Optimal Designs for Estimating the Interesting Part of a Dose-Effect Curve

A Bayesian quasi-likelihood design for identifying the minimum effective dose and maximum utility dose in dose-ranging studies

Most existing dose-ranging study designs focus on assessing the dose-efficacy relationship and identifying the minimum effective dose. There is an increasing interest in optimizing the dose based on the benefit-risk tradeoff. We propose a Bayesian quasi-likelihood dose-ranging design that jointly considers safety and efficacy to simultaneously identify the minimum effective dose and the maximum utility dose to optimize the benefit-risk tradeoff. The binary toxicity endpoint is modeled using a beta-binomial model. The efficacy endpoint is modeled using the quasi-likelihood approach to accommodate various types of data (e.g. binary, ordinal or continuous) without imposing any parametric assumptions on the dose-response curve. Our design utilizes a utility function as a measure of benefit-risk tradeoff and adaptively assign patients to doses based on the doses' likelihood of being the minimum effective dose and maximum utility dose. The design takes a group-sequential approach. At each interim, the doses that are deemed overly toxic or futile are dropped. At the end of the trial, we use posterior probability criteria to assess the strength of the dose-response relationship for establishing the proof-of-concept. If the proof-of-concept is established, we identify the minimum effective dose and maximum utility dose. Our simulation study shows that compared with some existing designs, the Bayesian quasi-likelihood dose-ranging design is robust and yields competitive performance in establishing proof-of-concept and selecting the minimum effective dose. Moreover, it includes an additional feature for further maximum utility dose selection.

Seamless phase II/III design: a useful strategy to reduce the sample size for dose optimization

BACKGROUND

The traditional more-is-better dose selection paradigm, originally developed for cytotoxic chemotherapeutics, can be problematic when applied to the development of novel molecularly targeted agents. Recognizing this issue, the US Food and Drug Administration initiated Project Optimus to reform the dose optimization and selection paradigm in oncology drug development, emphasizing the need for greater attention to benefit-risk considerations.

METHODS

We identify different types of phase II/III dose-optimization designs, classified according to trial objectives and endpoint types. Through computer simulations, we examine their operating characteristics and discuss the relevant statistical and design considerations for effective dose optimization.

RESULTS

Phase II/III dose-optimization designs are capable of controlling family-wise type I error rates and achieving appropriate statistical power with substantially smaller sample sizes than the conventional approach while also reducing the number of patients who experience toxicity. Depending on the design and scenario, the sample size savings range from 16.6% to 27.3%, with a mean savings of 22.1%.

CONCLUSIONS

Phase II/III dose-optimization designs offer an efficient way to reduce sample sizes for dose optimization and accelerate the development of targeted agents. However, because of interim dose selection, the phase II/III dose-optimization design presents logistical and operational challenges and requires careful planning and implementation to ensure trial integrity.

Optimal adaptive allocation using deep reinforcement learning in a dose-response study

Estimation of the dose-response curve for efficacy and subsequent selection of an appropriate dose in phase II trials are important processes in drug development. Various methods have been investigated to estimate dose-response curves. Generally, these methods are used with equal allocation of subjects for simplicity; nevertheless, they may not fully optimize performance metrics because of nonoptimal allocation. Optimal allocation methods, which include adaptive allocation methods, have been proposed to overcome the limitations of equal allocation. However, they rely on asymptotics, and thus sometimes cannot efficiently optimize the performance metric with the sample size in an actual clinical trial. The purpose of this study is to construct an adaptive allocation rule that directly optimizes a performance metric, such as power, accuracy of model selection, accuracy of the estimated target dose, or mean absolute error over the estimated dose-response curve. We demonstrate that deep reinforcement learning with an appropriately defined state and reward can be used to construct such an adaptive allocation rule. The simulation study shows that the proposed method can successfully improve the performance metric to be optimized when compared with the equal allocation, D-optimal, and TD-optimal methods. In particular, when the mean absolute error was set to the metric to be optimized, it is possible to construct a rule that is superior for many metrics.

Utility-Based Dose Selection for Phase II Dose-Finding Studies

BACKGROUND AND OBJECTIVES

Dose selection is a key feature of clinical development. Poor dose selection has been recognized as a major driver of development failure in late phase. It usually involves both efficacy and safety criteria. The objective of this paper is to develop and implement a novel fully Bayesian statistical framework to optimize the dose selection process by maximizing the expected utility in phase III.

METHODS

The success probability is characterized by means of a utility function with two components, one for efficacy and one for safety. Each component refers to a dose-response model. Moreover, a sequential design (with futility and efficacy rules at the interim analysis) is compared to a fixed design in order to allow one to hasten the decision to perform the late phase study. Operating characteristics of this approach are extensively assessed by simulations under a wide range of dose-response scenarios.

RESULTS AND CONCLUSIONS

Simulation results illustrate the difficulty of simultaneously estimating two complex dose-response models with enough accuracy to properly rank doses using an utility function combining the two. The probability of making the good decision increases with the sample size. For some scenarios, the sequential design has good properties: with a quite large probability of study termination at interim analysis, it enables to reduce the sample size while maintaining the properties of the fixed design.

Optimal dose-finding for efficacy-safety models

Dose-finding is an important part of the clinical development of a new drug. The purpose of dose-finding studies is to determine a suitable dose for future development based on both efficacy and safety. Optimal experimental designs have already been used to determine the design of this kind of studies, however, often that design is focused on efficacy only. We consider an efficacy-safety model, which is a simplified version of the bivariate Emax model. We use here the clinical utility index concept, which provides the desirable balance between efficacy and safety. By maximizing the utility of the patients, we get the estimated dose. This desire leads us to locally c -optimal designs. An algebraic solution for c -optimal designs is determined for arbitrary c vectors using a multivariate version of Elfving's method. The solution shows that the expected therapeutic index of the drug is a key quantity determining both the number of doses, the doses itself, and their weights in the optimal design. A sequential design is proposed to solve the complication of parameter dependency, and it is illustrated in a simulation study.

Design optimization for dose-finding trials: a review

Dose selection is one of the most difficult and crucial decisions to make during drug development. As a consequence, the dose-finding trial is a major milestone in the drug development plan and should be properly designed. This article will review the most recent methodologies for optimizing the design of dose-finding studies: all of them are based on the modeling of the dose-response curve, which is now the gold standard approach for analyzing dose-finding studies instead of the traditional ANOVA/multiple testing approach. We will address the optimization of both fixed and adaptive designs and briefly outline new methodologies currently under investigation, based on utility functions.

Implementing Optimal Designs for Dose-Response Studies Through Adaptive Randomization for a Small Population Group

In dose-response studies with censored time-to-event outcomes, D-optimal designs depend on the true model and the amount of censored data. In practice, such designs can be implemented adaptively, by performing dose assignments according to updated knowledge of the dose-response curve at interim analysis. It is also essential that treatment allocation involves randomization-to mitigate various experimental biases and enable valid statistical inference at the end of the trial. In this work, we perform a comparison of several adaptive randomization procedures that can be used for implementing D-optimal designs for dose-response studies with time-to-event outcomes with small to moderate sample sizes. We consider single-stage, two-stage, and multi-stage adaptive designs. We also explore robustness of the designs to experimental (chronological and selection) biases. Simulation studies provide evidence that both the choice of an allocation design and a randomization procedure to implement the target allocation impact the quality of dose-response estimation, especially for small samples. For best performance, a multi-stage adaptive design with small cohort sizes should be implemented using a randomization procedure that closely attains the targeted D-optimal design at each stage. The results of the current work should help clinical investigators select an appropriate randomization procedure for their dose-response study.

Feasibility of Exposure-Response Analyses for Clinical Dose-Ranging Studies of Drug Combinations

The exposure-response relationship of combinatory drug effects can be quantitatively described using pharmacodynamic interaction models, which can be used for the selection of optimal dose combinations. The aim of this simulation study was to evaluate the reliability of parameter estimates and the probability for accurate dose identification for various underlying exposure-response profiles, under a number of different phase II designs. An efficacy variable driven by the combined exposure of two theoretical compounds was simulated and model parameters were estimated using two different models, one estimating all parameters and one assuming that adequate previous knowledge for one drug is readily available. Estimation of all pharmacodynamic parameters under a realistic, in terms of sample size and study design, phase II trial, proved to be challenging. Inaccurate estimates were found in all exposure-response scenarios, except for situations where no pharmacodynamic interaction was present, with the drug potency and interaction parameters being the hardest to estimate. When previous knowledge of the exposure-response relationship of one of the monocomponents is available, such information should be utilized, as it enabled relevant improvements in parameter estimation and in correct dose identification. No general trends for classification of the performance of the tested study designs across different scenarios could be identified. This study shows that pharmacodynamic interactions models can be used for the exposure-response analysis of clinical endpoints especially when accompanied by appropriate dose selection in regard to the expected drug potencies and appropriate trial size and if information regarding the exposure-response profile of one monocomponent is available.

Simultaneous estimation of parameters in the bivariate Emax model

In this paper, we explore inference in multi-response, nonlinear models. By multi-response, we mean models with m > 1 response variables and accordingly m relations. Each parameter/explanatory variable may appear in one or more of the relations. We study a system estimation approach for simultaneous computation and inference of the model and (co)variance parameters. For illustration, we fit a bivariate Emax model to diabetes dose-response data. Further, the bivariate Emax model is used in a simulation study that compares the system estimation approach to equation-by-equation estimation. We conclude that overall, the system estimation approach performs better for the bivariate Emax model when there are dependencies among relations. The stronger the dependencies, the more we gain in precision by using system estimation rather than equation-by-equation estimation.

Designing dose finding studies with an active control for exponential families

Optimal design of dose-finding studies with an active control has only been considered in the literature for regression models with normally distributed errors and known variances, where the focus is on estimating the smallest dose that achieves the same treatment effect as the active control. This paper discusses such dose-finding studies from a broader perspective. We consider a general class of optimality criteria and models arising from an exponential family. Optimal designs are constructed for several situations and their efficiency is illustrated with examples.

Multiple-Objective Optimal Designs for Studying the Dose Response Function and Interesting Dose Levels

We construct an optimal design to simultaneously estimate three common interesting features in a dose-finding trial with possibly different emphasis on each feature. These features are (1) the shape of the dose-response curve, (2) the median effective dose and (3) the minimum effective dose level. A main difficulty of this task is that an optimal design for a single objective may not perform well for other objectives. There are optimal designs for dual objectives in the literature but we were unable to find optimal designs for 3 or more objectives to date with a concrete application. A reason for this is that the approach for finding a dual-objective optimal design does not work well for a 3 or more multiple-objective design problem. We propose a method for finding multiple-objective optimal designs that estimate the three features with user-specified higher efficiencies for the more important objectives. We use the flexible 4-parameter logistic model to illustrate the methodology but our approach is applicable to find multiple-objective optimal designs for other types of objectives and models. We also investigate robustness properties of multiple-objective optimal designs to mis-specification in the nominal parameter values and to a variation in the optimality criterion. We also provide computer code for generating tailor made multiple-objective optimal designs.

Improved precision of exposure-response relationships by optimal dose-selection. Examples from studies of receptor occupancy using PET and dose finding for neuropathic pain treatment

An understanding of the relationship between drug exposure and response is a fundamental basis for any dosing recommendation. We investigate optimal dose-selection for two different types of studies, a receptor occupancy study assessed by positron emission tomography (PET) and a dose-finding study in neuropathic pain treatment. For the PET-study, an inhibitory E-max model describes the relationship between drug exposure and displacement of a radioligand from specific receptors in the brain. The model has a mechanistic basis in the law of mass action and the affinity parameter (Ki PL ) is of primary interest. For optimization of the neuropathic pain study, the model is empirical and the exposure response curve itself is of primary interest. An alternative parameterization of the sigmoid Emax model was therefore used where the plasma concentration corresponding to the minimum relevant efficacy was estimated as a parameter. Optimal design methodology was applied using the D-optimal criterion as well as the Ds-optimal criterion where parameters of interest were defined. For the PET-study it was shown that the precision of Ki PL can be improved by inclusion of brain regions with both high and low receptor density and that the need for high doses is reduced when a brain region with low receptor density is included in the analysis. In the case of the neuropathic pain study it was shown that a Ds-optimal study design using the reparameterized Emax model can improve the precision in the minimum effective dose compared to a D-optimal design.

Implementation of maximin efficient designs in dose-finding studies

This paper considers the maximin approach for designing clinical studies. A maximin efficient design maximizes the smallest efficiency when compared with a standard design, as the parameters vary in a specified subset of the parameter space. To specify this subset of parameters in a real situation, a four-step procedure using elicitation based on expert opinions is proposed. Further, we describe why and how we extend the initially chosen subset of parameters to a much larger set in our procedure. By this procedure, the maximin approach becomes feasible for dose-finding studies. Maximin efficient designs have shown to be numerically difficult to construct. However, a new algorithm, the H-algorithm, considerably simplifies the construction of these designs. We exemplify the maximin efficient approach by considering a sigmoid Emax model describing a dose-response relationship and compare inferential precision with that obtained when using a uniform design. The design obtained is shown to be at least 15% more efficient than the uniform design.

The Add-Arm Design for Unimodal Response Curve with Unknown Mode

In a classical drop-loser (or drop-arm) design, patients are randomized into all arms (doses) and at the interim analysis, inferior arms are dropped. Therefore, compared to the traditional dose-finding design, this adaptive design can reduce the sample size by not carrying over all doses to the end of the trial or dropping the losers earlier. However, all the doses have to be explored. For unimodal (including linear or umbrella) response curves, we proposed an effective dose-finding design that allows adding arms at the interim analysis. The trial design starts with two arms, depending on the response of the two arms and the unimodality assumption; we can decide which new arms to be added. This design does not require exploring all arms (doses) to find the best responsive dose; therefore, it can further reduce the sample size from the drop-loser design by as much as 10-20%.

Dose finding when the target dose is on a plateau of a dose-response curve: comparison of fully sequential designs

Consider the problem of estimating a dose with a certain response rate. Many multistage dose-finding designs for this problem were originally developed for oncology studies where the mean dose-response is strictly increasing in dose. In non-oncology phase II dose-finding studies, the dose-response curve often plateaus in the range of interest, and there are several doses with the mean response equal to the target. In this case, it is usually of interest to find the lowest of these doses because higher doses might have higher adverse event rates. It is often desirable to compare the response rate at the estimated target dose with a placebo and/or active control. We investigate which of the several known dose-finding methods developed for oncology phase I trials is the most suitable when the dose-response curve plateaus. Some of the designs tend to spread the allocation among the doses on the plateau. Others, such as the continual reassessment method and the t-statistic design, concentrate allocation at one of the doses with the t-statistic design selecting the lowest dose on the plateau more frequently.

Two-stage designs for Phase 2 dose-finding trials

We propose a Bayesian adaptive two-stage design for the efficient estimation of the maximum dose or the minimum effective dose in a dose-finding trial. The new design allocates subjects in stage two according to the posterior distribution of the target dose location. Simulations show that the proposed two-stage design is superior to equal allocation and to a two-stage strategy where only one dose is left in the second stage.

Modelling and simulation in the pharmaceutical industry--some reflections

Modelling and simulation (M&S) is increasingly being applied in (clinical) drug development. It provides an opportune area for the community of pharmaceutical statisticians to pursue. In this article, we highlight useful principles behind the application of M&S. We claim that M&S should be focussed on decisions, tailored to its purpose and based in applied sciences, not relying entirely on data-driven statistical analysis. Further, M&S should be a continuous process making use of diverse information sources and applying Bayesian and frequentist methodology, as appropriate. In addition to forming a basis for analysing decision options, M&S provides a framework that can facilitate communication between stakeholders. Besides the discussion on modelling philosophy, we also describe how standard simulation practice can be ineffective and how simulation efficiency can often be greatly improved.

Adaptive dose-finding: Proof of concept with type I error control

We consider an adaptive dose-finding study with two stages. The doses for the second stage will be chosen based on the first stage results. Instead of considering pairwise comparisons with placebo, we apply one test to show an upward trend across doses. This is a possibility according to the ICH-guideline for dose-finding studies (ICH-E4). In this article, we are interested in trend tests based on a single contrast or on the maximum of multiple contrasts. We are interested in flexibly choosing the Stage 2 doses including the possibility to add doses. If certain requirements for the interim decision rules are fulfilled, the final trend test that ignores the adaptive nature of the trial (naïve test) can control the type I error. However, for the more common case that these requirements are not fulfilled, we need to take the adaptivity into account and discuss a method for type I error control. We apply the general conditional error approach to adaptive dose-finding and discuss special issues appearing in this application. We call the test based on this approach Adaptive Multiple Contrast Test. For an example, we illustrate the theory discussed before and compare the performance of several tests for the adaptive design in a simulation study.

Practical considerations for optimal designs in clinical dose finding studies

A key objective in the clinical development of a medicinal drug is the determination of an adequate dose level and, more broadly, the characterization of its dose response relationship. If the dose is set too high, safety and tolerability problems are likely to result, while selecting too low a dose makes it difficult to establish adequate efficacy in the confirmatory phase, possibly leading to a failed program. Hence, dose finding studies are of critical importance in drug development and need to be planned carefully. In this paper, we focus on practical considerations for establishing efficient study designs to estimate relevant target doses. We consider optimal designs for estimating both the minimum effective dose and the dose achieving a certain percentage of the maximum treatment effect. These designs are compared with D-optimal designs for a given dose response model. Extensions to robust designs accounting for model uncertainty are also discussed. A case study is used to motivate and illustrate the methods from this paper.

Dose finding - a challenge in statistics

A good understanding and characterization of the dose response relationship of any new compound is an important and ubiquitous problem in many areas of scientific investigation. This is especially true in the context of pharmaceutical drug development, where it is mandatory to launch safe drugs which demonstrate a clinically relevant effect. Selecting a dose too high may result in unacceptable safety problems, while selecting a dose too low may lead to ineffective drugs. Dose finding studies thus play a key role in any drug development program and are often the gate-keeper for large confirmatory studies. In this overview paper we focus on definitive and confirmatory dose finding studies in Phase II or III, reviewing relevant statistical design and analysis methods. In particular, we describe multiple comparison procedures, modeling approaches, and hybrid methods combining the advantages of both. An outlook to adaptive dose finding methods is also given. We use a real data example to illustrate the methods, together with a brief overview of relevant software.