Journal of Probability and Statistics

Volume 2018, Article ID 8654173, 11 pages

https://doi.org/10.1155/2018/8654173

## A Bayesian Adaptive Design in Cancer Phase I Trials Using Dose Combinations in the Presence of a Baseline Covariate

Biostatistics and Bioinformatics Research Center, Cedars-Sinai Medical Center 8700 Beverly Blvd, Los Angeles, CA 90048, USA

Correspondence should be addressed to Mourad Tighiouart; gro.shsc@trauoihgit.daruom

Received 3 May 2018; Revised 27 July 2018; Accepted 29 August 2018; Published 1 November 2018

Academic Editor: Ash Abebe

Copyright © 2018 Márcio Augusto Diniz et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A Bayesian adaptive design for dose finding of a combination of two drugs in cancer phase I clinical trials that takes into account patients heterogeneity thought to be related to treatment susceptibility is described. The estimation of the maximum tolerated dose (MTD) curve is a function of a baseline covariate using two cytotoxic agents. A logistic model is used to describe the relationship between the doses, baseline covariate, and the probability of dose limiting toxicity (DLT). Trial design proceeds by treating cohorts of two patients simultaneously using escalation with overdose control (EWOC), where at each stage of the trial, the next dose combination corresponds to the quantile of the current posterior distribution of the MTD of one of two agents at the current dose of the other agent and the next patient’s baseline covariate value. The MTD curves are estimated as function of Bayes estimates of the model parameters at the end of trial. Average DLT, pointwise average bias, and percent of dose recommendation at dose combination neighborhoods around the true MTD are compared between the design that uses the covariate and the one that ignores the baseline characteristic. We also examine the performance of the approach under model misspecifications for the true dose-toxicity relationship. The methodology is further illustrated in the case of a prespecified discrete set of dose combinations.

#### 1. Introduction

Despite the promise observed in preclinical experiments and initial high response rates, a large number of targeted drugs have not been successful in providing reproducible improvements in survival in patients with cancer when used as single agents. [1] In addition, targeted therapies do not work for every patient since they rely on the presence of the target. Therefore, chemotherapy and radiotherapy approaches are still the backbone of cancer treatment for tumors after surgical excision. These conventional cancer therapies may be combined with targeted agents to enhance treatment efficacy.

Statistical methodologies for designing phase I clinical trials for drug combinations have been studied extensively in the past decade [2–13]. These methods assume that the patient population is homogeneous of treatment tolerance and every patient should be treated at a dose combination corresponding to a predefined target probability of DLT (dose limiting toxicity). Therefore, an additional layer of complexity in specifying the dose-toxicity relationship given a baseline covariate is needed for drug combinations.

Strategies of drug allocation that accommodate individual patient needs have been used in [14–18] for single agent trials. Statistical designs allowing individualized maximum tolerable dose (MTD) determination in single agent cancer phase I trials have also been proposed and implemented in real trials by a number of authors for two groups with no prior knowledge of ordering [19, 20], for two prior ordered groups [21, 22] and two or more prior partially ordered groups [23, 24]. In general, ignoring the heterogeneity between groups can lead to higher toxicities in the most severely impaired group, statistical bias, and inefficiency of the MTD estimate for both groups.

In this work, we extend the design described by Tighiouart et al. [25] using escalation with overdose control (EWOC) principle [26], by treating cohorts of two patients simultaneously and accounting for patient baseline binary covariate. We assume that we do not have prior knowledge of the ordering between groups, but they will be ordered in the sense that the probability of toxicity for one group is always a constant shift from the probability of toxicity for the second group at the same dose. In this way, patients with different covariate values will have parallel MTD curves. This assumption is mathematically convenient and allows us to use parsimonious models due to the small sample size constraints in cancer phase I trials.

This paper is organized as follows. Section 2 will describe the dose-toxicity model and trial design for continuous dose levels. In Section 3, we evaluate the performance of the proposed method by assessing the safety of the trial design and the efficiency of the estimate of the MTD curve. The methodology is extended for discrete dose combinations in Section 4. Discussions will be presented in Section 5.

#### 2. Model

##### 2.1. Dose-Toxicity Model

We propose a parametric model to identify tolerable dose combinations of two synergistic drugs and [10–12, 25, 27] given a patient with a binary baseline covariate value of :where is the indicator of DLT, are the continuous dose levels of agents and , respectively, assuming values in , is a binary baseline covariate value, and is a known cumulative distribution function.

We assume partial ordering of the probability of DLT, i.e., it is a nondecreasing function of the dose of any one of the agents when the other one is held constant for and we also assume synergism between the two drugs. These assumptions are translated into constrains in the parameter space given by , and , respectively. The MTD for a patient with covariate value z is defined as the set of combinations such thatThe target probability of DLT, , is set relatively high when the DLT is a reversible or nonfatal condition, and low when it is life threatening. Using (1) and (2), the MTD isWe reparametrize model (1) to allow a more meaningful prior elicitation. Assuming that will be standardized to be in , , the probability of DLT at the minimum available doses of agents and for a patient with covariate value ; , the probability of DLT when the level of drug is , the level of drug is and ; , the probability of DLT when the level of drug is , the level of drug is and ; , the probability of DLT when the level of drug is , the level of drug is and ; and the interaction parameter . It follows thatNotice that implies that , . The MTD set given in (3) can be presented aswhere .

Let be the data after enrolling patients in the trial. The likelihood function under the reparametrization iswhere

##### 2.2. Prior and Posterior Distributions

We consider the priors , , , and conditional on , , , , and with mean and variance . As described in [25], vague priors are achieved by taking , while a vague Gamma prior is chosen with mean of 21 and variance of 540. The posterior distribution is given by,We used JAGS [28] to sample from the posterior distribution.

##### 2.3. Trial Design

The algorithm for dose escalation/deescalation is similar to one discussed in [11, 25] with the additional binary covariate information. It uses the EWOC principle [26] where at each stage of the trial, we seek a dose of one agent using the current posterior distribution of the MTD of the agent given the current dose of the other agent and the next patient’s baseline covariate value. For instance, if agent is held constant at level , the dose of agent is such that the posterior probability that exceeds the MTD of agent given the dose of agent and covariate value is bounded by a feasibility bound . Cohorts of two patients are enrolled simultaneously receiving different dose combinations. Specifically, the design proceeds as follows.(1)Let be the data from the first cohort of two patients such that each patient receives the same dose combination for .(2)In the second cohort of two patients, patient 3 receives dose and patient 4 receives dose . If or , is the th percentile of . Otherwise, patient 3 receives the minimum dose combination . If or , is the th percentile of . Otherwise, patient 4 receives the minimum dose combination . In general, the first time a patient is assigned to a given group defined by the binary covariate always receives the minimum dose combination no matter how many patients have been treated in the other group, as described in [20]. Here, is the posterior distribution of the MTD of agent given that the level of agent is and the baseline covariate value of patient 3 is , given the data . is defined similarly. and can be expressed in terms of , and .(3)In the -th cohort of two patients,(a)If is even, patient receives dose and patient receives dose , where and . Here, is the inverse cumulative distribution function of the posterior distribution, .(b)Similarly, if is odd, patient receives dose and patient receives dose , where and .(4)Repeat step (3), until patients are enrolled in the trial subject to the following stopping rule.

If the th percentile of or is less than 0 or greater than 1, the recommended dose for the next patient is 0 or 1, respectively. In steps (2) and (3) above, a dose escalation is further restricted to be no more than a prespecified fraction of the dose range of the corresponding agent.

*Stopping Rule*. It is sufficient to evaluate a stopping rule for safety at the minimum dose combination because of the partial ordering assumption. The probability of DLT of all doses for both agents will be higher than if the probability at the minimum dose is higher than .We stop enrollment to the trial if , i.e., if the posterior probability that the probability of DLT at the minimum available dose combination in the trial exceeds the target probability of DLT is high for . The design parameters and are chosen to achieve desirable model operating characteristics. At the completion of the trial, an estimate of the MTD curve for is obtained using (5) aswhere , , and are the posterior medians given the data .

#### 3. Simulation Studies

##### 3.1. Simulation Set-Up and Scenarios

We present four scenarios for the true MTD curves as shown in Figure 1. The first scenario (a) is a case where the two true MTD curves for two groups are parallel and close to the minimum doses with and equal to each other and slightly higher than ; the second scenario (b) is a case where the two true MTD curves for two groups are parallel but very close to each other; the third scenario (c) is a case where two true MTD curves for two groups are not parallel, and the last scenario (d) is a case where the two true MTD curves are parallel but lie far apart from each other and close to the maximum doses with and equal and largely lower than .