I trials while having many other duties to attend to. A natural approach to such a writing project is to write a how-to book. By the time I started this book project in the summer of 2008, I had helped design half a dozen CRM trials (three of which are included as examples in this book). In retrospect, I found some general patterns of how I calibrated the CRM parameters in these trials. These patterns, characterized collectively as a trial-and-error approach in Chapter 7, worked well in the sense that they gave reasonable operating characteristics to a design. However, it was time-consuming (weeks of simulation) and would require an intimate understanding of the CRM (I wrote a PhD dissertation on the CRM). I realized that some automation and step-by-step guidelines in this calibration process would be crucial and appreciated if the CRM was to be used on a regular basis by a wide group of statisticians. Chapters 7–10 try to address this need by breaking a CRM design into a list of design parameters, each of which is to be calibrated in a prescribed manner. Despite my pragmatic approach, I hope this book is not only a cookbook. I intend to provide a full coverage of the CRM. This book includes a comprehensive review of the CRM (Chapter 3) and elaborate properties of the CRM (Chapters 5 and 6). While this book is based on my previous publications on the CRM, I have introduced new material so as to present the CRM under a unified framework (Chapter 4). These chapters serve as the theoretical foundation of the calibration techniques presented in the later chapters. I also reflect on what not to do with the CRM (Chapter 12) and when not to use the CRM (Chapter 13). From a practical viewpoint, these not-to chapters are as important as, if not more important than, the how-to chapters, because they avoid abuses and pitfalls in applying the CRM. I believe that using the CRM in a wrong way or in the wrong trial is no better, or arguably worse, than falling back to the 3+3 algorithm. The time-to-event aspect of the toxicity endpoint has been a vii
- 12. viii PREFACE recurring concern in my previous CRM trials, and so is included as an extension of the CRM (Chapter 11). All in all, while this is not intended to be a cookbook, the inclusion of materials is based on their practical relevance. This book does not cover dose finding in all possible clinical settings. In fact, it has a singular focus on the simplest and the most common phase I trial setting, where the study endpoint is defined as a binary outcome and the subjects are assumed to come from a homogeneous population. I make no mention of the concerns with multiple toxicity and the gradation of severe toxicities. The topic of individualized dosing is omitted. While some basic ideas of dose finding using both efficacy and toxicity are outlined in Chapter 13, the discussion is brief and does not do full justice to this fast-growing area. All these are important topics in which I am intellectually interested. Their omission, however, is mainly due to my limited practical experience in dealing with these “nonstandard” situations in real dose finding studies; dealing with these issues simply from a methodological and theoretical viewpoint does not fit my intent of writing a practical book (although I think such a book is interesting in its own right and hope someone more qualified than I will deliver it). I do have a word or two to add from a methodological and theoretical viewpoint here, if not already alluded to in the book’s final chapter (Section 14.4, to be precise). First, a complete theoretical framework is crucial for these nonstandard methods to be successfully translated into actual practice. In this book, I try to explicate possible pathological behaviors (e.g., incoherence and rigidity) of some CRM modifications for the simplest setting; it is reasonable to infer that these pathologies will multiply for methods more complex than the CRM for the more complicated clinical settings. Solid theoretical investigation will help us navigate the potential pitfalls. I also hope the theoretical framework developed in this book for the simplest case will prove useful when extended to the complicated settings. Second, and more specifically, I think stochastic approximation offers partial solutions (albeit mostly theoretical) to many of these nonstandard dose finding settings. This is why I close this book with a chapter where I try to connect and compare the CRM with the rich stochastic approximation literature. The last points I just made give a hint about my methodological and theoretical interests. I hope that this book will in some way simulate research in the CRM and general dose finding methods, despite its practical nature. As I try to present the CRM and the dose finding criteria at a rigorous level, and to cover the CRM literature as comprehensively as possible, I also hope this book can serve as an introduction for those interested in doing research in this area. I taught a course on sequential experimentation at Columbia University from an early unpublished version of this book. This final manuscript is, in turn, adapted from the course notes, and is suitable for use in a course on sequential experimentation or clinical trials. There are several statistics books on dose finding. The two most popular ones are the edited volumes by Chevret [26] and by Ting [105]. Both give surveys of dose finding methods and are good introductions to the dose finding literature. By comparison, this book is a single-authored work on a specific dose finding method, which I think is necessary if we are to get down to the nuts and bolts of the method. By writing a book on the CRM, I do not imply that it is the best method out there.
- 13. PREFACE ix In fact, for the dose finding objective considered here, it is unlikely that there is one method that is best or optimal in a uniform sense. While some methods may work best under certain scenarios according to some criterion, the others are optimal under a different criterion. There have been numerous proposals in the last two decades. These proposals can be good alternatives against the 3+3 algorithm as long as they are calibrated properly. And, the CRM is one of these methods. Furthermore, the CRM has been worked out and discussed in the statistical and medical literature so extensively that I believe we are getting close to translating this method into practice. This book hopefully will be a catalyst in this translational process. I owe a debt of gratitude to Tom Cook, Bin Cheng, and an anonymous reviewer who have been generous with their time and given detailed comments on earlier versions of the book. I am grateful for Jimmy Duong for his help to maintain the R package ‘dfcrm’ (a companion software with this book). I would also like to thank Rick Chappell who introduced me to the CRM and clinical trials when I was a student at University of Wisconsin–Madison. This book would not be possible without his mentoring. Finally, my most heartfelt thanks go to my wife, Amy, for her support and enthusiasm during this writing process. New York October 2010
- 14. Contents I Fundamentals 1 1 Introduction 3 2 Dose Finding in Clinical Trials 7 2.1 The Maximum Tolerated Dose 7 2.2 An Overview of Methodology 10 2.3 Bibliographic Notes 15 2.4 Exercises and Further Results 16 3 The Continual Reassessment Method 17 3.1 Introduction 17 3.2 One-Stage Bayesian CRM 17 3.2.1 General Setting and Notation 17 3.2.2 Dose–Toxicity Model 17 3.2.3 Dose Labels 18 3.2.4 Model-Based MTD 20 3.2.5 Normal Prior on β 21 3.2.6 Implementation in R 21 3.3 Two-Stage CRM 22 3.3.1 Initial Design 22 3.3.2 Maximum Likelihood CRM 23 3.4 Simulating CRM Trials 25 3.4.1 Numerical Illustrations 25 3.4.2 Methods of Simulation 25 3.5 Practical Modifications 27 3.5.1 Dose Escalation Restrictions 27 3.5.2 Group Accrual 28 3.5.3 Stopping and Extension Criteria 30 3.6 Bibliographic Notes 31 3.7 Exercises and Further Results 31 xi
- 15. xii CONTENTS 4 One-Parameter Dose–Toxicity Models 33 4.1 Introduction 33 4.2 ψ-Equivalent Models 33 4.3 Model Assumptions† 36 4.4 Proof of Theorem 4.1† 40 4.5 Exercises and Further Results 40 5 Theoretical Properties 41 5.1 Introduction 41 5.2 Coherence 41 5.2.1 Motivation and Definitions 41 5.2.2 Coherence Conditions of the CRM 42 5.2.3 Compatibility 43 5.2.4 Extensions 45 5.3 Large-Sample Properties 46 5.3.1 Consistency and Indifference Interval 46 5.3.2 Consistency Conditions of the CRM 48 5.3.2.1 Home Sets 48 5.3.2.2 Least False Parameters 48 5.3.2.3 Main Result 49 5.3.2.4 A Relaxed Condition 49 5.3.3 Model Sensitivity of the CRM 51 5.3.4 Computing Model Sensitivity in R 53 5.4 Proofs† 54 5.4.1 Coherence of One-Stage CRM 54 5.4.2 Consistency of the CRM 55 5.5 Exercises and Further Results 56 6 Empirical Properties 57 6.1 Introduction 57 6.2 Operating Characteristics 57 6.2.1 Accuracy Index 57 6.2.2 Overdose Number 59 6.2.3 Average Toxicity Number 59 6.3 A Nonparametric Optimal Benchmark 60 6.4 Exercises and Further Results 62 II Design Calibration 63 7 Specifications of a CRM Design 65 7.1 Introduction 65 7.2 Specifying the Clinical Parameters 66 7.2.1 Target Rate θ 66 7.2.2 Number of Test Doses K 66 7.2.3 Sample Size N 66
- 16. CONTENTS xiii 7.2.4 Prior MTD ν0 and Starting Dose x1 67 7.3 A Roadmap for Choosing the Statistical Component 68 7.4 The Trial-and-Error Approach: Two Case Studies 69 7.4.1 The Bortezomib Trial 69 7.4.2 NeuSTART 71 7.4.3 The Case for an Automated Process 73 8 Initial Guesses of Toxicity Probabilities 75 8.1 Introduction 75 8.2 Half-width (δ) of Indifferent Interval 75 8.3 Calibration of δ 77 8.3.1 Effects of δ on the Accuracy Index 77 8.3.2 The Calibration Approach 78 8.3.3 Optimal δ for the Logistic Model 79 8.4 Case Study: The Bortezomib Trial 81 8.5 Exercises and Further Results 87 9 Least Informative Normal Prior 89 9.1 Introduction 89 9.2 Least Informative Prior 89 9.2.1 Definitions 89 9.2.2 Rules of Thumb 91 9.3 Calibration of σβ 93 9.3.1 Calibration Criteria 93 9.3.2 An Application to the Choice of ν0 93 9.3.3 Optimality Near σLI β 95 9.4 Optimal Least Informative Model 97 9.5 Revisiting the Bortezomib Trial 99 10 Initial Design 103 10.1 Introduction 103 10.2 Ordering of Dose Sequences 103 10.3 Building Reference Initial Designs 106 10.3.1 Coherence-Based Criterion 106 10.3.2 Calibrating Compatible Dose Sequences 107 10.3.3 Reference Initial Designs for the Logistic Model 109 10.4 Practical Issues 109 10.4.1 Sample Size Constraint 109 10.4.2 Dose Insertion† 112 10.5 Case Study: NeuSTART 113 10.6 Exercises and Further Results 115
- 17. xiv CONTENTS III CRM and Beyond 117 11 The Time-to-Event CRM 119 11.1 Introduction 119 11.2 The Basic Approach 119 11.2.1 A Weighted Likelihood 119 11.2.2 Weight Functions 120 11.2.3 Individual Toxicity Risks 122 11.3 Numerical Illustration 123 11.3.1 The Bortezomib Trial 123 11.3.2 Implementation in R 124 11.4 Enrollment Scheduling 125 11.4.1 Patient Accrual 125 11.4.2 Interim Suspensions 127 11.5 Theoretical Properties† 129 11.5.1 Real-Time Formulation 129 11.5.2 Real-Time Coherence 129 11.5.3 Consistency 130 11.6 Two-Stage Design 131 11.6.1 Waiting Window 131 11.6.2 Case Study: The Poly E Trial 132 11.7 Bibliographic Notes 135 11.8 Exercises and Further Results 136 12 CRM with Multiparameter Models 139 12.1 Introduction 139 12.2 Curve-Free Methods 139 12.2.1 The Basic Approach 139 12.2.2 Product-of-Beta Prior Distribution 140 12.2.3 Dirichlet Prior Distribution 143 12.2.4 Isotonic Design 144 12.3 Rigidity 146 12.3.1 Illustrations of the Problem 146 12.3.2 Remedy 1: Increase m 147 12.3.3 Remedy 2: Increase Prior Correlations 147 12.4 Two-Parameter CRM† 149 12.4.1 The Basic Approach 149 12.4.2 A Rigid Two-Parameter CRM: Illustration 150 12.4.3 Three-Stage Design 151 12.4.4 Continuous Dosage 153 12.5 Bibliographic Notes 154 12.6 Exercise and Further Results 154
- 18. CONTENTS xv 13 When the CRM Fails 155 13.1 Introduction 155 13.2 Trade-Off Perspective of MTD 155 13.2.1 Motivation 155 13.2.2 Maximum Safe Dose and Multiple Testing 156 13.2.3 A Sequential Stepwise Procedure 157 13.2.4 Case Study: The ASCENT Trial 159 13.2.5 Practical Notes 161 13.3 Bivariate Dose Finding 162 14 Stochastic Approximation 167 14.1 Introduction 167 14.2 The Past Literature 167 14.2.1 The Robbins-Monro Procedure 167 14.2.2 Maximum Likelihood Recursion 168 14.2.3 Implications on the CRM 169 14.3 The Present Relevance 170 14.3.1 Practical Considerations 170 14.3.2 Dichotomized Data 171 14.3.3 Virtual Observations 174 14.3.4 Quasi-Likelihood Recursion 175 14.4 The Future Challenge 176 14.5 Assumptions on M(x) and Y(x)† 177 14.6 Exercises and Further Results 178 References 179 Index 187
- 21. Chapter 1 Introduction The clinical development of a new drug or a new treatment proceeds through three phases of testing in human subjects. Phase I trials are small studies that evaluate safety and identify a safe dose range of the treatment. Once a dose range is cho- sen, its therapeutic efficacy will be examined in a phase II trial. Regimens that are shown promising in phase II trials will be moved to multi-institutional phase III clini- cal trials for randomized comparison to standard treatments. The ultimate goal of this entire process is to translate promising discoveries in the laboratory into new medical procedures that can be used in the general clinical settings. This division of clinical trials, however, may give an oversimplified picture of the actual drug development process. Often, several phase I-II trial sequels of a drug, possibly with minor vari- ations in the treatment schedule and patient populations, are needed before a phase III trial is warranted. This process is necessarily iterative rather than linear, as the phase I-II-III paradigm appears to suggest. In addition, the taxonomy of trials is not universal across disciplines, and may include finer divisions such as phase IA, IB, IIA, and IIB. The recent trend to combine phases of trials, the so-called combined phase I/II trials and seamless phase II/III trials, renders further refinement of the drug development process. This having been said, the phase I-II-III paradigm provides a conceptual frame- work for in-depth study of statistical methods. The subject matter of this book is dose finding using the continual reassessment method (CRM). The CRM [78] is among the first model-based designs for phase I cancer trials in which toxicity is the primary study endpoint. The role of toxicity in early-phase cancer trials had long been a sub- ject for discussion in the medical literature [93, 85]. In particular, for cytotoxic drugs, toxicity serves as evidence that the drug has reached a level that does harm not only to the cancer cells but also to a patient’s normal organs. In other words, a therapeutic dose is expected to cause a significant amount of severe but reversible toxicities in the cancer patient population. Therefore, a primary goal of phase I cancer trials is to identify the so-called maximum tolerated dose (MTD). For other disorders such as acute stroke and HIV, identifying the MTD is also a primary objective of early-phase safety studies (usually called phase IB trials). In addition, dose finding is important in phase II proof-of-concept trials where the goal is to identify a dose range with demonstrated biological activity. This objective is usually achieved through the esti- mation of the minimum effective dose (MED) [106, 27]. From a statistical viewpoint, 3
- 22. 4 INTRODUCTION the MTD in safety studies and the MED in efficacy studies can be formulated in an analogous way. Therefore, this book is relevant to the design of phase I and II dose finding trials. Under the modernized paradigm, the dose finding principles discussed here also address the design issues in the combined phase I/II trials, in which both the safety and the efficacy endpoints are considered as co-primary (cf. Section 13.3). Another advantage of dividing the drug development process into phases is that by doing so, we can set a clear and manageable benchmark to achieve in a particular study. This entails a clearly defined set of study endpoints and an interpretable study objective. Since clinical trials are conducted in human subjects, each benchmark is to be reached within certain ethical constraints. In particular, in phase I dose finding studies, randomization is not entirely appropriate because it may expose subjects to excessively high and toxic doses without sufficiently testing the lower doses. (Some would also argue randomization exposes subjects to low and inefficacious doses, although this aspect is apparently not as alarming.) We illustrate these points using a bortezomib dose finding trial [62]. Bortezomib is a proteasome inhibitor with proven activity in lymphoma. In the trial, bortezomib was given in combination with the standard chemotherapy as a first-line treatment for patients with diffuse large B cell or mantle cell non-Hodgkin’s lymphoma. Each patient would receive up to six 21-day cycles of the treatment combination. Table 1.1 describes the five dose schedules of bortezomib tested in the trial. The primary safety concerns related to bortezomib were neuropathy, low platelet count, and symptomatic non-neurologic or non-hematologic toxicity. Toxicity was graded according to the National Cancer Institute Common Terminology Criteria for Adverse Events [71], with grade 3 or higher defined as dose limiting. Generally, a grade 3 toxicity is se- vere but can be resolved by symptomatic treatment, whereas a grade 4 toxicity is irreversible; toxic death due to the treatment is invariably defined as grade 5. The primary endpoint of each patient was the indicator of whether any dose-limiting tox- icity (DLT) was experienced at any time during the six cycles. The objective of the trial was to determine the MTD, defined as a dose associated with a 25% DLT rate. Table 1.1 gives the number of patients and the number of DLTs per dose in the borte- zomib trial. The data show strong evidence that the highest dose is adequately safe: we pool the observations in dose levels 4 and 5 by assuming an increasing dose– toxicity relationship; based on 1 DLT out of 16 patients, we obtain a 95% confidence upper bound of 0.26 for the DLT probability. Table 1.1 The bortezomib trial [62]: dose schedules of bortezomib, sample size (n), and the number of DLT (z) at each dose Level Dose and schedule within cycle n z 1 0.7 mg/m2 on day 1 of each cycle 0 0 2 0.7 mg/m2 on days 1 and 8 of each cycle 0 0 3 0.7 mg/m2 on days 1 and 4 of each cycle 4 0 4 1.0 mg/m2 on days 1 and 4 of each cycle 9 1 5 1.3 mg/m2 on days 1 and 4 of each cycle 7 0
- 23. 5 While simple analyses are usually adequate to address the primary scientific questions in a phase I study, the summary statistics in Table 1.1 ignore how the data were collected. Figure 1.1 shows the dose assignments of the trial in chronological order. The trial started at level 3, a dose schedule that the investigators believed to be safe to treat patients. Escalation to the next higher dose occurred after four patients had been followed for several weeks without signs of toxicity, and another escalation took place after three following patients. Shortly after the eighth patient entered the trial at the highest dose, patient 7 at dose level 4 experienced a DLT, thus leading to a deescalation for the ninth patient. Subsequent patients were enrolled in a staggered fashion, allowing months to pass before reescalating to the highest level. A central feature of this dose assignment scheme is its outcome adaptiveness. Specifically, in the bortezomib trial, the dose assignments were made in accordance with the time-to- event continual reassessment method (TITE-CRM), an extension of the CRM to be discussed in Chapter 11. For ethical reasons, most dose finding trials are conducted in an outcome-adaptive manner, so that the dose assignment of the current patient depends on those of the previous patients. As such, the focus of this book is the de- sign (as opposed to analysis) of a dose finding study using the CRM and its variants. 0 2 4 6 8 10 12 Calendar time since entry of patient 1 (months) Dose level 1 2 3 4 5 12 34 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 7 O 2 D Figure 1.1 Dose assignments in the bortezomib trial. Each number indicates a patient: An unmarked number represents the patient’s entry time; a number marked with “O” indicates the time when a DLT occurs, and “D” indicates the time of dropout. Vertical positions of some numbers are jittered for clarification.
- 24. 6 INTRODUCTION This book is organized into three parts. Part I (Chapters 2–6) contains the back- ground and introductory material of the CRM. Specifically, Chapter 2 provides the clinical background, outlines the problem of dose finding in the context of several real trial examples, and reviews the dose finding literature. Chapter 3 introduces the basic approach of the CRM and presents its major modifications. The method will be developed along with a description of an R package ‘dfcrm’. Chapter 4 presents a unified framework for dose–toxicity models used in the CRM. Chapters 5 and 6, respectively, discuss the theoretical and empirical properties of the CRM. The ob- jective of Part I is for the readers to develop a basic understanding of the CRM and be able to implement the method using a simple R code. Readers familiar with the basic CRM methodology are also encouraged to review the materials, as they are reorganized and presented in a unified framework in this book. Part II (Chapters 7–10) details the calibration process of the CRM based on the notation and the theory introduced in Part I. Chapter 7 introduces a system of design parameters involved in the CRM, and classifies them into two categories: clinical parameters and model parameters. The subsequent chapters then present fine-tuning techniques of the model parameters: the initial guesses of the toxicity probabilities (Chapter 8), the prior distribution of the model parameter (Chapter 9), and the initial design of a two-stage CRM (Chapter 10). The objective of Part II is for the readers to develop the ability to design a “good” CRM trial within a reasonable timeline. Part III (Chapters 11–14) contains a variety of advanced topics related to the CRM. Chapter 11 presents the TITE-CRM to deal with situations in which the toxic- ity outcome is defined with respect to a nontrivial duration. Chapter 12 gives a critical review of CRM using multiparameter models. Chapter 13 considers situations where the CRM is an inappropriate design, and puts forward some alternatives. Chapter 14 connects the CRM and modern dose finding trials to the large literature of stochastic approximation. The objective of Part III is to stimulate further research in the CRM and general dose finding methodology. The materials in this book are presented at a level that requires college algebra and some basic calculus concepts. Sections marked with “†” in the table of contents contain technical details that may be skipped without affecting the reading of the other chapters. Exposition in the book will be supplemented by illustrations of the usage of R functions in the ‘dfcrm’ package. While some basic knowledge of R will enhance the reading experience, proficiency in R is not required. Interested readers can find out more information about R from the Comprehensive R Archive Network (CRAN) [83] at http://www.r-project.org.
- 25. Chapter 2 Dose Finding in Clinical Trials 2.1 The Maximum Tolerated Dose The primary objective of phase I trials of a new anticancer drug is to assess the toxic side effects of the drug and to recommend a dose for the subsequent phase II trials. This recommended dose is typically the maximum test dose that does not exceed an acceptable level of toxicity, the so-called maximum tolerated dose (MTD). Traditional chemotherapy takes the cytotoxic therapeutic mechanism under which toxicity may be viewed as a surrogate for anti-tumor activity. Toxicity, therefore, is in a sense a desirable endpoint, so the trial objective is to find a dose that is associated with a given level of toxicity probability. Also in this sense, this MTD is presumed optimal in the absence of information about efficacy and clinical response. Definition 2.1 (MTD—surrogacy perspective). In a trial with K test doses, let pk denote the toxicity probability associated with dose level k for k = 1,...,K. The MTD from a surrogate-for-efficacy perspective, denoted by ν, is defined as the dose level with toxicity probability closest to a prespecified target probability θ, that is, ν ≡ argmink |pk −θ|. Since the late 1980s, most dose finding designs have been proposed to specifically address dose finding of cytotoxic drugs in patients with solid tumors and other forms of malignancies. The bortezomib trial in lymphoma patients introduced in Chapter 1 is one such example. As a result, the phase I method literature has focused on the surrogacy definition of MTD. See Exercise 2.2 for an alternative definition of MTD from a surrogacy perspective. For noncytotoxic target anticancer agents and treatments for other diseases such as acute ischemic stroke, toxicity does not play a therapeutic role, but safety remains the primary concern in the early drug development phase. For these agents, it is still useful to define an upper safety limit of the dose range for further clinical research. Furthermore, under the assumption that efficacy of a drug increases with dose, there is merit in pushing the dose as high as safety will permit. In view of this trade- off between safety and efficacy, one may seek to maximize the dose administered to patients subject to toxicity constraints. This may lead to a slightly different dose recommendation for the next study phase: Definition 2.2 (MTD—trade-off perspective). In a trial with K test doses, let pk 7
- 26. 8 DOSE FINDING IN CLINICAL TRIALS denote the toxicity probability associated with dose level k for k = 1,...,K. The MTD from a trade-off-for-efficacy perspective, denoted by γ, is defined as the largest dose level with toxicity probability no greater than a prespecified threshold θ, that is, γ ≡ max{k : pk ≤ θ}. Both the surrogacy and the trade-off perspectives define the MTD with respect to a target toxicity rate θ, and as such formulate dose finding as a percentile estimation problem. However, from a statistical viewpoint, it is generally easier to estimate ν than γ (if ν is well defined) by making use of monotonicity of the dose–toxicity curve. Therefore, it is in some cases pragmatic to take ν as the operative objective of a trial, even though the toxicity endpoint is generally not a surrogate for efficacy for noncytotoxic drugs. In practice, it is important to discern the appropriate objective for a given clinical setting, and the degree of tolerance in terms of the target θ. Apparently, these decisions need to be made on a trial-by-trial basis. Example 2.1 (acute ischemic stroke). A number of statins, when administered early after stroke in animal models, have demonstrated neuroprotective effects in a dose-dependent manner, with the greatest effects at the highest doses. The NeuSTART (Neuroprotection with Statin Therapy for Acute Recovery Trial) drug development program aimed to translate preclinical research and test the role of high- dose statins in stroke patients. In a phase IB dose finding study under the NeuSTART program [34], high-dose lovastatin was given to patients for 3 days after stroke fol- lowed by a standard dose for 27 days. The primary safety concerns for giving high- dose lovastatin included elevated liver enzyme; a toxicity was said to occur if the peak enzyme levels at any posttreatment time points exceeded a prespecified threshold. There were five test doses in the NeuSTART and the trial objective was to identify a dose with toxicity rate closest to 10%, that is, θ = 0.10. Example 2.2 (cancer prevention). Polyphenon E (Poly E) is a tea catechin extract that is thought to block tumor promotion by inhibiting cell proliferation and inducing cell cycle arrest and apoptosis. The multiple mechanisms of Poly E make it a good candidate agent for chemoprevention. On the other hand, the agent has been shown toxic and causing mortality in female beagle dogs after an overnight fast. Toxicity generally involved the gastrointestinal (GI) system, producing vomiting and damage to the lining of the GI tract, with hemorrhage and necrosis apparent at autopsy. A Poly E trial was conducted in women with a history of hormone receptor-negative breast cancer. Three different doses of Poly E were administered to subjects over 6 months. The study objective was to find the MTD defined as a dose that causes 25% DLT during the six-month period, where a DLT here meant any grade 2 or higher toxicity that would persist for at least one week or requires stopping the study drug. As a secondary objective, this study included biologic correlates such as tissue- based biomarkers and mammography. Therefore, some subjects were randomized to receive a placebo, to which the identified MTD would be compared. Example 2.3 (early neuro-rehabilitation). Early rehabilitation was conjectured to
- 27. THE MAXIMUM TOLERATED DOSE 9 enhance recovery in stroke patients. On the other hand, premature physical ther- apy might cause neurologic worsening, cardiac complication, or even death in the short term. Based on historical data, the adverse event rate is estimated to be 25% in the untreated patients during the first 4 days after stroke. The ASCENT (Acute Stroke Collaboration with Early Neurorehabilitation Therapy) trial was a clinical trial of early physical therapy (PT) in stroke patients, its objective being to identify the largest PT dose that could be instituted without causing adverse events in excess of the 25% background rate. Table 2.1 displays the six PT regimens in ASCENT. In this example, a dose is composed of the timing and duration of therapy. Table 2.1 Physical therapy (PT) regimen in ASCENT Regimen Minutes of PT on Total PT dose Day 2 Day 3 Day 4 (minutes) 1 0 0 30 30 2 0 30 30 60 3 30 30 30 90 4 30 30 60 120 5 30 60 60 150 6 60 60 60 180 In the NeuSTART in Example 2.1, liver enzyme elevation is expected in about 3% of the stroke population, which consists mainly of the elderly. However, since the toxicity endpoint is quite mild and reversible upon drug withdrawal, a toxicity rate higher than the background would be tolerated for the potential benefit in efficacy, and it would also seem reasonable to accept a dose above the target as long as the toxicity probability at this dose was close to θ. Therefore, Definition 2.1, that is, ν was chosen as the operative MTD objective for the trial, even though in this case, the surrogacy perspective is far from reality: Liver enzyme elevation was by no means a surrogate for any clinical benefits in stroke patients. Similar argument may be made for using ν as the objective in the Poly E trial; cf. Exercise 2.1. On the other hand, in the ASCENT trial in Example 2.3, functional recovery due to early physical therapy may not warrant an elevated adverse event rate, and the target rate θ should be set at the background 25%. In this situation, Definition 2.2 appears more appropriate than Definition 2.1. For one thing, since we expect that the dose–toxicity probability is about 25% for doses below and up to the MTD, the objective ν is not uniquely defined. Figure 2.1c displays the plausible shape of the dose–toxicity curve in the ASCENT trial, under which γ is still well defined. Gener- ally, the use of ν requires a strictly increasing dose–toxicity relationship around the MTD, whereas γ always exists regardless of the shape of the dose–toxicity curve. Figure 2.1 also shows plausible dose–toxicity curves for the bortezomib trial and the Poly E trial, where the MTD is defined with respect to a 25% target toxicity rate. In the bortezomib trial, since untreated lymphoma patients will be at no risk of getting a grade 3 neuropathy, the probability (y-intercept) in Figure 2.1a approaches zero as
- 28. 10 DOSE FINDING IN CLINICAL TRIALS Dose Toxicity probability MTD 0 0.25 0.5 0.75 1 (a) Bortezomib trial Dose Toxicity probability MTD 0 0.25 0.5 0.75 1 (b) Poly E trial Dose Toxicity probability MTD 0 0.25 0.5 0.75 1 (c) ASCENT Figure 2.1 Plausible shapes of dose–toxicity curve on a conceptual continuous dose range in three studies and the corresponding MTD, defined as a dose associated with a 25% target toxicity rate. Definition 2.1 is used to define MTD in (a) and (b), and Definition 2.2 in (c). dose decreases. In contrast, because low-grade GI toxicities are not uncommon in subjects with a history of cancer in the Poly E trial, there is a nonzero intercept in Figure 2.1b. However, it is generally believed that the dose–toxicity curve will be strictly increasing around the 25th percentile in both trials, and thus ν is well defined. It is easy to verify that ν = γ when the dose–toxicity curve is a continuous and strictly increasing function of dose. Thus, if a continuum of test doses is available for the trial, it will be a practical choice to use ν as the operative objective, because estimating ν is generally easier than estimating γ. On the other hand, most trials in practice allow only a discrete number of test doses; in which case, Definition 2.1 may yield a slightly more aggressive recommendation than Definition 2.2, because ν ≥ γ. Therefore, the choice between ν and γ depends on whether it is acceptable to be (slightly) more aggressive than the target θ, given the clinical factors such as the nature of treatment, the severity of the disease, and the seriousness of the study endpoint. As alluded to earlier, the CRM is motivated by applications in cancer trials with a surrogacy view. Thus, the book will naturally focus on the estimation of ν using the CRM. However, Chapter 13 will explore situations in which the CRM is not applicable, and introduce an alternative approach that uses γ as trial objective. Also, as the MED in efficacy trials may be defined analogously to ν, the CRM may be applicable to dose finding in phase II efficacy trials. However, this book will focus on the MTD finding by the CRM according to the method’s originally intended use; the design strategy for the MED can be derived by analogy. 2.2 An Overview of Methodology This section gives a brief overview of the development of the dose finding literature since the late 1980s so as to put the CRM in a historical light. Limited by the scope of this book, the review will be cursory. Interested readers can find additional topics in the edited volume by Chevret [26] and the article by Le Tourneau et al. [59].
- 29. AN OVERVIEW OF METHODOLOGY 11 The 3+3 algorithm. Traditionally, a 3+3 algorithm is used to dictate dose escalation and to approach the eventual recommended dose. The method starts the trial at a low dose (e.g., one-tenth of LD10 in mice) and escalates after every three to six patients per dose; the recommended dose is defined as the largest dose with fewer than two patients experiencing a predefined DLT during the first course of treatment. Table 2.2 describes the dose escalation rules of this algorithm. In practice, there may be slight variations from institution to institution. For example, in order to obtain preliminary information about efficacy of the drug, it is common to treat additional patients (usually 6 to 12) at the identified MTD. Table 2.2 Escalation rules at a given dose with cumulative sample size n and total number z of DLT in accordance with the 3+3 algorithm n z Action 3 0 Escalate to the next higher dose 3 1 Treat three additional patients at the current dose 6 1 Escalate to the next higher dose 3 or 6 ≥ 2 Stop escalation and terminal triala aThe MTD is estimated by the dose immediately below the terminating dose. The 3+3 algorithm has historically been the most widely used phase I trial design. Its main advantage is simplicity. Since the dose escalation rules can be tabulated before a trial starts, the clinical investigators can make dose decisions during a trial without help from a statistician. However, the algorithm has two major shortcomings. First, due to a low starting dose and the conservative escalation scheme, the 3+3 algorithm tends to treat many patients at low and inefficacious doses. Since phase I cancer trials typically enroll patients as subjects, as opposed to healthy volunteers, there is an intent to treat the subjects at a therapeutic dose that is likely higher than the lowest test dose. As such, the algorithm is discordant with the therapeutic intent of these trials [84, 55]. Second, the 3+3 algorithm has no statistical justification. There is no intrinsic property in the method to stop escalation at any given percentile, and thus the distribution of the recommended MTD depends arbitrarily on the underlying dose–toxicity curve and the number of test doses [97]. This deficiency is due to the fact that there is no correspondence between the method and any quantitative definition of the MTD, as the 3+3 algorithm does not involve an explicit choice of the target θ. As poor dose selection in the early phase will likely be carried over to its subsequent developmental phases, the use of the 3+3 algorithm will have lingering financial implications and adverse scientific consequences. And thus, the simplicity of the method does not justify its widespread application. Stochastic approximation. Although discussions on phase I designs for cancer trials can be traced back to the 1960s [93], a formal statistical formulation of the MTD appeared at a much later time. Among the earliest discussions was Anbar [2], who in 1984 proposed the use of stochastic approximation [86] in phase I trials. The
- 30. 12 DOSE FINDING IN CLINICAL TRIALS procedure assigns dose sequentially by the recursion xi+1 = xi − 1 ib (Yi −θ) (2.1) for some prespecified constant b > 0, where xi is the dose assigned to patient i, and Yi is the toxicity indicator. The stochastic approximation is a nonparametric method in that it does not assume any parametric structure on the dose–toxicity relationship. Let π(x) = Pr(Yi = 1 | x) denote the probability of toxicity at dose x. Under very mild assumptions of π(x), the dose sequence {xi} generated by (2.1) will converge with probability 1 to a dose x∗ such that π(x∗) = θ. However, the use of recursion (2.1) implicitly assumes a continuum of doses is available for testing in the trial. This is not always feasible in practice. In many situations such as the bortezomib trial, there may be no natural scale of dosage; rather, “dose” is composed of drug dosage and treatment schedule. There are also other difficulties from a statistical viewpoint. First, the stochastic approximation has been shown to be inferior to model-based methods such as the maximum likelihood recursion for binary data [112, 50]. Second, the choice of the constant b in (2.1) has a large impact on the performance of the procedure. As a consequence, the stochastic approximation has seldom been used in dose finding trials. We will return to this in Chapter 14. Up-and-down designs. Subsequent to Anbar’s work [2], Storer [98] considered the up-and-down schemes originally described by Wetherill [110]. An example of the up-and-down design, which Storer called design D, is to enroll patients in groups of three; then, escalate dose for the next group if there is no toxicity in the most recent group, deescalate if there is more than one toxicity in the group, and stay at the same level if one of three patients has toxicity. By Markov chain representation, design D can be shown to sample around a dose that causes toxicity with a probability θ = 0.33. The group size and the decision rules in the up-and-down schemes can apparently be modified to accommodate other target θ; for example, for a target θ = 0.20, one will intuitively enroll patients in groups of five. Also, combinations of schemes can be applied in stages. Storer, in particular, suggested using a group size of one initially in the trial and switching to design D upon the first observed toxicity. The idea here is to move the trial quickly through the low doses so as to avoid treating many patients at low and inefficacious doses. Durham et al. [32] proposed a randomized version of the up-and-down rule for any target at or below the 50th percentile, that is, θ ≤ 0.50. The design deescalates the dose for the next patient if the current patient has toxicity, and escalates according to a biased coin with probability θ/(1−θ) if there is none. The method is thus called a biased coin design. In a trial that uses design D or the biased coin design for dose escalation, when the enrollment is complete, we may naturally estimate the dose–toxicity curve using logistic regression [98] or isotonic regression [99], and estimate the MTD by the 100θth percentile of the fitted curve. The consistency conditions for the estimation of the MTD hold, because the design points {xi} “spread out” under this type of random walk sampling plans. That is, technically, it can be shown that the number of patients
- 31. AN OVERVIEW OF METHODOLOGY 13 treated at each dose will grow indefinitely as sample size increases. It can also be shown by the properties of random walk that the asymptotic distribution of dose allocation has a mode near the target MTD. However, from a design viewpoint, the “memoryless” property of random walk may cause ethical difficulties: Since these up-and-down rules make dose decisions based only on the most recent patient or group of patients at the current dose, previously accrued data are ignored and a trial will likely reescalate to a dose that appears toxic. Model-based designs. In brief, a model-based design makes dose decisions based on a dose–toxicity model, which is being updated repeatedly throughout a trial as data are accrued. The continual reassessment method (CRM), proposed by O’Quigley et al. [78] in 1990, is the first model-based design in the modern dose finding literature. Several model-based designs proposed since 1990 share a similar notion with the CRM. One example is the escalation with overdose control (EWOC) design [4], which takes the continual reassessment notion but estimates the MTD with respect to an asymmetric loss function that places heavier penalties on over- dosing than underdosing; see (2.2) below. A list of model-based methods is given in Section 2.3. Most model-based methods take the myopic approach by which dose assignment is optimized with respect to the next immediate patient without regard to the future patients. For example, the EWOC at each step minimizes the Bayes risk with respect to the loss function: xi+1 = argmin x Ei α(ν −x)+ +(1 −α)(x−ν)+ (2.2) where x+ = max(x,0) and Ei(·) denotes expectation computed with respect to the posterior distribution of the MTD ν, given the first i observations. As the EWOC is intended to control overdose, the loss function (2.2) should be specified with a value of α, that is, α 0.50. More recently, Bartroff and Lai [6] take a stochastic optimization approach that minimizes the global risk and propose to choose the doses {x1,x2,...,xN} sequentially so as to mininize E ( N ∑ i=1 α(ν −xi)+ +(1 −α)(xi −ν)+ ) , where the expectation is taken with respect to the joint distribution of xis and ν. Such sequential optimization is implemented by backward induction and requires dy- namic programming which can be computationally intensive. However, this approach presents a new direction for model-based design and warrants further research. The model-based approach facilitates borrowing strength from information across doses through the parametric assumptions on the dose–toxicity curve. This is especially important in early-phase dose finding trials where sample sizes are small and informational content is low [41]. In addition, since these methods allow starting a trial at a dose higher than the lowest level, the in-trial allocation tends to concentrate around the target dose. Many model-based designs take a Bayesian approach. They update sequentially
- 32. 14 DOSE FINDING IN CLINICAL TRIALS the uncertainty about the dose–toxicitycurve with respect to the posteriordistribution given the interim data. This approach is by nature automated, so far as the posterior computations can be efficiently programmed and reproduced. An advantage of such automation is that a carefully calibrated dose–toxicity model can handle unplanned contingencies in a manner that is coherent with the trial objective. In contrast, we may imagine the predicament arising with the 3+3 rule in a trial where there are two toxic outcomes among 7 patients at a dose; this contingency can be caused by overaccrual due to administrative delays, and is not uncommon in practice. Several practical difficulties may hinder the use of a model-based design. First, there is skepticism among the clinicians because of the “blackbox” approach of these designs. Second, there is the perception that the success of the method is sensitive to the choice of the dose–toxicity model. Third, as a model-based design requires specialized computations, the clinical team will need to interact regularly with the study statistician for interim dose decisions. Such interaction may be perceived as adding unnecessary burdens on both parties, in light of the fact that the standard 3+3 algorithm requires minimal statistical inputs. This book attempts to address the second difficulty with a focus on the CRM. By theoretical and empirical arguments, we will see that the method’s performance does not depend on correctness of the model specification. This book also partially addresses the third difficulty by illustrating how a CRM design can be calibrated and implemented in practice. The ultimate goal is to alleviate the statistician’s burden during the planning stage and the conduct of the trial. This endeavor is facilitated by availability of software. This book focuses on the R package ‘dfcrm’ for the CRM and its major variants. Software for some model-based designs is also available to public access; see Table 2.3. Table 2.3 Some software links for model-based approaches Description Section Reference Escalation with overdose control (EWOC)a 2.2 [4] Late-onset toxicity monitor using predicted risksb 11.4.2 [8] Modified CRM on a continuous dosage rangec 12.4.4 [81] Phase I/II dose finding based on efficacy and toxicityb 13.3 [102] ahttp://www.sph.emory.edu/BRI-WCI/ewoc.html bhttp://biostatistics.mdanderson.org/softwaredownload chttp://www.cancerbiostats.onc.jhmi.edu/software.cfm Algorithm-based designs. Because of the above-mentioned difficulties with the model-based designs, there seems to be a renewed interest in the algorithm-based designs since the late 1990s. Generally, an algorithm-based design prescribes a set of escalation rules for any given dose without regard to the outcomes at other doses. As a result of the independence among observations across doses, the rules can be tabulated and made accessible to the clinical investigators before a trial starts. The 3+3 algorithm is the most prominent example of algorithm-based designs. Efforts have recently been made to extend this traditional method so as to obtain
- 33. BIBLIOGRAPHIC NOTES 15 well-defined statistical properties. In particular, Cheung [18] formulates dose finding as a multiple testing problem and introduces a class of stepwise test procedures that operate in a manner similar to the 3+3 algorithm. This approach has practical ap- peal because clinicians are familiar with the 3+3 algorithm. Chapter 13 gives further details of the stepwise procedures. Ji et al. [49] propose a class of up-and-down designs that make dose decisions based on the posterior toxicity probability intervals. Specifically, the parameter space of the toxicity probability pk at dose k is partitioned into three sets: for some prespec- ified constants K1,K2 0, Θk,E = {pk −θ −K1σk} Θk,S = {−K1σk ≤ pk −θ ≤ K2σk} Θk,D = {pk −θ K2σk} so that Θk,E ∪Θk,S ∪Θk,D = [0,1], where σk is the posterior standard deviation of pk. Escalation from a current dose, say dose k, will take place if the posterior probability of Θk,E is largest among the three sets. Similarly, deescalation occurs if Θk,D is the most probable event according to the posterior distribution of pk. Otherwise, the next dose will remain at level k. An important difference between this design and the random walk up-and-down is that the posterior interval uses all observations accrued to a dose to make a dose decision and avoids the memoryless problem of a random walk design. 2.3 Bibliographic Notes The practice and design of phase I trials are discussed in the medical community by Schneiderman [93], Carbone et al. [13], Ratain et al. [85, 84], and Kurzrock and Benjamin [55]. Korn [53] discusses the relevance of MTD in noncytotoxic targeted cancer treatments. Discussion of dose finding strategies in the other disease areas is relatively sporadic. Fisher et al. [38] present some phase I and II trial designs in the context of acute stroke MRI trials, and make a case against the use of the 3+3 algorithm in the phase I safety studies. Cheung et al. [23] make analogous arguments for early-phase trials in patients with amyotrophic lateral sclerosis. The dose finding design of NeuSTART is reported in Elkind et al. [34] as a case study. Robbins and Monro [86] introduce the first stochastic approximation method, which has been studied extensively and has motivated a large number of subsequent modifications. (See for example Sacks [92], Venter [107], Lai and Robbins [57, 58], and Wu [112, 113].) In the more recent literature, Lai [56] gives a thorough review of the advances of stochastic approximation. Cheung [19] draws a specific tie between this area and modern dose finding methods. Lin and Shih [64] study the operating characteristics of a class of A+B designs that include the 3+3 algorithm as a special case. The theoretical properties of the biased coin design are established by Durham and colleagues [29, 30, 31]. Several model-based designs have been proposed since the 1990s. These include the Bayesian decision-theoretic design [111], the logistic dose-ranging strategy [70],
- 34. 16 DOSE FINDING IN CLINICAL TRIALS and the Bayesian c-optimal design [44]. The CRM has generated a large literature and will be reviewed in the next chapter. Chapter 12 will comment on two CRM-like dose finding designs: the curve-free method [40] and the isotonic design [63]. For the EWOC, Zacks et al. [117] prove that the method is Bayesian-feasible, Bayesian- optimal, and consistent under the assumption that the specified dose–toxicity model is correct. These Bayesian criteria are introduced in the previous work by Eichhorn and Zacks [33]. 2.4 Exercises and Further Results Exercise 2.1. Discuss the MTD objective (surrogacy versus trade-off perspectives) for cancer prevention in the context of the Poly E trial. Exercise 2.2. Definition 2.1 formulates dose finding as estimating a percentile on a dose–toxicity curve. Another possible alternative to define the MTD according to the surrogacy perspective is ν′ = argmin k |π−1 (pk)−π−1 (θ)|. Show that |ν − ν′| ≤ 1, that is, the two definitions can differ by at most one dose level. Discuss why ν′ may not be applicable for the bortezomib trial. Exercise 2.3. By computer simulations, generate the outcomes of a trial using the 3+3 algorithm with K = 5 doses and true toxicity probabilities p1 = 0.02, p2 = 0.04, p3 = 0.10, p4 = 0.25, and p5 = 0.50. Observe the recommended MTD. Repeat the simulations 1000 times, and record the distribution of the recommended MTD in the 1000 simulated trials.
- 35. Chapter 3 The Continual Reassessment Method 3.1 Introduction In this book, we consider two types of dose finding strategies using the continual reassessment method (CRM): a one-stage design that necessitates the use of Bayesian CRM (Section 3.2) and a two-stage design (Section 3.3). This chapter outlines the basic CRM approach and introduces the necessary notation for further development in the later chapters. Section 3.4 presents some simulation outputs of the CRM to illustrate how the method may operate in practice. Section 3.5 reviews some common modifications of the CRM. Section 3.6 gives key references in the CRM literature. 3.2 One-Stage Bayesian CRM 3.2.1 General Setting and Notation Consider a trial with K test doses with numerical labels d1,...,dK. In a dose finding trial, patients are enrolled in small groups of size m ≥ 1. Let xi ∈ {d1,...,dK} denote the dose assigned to the ith group of patients, so that patients in the same group receive the same dose. In what follows, we first consider a fully sequential enrollment plan (i.e., m = 1), where we observe a binary toxicity outcome Yi from the ith patient, and postulateYi as a Bernoulli variable with toxicity probability π(xi), where π(x) is a monotone increasing function in x. We will consider group accrual enrollment, that is, m 1, in Section 3.5.2. In accordance with Definition 2.1, the trial objective is to identify the dose level ν ∈ {1,...,K} that is associated with a toxicity probability θ. 3.2.2 Dose–Toxicity Model The CRM assumes a dose–toxicity model F(x,β); that is, the true dose–toxicity curve π(x) is postulated to be F(x,β0) for some true parameter value β0. Generally, the CRM does not require F to be a correct model for π, and β0 may then be viewed as a “least false” value. Briefly, we require F(x,β) to be strictly increasing in the dose x, in addition to some regularity conditions. Details of the assumptions will be given in Chapters 4 and 5. The two most commonly used models in the CRM literature are the empiric function F(x,β) = xβ for 0 x 1 (3.1) 17
- 36. 18 THE CONTINUAL REASSESSMENT METHOD and a one-parameter logistic function F(x,β) = exp(a0 +βx) 1 +exp(a0 +βx) for −∞ x ∞ where the intercept a0 is a fixed constant. Another common dose–toxicity model is the hyperbolic tangent function [78, 67] F(x,β) = tanhx+1 2 β for −∞ x ∞. To ensure an increasing dose–toxicity relationship, the parameter β in these models is restricted to taking on positive values. The positivityconstraint could present some difficulty in estimation, especially when the sample size is small. Hence, it is useful to consider the following parameterization: empiric: F(x,β) = xexp(β) for 0 x 1 (3.2) logistic: F(x,β) = exp{a0 +exp(β)x} 1 +exp{a0 +exp(β)x} for −∞ x ∞ (3.3) hyperbolic tangent: F(x,β) = tanhx+1 2 exp(β) for −∞ x ∞ (3.4) under which the parameter β is free to take on any real values while F(x,β) is strictly increasing. The original formulation of the CRM uses a Bayesian approach by which the model parameter β is assumed random and follows a prior distribution G(β). We will focus on the normal prior distribution, that is, β ∼ N(β̂0,σ2 β ) where β̂0 and σ2 β are, respectively, the prior mean and variance. 3.2.3 Dose Labels An important point about the CRM is that the numerical dose labels d1,...,dK are not the actual doses administered, but rather are defined on a conceptual scale that represents an ordering of the risks of toxicity. Consider the dose schedules used in the bortezomib trial (Table 1.1). The first three levels prescribe bortezomib at a fixed dose 0.7 mg/m2 with increasing frequency, whereas the next two levels apply the same frequency with increasing bortezomib dose. While it is reasonable to assume that the toxicity risk increases with each level, there is no natural unit for dose (e.g., mg/m2) in this application. Similarly, in the ASCENT trial (Example 2.3), “dose” is a composite of timing and duration of physical therapy given. In these examples, it is artificial to assume the dose–toxicity curve π(x) is well defined on a continuous
- 37. ONE-STAGE BAYESIAN CRM 19 dose range. Instead, one will have access only to a discrete set of increasing doses. As the CRM operates on a discrete set of dose levels, a physical interpretation for the dose labels d1,...,dK is not required, as long as they constitute a strictly increasing sequence. In practice, to ensure monotonicity, the label dk can be obtained by substituting the initial guess of toxicity probability p0k for dose level k into the dose–toxicity model, that is, solving p0k = F(dk,β̂0). (3.5) The set of initial guesses {p0k} is sometimes called the ‘skeleton’ of the CRM, and is a strictly increasing sequence, that is, p01 p02 ··· p0K. Suppose there is a prior belief that dose level ν0 is the MTD. We may set the initial guess p0,ν0 = θ. (3.6) Consider, for instance, a trial with K = 5 dose levels and a target probabilityθ = 0.25. Suppose we use the logistic function (3.3) with a0 = 3 and prior mean β̂0 = 0, and we believe that ν0 = 3 is the prior MTD such that p03 = 0.25. Then we can solve 0.25 = exp{3 +exp(0)d3} 1 +exp{3 +exp(0)d3} and obtain d3 = −4.10. Suppose further that p01 = 0.05, p02 = 0.12, p04 = 0.40, and p05 = 0.55. Then in the same manner, we obtain d1 = −5.94,d2 = −4.99, d4 = −3.41, and d5 = −2.80. Table 3.1 shows the dose labels obtained by back- ward substitution (3.5) under various dose–toxicity functions using the same skele- ton. Note that the range of dk varies with the model, and may take on negative values. However, it is always true that d1 ··· dK. Table 3.1 Dose labels via backward substitution under four CRM models for K = 5 with p01 = 0.05, p02 = 0.12, p03 = 0.25, p04 = 0.40, p05 = 0.55, and prior mean β̂0 = 0 Model d1 d2 d3 d4 d5 Empiric (3.2) 0.05 0.12 0.25 0.40 0.55 Logistic (3.3) with a0 = 0 −2.94 −1.99 −1.10 −0.41 0.20 Logistic (3.3) with a0 = 3 −5.94 −4.99 −4.10 −3.41 −2.80 Hyperbolic tangent (3.4) −1.47 −1.00 −0.55 −0.20 0.10 The backward substitution (3.5) ensures the dose–toxicity model F provides an exact fit over the initial guesses of toxicity probabilities, which ideally should reflect the clinicians’ prior beliefs. This is a crucial step particularly because of the use of underparameterized (one-parameter) model. In practice, it is often unrealistic for the clinicians to provide reliable guesses for all test doses prior to a study. Rather, we take the approach by which the skeleton {p0k} is numerically calibrated to yield good design’s operating characteristics. We shall return to this in Chapter 8.
- 38. 20 THE CONTINUAL REASSESSMENT METHOD 3.2.4 Model-Based MTD The CRM starts a trial by treating the first patient at the prior MTD ν0, that is, x1 = dν0. By (3.6), this starting dose is the dose initially believed to have toxicity probability (closest to) θ. Each subsequent xi is determined sequentially based on the previous observation history Hi = {(xj,Yj) : j i} for i ≥ 2. A CRM design D1 can be viewed as a function defined on the increasing Hi. The basic idea is to treat the next patient at the model-based MTD estimate, given Hi. Precisely, xi = D1(Hi) = argmin dk |F(dk,β̂i−1)−θ| (3.7) where β̂i−1 = R ∞ −∞ βLi−1(β)dG(β) R ∞ −∞ Li−1(β)dG(β) is the posterior mean of β given Hi and Li−1(β) = i−1 ∏ j=1 {F(xj,β)}Yj {1 −F(xj,β)}1−Yj (3.8) is the binomial likelihood. The assignment (3.7) continues in a sequential fashion until a prespecified sample size N is reached. For the CRM, the final MTD estimate is given by xN+1 = D1(HN+1), that is, a dose would have been given to the (N +1)st patient enrolled to the trial. In other words, the CRM attempts to treat the next patient at the current best guess of the MTD, a dose with toxicity probability estimated to be closest to the target θ. The motivation of this algorithm is to correct for the deficiency of the 3+3 algorithm, which treats the majority of the subjects at low and inefficacious doses. While this is ethically sound on a conceptual level, there may be various ways to calculate the “best” dose on the implementation level. The model-based MTD (3.7) is obtained by estimating pk = π(dk) with a plug-in estimate F(dk,β̂i−1). An alternative MTD estimate is D∗ 1 (Hi) = argmin dk |Ei−1{F(dk,β)} −θ| (3.9) where Ei−1(·) denotes expectation computed with respect to the posterior given Hi, that is, Ei−1{F(dk,β)} = R ∞ −∞ F(dk,β)Li−1(β)dG(β) R ∞ −∞ Li−1(β)dG(β) . The MTD estimate (3.9) involves the computation of K integrals at each interim, and is a more formal Bayesian estimate of pk than (3.7). In the early CRM literature, the plug-in estimate (3.7) emerged to be the convention because of its computational ease (although computational consideration is of much less importance today). Also, the estimate (3.9) as a formal Bayesian estimate is conceptually advantageous only when F is a correct model for π. As we will see in Chapter 5, an attractive feature of the model-based CRM is that its performance does not rely on correct specification of F. Hence, we will focus on the plug-in CRM (3.7), which is studied more thoroughly and systematically than the other estimators in the literature.
- 39. ONE-STAGE BAYESIAN CRM 21 3.2.5 Normal Prior on β Now, suppose that β has a normal prior distribution with mean β̂0. For the logistic function (3.3), applying backward substitution (3.5) gives dk = logit(p0k)−a0 exp(β̂0) where logit(p) = log{p/(1 − p)}. As a result, we have F(dk,β) = exp h a0 +exp(β −β̂0){logit(p0k)−a0} i 1 +exp h a0 +exp(β −β̂0){logit(p0k)−a0} i. Since F(dk,β) depends on the parameter β only via β − β̂0, which is mean zero normal, we may arbitrarily set β̂0 = 0 without affecting the computation. The logistic model (3.3) is therefore invariant to the mean of a normal prior distribution. This invariance property holds for the general class of dose–toxicity models described in Chapter 4. 3.2.6 Implementation in R The R package ‘dfcrm’ consists of functions for the implementation and the design of the Bayesian CRM using the empiric (3.2) and logistic (3.3) models. In particular, the function crm takes cumulative patient data and returns a dose for the next patient according to the model-based estimate (3.7). ### Return the recommended dose level for patient 6 ### based on data from five patients library(dfcrm) p0 - c(0.05,0.12,0.25,0.40,0.55) # initial guesses theta - 0.25 # target toxicity rate y - c(0, 0, 1, 0, 0) # toxicity indicators lev - c(3, 5, 5, 3, 4) # dose levels fooB - crm(p0,theta,y,lev,model=logistic,intcpt=3) fooB$estimate # posterior mean of beta [1] 0.2794614 fooB$mtd [1] 4 fooB$doses # Dose labels [1] -5.944439 -4.992430 -4.098612 -3.405465 -2.799329 The above R code illustrates the usage of crm for a trial with K = 5 test doses and a target θ = 0.25; the function is applied to observations from the first 5 subjects who receive dose levels 3, 5, 5, 3, and 4, where the third patient has a toxic outcome. The one-parameter logistic function (3.3) is used when the argument model is specified
- 40. 22 THE CONTINUAL REASSESSMENT METHOD as “logistic”. The default intercept value is 3, and can be modified by the argument intcpt; thus, the specification “intcpt=3” in the above illustration is redundant. When no value for the argument model is provided, crm will use the empiric model (3.2) as the default. The function computes the posterior mean of β by using a normal prior with mean 0. The prior standard deviation is specified by the argument scale; if scale is not specified (as in the last illustration), the default value is √ 1.34. One-stage Bayesian CRM The one-stage Bayesian CRM requires the specification of an array of design parameters which can be classified into clinical parameters (or clinician-input parameters) and model parameters. Planning and implementation of the method in a dose finding trial takes three steps: 1. Setting the clinical parameters: • Target toxicity probability θ • Number of test doses K • Prior MTD ν0 • Sample size N 2. Calibrating the model parameters: • Functional form of the dose–toxicity model F(·,β) • Skeleton {p0k} and hence the dose labels dks via backward substitution • Prior distribution G(β) of β 3. Execution: Treat the first patient at ν0, and the subsequent patients at the most recent model-based MTD (3.7) as data accrue throughout the trial. 3.3 Two-Stage CRM 3.3.1 Initial Design There are two practical difficulties associated with the use of the one-stage Bayesian CRM. First, treating the first patient at the prior MTD rather than the lowest dose level may raise safety concerns. Second, the use of a prior distribution G(β) may be viewed as subjective and arbitrary. To address the first difficulty, several authors suggest starting a CRM trial at the lowest dose, and applying dose escalation restrictions when the model-based MTD appears aggressive. This approach can be represented by a two-stage CRM, defined as follows. First, specify an initial design as a predetermined nondecreasing dose sequence {xi,0} such that xi−1,0 ≤ xi,0. Then a two-stage CRM D2(Hi) is defined as D2(Hi) = xi,0, if Yj = 0 for all j i, D1(Hi) if Yj = 1 for some j i. (3.10) In other words, the initial design is in effect until the first observed toxicity; once a
- 41. TWO-STAGE CRM 23 toxic outcome is observed, the trial turns to the model-based CRM for dose assign- ments. Because the 3+3 algorithm is familiar, there is an inclination to consider the “group-of-three” initial design by which escalation takes place after every group of three nontoxic outcomes. That is, x1,0 = x2,0 = x3,0 = d1, x4,0 = x5,0 = x6,0 = d2, x7,0 = x8,0 = x9,0 = d3 ... and so on. There is, however, no clear justification for using the group-of-three rule apart from convention. In fact, this initial design is sometimes not in line with the motivation of the CRM. We will study the calibration of the initial dose sequences in Chapter 10. 3.3.2 Maximum Likelihood CRM In response to the difficulty associated with the subjectivity of the Bayesian approach, we may use maximum likelihood estimation in conjunction with the CRM [80]. The idea is simple and analogous to the Bayesian CRM: with data observed in the first i −1 patients, the dose for the next subject is computed as xi = D̃1(Hi) = argmin dk |F(dk,β̃i−1)−θ| (3.11) where β̃i−1 = argmaxβ Li−1(β) is the maximum likelihood estimate (mle) of β for given Hi. The R function crm also implements the maximum likelihood CRM through the specification of the argument method: ### Compute the recommended dose level for patient 6 ### using maximum likelihood CRM on the same data ### i.e., same values of p0, theta, y, lev fooL - crm(p0,theta,y,lev,model=logistic,method=mle) fooL$estimate # mle of beta [1] 0.3142946 fooL$mtd [1] 5 The function crm evaluates the maximum likelihood estimate of MTD (3.11) through the specification of the method argument as “mle”. When method is not specified, the Bayesian CRM is assumed. Using (3.11) presupposes the existence of mle of β. For a one-parameter model F, the mle β̃i−1 exists if and only if there is heterogeneity in the toxicity outcomes among patients, that is, Yj = 0 and Yj′ = 1 for some j, j′ i. In the last R illustration, there is one toxic outcome out of 5 patients, and thus β̃5 exists. In general, when one plans to use the maximum likelihood CRM, it is necessary to consider a two-stage design, which can be formed by replacing D1(Hi) with D̃1(Hi) in (3.10). The posterior mean β̂i−1 and the mle β̃i−1 are generally different, thus leading
- 42. 24 THE CONTINUAL REASSESSMENT METHOD to potentially different dose recommendations. For example, in the above R codes, the Bayesian CRM chooses dose level 4 for the sixth patient, whereas the maximum likelihood CRM chooses level 5. However, the posterior mean β̂5 = 0.279 and mle β̃5 = 0.314 are only slightly different. Looking at the model-based estimates of the toxicity probabilities reveals that both dose levels 4 and 5 are roughly equally apart from the target θ = 0.25: round(fooB$ptox,digits=2) # Posterior toxicity rates [1] 0.01 0.03 0.08 0.18 0.33 round(fooL$ptox,digits=2) # Maximum likelihood estimates [1] 0.01 0.02 0.07 0.16 0.30 The difference in the estimation of β will diminish as sample size increases; and eventually, the choice of the estimation method per se will have minimal impact on the dose assignments. However, the estimation method may have different implied performance due to the fact that the maximum likelihood CRM always requires a two-stage strategy whereas Bayesian CRM is typically used as a one-stage design. Two-stage CRM The two-stage CRM requires the specification of an initial design as part of the design parameters in addition to those required in the one-stage CRM. Planning and implementation of a two-stage CRM in a dose finding trial takes four steps: 1. Setting the clinical parameters: • Target toxicity probability θ • Number of test doses K • Prior MTD ν0 • Sample size N 2. Calibrating the model parameters: • Functional form of the dose–toxicity model F(·,β) • Skeleton {p0k} and hence the dose labels dks via backward substitution† • Prior distribution G(β) of β, if Bayesian CRM is used 3. Specifying an initial dose sequence: x1,0 ≤ x2,0 ≤ ... ≤ xN,0. 4. Execution: Treat patients initially according to {xi,0}. Upon the first observed toxic outcome, treat the subsequent patient at the most recent model-based MTD (3.7) or (3.11) as data accrue throughout the trial. †Backward substitution for the maximum likelihood CRM can be carried out as in (3.5) using an arbitrary initial value β̃0 without affecting the dose calculations. More details will be given in Chapter 4.
- 43. SIMULATING CRM TRIALS 25 3.4 Simulating CRM Trials 3.4.1 Numerical Illustrations Computer simulation is a primary tool for evaluating the aggregate performance of the method. To get a sense of how the CRM works, it is also useful to examine individual simulated trials. Figure 3.1 shows the outcomes of simulated trials using a one-stage and two-stage CRM for a trial with target θ = 0.25 and K = 5 under the true dose–toxicity scenario p1 = 0.02, p2 = 0.04, p3 = 0.10, p4 = 0.25, p5 = 0.50. (3.12) Hence, dose level 4 is the true MTD. The dose assignments by the two methods in the figure are quite different, although both select the correct MTD (dose level 4) based on the 20 simulated subjects. The one-stage CRM, taking advantage of a high starting dose, treats the majority of the subjects at the MTD but also seven patients at an overdose. In contrast, the two-stage CRM takes a conventional escalation approach initially and potentially treats many patients at low doses. As a result, the one-stage CRM causes two toxic outcomes (patients 3 and 9) more than the two-stage CRM. It is debatable whether the one-stage CRM is unsafe. On the one hand, the CRM has been criticized on account of ethical concerns, as it is shown to cause more toxic outcomes than the standard 3+3 method under some dose–toxicity scenarios [54]. On the other hand, overdosing is not necessarily a worse mistake than underdosing when treating patients with severe diseases such as cancer. Rosa et al. [89] describe a case study in which starting at a low dose (per the 3+3 algorithm) causes an ethical dilemma when consenting a cancer patient. Overall, the risk–benefit trade-off should be considered on a case-by-case basis. While the bortezomib trial in Chapter 1 started at the third level, the NeuSTART (Example 2.1) exercised caution via a low starting dose. In addition, the one-stage CRM in Figure 3.1 causes a 25% observed toxicity rate (5 out of 20), which is on target. From a numerical viewpoint, the two-stage design appears to be overconservative in this particular simulated trial. 3.4.2 Methods of Simulation When simulating toxicity outcomes in a trial, each patient may be viewed to be car- rying a latent toxicity tolerance ui that is uniformly distributed on the interval [0,1]. If the uniform variate is smaller than the true toxicity probabilityof the dose assigned to the patient, the patient has a toxic outcome; otherwise, the patient does not have a toxic outcome. That is, Yi = 1 if ui ≤ π(xi) 0 otherwise Table 3.2 displays the latent tolerance of the 20 simulated patients used in the simulated trials in Figure 3.1. Consider, for example, patient 1 who receives dose level 3 according to the one-stage CRM (left panel of the figure). He does not have a toxic outcome because the tolerance u1 = .571 p3 = .10. Consequently, using the
- 44. 26 THE CONTINUAL REASSESSMENT METHOD 5 10 15 20 1 2 3 4 5 Patient number Dose level x x x x x o o o oo oo o ooooo oo One−stage CRM 5 10 15 20 1 2 3 4 5 Patient number Dose level x x x ooo ooo ooo oo oooo oo Two−stage CRM Figure 3.1 Simulated trials using the one-stage and two-stage Bayesian CRM in 20 subjects. The logistic model (3.3) with a0 = 3 in Table 3.1 is used with β ∼ N(0,1.34) a priori. For the two-stage CRM, the initial design escalates according to the group-of-three rule. Each point represents a patient, with “o” indicating no toxicity and “x” indicating toxicity. same CRM model as in Figure 3.1, we obtain the posterior mean β̂1 = 0.60, which implies x2 = 5. Table 3.2 also gives the numerical outputs of the one-stage CRM based on these 20 simulated patients. Table 3.2 A simulated CRM trial through latent toxicity tolerance of 20 simulated patients i xi π(xi) ui yi β̂i i xi π(xi) ui yi β̂i 1 3 0.10 .571 0 0.60 11 5 0.50 .321 1 0.25 2 5 0.50 .642 0 0.93 12 4 0.25 .099 1 0.15 3 5 0.50 .466 1 0.04 13 4 0.25 .383 0 0.18 4 3 0.10 .870 0 0.18 14 4 0.25 .995 0 0.21 5 4 0.25 .634 0 0.28 15 4 0.25 .628 0 0.24 6 4 0.25 .390 0 0.34 16 4 0.25 .346 0 0.26 7 5 0.50 .524 0 0.41 17 4 0.25 .919 0 0.28 8 5 0.50 .773 0 0.47 18 4 0.25 .022 1 0.21 9 5 0.50 .175 1 0.31 19 4 0.25 .647 0 0.22 10 5 0.50 .627 0 0.35 20 4 0.25 .469 0 0.24 In a real trial, the latent tolerance ui is not observable. In computer simulation, on the other hand, toxicity tolerance can be easily generated and is a useful tool to make different designs comparable in experiments where the dose assignments are made adaptively. For example, the same latent tolerance sequence in Table 3.2 can be used to generate the two-stage CRM in Figure 3.1 (right panel) so that both the
- 45. PRACTICAL MODIFICATIONS 27 one-stage and two-stage designs are treating the same patients, while the patients are not necessarily treated at the same doses. The concept of toxicity tolerance is also instrumental to the construction of a nonparametric optimal design. We will return to this in Chapter 5. The function crmsim in the ‘dfcrm’ package can be used to simulate multiple CRM trials under a given dose–toxicity curve. The following R code runs 10 trials using the one-stage Bayesian CRM specified in Figure 3.1 under the dose–toxicity scenario (3.12): ### Generate 10 CRM trials theta - 0.25 PI - c(0.02,0.04,0.10,0.25,0.50) # True MTD = 4 N - 20 # sample size x0 - 3 # starting dose level foo10 - crmsim(PI,p0,theta,N,x0,nsim=10,model=logistic,restrict=F) simulation number: 1 simulation number: 2 simulation number: 3 simulation number: 4 simulation number: 5 simulation number: 6 simulation number: 7 simulation number: 8 simulation number: 9 simulation number: 10 foo10$MTD # Display the distribution of recommended MTD [1] 0.0 0.0 0.1 0.8 0.1 In this illustration, dose level 4 is selected as the MTD in 8 of the 10 simulated trials. 3.5 Practical Modifications 3.5.1 Dose Escalation Restrictions The one-stage CRM in Figure 3.1 assigns dose level 5 to the second patient after the nontoxic outcome in the first patient who receives dose level 3. This escalation may raise safety concerns because a high dose is tested without testing an intermediate dose level. Several authors have noted the potential problem with dose skipping by the CRM, and proposed the restricted CRM by imposing an escalation restriction: The dose level for the next patient cannot be more than one level higher than that of the current patient. Likewise, it is possible for the unrestricted CRM to skip doses in deescalation; cf. patient 4 under the one-stage CRM in Figure 3.1. While skipping doses in deescalation has not been perceived to be as problematic or unsafe, we may apply a similar restriction that the dose level for the next patient cannot be more than one level lower than that of the current patient. At any rate, these restrictions against dose skipping will typically be applied, if ever, only to the first few patients because
- 46. 28 THE CONTINUAL REASSESSMENT METHOD the change in the model-based estimates diminishes as the number of observations increases (try plotting the sequence of β̂i in Table 3.2). Another pathologyin escalation is illustratedin the two-stage CRM in Figure 3.1: the dose for patient 13 is escalated from that of patient 12, who has a toxic outcome. Such an escalation is called incoherent as it puts patient 13 at undue risk of toxicity in light of the outcome in the previous patient [16]. To avoid the potential incoherent moves by the CRM, we may apply the restrictions: Coherence in escalation: If a toxicity is observed in the current patient, then the dose level for the next patient cannot be higher than that of the current patient. Coherence in deescalation: If the current patient does not experience toxicity, then the dose level for the next patient cannot be lower than that of the current patient. It is noteworthythat there is no incoherent move by the one-stage CRM in Figure 3.1. That is, escalation occurs only after a nontoxic outcome, and deescalation after a toxic outcome. In fact, the one-stage CRM will never induce an incoherent move— even if no restriction is applied. This property indicates that the model-based CRM is doing the right thing ethically. In contrast, an unrestricted two-stage CRM may yield incoherent escalation as seen in Figure 3.1. It turns out that the problem is due to poor calibration of the initial design. In this particular example, the group-of-three escalation sequence is not an appropriate choice. This topic will be further discussed in Chapters 5 and 10. Figure 3.2 shows the outcomes of the restricted CRM trials using the same model specifications as in Figure 3.1 and the same 20 patients as in Table 3.2. The dose assignment patterns are similar to that of the unrestricted counterparts. In general, the dose escalation restrictions have little impact on the CRM’s aggregate behaviors in terms of the probability of correctly selecting the MTD and the average toxicity number; try Exercise 3.4. Instead, these restrictions modify the pointwise properties of the design: a pointwise property of a dose finding design concerns the behaviors of individual outcome sequences (“points”). The pointwise properties of a design should agree with principles deemed sensible to clinicians. Violations of any such principles (e.g., making incoherent moves) will be perceived as worse than showing poor statistical properties. After all, only a single outcome sequence is observed in an actual trial. The aggregate operating characteristics are comparatively abstract to clinicians. In the rest of this book, unless specified otherwise, we assume that the restriction of no dose skipping in escalation and the coherence restrictions are in effect. This is the assumption used in the function crmsim if no value is given to the argument restrict. 3.5.2 Group Accrual The fully sequential CRM follows the current patient over an evaluation period, called an observation window, before enrolling the next patient. A long observation
- 47. PRACTICAL MODIFICATIONS 29 5 10 15 20 1 2 3 4 5 Patient number Dose level x x x x x o o ooo oo o ooooo oo Restricted one−stage CRM 5 10 15 20 1 2 3 4 5 Patient number Dose level x x x ooo ooo ooo oo o oo o oo Restricted two−stage CRM Figure 3.2 Simulated trials using the one-stage and the two-stage Bayesian CRM with dose escalation restrictions. Each point represents a patient, with “o” indicating no toxicity and “x” indicating toxicity. window (e.g., 6 months) will likely result in repeated interim accrual suspensions, impose excessive administrative burdens, and cause long trial duration. Goodman et al. [43] address this problem by assigning m 1 patients at a time to each dose, and updating the model-base estimate (3.7) between every group of m patients. Figure 3.3 shows the outcomes of the simulated trials by group accrual CRM, using the models in Figure 3.1 and the same 20 patients in Table 3.2. In the left panel of Figure 3.3, the model-based MTD estimate (3.7) is updated after every m = 2 patients. The dose assignment pattern is similar to that of the fully sequential CRM in Figure 3.1: half of the study subjects receive the MTD. The use of a larger group size with m = 4, as shown in the right panel of the figure, gives the CRM fewer occasions to adapt to the previous outcomes. This may partly explain the findings that the accuracy of MTD estimation decreases as a large group size is used, when the assumed model F is not a correct specification of the true dose–toxicity curve [43]. Therefore, although a large group size reduces the number of interim calculations and shortens trial duration, it also undermines the advantage of the adaptiveness of the CRM. When the observation window is long in comparison to the recruitment period, the reduction in trial duration by group accrual may not be adequate. Suppose, for instance, each patient in the current group is to be followed for 6 months before the next group is enrolled in Figure 3.3. Assuming instantaneous patient availability, a group CRM with size 4 will take roughly 24 months to enroll 20 patients, whereas a group size of 2 will take 54 months. We will return to consider clinical settings with a long observation window due to late toxicities in Chapter 11, where we introduce the time-to-event continual reassessment method (TITE-CRM) as an alternative to the group accrual CRM.
- 48. 30 THE CONTINUAL REASSESSMENT METHOD 5 10 15 20 1 2 3 4 5 Patient number Dose level x x xx x oo oo o oo o ooooo oo Group CRM (size 2) 5 10 15 20 1 2 3 4 5 Patient number Dose level x xx x x oooo oooo o oooo o o Group CRM (size 4) Figure 3.3 Simulated trials using the one-stage group accrual CRM. The MTD estimate is updated after every two patients in the left panel, and every four patients in the right panel. Dose escalation restrictions are applied. Each point represents a patient, with “o” indicating no toxicity and “x” indicating toxicity. 3.5.3 Stopping and Extension Criteria It is common to have provisions for early termination in clinical trials for economic and ethical reasons. The standard 3+3 design for dose finding trials stops a trial when two toxic outcomes are observed at a dose. In contrast, the original CRM works with a fixed sample size N. From a practical viewpoint, neither the economic nor ethical reason is sufficiently compelling for stopping a CRM trial early. Economically, the sample size of a phase I trial is already small, and there is not much room to reduce the sample size unless the early stopping rules are liberal. Ethically, the CRM is expected to converge to the MTD as the trial accrues patients. Therefore, patients enrolled towards the end of the trial will likely receive a good dose, and thus there will be a reduced ethical imperative to stop the trial. On the other hand, there may be good reasons for extending enrollment with the CRM in some situations. Goodman et al. [43] consider continuing a CRM trial beyond N until the recommended MTD has at least a certain number of patients assigned to it. This is to avoid the case where only few patients are treated at the recommended MTD. Consider the two-stage CRM in Figure 3.2 (right panel). Based on the 20 observations shown in the figure, the recommended MTD is dose level 5, which is one level higher than the dose given to the final patients, and has only had 5 patients with an observed toxicity rate of 40%. These are signs that the recommended MTD is not adequately assessed. Figure 3.4 shows the outcomes of an extended CRM trial with a minimum sample size criterion: the trial will go beyond 20 patients until at least 9 patients have been treated at the recommended MTD. As a result of this modification, the trial continues for another 7 patients. With a total of 27
- 49. BIBLIOGRAPHIC NOTES 31 subjects, there is evidence that dose level 5 exceeds the acceptable toxicity level (44% observed toxicity rate) and the final MTD is dose level 4. 0 5 10 15 20 25 1 2 3 4 5 Patient number Dose level x x x x x x ooo ooo ooo oo o oo o oo oo oo Restricted two−stage CRM with minimum MTD size 9 Figure 3.4 Simulated trials using the restricted two-stage CRM with a minimum sample size criterion. Dose escalation restrictions are applied. Each point represents a patient, with “o” indicating no toxicity and “x” indicating toxicity. 3.6 Bibliographic Notes O’Quigley and Chevret [75] and Chevret [25] examine the operating characteristics of the one-stage Bayesian CRM by simulation study. Moller [67] and O’Quigley and Shen [80] study the two-stage CRM; the latter also propose the use of maximum likelihood estimation. Several authors, including Faries [36], Korn et al. [54], and Goodman et al. [43], note the potential problem with dose skipping by the CRM and propose to modify the CRM to limit escalation by no more than one level at a time. Faries [36] also suggests enforcing coherence in escalation by restriction; the notion of coherence is subsequently formalized in Cheung [16]. Ahn [1] compares various variants of the CRM by simulations. Early stopping of the CRM is considered in Heyd and Carlin [45] and O’Quigley and Reiner [79]. A recent overview of the CRM is given in Garrett-Mayer [39] and Iasonos et al. [48]. 3.7 Exercises and Further Results Exercise 3.1. Apply backward substitution (3.5) to the empiric function (3.2) with β̂0 = 0, and show that F(dk,β) = p exp(β) 0k for a given skeleton {p01,..., p0K}.
- 50. 32 THE CONTINUAL REASSESSMENT METHOD Exercise 3.2. Use the toxicity tolerance in Table 3.2 to verify the outcomes of the two-stage CRM in Figure 3.1. Exercise 3.3. The function crmsim can implement the two-stage CRM by speci- fying the argument x0 by the initial dose sequence, whose length should equal the specified sample size N. Modify the R code in Section 3.4.2 to run 20 simulated trials using the two-stage CRM in Figure 3.1 with a group-of-three initial rule. Exercise 3.4. By default, the function crmsim runs the restricted CRM with group size 1. These options can be modified through the arguments restrict and mcohort. Usage of these options is documented in the reference manual of ‘dfcrm’ at the R Web site: http://www.r-project.org. Run simulations, and examine the effects of escalation restrictions on the distribution of recommended MTD under dose–toxicity curve (3.12).
- 51. Chapter 4 One-Parameter Dose–Toxicity Models 4.1 Introduction One-parameter functions are predominantly used in the CRM literature to model the dose–toxicity relationship. An introduction on the use of one-parameter functions is in order—for two reasons. First, these functions are underparameterized and cannot be expected to produce a realistic fit on the entire dose range. Rather, they should be flexible enough to provide a reasonable approximation locally around the target dose. Second, the use of one-parameter models is not common in the other statistical areas, and modeling in the context of CRM is quite different from the conventional approach. This chapter presents a unified framework for CRM models. Section 4.2 introduces a class of dose–toxicity functions that includes the most commonly used CRM models. The required assumptions on a one-parameter CRM model are given in Section 4.3. Proof of the main result in this chapter is given in Section 4.4. 4.2 ψ-Equivalent Models Consider the following class of dose–toxicity functions: F(x,β) = ψ{c(β)h(x)} (4.1) where the parameter β is a scalar, and the functions ψ, c, and h are strictly monotone and known. This class of functions includes the most commonly used dose–toxicity curves in the CRM literature. For example, the empiric function (3.2) corresponds to ψ(z) = exp(z), c(β) = exp(β), h(x) = log(x) (4.2) so that F(x,β) = exp{exp(β) log(x)} = xexp(β) . Also, the logistic function (3.3) with a fixed intercept a0 is represented by ψ(z) = exp(a0 +z) 1 +exp(a0 +z) , c(β) = exp(β), h(x) = x. Recall that the dose labels in the CRM are obtained by backward substitution so that F(dk,β̂0) = p0k, where p0k is the initial guess of toxicity probability for dose 33
- 52. 34 ONE-PARAMETER DOSE–TOXICITY MODELS level k, and that β̂0 is the prior mean of β in a Bayesian CRM or an initial value of β in a maximum likelihood CRM. Therefore, a CRM model is defined by the function form F(·,β) of the dose–toxicity curve, the skeleton {p01,..., p0K}, and the initial value β̂0. (However, see Theorem 4.1 below.) Denote Fk(β) = F(dk,β) for k = 1,...,K. Under (4.1), the ψ-representation of a CRM model can be written as Fk(β) = ψ{c(β)h(dk)} = ψ ( c(β) c(β̂0) ψ−1 (p0k) ) (4.3) which depends on the dose index k via the initial guess p0k. In order for the model to distinguish between doses, therefore, the skeleton needs to be specified as a strictly increasing sequence. Furthermore, the representation (4.3) does not depend on the function h(x). Thus, two CRM models are identical if their dose–toxicity functions are represented by the same ψ(z) and c(β). For example, the hyperbolic tangent function (3.4) corresponds to ψ(z) = exp(z) and c(β) = exp(β) with h(x) = log{(tanhx+1)/2}. Since the function is represented by the same ψ(z) and c(β) as in (4.2), the CRM model defined by the hyperbolic tangent function is identical to that by the empiric function. The ψ-representation (4.3) of both functions is Fk(β) = exp ( exp(β) exp(β̂0) log(p0k) ) = p exp(β−β̂0) 0k . Example 4.1. The logistic function with a fixed slope a1 0: F(x,β) = exp(β +a1x) 1 +exp(β +a1x) (4.4) corresponds to ψ(z) = z/(1 +z) and c(β) = exp(β), which are free from the choice of a1. After backward substitution, the model can be represented by Fk(β) = exp(β −β̂0)p0k 1 − p0k +exp(β −β̂0)p0k . (4.5) The function (4.4) leads to the same CRM model regardless of the value of a1. One may therefore set a1 = 1 arbitrarily. Example 4.1 illustrates how the ψ-representation can be used to identify identical CRM models, and reduce unnecessary comparisons of different functions (e.g., a1). Definition 4.1 (ψ-equivalent models). Two CRM models under class (4.1) are said to be ψ-equivalent if their dose–toxicity functions can be represented by the same ψ(z).
- 53. ψ-EQUIVALENT MODELS 35 While ψ-equivalent models are not necessarily identical, the difference arises only as a result of different parameterizations. For example, another form (3.1) of the empiric function, F(x,β) = xβ , can be represented by ψ(z) = exp(z) and c(β) = β, and is ψ-equivalent to (3.2). While (3.1) and (3.2) are not equivalent, two functions differ only in terms of how the parameter appears in the functions. Hence, both functions will lead to identical estimation of the toxicity probabilities when maximum likelihood estimation is used. Theorem 4.1. Suppose that F (1) k (β) and F (2) k (φ) are derived from two ψ-equivalent models and the same skeleton {p0k}. Let β̃i−1 and φ̃i−1 be the respective maximum likelihood estimates of β and φ for the two models given the observation history Hi. Then F (1) k (β̃i−1) = F (2) k (φ̃i−1) for all k. In words, Theorem 4.1 states that the maximum likelihood CRM is invariant within a ψ-equivalent class of CRM models. The proof is an extension of the invariance property of maximum likelihoodestimation, and is given in Section 4.4. Interestingly, Theorem 4.1 holds regardless of the initial values of β and φ, thus implying that we can arbitrarily choose the initial values, β̂0 and φ̂0, used in the backward substitution step (3.5). Without loss of generality, from now on, we will set the initial values so that c(β̂0) = c(φ̂0) = 1. Example 4.2. The dose–toxicity function F(x,φ) = φx2 a2 +φx2 for some a2 0 (4.6) can be represented by ψ(z) = z/(1 +z) with c(φ) = φ and h(x) = x2/a2. Therefore, the CRM model generated by (4.6) is ψ-equivalent to that by the logistic function with a fixed slope; cf., equation (4.4) in Example 4.1. That is, both models will yield identical dose assignments if maximum likelihood CRM is used. In contrast, the Bayesian CRM (3.7) is not invariant among ψ-equivalent models. The model (4.6) in Example 4.2 can be represented by Fk(φ) = φ/φ̂0 p0k 1 − p0k + φ/φ̂0 p0k (4.7) where φ̂0 is the prior mean of φ. Bayesian CRM using models (4.5) and (4.7) would lead to identical posterior computations if exp(β −β̂0) and φ/φ̂0 had the same prior distribution. However, this is impossible because E0(φ/φ̂0) = 1 by definition of φ̂0, whereas E0 n exp(β −β̂0) o exp n E0(β −β̂0) o = 1
- 54. 36 ONE-PARAMETER DOSE–TOXICITY MODELS by Jensen’s inequality. This example illustrates that parameterization via c(β) can be an important consideration when the Bayesian CRM (3.7) is used. When c(β) = exp(β), the ψ-representation (4.3) of a CRM model depends on the parameter β only via β −β̂0. If the prior distribution of β is normal, the centered variable β −β̂0 is always mean zero normal. Therefore, the posterior computations will be identical regardless of the specified prior mean β̂0, and the Bayesian CRM is invariant to the mean of a normal prior. In general, this invariance property holds for Bayesian CRM when model (4.1) is used and the prior distribution of β constitutes a location-scale family (Lehmann, 1983, page 20). In this book, we will focus on the model class (4.1) with c(β) = exp(β) and a normal prior distributionfor β, so that the model parameters of the Bayesian CRM are the function ψ(z), the skeleton {p0k}, and the variance σ2 β of the normal prior. Table 4.1 gives some simple examples of ψ(z) and the corresponding CRM models. The calibration of {p0k} will be detailed in Chapter 8, and specification of σ2 β in Chapter 9. 4.3 Model Assumptions In this section, we state the regularity conditions on the one-parameter CRM model Fk(β) assumed in this book. The role of these conditions in various theoretical results will be discussed in the following chapters. In this chapter, we focus on the intuition behind the conditions and their implications. A practical point: all these assumptions are verifiable for any given Fk(β), and thus can serve as preliminary conditions to remove certain models from consideration for use. Condition 4.1. F(x,β) is strictly increasing in x for all β. Condition 4.2. Fk(β) is monotone in β in the same direction for all k. Conditions 4.1 and 4.2 are satisfied by many dose–toxicity functions. In particular, the model class (4.1) satisfies these conditions when either h(dk) 0 for all k or 0 for all k. It is easy to verify that the logistic model with a0 = 0 in Table 3.1 does not satisfy Condition 4.2 because h(dk) 0 for k ≤ 4 but h(d5) 0. Thus, this model should not be considered for use. Graphically, Condition 4.2 implies that the family of curves induced by F(x,β) do not cross each other; see Figure 4.1. For the next condition, we first define gi j(β) = {1 −Fi(β)}/{1 −Fj(β)}. Condition 4.3. The derivatives F′ k(β) and g′ i j(β) exist and g′ i j(β)F′ k (β) ≤ 0 for all k and i j. Condition 4.3 can be equivalently stated as |F′ i (β)| ≥ |F′ j(β)|{1 −Fi(β)}|/{1 −Fj(β)} for all i j, which puts a lower bound on |F′ i (β)| relative to |F′ j(β)|. Using the fact that log{1 − Fk(β)} = R β −∞[−F′ k(φ)/{1 − Fk(φ)}]dφ when F′ k (φ) 0 for all k, one can show that Condition 4.3 implies Condition 4.1 and hence is a stronger condition.
- 55. Exploring the Variety of Random Documents with Different Content
- 56. THE BIÈVRE TANNERIES Etching by Martial dealers have set up their huts; and hovels line strange streets made with the clearings of other streets. Once, these spacious grounds were one stretch of flower gardens and market gardens watered by the Bièvre. In a most interesting book, somewhat forgotten now, Alfred Delvau tells us much of the former history, under Louis- Philippe, of the Saint-Marceau faubourg, the Butte-aux-Cailles, the Rue Croulebarde, and also the Rue du Champ-de-l'Alouette, in which last street the Shepherdess of Ivry was murdered, the crime by its bizarre character producing a deep impression in the Capital in 1827. It was a public-house waiter, Honoré Ulbach, who had stabbed a girl, Aimée Millot by name; she, as a keeper of goats, was popular at Ivry. Every day, she was to be seen, with a large straw hat on her head and a book in her hand, tending her mistress's goats. The Shepherdess of Ivry she was called in the neighbourhood; in 1827, there were still shepherdesses in Paris! The trial that followed excited the whole town; the crime was one of love and jealousy; the victim was nineteen; she was virtuous and a shepherdess; women cursed the murderer, even while pitying him perhaps, wrote the newspapers of the time; and even the giraffe
- 57. THE BIÈVRE ABOUT 1900—THE VALENCE MILL-RACE Schaan, pinxit (Carnavalet Museum) but recently arrived at the King's Garden was neglected for the Ivry drama. On the 27th of July, Ulbach, who seems to have been half-mad, was condemned to death; and, at four o'clock in the evening on the 10th of September, he was executed on the Grève Square. A Municipal Crèche, in the Rue des Gobelins, occupies, at No. 3, a fine Louis XIII. mansion, once inhabited by the Marquis of Saint- Mesme, a lieutenant- general and the husband of Elizabeth Gobelin, close to a handsome lordly-looking building which in the quarter bears the name of Queen Blanche's Mansion. The legend attaching to the latter is false, affirms Monsieur Beaurepaire, the learned and amiable librarian of the City of Paris. It was, he says, simply Catherine d'Hausserville's home, where Charles VI. was nearly burnt alive during the performance of a ballet, his fancy dress having caught fire. The edifice, with its noble appearance, forms a strange contrast in this poor yet picturesque district.
- 58. Another fine mansion, in the Rue Scipio, is the one built by Scipio Sardini, in the reign of Henri III., with terra-cotta medallions, rare Parisian specimens of the exceedingly pretty decoration that pleases us so much at Florence, Pisa, and Verona. This Scipio Sardini was a peculiar man, and his story deserves to be told. Of Tuscan origin, he came to France after the death of Henri II., just when Catherine de Medici seized the reins of power. Amiable, witty, ingratiating, a great financier, clever in his enterprises, and unscrupulous, he quickly gained a preponderant position in the frivolous, dissolute, mirth- loving Court. He excelled in combining business and pleasure. An illustrious marriage seemed to him essential to people's forgetting his low origin and the rapid rise of his fortunes. He married the fair Limeuil, one of the most seductive beauties of the Queen's flying squadron—All of them capable of setting the whole world on fire, said Brantôme. This attractive person had been successively courted by the most noble lords of the Court before effecting the conquest of Condé, by whom she had a child. At Dijon, during one of the Queen's receptions, Mademoiselle de Limeuil was taken ill and was delivered of a boy. It is inexplicable, writes Mézeray, that such a prudent woman should have so miscalculated. There was a scandal; the Queen Mother was indignant; the fair Isabella was imprisoned; but Condé who was still amorous, succeeded in effecting her escape. The Protestants, however, were on the watch, and induced their leader to give up his too compromising mistress. Then it was that Scipio Sardini came forward, the richest man of the period, the King's banker, as also the nobles' and clergy's. He managed to get himself accepted; the marriage took place; and he settled in this pretty mansion that we still admire, and that is mentioned by Sauval as one of the most beautiful in Paris, amidst vineyards, orchards, and fields bordering on the Bièvre. There he lived, surrounded by luxury, works of art, books and flowers, and died there about 1609. As early as 1636, the mansion was converted into a hospital, which in 1742 was once more transformed, this time into a bakery. To-day, it is the Bakery of the City of Paris Hospitals.
- 59. Let us keep along by the Wine Market, and, before crossing to the right bank of the river, respectfully pause on the Stockade Bridge, close to the small monument erected to the famous sculptor Barye by his admirers,—to the great Barye who, misunderstood and mocked, sold up by his creditors, often came in the evening, after leaving his modest studio on the Célestins Quay, to forget his sufferings and muse in this same place before the splendid panorama of Paris crowned by the grand silhouette of the Panthéon. Here, too, is one of the City's best views. Nothing is more relative than an impression felt. To certain minds in love with the Past, this or that ruin is much more affecting than the most modern palace; it is the same with streets, houses, and pavements. An exquisite hour to call up the soul of old Paris is at twilight. The colour peculiar to each object has melted into the general shades and tints spread by the day which is departing and the night which comes. Delicate lace-work outlines stand out against the sky, while huge violet, black, and blue masses of atmosphere bathe whole streets in fathomless mystery. Then thought awakens, souvenirs revive and grow clear; scenes are lived through again of which these streets and houses were the silent witnesses. One hears cries of fury or of joy; drums beat, bells ring, groups pass singing 'mid these dream visions that rise again! In order to enjoy such an experience no better spot could be chosen than the Stockade Bridge, which, with its barrier of black beams, as it were shuts off to the east Paris of the olden days. The City slumbers in the calm of evening, the smoke curls lazily up. Afar sound bells; swallows sweep crying in the air embalmed by falling night; noises ascend vague and weird, interpreted according
- 60. to the fancy of one's musings. All life seems to sleep; the soul of the past awakes. It is the hour desired. THE CONSTANTINE BRIDGE AND STOCKADE Etching by Martial THE PONT ROYAL IN 1800 Boilly, pinxit (Carnavalet Museum)
- 61. THE LESDIGUIÈRES MANSION THE RIGHT BANK OF THE RIVER The Arsenal quarter, built over the site of the two Royal Palaces—the Saint-Paul mansion, the Tournelles palace—and the soil of the Louviers Isle, joined to the river bank in 1843, serve as a natural transition from the old to modern Paris. Notwithstanding its warlike name, the Arsenal quarter is one of the most peaceful parts of the Capital. Centuries ago, the palaces disappeared that brought it its wealth, life and movement. On their ruins and their huge gardens, humble, tranquil streets have been made: the Rue de la Cerisaie, where Marshal Villeroy received Peter the Great in the sumptuous Zamet mansion; the Rue Charles V., where once was the elegant home of the Marchioness de Brinvilliers, now at No. 12, premises in which a white-capped sister-of-charity distributes cod-liver oil and woollen socks to poor, suffering children; the Rue des Lions-Saint-Paul; the Rue Beautreillis, where Victorien Sardou was born; near there the great Balzac dwelt. I was then living, he says in his admirable Facino Cane, in a small street you
- 62. probably don't know, the Rue de Lesdiguières. It commences at the Rue Saint-Antoine, opposite a fountain near the Place de la Bastille, and issues in the Rue de la Cerisaie. Love of knowledge had driven me into a garret, where I worked during the night, and spent the day in a neighbouring library, that of Monsieur. When it was fine, I took rare walks on the Bourdon Boulevard. This modest Rue de Lesdiguières still exists in part; on the site occupied by Nos. 8 and 10, could be seen, a few years ago, one of the containing walls of the Bastille; narrow houses have been stuck against it; and, at No. 10, it is the very wall of the old Parisian fortress which constitutes the back of the porter's lodge! What a destiny for a prison wall! Of what was once the Arsenal only the mansion of the Grand Master is left; it is, at present, the Arsenal Library—formerly called, as Balzac says, the Library of Monsieur. It used to be a fine dwelling, the home of Sully, and possesses priceless books and autographs, and most valuable writings. In a coffer, covered with flower-de-luces, may be admired Saint Louis's book of hours, side by side with a fragment of his royal mantle, the blue silk of it, worn with time, being strewn with golden flower-de-luces; the old book bears this venerable inscription: It is the psalter of Monseigneur Loys, once his mother's; and was taken from the scattered treasures of the Sainte- Chapelle. Then there is Charles the Fifth's Bible with the King's writing on it: This book (belongs) to me, the King of France; and a missal, each leaf of which is framed with an incomparable garland due to the brush of the master of flowers, a great artist whose name is unknown to us. Besides, there are rare manuscripts, marvellous bindings, unique editions, romances of chivalry, classics, poets of every age, complete in this fine palace; together with Latude's letters, the box that served for his ridiculous attempt against Madame de Pompadour; and, near them, the cross- examination of the Marchioness de Brinvilliers, and the death- certificate of the Man in the Iron Mask; Henri IV.'s love-letters too, with his kisses sent to the Marchioness de Verneuil, and the documents relating to the affair of the Necklace. How many more things in addition...!
- 63. Let us add that the curators—Henri Martin, so learned and obliging, Funck-Brentano, the exquisite historian of the Bastille, the picturesque relater of all its dramas. Sheffer and Eugène Muller are not only scholars needing no praise but most courteous and genial men—and you will quite understand why the Arsenal is one of the few corners in Paris where it is delightful to go and work or to saunter about. Indeed, it is a tradition of the house. Nodier, good old Nodier, who was one of Monsieur de Bornier's predecessors and a predecessor also of J. M. de Heredia, the master who has so recently gone from us, Nodier, the admirable author of the Trophées, had succeeded in making the Arsenal the centre of literary and artistic Paris. Hugo, Lamartine, de Musset, Balzac, Méry, de Vigny, and Fr. Soulié used to meet there; and fine verses were said while regarding the sun glow with red flame behind the towers of Notre Dame. The towers of Notre Dame his name's great H composed! wrote Vacquerie. Of the Bastille nothing remains except a few stones which formed the substructure of one of the old towers; and these have been carefully removed to the Célestins Quay, along the Seine, where they are visible to-day. In vain, therefore, would any one now seek for a vestige of the sombre fortress over which so many legends hovered. Latude's great shade itself would hardly locate the spot; and yet how full Paris history is of this traditional Bastille, which the people, amazed with their easy victory, could not tire of visiting after the 15th of July 1789. Such was their curiosity and such their eagerness that Soulès, the governor appointed by the Parisian municipality, was compelled to stop the visits, on the curious ground that such damage had already been done to the fortress by visitors that more than 200,000 livres would be required to repair it. Repair the Bastille! The souvenir manuscripts of Paré tell us the fury excited by this strange pretension in Danton, sergeant of a section of the National Guard, who, with his company, was turned back by the order.
- 64. COMMEMORATIVE BALL ON THE RUINS OF THE BASTILLE Danton had himself admitted into the presence of the unfortunate Soulès, seized him by the collar and dragged him to the Town Hall; the prohibition was removed; and Citizen Palloy was thenceforth allowed to exploit the celebrated State prison. The stones were hewn and cut into images of the fortress and dedicated to the various departments and assemblies, or into commemorative slabs intended to rouse people's courage. Palloy cut up the leads into medals, and made rings with the iron chains; out of the marble he manufactured games of dominoes, and had the delicate thought to offer one of these games to the young Dauphin to inspire him with the horror of tyranny. Balls were held on the site of the Bastille. Wine flowed, fiddles were scraped, and printed calicoes of that period show us the ruins of the old Parisian citadel surmounted with this inscription: Dancing here. The huge space left vacant by the demolition had to be filled up. Napoleon I., whose artistic conceptions were sometimes disconcerting, had constructed there, in 1811, by Alavoine, a strange sort of fountain of bizarre
- 65. Dancing here From a coloured engraving of the eighteenth century THE SENS MANSION ABOUT appearance: it was a colossal elephant, twenty-four metres high, which spouted water from its trunk. Built temporarily in plaster and mud, the elephant quickly crumbled away under the action of weather and rain; and soon became a lamentable débris surrounded with disjointed planks. The urchins of the district made it the scene of Homeric struggles; but the real familiars were the rats that had made their home inside the structure, so that, when the demolition began, regular battues had to be organised with men and dogs; and, for months, these dreaded rodents infested the terrorised quarter. In 1840, the present column was erected; since then, the genius of Liberty has poised over Paris his airy foot, and Barye's fine lion watches over the repose of the victims of 1830 that are interred within the crypt of the monument. The Rue Saint-Antoine contains certain handsome mansions: the Cossé mansion, where Quélus died; the Mayenne and Ormesson mansion, built by du Cerceau on the remains of the Saint-Paul mansion and Germain Pilon's studio; the Sully mansion, whose noble front was not long ago mutilated. Hard by, at the corner of the Rue du Figuier and the picturesque Rue de l'Hôtel de Ville, which latter used to be the Rue de la Mortellerie, stands what is left of the Sens mansion, the only specimen, together with the Cluny Museum, of what private architecture was in the fifteenth century. After being
- 66. 1835 From a lithograph by Rouargue inhabited by Princes of the Church, Bishops, Cardinals, and also by Marguerite de Valois (Queen Margot), the Sens mansion fell on evil days. It became the Diligence Office; and from its courtyard is said to have started the famous courier whose murder was attributed to Lesurques, the unfortunate Lesurques popularised by the well-known drama performed at the Ambigu, which caused so many tears to flow. In more recent times, the Hôtel de Sens derogated further still. It became a manufactory of sweets! At No. 5 of the Rue du Figuier, we meet with a draw-well, the top of which is finely sculptured; the spot brings back the memory of Rabelais, the admirable Rabelais, who died quite near, in the Rue des Jardins. At No. 15, opened the sixteenth-century door through which the actors of the illustrious theatre established on the ancient site of the Jeu de Paume de la Croix-Noire, proceeded to their private stage-room. It was before this door that Molière was arrested and taken to the Châtelet, because he owed 142 livres to Antoine Fausseur, master-chandler, his purveyor of light. Let us cross the Place de la Bastille and go down the Rue du Faubourg-Saint-Antoine. There, at No. 115, in front of an old eighteenth-century house, the Deputy Baudin was killed against a barricade, on the 3rd of December 1851. At No. 303, in the reign of Napoleon I., stood Dr. Dubuisson's private hospital, where General Malet was confined. There he hatched the prodigious plot the disconcerting history of which we intend shortly to relate. Farther on, near the Rue de Montreuil, we pass by the remains of Réveillon's wall-paper stores, pillaged on the 17th of April 1789; it was one of the preludes of the Revolution. Last of all, at No. 70, in the Rue de Charonne, Dr. Belhomme's private hospital stood, which was used as a special prison under the Revolution. Only those were admitted who could pay and pay well. The irrefutable memoirs of Monsieur de Saint-Aulaine reveal to us a
- 67. Belhomme familiar, cynical, exacting his fees and thouing Duchesses short of money who haggled with him on the question of their life. The most amiable of historians, my excellent friend G. Lenôtre, whom it is always necessary to quote when facts of the Revolutionary epoch are in question, has reconstituted the terrible and surprising story of the Belhomme institution where they laughed, danced, or even flirted under the dread eye of Fouquier- Tinville; and has related, with his habitual documentation, the bizarre liaison of the Duchess of Orléans, widow of Louis-Philippe Egalité, with Rouzet, the Conventional, buried later at Dreux under the name of the Count de Folmon in the Orléans family vault. Pursuing our way and passing by the Church of Sainte Marguerite, in which Louis XVIII. was interred ... or his double, we reach the barrier of the Throne (the Throne overthrown, people said in 1793). The scaffold, which had temporarily quitted the Revolution Square, was put up here during the most terrible period of the Terror, and the great batches were executed upon it. In six weeks, 1300 victims perished, among them, André Chénier, the Baron de Trenck, the Abbess of Montmorency, Cécile Renaud, Madame de Sainte- Amaranthe, the poet Roucher, and many others. The bodies of these unfortunate people, stripped of their clothing, were loaded each evening on covered waggons, with their severed heads between their legs; and the horrible vehicle, dripping with blood along the road, was tipped into some pit dug at the bottom of the Picpus Convent Gardens, where still exists the cemetery of those that were executed during the Revolution. Retracing our steps, we arrive at No. 9 of the Rue de Reuilly; here was once the Hortensia Tavern, kept in 1789 by the famous Santerre, a major in the National Guard. The house has not much changed; at present, however, it is a girls' boarding-school which occupies the large rooms where the thundering General organised those terrible descents on Paris and launched those dreadful battalions of the faubourg that terrorised even the Convention itself.
- 68. THE PROVOST HUGUES AUBRYOT'S MANSION CHARLEMAGNE'S COURTYARD AND PASSAGE IN 1867 Drawn by A. Maignan On the other side of the Place de la Bastille, in the Rue Saint-Antoine, near Saint Paul's Church, is the Charlemagne Passage, most picturesque by reason of the old souvenirs it contains and the strange population it harbours: chair- menders, mattress-carders, milk-women, open-air flower-women gather round the ruin of the charming mansion which, under Charles V., was the sumptuous abode of the provost, Hugues Aubryot. The front, which is still remarkable and fine-looking, is an astonishing contrast to the poor, low houses that huddle round it. Fowls peck at the foot of the fifteenth-century turrets, which enclose a handsome staircase; and patched linen dries on iron wire stretched between the caryatide windows of the seventeenth century, replacing those behind which once mused the Duke d'Orléans and the Duke de Berri, as also, in 1409, Jean de Montaigu, beheaded for sorcery! who were formerly illustrious guests in this elegant dwelling. And now, let us stop at the Vosges Square on the other side of the Bastille. It is another rare nook of our old City, which, through the centuries, has preserved its ancient character very nearly intact. The houses there, in Louis XIII. style, have not changed. The scenery has remained the same. The Précieuses could take their favourite walks there; and those punctilious in honour might draw their sword, as in the time of Richelieu and the Edict-malcontents; only the public of spectators would be quite different. The fine ladies of the country hight Tender, the Cydalises and Aramynthas, the lords once living in those noble dwellings, they who, on the 16th of March 1612, were
- 69. present at the tournament given by the Queen Regent, Marie de Médici, in honour of the peace concluded with Spain, or they who proceeded in grand coaches to the fair Marion de Lorme's or to Madame de Sévigné's, are to-day replaced by petty annuitants, modest shopkeepers retired from business and pensioned-off officers. Humble charwomen work at their tasks in the spots where Mazarin's nieces paused in their sedan-chairs; and the numerous Jews that live in the quarter meet there on Saturdays. It is a curious spectacle to see these men and women of strongly marked type betaking themselves to the Synagogue, which is near a partially subsisting eighteenth-century mansion still bearing delicate decorations, but at present occupied by a butcher, in the Rue du Pas-de-la-Mule. Not a few old men wear the long gaberdine, their hair in corkscrew curls, and earrings in their ears. Velvet-eyed girls coifed with bands, wonderfully handsome and peculiarly dressed, assemble there on certain religious feast-days. It is a strange evocation; 'twould seem that in these peaceful quarters biblical traditions have been preserved in some Jewish families. THE PLACE ROYALE ABOUT 1651 (NOW THE VOSGES SQUARE) Israël, del. The old-time animation, however, is an exception. The Vosges Square, once the Place Royale, where Richelieu lived and Fronsac,
- 70. Chabannes, Marshal de Chaulnes, Rohan-Chabot, Rotrou, Dangeau, Canillac, the Prince de Talmont and Mademoiselle du Châtelet, where Madame de Sévigné was born, where the tragic actress Rachel dwelt, and Théophile Gautier and Victor Hugo, is to-day completely neglected; and this delightful Paris nook, where so much wit was spent, such fine ladies rivalled in grace and elegance and so many exquisites drew their swords, is now nothing but a large, lonely garden, provincial and melancholy, frequented almost exclusively by the pupils of neighbouring boarding-schools, who play there at prisoners' base, and leap-frog, beneath the debonair shadow of Louis XIII.'s statue, with its philosophic frame of a Punch-and-Judy show and a chair-woman's stall. In the ancient Rue Culture-Sainte-Catherine (at present called the Rue de Sévigné) on the site now occupied by No. 11, formerly stood the Marais theatre, built with money provided by Beaumarchais. In 1792, the Guilty Mother was performed there, for the benefit, said the play-bill, of the first soldier who shall send citizen Beaumarchais an Austrian's ear. The modern building is a modest private-bath establishment, with a small garden in front in which grow some spindle-trees—in boxes, and which is adorned with silvered balls. The huge wall, all grim and grey, backing the slightly-built bath establishment, is the old wall of the Force Prison, where, on a post at the corner of the Rue des Balais, Madame de Lamballe was executed, where also Madame de Tallien was transferred, and Princess de Tarente was confined, the latter, the grandmother of the kind, courteous and learned Duke de la Trémoïlle, who had only to dip into his incomparable family archives to give us the most precious documents of French history, and to whom we are indebted for those picturesque and exciting Souvenirs of Madame de Tarente, one of the most valuable narrations by an eye-witness of the Revolutionary period. The Carnavalet mansion, Madame de Sévigné's dear Carnavalette, is close by, as also the ancient Le Peletier-Saint-Fargeau mansion, to- day the City of Paris Library. It is a fine, large building of noble appearance, which contains wonderful books, maps, plans and
- 71. manuscripts. The written history of Paris is there; and all workers know the pretty, sculpture-ornamented room of Monsieur le Vayer, the erudite, obliging Curator of these fine collections. Messieurs Poète, Beaurepaire, Jacob, Jarach and Wilhem, in the Library; Messieurs Pètre and Stirling in the History room are the wise and welcoming hosts of this admirable Parisian Library. All this Marais quarter, indeed, contains sumptuous mansions, not one of which, alas! has been respected. All are given over to business and manufacturing. The Lamoignon mansion is occupied by glass-polishers and garden-seatmakers; the Albret mansion by a bronze lamp-dealer; those of Tallard, Maulevrier, Sauvigny, Brevannes, Epernon, c., are still standing, but in what a state! The Rue des Nonnains-d'Hyères offers us its curious bass-relief, in painted stone, representing a knife-grinder in eighteenth-century costume. In 1748, a Madame de Pannelier kept a wit-office in this same street; Lalande, Sautereau, Guichard, Leclerc de Merry used to attend meetings there. They were held on Wednesdays, and were preceded by an excellent dinner. The tradition has happily been preserved in Paris. In the Rue François-Miron, one sees a spacious, handsome mansion with circular pediment, escutcheons and garlands. It is the Beauvais mansion, built by Le Pautre in 1658. To look at it now, old and in a dull street, one would hardly think that the coaches of Louis XIV.—King Sun—had passed under the dark vault of the entrance gate and that, from the top of the central pavilion balcony, Queen Anne of Austria, in company with the Queen of England, Cardinal Mazarin, Marshal de Turenne and other illustrious nobles, had watched her son Louis XIV. and her daughter- in-law, the new Queen Marie-Thérèse of Austria, go by as they made, through Saint-Antoine's Gate, their solemn entry into Paris on the 26th of August 1660![3] On account of its picturesque aspect and the fine mansions it contains, the Rue Geoffroy-l'Asnier is one of the most curious in Paris. At No. 26 stands the Châlons-Luxembourg mansion, with its
- 72. THE RUE GRENIER- monumental door and wonderful knocker. At the bottom of the courtyard is an exceedingly elegant Louis XIII. pavilion in brick and stone, and of delicate proportions. The mansion was built for the second Constable of Montmorency, and though it is quite lost in this gloomy quarter, it maintains its proud bearing. After the Revolution, this street, whence nearly all the owners of houses had emigrated, if they had not been guillotined, was completely stripped of its former splendour. Petty annuitants, small clerks, and poor people took up their abode in the abandoned buildings. Grass grew in the streets; many of the dwellings had been sold as national property; and the Rue Geoffroy-l'Asnier underwent the common fate; it became democratic. Between this street and the neighbouring Rue des Barres, one is surprised to see a sort of fissure so narrow that two persons would find it difficult to walk abreast through it, a sort of corridor along which the wind sweeps past dilapidated, leaning houses on either side. It is the Rue Grenier-sur-l'Eau, wretched and dirty enough, but quaint, with the glorious tower of Saint-Gervais-Saint-Protais in the background, rising and standing out against the sky. The proper moment to take a look at the sinister little Rue des Barres is on a stormy night, behind the church of Saint-Gervais. It is then easy to imagine what this quiet quarter must have been like when, on the 9th of Thermidor, about eleven in the evening, 'mid torch-lights, calls to arms, the noise of the tocsin and shouts of the multitude, the dead body of Lebas was brought thither, and, on a chair, Augustin
- 73. SUR-L'EAU IN 1866 Drawn by A. Maignan THE SAINT-PAUL PORT Water-colour by Boggs (G. Cain Collection) Robespierre, who had broken his thighs in leaping from one of the Town Hall windows. The dead man and the dying man were dragged to the Barres mansion transformed into a Sectional Committee Tribunal. On the morrow Lebas was buried, and Robespierre was carried before the Committee of Public Safety, who sent him to the scaffold. The Rue des Barres descends to the Seine, near the old Town Hall Quay, where the big, flat boats laden with apples, stones, or sand take their moorings. Into it opens one of the exits of the charming Church of Saint- Gervais, whose fine painted windows, masterpieces of Pinaigrier and Jean Cousin, were almost totally destroyed twenty years ago by an explosion of dynamite. Against the church walls, in the laicised ruins of an ancient chapel, a sweet manufacturer has installed his alembics and copper pans; and it is a curious sight to see the lighted fires of this strange kitchen beneath these antique Gothic arches, between these blackened pillars still bearing traces of the candles that once burned in front of the holy images, on a ground formerly used for burying and even now concealing bones. The out-offices of the old church still remain,
- 74. THE BARBETT MANSION The Rue Paradis-des- Francs-Bourgeois and the Rue Vieille-du- Temple in 1866 Drawn by A. Maignan wonderfully picturesque, and open into the Rue François-Miron, No. 2, on the left of the entrance portal of the church, between a laundress's establishment and a furniture-remover's premises! On one side, the little Rue de l'Hôtel- de-Ville brings us to the Rue Vieille- du-Temple, where we can admire, at No. 47, what is left of the quaint mansion of the Dutch Ambassadors, where Monsieur Caron de Beaumarchais and Madame his spouse, as an almanac of 1787 called them, established in 1784 a Provident Institution for poor nursing mothers. Indeed, it was for the benefit of this undertaking that the fiftieth performance of the Mariage de Figaro was given. Farther on, to the right, at the corner of the Rue des Francs- Bourgeois, stands the pretty turret built about 1500 for Jean Hérouet; and, last of all, the fine Rohan palace, which to-day is the National Printing House. This last is a noble and spacious building which the elegant Cardinal that once lived in it took pleasure in sumptuously decorating. A masterpiece may be seen there, the Horses of Apollo, in a wonderful bass-relief by Pierre Le Lorrain. The saloon of the Apes, by Huet, is charming, and the private room of Monsieur Christian, the witty and learned Director of the National Printing House, contains a beautiful Caffieri time-piece. Why must, alas! this fine palace be condemned soon to disappear? The Rohan mansion is to be demolished, and the State will commit the sacrilege! May the endeavours of lovers of
- 75. Paris succeed in preserving for us this precious vestige of a past that each day removes farther from us! A cabman whose astonishment must have been great was a certain George who, on the 22nd of October 1812, at half-past eleven in the evening, amid a driving rain that turned the miry soil of Saint-Peter's pudding-bag (now the Villehardouin blind alley) into a veritable bog, saw get out of his cab, near the Rue Saint-Gilles, a completely naked man, with his uniform under his arm—a soldier whom, twenty minutes before, he had picked up in the Louvre Square. This strange passenger was Corporal Rateau, proceeding to the appointment made with him by General Malet, inside Dr. Dubuisson's private hospital and asylum, 303 Faubourg-Saint-Antoine, where the latter was confined by the authorities. In his haste to put on the fine uniform of an orderly officer, which was ready for him in exchange for his own, Rateau had undressed in the cab; and up the dark staircase of the gloomy house in the gloomy street he rushed with absolutely nothing on. The little house still exists, wretched and dingy-looking, where Malet appointed to meet his accomplices, on the third floor in the abode of the Abbé Cajamanos, an old bewildered Spanish priest who had quitted the Bicêtre asylum. This adventure of General Malet's is both prodigious and disconcerting. For, in 1812, at the moment when Napoleon seemed to be at the summit of his power, Malet, in a sort of dungeon, with the help of five or six obscure assistants, an old priest with hardly any knowledge of French, a half-pay officer, an almost illiterate sergeant and a few other hare-brained people, had been able, even while confined, watched and suspected, to combine everything, prepare everything, so that the report of the Emperor's death might be believed—the Emperor being absent in the icy steppes of Russia, and no news arriving from him. And his calculations were justified. All the Imperial functionaries, from Savary, the head of the police, down to Frochot, the Prefect of the Seine, accepted General Malet's allegations, without testing or discussing them. Especially, all
- 76. believed his fine promises; and it is hard to say where the hoaxer would have stopped if an officer, simply obeying his orders, had not refused to be gained over with fine words, and asked for proofs. Malet, being taken aback, grew impatient, and replied with a pistol- shot. Major Doucet forthwith arrested him, and the comedy ended in a tragedy. All the more haste was made to get rid of the organisers of this plot, which had so nearly succeeded, as it was necessary to suppress as quickly as possible their awkward testimony to such cowardice, lying, and compromise. The poor dwelling in the Villehardouin blind alley was searched by all the Paris police; papers, uniforms, cocked hats, and swords were fished out of the little well, still existing, into which they had been wildly thrown. In a few hours, Malet, Lahorie, Rateau, and Guidal were tried, condemned, and executed. The replies of the General to the Tribunal that so summarily judged him were home-thrusts. Asked (somewhat late) who were his accomplices: All of you, he said, if I had succeeded! Taken to the wall of evil memory in the plain of Grenelle, he insisted on giving the firing-order to the execution-platoon; and, as if he had been on the drill-ground, made the soldiers repeat the aiming movement, which had not been carried out with military precision. Rateau, who, as a matter of fact, had understood nothing of this strange drama, in which he had been one of the most picturesque confederates, is said to have died in crying: Long live the Emperor! Between the Archives and the Rue Sainte-Croix-de-la-Bretonnerie, there was once a large monastery, which, in 1631, became the property of the Carmelite Billettes,—the name being derived from an ornament worn by these monks on their gowns. The Revolution suppressed the monastery; but the small cloister has come down to us with its charming proportions and its monastic cosiness. To-day, it is a Town School, and the neighbouring church is devoted to Protestant worship.
- 77. THE RUE DE VENISE Water-colour by Truffaut (Carnavalet Museum) The Rue de Venise, one of the most ancient Paris streets, is not far away. It is now a low, bad-smelling lane inhabited by vagabonds of both sexes. Women, whose age it is impossible to tell, trail and traipse in front of alleys within which loom greasy, black staircases. Mended linen hangs from the windows; acrid smoke issues from between thick bars protecting old mansions now degenerated into mere dens, defended, however, by heavy doors studded with rusty nails. It is hideous, yet quaint, as indeed all this quarter, which is made up besides of the Rue Pierre-au-Lard, the Rue Brise-Miche, and the Rue Taille-Pain; not forgetting Saint-Merri's cloister, the name being that of the old church whose tocsin so often sounded the alarm during the riots in the reign of Louis-Philippe. At the least popular excitement, this inextricable labyrinth of small streets used to bristle with barricades. At the crossing of the Rue Saint-Martin and the Rue Aubry-le-Boucher was raised the terrible barricade defended by Jeanne and his intrepid companions. Following on the burial of General Lamarque, who died while pressing to his lips the sword offered to him by the Bonapartist
- 78. officers of the Hundred Days, an immense revolutionary movement had galvanized Paris. The old soldiers of the Empire, the survivors of the Terror and those of 1830, allied in their common hatred of Louis- Philippe's government, had joined the malcontents of all parties and the members of the then numerous secret societies. In the evening of the 5th of June 1832, the centre of Paris was covered with barricades; and both troops and National Guard had been obliged to reconquer, one by one, the positions that had been lost. Slaughter had been going on the whole night. When the dawn of the 6th of June tinged the house-roofs with pink, the large Saint-Merri barricade was seen to be holding out; its defenders, a handful of heroic men, had sworn to bury themselves under its ruins. Already they had repulsed ten furious assaults; now they were awaiting death; and the loud tones of the Saint-Merri tocsin, unceasingly sounding above their heads, seemed to be tolling their funeral knell! Part of the Paris army had to be utilised to vanquish these dauntless insurgents. Firing went on from windows, cellars, the pavement. Round the barricades, dead bodies of National Guards and soldiers, riddled with balls, crushed beneath blocks of stone hurled from roof- tops, testified to the frightful savagery of this intestine struggle. For long afterwards, the ground was red with blood! What numbers of balls and bullets, what quantities of grapeshot all these old house- fronts have received in the haphazard of riots, frequent during the reign of Louis-Philippe. The drums no sooner beat than the citizens armed and hurried to defend order ... or to attack it; anxious women, cowering behind closed shutters, watched for the biers. Things resumed their ordinary course immediately the disorder was over; the insurgent hobnobbed with the honest National Guard whom he had aimed his gun at on the day before. Sometimes, however, grudges remained. My parents knew an old woman, living in the Rue Saint-Merri, who, for forty years after 1836, never passed without trembling by the door of the tenant underneath her flat. As people were surprised at
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