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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