![Figure 2: Probability of achieving the targeted interval, P (πj,k ∈ [θ − δ; θ + δ]) at the end of the trial for each combination. The dashed vertical line corresponds to the highest probability, which determines the combination selected as the MTD ((2, 3) in this example).](/figures/figure-2-probability-of-achieving-the-targeted-interval-p-pj-xvyczby5.webp)
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A Bayesian dose-nding design for drug combination
clinical trials based on the logistic model
Marie-Karelle Riviere, Ying Yuan, Frédéric Dubois, Sarah Zohar
To cite this version:
Marie-Karelle Riviere, Ying Yuan, Frédéric Dubois, Sarah Zohar. A Bayesian dose-nding design for
drug combination clinical trials based on the logistic model. Pharmaceutical Statistics, Wiley, 2014,
13 (4), pp.247-257. �10.1002/pst.1621�. �hal-01298657�
A Bayesian dose-finding design for drug combination
clinical trials based on the logistic model
Marie-Karelle Riviere
(a,b)
, Ying Yuan
(c)
, Fr
´
ed
´
eric Dubois
(b)
and Sarah Zohar
(a)
(a) INSERM, UMRS 1138, Equipe 22, Centre de Recherche des Cordeliers, Universit
´
e Paris 5, Universit
´
e Paris 6, Paris, France
(b) IRIS (Institut de Recherches Internationales Servier), Suresnes, France
(c) Department of Biostatistics, The University of Texas MD Anderson Cancer Center, Houston, TX 77030, U.S.A.
Abstract
In early phase dose-finding cancer studies, the objective is to determine the maximum tolerated
dose, defined as the highest dose with an acceptable dose-limiting toxicity rate. Finding this dose
for drug-combination trials is complicated due to drug-drug interactions, and many trial designs
have been proposed to address this issue. These designs rely on complicated statistical models that
typically are not familiar to clinicians, and are rarely used in practice. The aim of this paper is
to propose a Bayesian dose-finding design for drug combination trials based on standard logistic
regression. Under the proposed design, we continuously update the posterior estimates of the model
parameters to make the decisions of dose assignment and early stopping. Simulation studies show
that the proposed design is competitive and outperforms some existing designs. We also extend our
design to handle delayed toxicities.
Keywords: Bayesian inference; Dose-finding; Drug combination; Oncology; Phase I trial.
1
1 Introduction
For oncologists, the objective of phase I dose-finding studies is to determine the maximum tolerated
dose (MTD), defined as the highest dose with a relatively acceptable dose-limiting toxicity (DLT)
[1, 2]. DLT is usually defined as a toxicity of grade 3 or higher according to the U.S. National
Cancer Institute toxicity criteria [3]. In practice, patients included in phase I clinical cancer trials
have already been heavily pre-treated and in many cases no alternative therapeutic options are
available to them. For cytotoxic anti-cancer drugs, a dose-toxicity effect is assumed whereby the
higher the dose, the greater the risk of DLT and the greater the efficacy.
Dose finding for drug combination trials is more difficult than that for conventional single-
agent trials due to complicated drug-drug interactions. Moreover, when combining several agents,
the order of the toxicity probabilities is not fully known. Should investigators wish to gradually
increase the acceptable level of toxicity during the trial, the appropriate order in which the doses for
the various drugs in the combination should be increased would be of great interest. For instance,
when combining two cytotoxic agents, it remains difficult to decide how to escalate or de-escalate
the dose combination, even when a partial ordering is known [4, 5].
Recently, many phase I dose-finding designs have been proposed for drug combination trials.
Thall et al. proposed a Bayesian dose-finding method based on a six-parameter model [6]. Wang
and Ivanova developed a 3-parameter model-based method in which the parameters are estimated
using Bayesian inference [7]. Mandrekar et al. proposed an approach incorporating the toxicity and
efficacy of each agent into the identification of an optimal dosing region for the combination by us-
ing a continuation ratio model to separate each agent’s toxicity and efficacy curves [8, 9]. Yin and
Yuan developed a Bayesian adaptive design based on latent 2x2 tables in which the combination’s
toxicity probabilities in the two-dimensional space are estimated using a Gumbel-type model [10].
Yin and Yuan extended their method by changing to a copula-type model to simulate the effect of
two or more drugs in combination [11]. Bailey et al. introduced a second agent as a covariate in
a logistic model [12]. Wages et al. considered an approach based on the continual reassessment
method and taking into account different orderings with partial order between combinations. In this
case, the MTD is estimated for the order associated with the highest model-selection criterion [5].
Most of these existing designs rely on complicated statistical models that typically are not familiar
to clinicians, which hinder their acceptance and application in practice. In addition, the perfor-
mance of these designs seems comparable and there is no consensus which design should be used
[13]. As a result, many of the dose-finding clinical trials conducted to evaluate drug combinations
still use the conventional “3+3” approach, which was developed for single agents and was shown
to be inefficient in terms of dose identification [14, 15, 16, 17, 18].
In this paper, we propose an Bayesian dose-finding design for drug combination trials based
on standard logistic regression. Under the Bayesian paradigm, data monitoring, early stopping
and dose assignment occur continuously throughout the trial by updating the posterior estimates of
the model parameters. Simulation studies show that in general the proposed design provide better
performance than some existing designs. To facilitate the use of the proposed design by clinicians,
R package will be developed for implementing the new design.
This manuscript is laid out as follows: in Section 2, we propose the simple statistical method
for modeling the toxicity probabilities of the drug combination under evaluation. Moreover, we
present the likelihood function and the prior specifications for the unknown parameters. In this
section, we also describe our allocation and dose-finding method, as well as propose stopping rules.
2
We conduct extensive simulation studies to examine the operating characteristics of our design in
Sections 3 and 4, and conclude with a discussion in Section 5.
2 Methods
2.1 Statistical method for combination evaluation (LOGISTIC)
Dose-combination model
Let there be a two-drug combination used in a phase I dose-finding clinical trial for which the
dose-toxicity relationship is monotonic and increases with the dose levels. Let (j, k) denote the
dose level of a combination in which j refers to Agent 1 (j = 1, . . . , J), and k refers to Agent 2
(k = 1, . . . , K). Y
i
is a Bernoulli random variable, denoting the toxicity that is equal to 1 if DLT
occurs in patient i and 0 otherwise (i = 1, . . . , N). We assume that n
j,k
patients are allocated at
combination (j, k) and that a total of t
j,k
DLTs are observed for that combination. We define π
j,k
as
the toxicity probabilities of combination (j, k), θ as the target probability of toxicity, and p
1
, ..., p
J
and q
1
, ..., q
K
as the respective prior toxicity probabilities of Agent 1 and Agent 2 taken alone. For
simplicity purposes, we refer to the selected combination as the MTD in order to maintain the same
designation as in single-agent trials. In this manuscript, we focus on finding one MTD at the end
of the trial. We note that in some cases, it is of interest to find multiple MTDs that can be further
tested in phase II trials, see Yuan and Yin [19] and Ivanova and Wang [20] for related designs.
Let u
j
and v
k
denote the “effective” or standardized doses ascribed to the jth level of Agent 1
and k level of Agent 2, respectively. We model the drug combination-toxicity relationship using a
4-parameter logistic model, as follows:
logit(π
j,k
) = β
0
+ β
1
u
j
+ β
2
v
k
+ β
3
u
j
v
k
, (1)
where β
0
, β
1
, β
2
and β
3
are unknown parameters that represent the toxicity effect of Agent 1 (β
1
),
that of Agent 2 (β
2
) and that of the interaction between the two agents (β
3
). These parameters are
defined such that β
1
> 0 and β
2
> 0, ensuring that the toxicity probability is increasing with the
increasing dose level of each agent alone, ∀k, β
1
+ β
3
v
k
> 0 and ∀j, β
2
+ β
3
u
j
> 0, ensuring that
the toxicity probability is increasing with the increasing dose levels of both agents together, and
intercept −∞ < β
0
< ∞. The standardized dose of two agents are defined as u
j
= log
(
p
j
1−p
j
)
and v
k
= log
(
q
k
1−q
k
)
, where p
j
and q
k
are the prior estimates of the toxicity probabilities of the jth
dose level of Agent 1 and the kth dose level of Agent 2, respectively, when they are administered
individually as a single agent. Before two agents can be combined, each of them typically have
been thoroughly investigated individually. Therefore, there is often rich prior information on p
j
’s
and q
k
’s and their values can be readily elicited from physicians. Using the prior information to
define standardized dose has been widely used in dose-finding trial designs, and the most well
known example perhaps is the skeleton of the continuous reassessment method (CRM) [21] with a
logistic model. Research has shown that this approach improves the estimation stability and trial
performance [2, 22]. Our definition of u
j
and v
k
can be viewed as an extension of the skeleton of
the CRM (with a logistic model) to drug-combination trials.
3
Likelihood and posterior inference
Under the proposed model, the likelihood is simply a product of the Bernoulli density, given by
L(β
0
, β
1
, β
2
, β
3
|data) ∝
J
∏
j=1
K
∏
k=1
π
t
j,k
j,k
(1 − π
j,k
)
n
j,k
−t
j,k
.
We assume the prior distributions of the model parameters are independent. For β
0
and β
3
, we
assign a vague normal prior N(0, 10) centered at 0 to indicate that a priori we do not favor either
positive or negative values for these parameters and let the observed data speak for themselves
through posteriors. For β
1
and β
2
, we assume an informative prior distribution Exp(1) centered
at 1, as those parameters should not be too far from 1. Then, the joint posterior distribution of
parameters β
0
, β
1
, β
2
and β
3
is given by
f(β
0
, β
1
, β
2
, β
3
|data) ∝ L(β
0
, β
1
, β
2
, β
3
|data)f(β
0
)f(β
1
)f(β
2
)f(β
3
). (2)
We sample this posterior distribution using Gibbs sampler, which sequentially draws each of the
parameters from their full conditional distributions (see Appendix). These full conditional dis-
tributions do not have closed forms, and we use the Adaptive Rejection Metropolis Sampling
(ARMS) method [23] to sample them. In order to impose the constrain that ∀k, β
1
+ β
3
v
k
> 0
and ∀j, β
2
+ β
3
u
j
> 0, at each iteration of Gibbs sampling, if sampled β
1
, β
2
and β
3
fail to satisfy
the constraint, we re-sample β
1
, β
2
and β
3
. Let (β
(ℓ)
0
, β
(ℓ)
1
, β
(ℓ)
2
, β
(ℓ)
3
)
ℓ=1,...,L
denote the L posterior
samples obtained from Gibbs sampler, the posterior toxicity probabilities can be estimated using
Monte Carlo by:
˜π
j,k
=
1
L
L
∑
ℓ=1
exp(β
(ℓ)
0
+ β
(ℓ)
1
u
j
+ β
(ℓ)
2
v
k
+ β
(ℓ)
3
u
j
v
k
)
1 + exp(β
(ℓ)
0
+ β
(ℓ)
1
u
j
+ β
(ℓ)
2
v
k
+ β
(ℓ)
3
u
j
v
k
)
.
Dose finding algorithm and determination of the MTD
During the trial conduct, we use the dose-finding algorithm proposed by Yin and Yuan [11, 10] to
determine dose escalation and deescaltion, and propose a different criterion for MTD selection at
the end of the trial. In our design, similar to Yin and Yuan [11], we restrict dose escalation and de-
escalation one level at a time (i.e, we do not allow dose escalate or de-escalate along the diagonal)
based on the practical consideration that physicians are conservative and typically do not allow
two agent to escalate at the same time for patient safety. Nevertheless, we note that Sweeting and
Mander [24] showed that diagonal escalation strategy may be more efficient in reaching the target
toxicity level quicker with fewer patients treated at sub-optimal doses and have a higher percentage
of correct selection at the end of the trial. We take this strategy in our start-up phase described later.
Let c
e
be the probability threshold for dose escalation and c
d
the probability threshold for dose
de-escalation. We require c
e
+c
d
> 1 to avoid that the decisions of dose escalation and deescalation
occur at the same time. Our dose-finding algorithm can be described as follows:
• If the current combination is (j, k) and P (π
j,k
< θ|data) > c
e
,
4