A Note on the Required Sample Size of Model-Based Dose-Finding Methods for Molecularly Targeted Agents

Research Article

Austin Biom and Biostat. 2016; 3(1): 1032.

A Note on the Required Sample Size of Model-Based Dose-Finding Methods for Molecularly Targeted Agents

Sato H1*†, Hirakawa A2*† and Hamada C3

1Biostatistics Group, Center for Product Evaluation, Pharmaceuticals and Medical Devices Agency, Japan

2Statistical Analysis Section, Center for Advanced Medicine and Clinical Research, Nagoya University Hospital, Japan

3Department of Information and Computer Technology, Tokyo University of Science, Japan †These authors contributed equally to this work.

*Corresponding author: Sato H, Biostatistics Group, Center for Product Evaluation, Pharmaceuticals and Medical Devices Agency, 3-3-2 Kasumigaseki, Chiyodaku, Tokyo 100-0013, Japan

Received: November 07, 2016; Accepted: December 05, 2016; Published: December 14, 2016

Abstract

Some Molecularly Targeted Agents (MTAs) exhibit non-monotonic patterns in the dose-response relationships. Although many model-based dose-finding methods to account for such patterns have been proposed, the required sample size to determine the true Optimal Dose (OD) has not been adequately investigated. A little knowledge of the required sample size might potentially prevent wide-ranging application of model-based dose-finding methods in practice. In this study, we focus on three model-based dose-finding methods that accommodate non-monotonic patterns in the dose-efficacy relationship, and discuss the required sample sizes under various conditions, using simulation studies. We found that the selection rate of the true OD did not necessarily improve as the sample size increased. Based on the results of our simulation studies, we provide notes and guidelines on sample size determination when using model-based dose-finding methods for MTAs.

Keywords: Change-point model; Sample size; Dose-finding; Oncology; Phase I

Abbreviations

AR: Adaptive Randomization; CP: Change Point; CP method: dose-finding method proposed by Sato et al.; CRM: Continual Reassessment Method; MCMC: Markov Chain Monte Carlo; MTD: Maximum Tolerated Dose; MTAs: Molecularly Targeted Agents; OD: Optimal Dose; TC method: dose-finding method proposed by Thall and Cook; WMD: Weighted Mahalanobis Distance; WT method: dose-finding method proposed by Wages and Tait

Introduction

The objective of phase I oncology trials is generally to determine the Maximum Tolerated Dose (MTD). This is defined as the highest dose level that can be administered to patients with clinically acceptable toxicity. The dose-finding methods for determining the MTD are roughly categorized into two groups, model-based and rule-based methods. Rule-based methods, such as the 3+3 design, are widely used in practice, but the lack of statistical rationale and low accuracy of determining the true MTD are often problematic. Many model-based dose-finding methods, such as the Continual Reassessment Method (CRM) [1], assume that the probabilities of toxicity and efficacy of an agent increase monotonically as the dose of the agent increases; therefore, dose escalation or de-escalation is commonly based solely on toxicity outcome. Such methods outperform rule-based methods in many cases [2-4].

Some Molecularly Targeted Agents (MTAs) exhibit nonmonotonic patterns in dose-efficacy relationships. Therefore, the model-based dose-finding method based on the above-mentioned assumptions may not be reasonable for determining the Optimal Dose (OD) of MTAs. To account for non-monotonic patterns in the dose-efficacy relationships of MTAs, dose-finding methods that account for both toxicity and efficacy outcomes are required. Such methods generally determine the OD based on toxicity and efficacy outcomes. The OD is often considered to be the dose level with the maximum efficacy probability among the dose levels with toxicity probabilities lower than a pre-specified value (e.g., 30 or 40%), although the definition of the OD varies depending on the individual method proposed. Many researchers have developed dose-finding methods based on toxicity and efficacy outcomes for single-agent or two-agent combination phase I trials [5-11]. Thall and Cook [6] proposed using the Gumbel model [12] to capture the relationship between the bivariate binary toxicity and efficacy outcomes (termed the TC method). They used a quadratic model for the dose-efficacy relationship in order to consider a non-monotonic pattern. Wages and Tait [11] proposed using a power model for the binary efficacy and toxicity outcomes (termed the WT method). They assumed some class of working model for the efficacy outcome and used model selection techniques to allow greater flexibility in modeling the doseefficacy relationship. Recently, we developed a new dose-finding method using the Change-Point (CP) logistic model for single MTA trials (termed the CP method) [13]. Specifically, we developed a doseefficacy model, the parameters of which are allowed to change in the vicinity of the change point of the dose level, in order to address non-monotonic patterns of the dose-efficacy relationship. The change point is defined as the dose that maximizes the log-likelihood of the assumed dose-efficacy and dose-toxicity models.

Although many useful dose-finding methods have been proposed that account for non-monotonic patterns of the dose-efficacy relationship for MTAs, the required sample size for determining true the OD using these methods has not been adequately investigated. For instance, the selection rate for the true OD is generally evaluated using fixed sample sizes in simulation studies [6,11,13], but the required sample size to achieve the target selection rate for the true OD is not. Thus, little is known about the required sample size for the existing dose-finding methods for MTA. It is useful for investigators to provide the required sample sizes to use novel model-based dosefinding methods under various conditions (e.g., number of dose levels evaluated and prior distribution for model parameters). In this study, we focus on the three model-based dose-finding methods that can be used for MTA (i.e., the CP, TC, and WT methods), and discuss the required sample size to determine the true OD under various conditions, using simulation studies. Based on the results of the simulation studies, we provide notes and guidelines for determining the sample size for model-based dose-finding methods for MTAs.

This paper is organized as follows: in the next section, we provide an overview of the three dose-finding methods. The simulation studies are described in the third section, and we discuss the determination of the required sample size and provide guidelines for determining the sample size for model-based dose-finding methods for MTAs in the fourth section.

Dose-Finding Methods Used

An adaptive dose-finding method for a MTA using the Change-Point model (CP method)

Let YEi and YTi denote binary efficacy and toxicity outcomes for the ith of N patients, respectively. YEi(orYTi)= 1 indicates that efficacy (or toxicity) is observed, and YEi(orYTi) = 0 otherwise. Following Islam et al. [14], the joint probabilities for YEi and YTi are given in Table 1.