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The optimal dose for treating patients with a molecularly targeted agent may differ according to the patient's individual characteristics, such as biomarker status. In this article, we propose a Bayesian phase I/II dose-finding design to find the optimal dose that is personalized for each patient according to his/her biomarker status. To overcome the curse of dimensionality caused by the relatively large number of biomarkers and their interactions with the dose, we employ canonical partial least squares (CPLS) to extract a small number of components from the covariate matrix containing the dose, biomarkers, and dose-by-biomarker interactions. Using these components as the covariates, we model the ordinal toxicity and efficacy using the latent-variable approach. Our model accounts for important features of molecularly targeted agents. We quantify the desirability of the dose using a utility function and propose a two-stage dose-finding algorithm to find the personalized optimal dose according to each patient's individual biomarker profile. Simulation studies show that our proposed design has good operating characteristics, with a high probability of identifying the personalized optimal dose. Supplementary materials for this article are available online.

Much of the statistical methodology underlying the experimental design of phase 1 trials in oncology is intended for studies involving a single cytotoxic agent. The goal of these studies is to estimate the maximally tolerated dose, the highest dose that can be administered with an acceptable level of toxicity. A fundamental assumption of these methods is monotonicity of the dose–toxicity curve. This is a reasonable assumption for single‐agent trials in which the administration of greater doses of the agent can be expected to produce dose‐limiting toxicities in increasing proportions of patients. When studying multiple agents, the assumption may not hold because the ordering of the toxicity probabilities could possibly be unknown for several of the available drug combinations. At the same time, some of the orderings are known and so we describe the whole situation as that of a partial ordering. In this article, we propose a new two‐dimensional dose‐finding method for multiple‐agent trials that simplifies to the continual reassessment method (CRM), introduced by O'Quigley, Pepe, and Fisher (1990, Biometrics 46, 33–48), when the ordering is fully known. This design enables us to relax the assumption of a monotonic dose–toxicity curve. We compare our approach and some simulation results to a CRM design in which the ordering is known as well as to other suggestions for partial orders.

We present an adaptive Bayesian method for dose‐finding in phase I/II clinical trials based on trade‐offs between the probabilities of treatment efficacy and toxicity. The method accommodates either trinary or bivariate binary outcomes, as well as efficacy probabilities that possibly are nonmonotone in dose. Doses are selected for successive patient cohorts based on a set of efficacy–toxicity trade‐off contours that partition the two‐dimensional outcome probability domain. Priors are established by solving for hyperparameters that optimize the fit of the model to elicited mean outcome probabilities. For trinary outcomes, the new algorithm is compared to the method of Thall and Russell (1998, Biometrics54, 251–264) by application to a trial of rapid treatment for ischemic stroke. The bivariate binary outcome case is illustrated by a trial of graft‐versus‐host disease treatment in allogeneic bone marrow transplantation. Computer simulations show that, under a wide rage of dose‐outcome scenarios, the new method has high probabilities of making correct decisions and treats most patients at doses with desirable efficacy–toxicity trade‐offs.

In contrast with typical Phase III clinical trials, there is little existing methodology for determining the appropriate numbers of patients to enroll in adaptive Phase I trials. And, as stated by Dennis Lindley in a more general context, u [t]he simple practical question of 'What size of sample should I take' is often posed to a statistician, and it is a question that is embarrassingly difficult to answer.\" Historically, simulation has been the primary option for determining sample sizes for adaptive Phase I trials, and although useful, can be problematic and time-consuming when a sample size is needed relatively quickly. We propose a computationally fast and simple approach that uses Beta distributions to approximate the posterior distributions of DLT rates of each dose and determines an appropriate sample size through posterior coverage rates. We provide sample sizes produced by our methods for a vast number of realistic Phase I trial settings and demonstrate that our sample sizes are generally larger than those produced by a competing approach that is based upon the nonparametric optimal design.

We propose a hierarchical model for the probability of dose-limiting toxicity (DLT) for combinations of doses of two therapeutic agents. We apply this model to an adaptive Bayesian trial algorithm whose goal is to identify combinations with DLT rates close to a prespecified target rate. We describe methods for generating prior distributions for the parameters in our model from a basic set of information elicited from clinical investigators. We survey the performance of our algorithm in a series of simulations of a hypothetical trial that examines combinations of four doses of two agents. We also compare the performance of our approach to two existing methods and assess the sensitivity of our approach to the chosen prior distribution.

We consider the problem of testing for a dose-related effect based on a candidate set of (typically nonlinear) doseresponse models using likelihood-ratio tests. For the considered models this reduces to assessing whether the slope parameter in these nonlinear regression models is zero or not. A technical problem is that the null distribution (when the slope is zero) depends on non-identifiable parameters, so that standard asymptotic results on the distribution of the likelihood-ratio test no longer apply. Asymptotic solutions for this problem have been extensively discussed in the literature. The resulting approximations however are not of simple form and require simulation to calculate the asymptotic distribution. In addition, their appropriateness might be doubtful for the case of a small sample size. Direct simulation to approximate the null distribution is numerically unstable due to the non identifiability of some parameters. In this article, we derive a numerical algorithm to approximate the exact distribution of the likelihood-ratio test under multiple models for normally distributed data. The algorithm uses methods from differential geometry and can be used to evaluate the distribution under the null hypothesis, but also allows for power and sample size calculations. We compare the proposed testing approach to the MCP-Mod methodology and alternative methods for testing for a dose-related trend in a dose-finding example data set and simulations.

An outcome-adaptive Bayesian design is proposed for choosing the optimal dose pair of a chemotherapeutic agent and a biological agent used in combination in a phase I/II clinical trial. Patient outcome is characterized as a vector of two ordinal variables accounting for toxicity and treatment efficacy. A generalization of the Aranda-Ordaz model (1981, Biometrika 68, 357-363) is used for the marginal outcome probabilities as functions of a dose pair, and a Gaussian copula is assumed to obtain joint distributions. Numerical utilities of all elementary patient outcomes, allowing the possibility that efficacy is inevaluable due to severe toxicity, are obtained using an elicitation method aimed to establish consensus among the physicians planning the trial. For each successive patient cohort, a dose pair is chosen to maximize the posterior mean utility. The method is illustrated by a trial in bladder cancer, including simulation studies of the method's sensitivity to prior parameters, the numerical utilities, correlation between the outcomes, sample size, cohort size, and starting dose pair.

This article considers the topic of finding prior distributions when a major component of the statistical model depends on a nonlinear function. Using results on how to construct uniform distributions in general metric spaces, we propose a prior distribution that is uniform in the space of functional shapes of the underlying nonlinear function and then back‐transform to obtain a prior distribution for the original model parameters. The primary application considered in this article is nonlinear regression, but the idea might be of interest beyond this case. For nonlinear regression the so constructed priors have the advantage that they are parametrization invariant and do not violate the likelihood principle, as opposed to uniform distributions on the parameters or the Jeffrey’s prior, respectively. The utility of the proposed priors is demonstrated in the context of design and analysis of nonlinear regression modeling in clinical dose‐finding trials, through a real data example and simulation.

Background. Absent adaptive, individualized dose-finding in early-phase oncology trials, subsequent 'confirmatory' Phase III trials risk suboptimal dosing, with resulting loss of statistical power and reduced probability of technical success for the investigational therapy. While progress has been made toward explicitly adaptive dose-finding and quantitative modeling of dose-response relationships, most such work continues to be organized around a concept of 'the' maximum tolerated dose (MTD). The purpose of this paper is to demonstrate concretely how the aim of early-phase trials might be conceived, not as 'dose-finding', but as dose titration algorithm (DTA)-finding. Methods. A Phase I dosing study is simulated, for a notional cytotoxic chemotherapy drug, with neutropenia constituting the critical dose-limiting toxicity. The drug's population pharmacokinetics and myelosuppression dynamics are simulated using published parameter estimates for docetaxel. The amenability of this model to linearization is explored empirically. The properties of a simple DTA targeting neutrophil nadir of 500 cells/mm 3 using a Newton-Raphson heuristic are explored through simulation in 25 simulated study subjects. Results. Individual-level myelosuppression dynamics in the simulation model approximately linearize under simple transformations of neutrophil concentration and drug dose. The simulated dose titration exhibits largely satisfactory convergence, with great variance in individualized optimal dosing. Some titration courses exhibit overshooting. Conclusions. The large inter-individual variability in simulated optimal dosing underscores the need to replace 'the' MTD with an individualized concept of MTD i . To illustrate this principle, the simplest possible DTA capable of realizing such a concept is demonstrated. Qualitative phenomena observed in this demonstration support discussion of the notion of tuning such algorithms. Although here illustrated specifically in relation to cytotoxic chemotherapy, the DTAT principle appears similarly applicable to Phase I studies of cancer immunotherapy and molecularly targeted agents.

During development of a drug, typically the choice of dose is based on a Phase II dose-finding trial, where selected doses are included with placebo. Two common statistical dose-finding methods to analyze such trials are separate comparisons of each dose to placebo (using a multiple comparison procedure) or a model-based strategy (where a dose–response model is fitted to all data). The first approach works best when patients are concentrated on few doses, but cannot conclude on doses not tested. Model-based methods allow for interpolation between doses, but the validity depends on the correctness of the assumed dose–response model. Bretz et al. (2005, Biometrics 61, 738–748) suggested a combined approach, which selects one or more suitable models from a set of candidate models using a multiple comparison procedure. The method initially requires a priori estimates of any non-linear parameters of the candidate models, such that there is still a degree of model misspecification possible and one can only evaluate one or a few special cases of a general model. We propose an alternative multiple testing procedure, which evaluates a candidate set of plausible dose–response models against each other to select one final model. The method does not require any a priori parameter estimates and controls the Type I error rate of selecting a too complex model.