A simple technique to evaluate model sensitivity in the continual reassessment method

The continual reassessment method (CRM) is a sequential design used in phase I cancer trials to determine the maximal dose with acceptable toxicity. It has been established that the CRM is consistent under model misspecification but not generally. When the method does not converge to the target percentile, some dose-response models will be more sensitive than others in terms of how close the converged recommendation is to the target. In this article, we interpret the main condition under which the CRM is consistent and apply it to evaluate the sensitivity of the model used with the CRM. The technique presented is found to be a useful supplement to simulation when planning a phase I trial.     

A curve-free method for phase I clinical trials

Consider the problem of finding the dose that is as high as possible subject to having a controlled rate of toxicity. The problem is commonplace in oncology Phase I clinical trials. Such a dose is often called the maximum tolerated dose (MTD) since it represents a necessary trade-off between efficacy and toxicity. The continual reassessment method (CRM) is an improvement over traditional up-and-down schemes for estimating the MTD. It is based on a Bayesian approach and on the assumption that the dose-toxicity relationship follows a specific response curve, e.g., the logistic or power curve. The purpose of this paper is to illustrate how the assumption of a specific curve used in the CRM is not necessary and can actually hinder the efficient use of prior inputs. An alternative curve-free method in which the probabilities of toxicity are modeled directly as an unknown multidimensional parameter is presented. To that purpose, a product-of-beta prior (PBP) is introduced and shown to bring about logical improvements. Practical improvements are illustrated by simulation results.     

Cancer phase I trial design using drug combinations when a fraction of dose limiting toxicities is attributable to one or more agents

Drug combination trials are increasingly common nowadays in clinical research. However, very few methods have been developed to consider toxicity attributions in the dose escalation process. We are motivated by a trial in which the clinician is able to identify certain toxicities that can be attributed to one of the agents. We present a Bayesian adaptive design in which toxicity attributions are modeled via copula regression and the maximum tolerated dose (MTD) curve is estimated as a function of model parameters. The dose escalation algorithm uses cohorts of two patients, following the continual reassessment method (CRM) scheme, where at each stage of the trial, we search for the dose of one agent given the current dose of the other agent. The performance of the design is studied by evaluating its operating characteristics when the underlying model is either correctly specified or misspecified. We show that this method can be extended to accommodate discrete dose combinations.     

Curve-free and model-based continual reassessment method designs

Gasparini and Eisele (2000, Biometrics 56, 609 615) present a development of the continual reassessment method of O'Quigley, Pepe, and Fisher (1990, Biometrics 46, 33-48). They call their development a curve-free method for Phase I clinical trials. However, unless we are dealing with informative prior information, then the curve-free method coincides with the usual model-based continual reassessment method. Both methods are subject to arbitrary specification parameters, and we provide some discussion on this. Whatever choices are made for one method, there exists equivalent choices for the other method, where " equivalent" means that the operating characteristics (sequential dose allocation and final recommendation) are the same. The insightful development of Gasparini and Eisele provides clarification on some of the basic ideas behind the continual reassessment method, particularly when viewed from a Bayesian perspective. But their development does not lead to a new class of designs and the comparative results in their article, indicating some preference for curve-free designs over model-based designs, are simply reflecting a more fortunate choice of arbitrary specification parameters. Other choices could equally well have inversed their conclusion. A correct conclusion should be one of operational equivalence. The story is different for the case of informative priors, a situation that is inherently much more difficult. We discuss this. We also mention the important idea of two-stage designs (Moller, 1995, Statistics in Medicine 14, 911-922; O'Quigley and Shen, 1996, Biometrics 52, 163-174), arguing, via a simple comparison with the results of Gasparini and Eisele (2000), that there is room for notable gains here. Two-stage designs also have an advantage of avoiding the issue of prior specification altogether.     

Continual reassessment method for partial ordering

Summary 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.     

The continual reassessment method for multiple toxicity grades: a Bayesian quasi-likelihood approach

We consider the case of phase I trials for treatment of cancer or other severe diseases in which grade information is available about the severity of toxicity. Most dose allocation procedures dichotomize toxicity grades based on being dose limiting, which may not work well for severe and possibly irreversible toxicities such as renal, liver, and neurological toxicities, or toxicities with long duration. We propose a simple extension to the continual reassessment method (CRM), called the Quasi-CRM, to incorporate grade information. Toxicity grades are first converted to numeric scores that reflect their impacts on the dose allocation procedure, and then incorporated into the CRM using the quasi-Bernoulli likelihood. A simulation study demonstrates that the Quasi-CRM is superior to the standard CRM and comparable to a univariate version of the Bekele and Thall method (2004, Journal of the American Statistical Association 99, 26-35). We also present sensitivity analysis of the new method with respect to toxicity scores, and discuss practical issues such as extending the simple algorithmic up-and-down designs.     

Continual reassessment designs with early termination

The continual reassessment method (CRM) is an increasingly popular approach for estimating the maximum tolerated dose (MTD) in phase I dose finding studies. In its original formulation, the scheme is based on a fixed sample size. Many experimenters feel that, whenever possible, it may be advantageous to bring these trials to an early halt and thus reduce average sample size required to complete the study. To address this issue a stopping rule has been proposed (O'Quigley and Reiner, 1998) based on the idea that continuing the study would not lead to a change in recommendation with high probability. The rule, based on precise probabilistic calculation, is quite involved and not straightforward to implement. A much simpler rule can be constructed based on the idea of having settled at some level. In this work we investigate more deeply the essential ingredients behind these rules and consider more closely their operating characteristics.     

Sequential designs for phase I clinical trials with late-onset toxicities

Traditional designs for phase I clinical trials require each patient (or small group of patients) to be completely followed before the next patient or group is assigned. In situations such as when evaluating late-onset effects of radiation or toxicities from chemopreventive agents, this may result in trials of impractically long duration. We propose a new method, called the time-to-event continual reassessment method (TITE-CRM), that allows patients to be entered in a staggered fashion. It is an extension of the continual reassessment method (CRM; O'Quigley, Pepe, and Fisher, 1990, Biometrics 46, 33-48). We also note that this time-to-toxicity approach can be applied to extend other designs for studies of short-term toxicities. We prove that the recommended dose given by the TITE-CRM converges to the correct level under certain conditions. A simulation study shows our method's accuracy and safety are comparable with CRM's while the former takes a much shorter trial duration: a trial that would take up to 12 years to complete by the CRM could be reduced to 2-4 years by our method.     

Dose-Finding Based on Bivariate Efficacy-Toxicity Outcome Using Archimedean Copula

In dose-finding clinical study, it is common that multiple endpoints are of interest. For instance, efficacy and toxicity endpoints are both primary in clinical trials. In this article, we propose a joint model for correlated efficacy-toxicity outcome constructed with Archimedean Copula, and extend the continual reassessment method (CRM) to a bivariate trial design in which the optimal dose for phase III is based on both efficacy and toxicity. Specially, considering numerous cases that continuous and discrete outcomes are observed in drug study, we will extend our joint model to mixed correlated outcomes. We demonstrate through simulations that our algorithm based on Archimedean Copula model has excellent operating characteristics.     

Statistical properties of the traditional algorithm-based designs for phase I cancer clinical trials

Although there are several new designs for phase I cancer clinical trials including the continual reassessment method and accelerated titration design, the traditional algorithm-based designs, like the '3 + 3' design, are still widely used because of their practical simplicity. In this paper, we study some key statistical properties of the traditional algorithm-based designs in a general framework and derive the exact formulae for the corresponding statistical quantities. These quantities are important for the investigator to gain insights regarding the design of the trial, and are (i) the probability of a dose being chosen as the maximum tolerated dose (MTD); (ii) the expected number of patients treated at each dose level; (iii) target toxicity level (i.e. the expected dose-limiting toxicity (DLT) incidences at the MTD); (iv) expected DLT incidences at each dose level and (v) expected overall DLT incidences in the trial. Real examples of clinical trials are given, and a computer program to do the calculation can be found at the authors' website approximately linyo" locator-type="url">http://www2.umdnj.edu/ approximately linyo.     

On the use of nonparametric curves in phase I trials with low toxicity tolerance

Gasparini and Eisele (2000, Biometrics 56, 609-615) propose a design for phase I clinical trials during which dose allocation is governed by a Bayesian nonparametric estimate of the dose-response curve. The authors also suggest an elicitation algorithm to establish vague priors. However, in situations where a low percentile is targeted, priors thus obtained can lead to undesirable rigidity given certain trial outcomes that can occur with a nonnegligible probability. Interestingly, improvement can be achieved by prescribing slightly more informative priors. Some guidelines for prior elicitation are established using a connection between this curve-free method and the continual reassessment method.     

Time‐to‐event continual reassessment method incorporating treatment cycle information with application to an oncology phase I trial

Delayed dose limiting toxicities (i.e. beyond first cycle of treatment) is a challenge for phase I trials. The time‐to‐event continual reassessment method (TITE‐CRM) is a Bayesian dose‐finding design to address the issue of long observation time and early patient drop‐out. It uses a weighted binomial likelihood with weights assigned to observations by the unknown time‐to‐toxicity distribution, and is open to accrual continually. To avoid dosing at overly toxic levels while retaining accuracy and efficiency for DLT evaluation that involves multiple cycles, we propose an adaptive weight function by incorporating cyclical data of the experimental treatment with parameters updated continually. This provides a reasonable estimate for the time‐to‐toxicity distribution by accounting for inter‐cycle variability and maintains the statistical properties of consistency and coherence. A case study of a First‐in‐Human trial in cancer for an experimental biologic is presented using the proposed design. Design calibrations for the clinical and statistical parameters are conducted to ensure good operating characteristics. Simulation results show that the proposed TITE‐CRM design with adaptive weight function yields significantly shorter trial duration, does not expose patients to additional risk, is competitive against the existing weighting methods, and possesses some desirable properties.     

Efficiency of New Dose Escalation Designs in Dose-Finding Phase I Trials of Molecularly Targeted Agents

Background

Statistical simulations have consistently demonstrated that new dose-escalation designs such as accelerated titration design (ATD) and continual reassessment method (CRM)-type designs outperform the standard “3+3” design in phase I cancer clinical trials.

Methods

We evaluated the actual efficiency of different dose escalation methods employed in first-in-human phase I clinical trials of targeted agents administered as single agents published over the last decade.

Results

Forty-nine per cent of the 84 retrieved trials used the standard “3+3” design. Newer designs used included ATD in 42%, modified CRM [mCRM] in 7%, and pharmacologically guided dose escalation in 1%. The median numbers of dose levels explored in trials using “3+3”, ATD and mCRM designs were 6, 8 and 10, respectively. More strikingly, the mean MTD to starting dose ratio appeared to be at least twice as high for trials using mCRM or ATD designs as for trials using a standard “3+3” design. Despite this, the mean number of patients exposed to a dose below the MTD was similar in trials using “3+3”, ATD and mCRM designs.

Conclusion

Our results support a more extensive implementation of innovative dose escalation designs such as mCRM and ATD in phase I cancer clinical trials of molecularly targeted agents.     

Bayesian optimal designs for Phase I clinical trials

A broad approach to the design of Phase I clinical trials for the efficient estimation of the maximum tolerated dose is presented. The method is rooted in formal optimal design theory and involves the construction of constrained Bayesian c- and D-optimal designs. The imposed constraint incorporates the optimal design points and their weights and ensures that the probability that an administered dose exceeds the maximum acceptable dose is low. Results relating to these constrained designs for log doses on the real line are described and the associated equivalence theorem is given. The ideas are extended to more practical situations, specifically to those involving discrete doses. In particular, a Bayesian sequential optimal design scheme comprising a pilot study on a small number of patients followed by the allocation of patients to doses one at a time is developed and its properties explored by simulation.     

Dose finding for continuous and ordinal outcomes with a monotone objective function: a unified approach

Summary .  In many phase I trials, the design goal is to find the dose associated with a certain target toxicity rate. In some trials, the goal can be to find the dose with a certain weighted sum of rates of various toxicity grades. For others, the goal is to find the dose with a certain mean value of a continuous response. In this article, we describe a dose-finding design that can be used in any of the dose-finding trials described above, trials where the target dose is defined as the dose at which a certain monotone function of the dose is a prespecified value. At each step of the proposed design, the normalized difference between the current dose and the target is computed. If that difference is close to zero, the dose is repeated. Otherwise, the dose is increased or decreased, depending on the sign of the difference.     

Estimating the probability of toxicity at the recommended dose following a phase I clinical trial in cancer.

The problem of point and interval estimation following a Phase I trial, carried out according to the scheme outlined by O'Quigley, Pepe, and Fisher (1990, Biometrics 46, 33-48), is investigated. A reparametrization of the model suggested in this earlier work can be seen to be advantageous in some circumstances. Maximum likelihood estimators, Bayesian estimators, and one-step estimators are considered. The continual reassessment method imposes restrictions on the sample space such that it is not possible for confidence intervals to achieve exact coverage properties, however large a sample is taken. Nonetheless, our simulations, based on a small finite sample of 20, not atypical in studies of this type, indicate that the calculated intervals are useful in most practical cases and achieve coverage very close to nominal levels in a very wide range of situations. The relative merits of the different estimators and their associated confidence intervals, viewed from a frequentist perspective, are discussed.     

Sequential designs for ordinal phase I clinical trials

Sequential designs for phase I clinical trials which incorporate maximum likelihood estimates (MLE) as data accrue are inherently problematic because of limited data for estimation early on. We address this problem for small phase I clinical trials with ordinal responses. In particular, we explore the problem of the nonexistence of the MLE of the logistic parameters under a proportional odds model with one predictor. We incorporate the probability of an undetermined MLE as a restriction, as well as ethical considerations, into a proposed sequential optimal approach, which consists of a start‐up design, a follow‐on design and a sequential dose‐finding design. Comparisons with nonparametric sequential designs are also performed based on simulation studies with parameters drawn from a real data set.     

A statistical framework for quantile equivalence clinical trials with application to pharmacokinetic studies that bridge from HIV-infected adults to children

SUMMARY: Bridging clinical trials are sometimes designed to evaluate whether a proposed dose for use in one population, for example, children, gives similar pharmacokinetic (PK) levels, or has similar effects on a surrogate marker as an established effective dose used in another population, for example, adults. For HIV bridging trials, because of the increased risk of viral resistance to drugs at low PK levels, the goal is often to determine whether the doses used in different populations result in similar percentages of patients with low PK levels. For example, it may be desired to evaluate that a proposed pediatric dose gives approximately 10% of children with PK levels below the 10th percentile of PK levels for the established adult dose. However, the 10th percentile for the adult dose is often imprecisely estimated in studies of relatively small size. Little attention has been given to the statistical framework for such bridging studies. In this article, a formal framework for the design and analysis of quantile-based bridging studies is proposed. The methodology is then developed for normally distributed outcome measures from both frequentist and Bayesian directions. Sample size and other design considerations are discussed.     

Continual reassessment method for ordered groups

We investigate the two-group continual reassessment method for a dose-finding study in which we anticipate some ordering between the groups. This is a situation in which, for either group, we have little or almost no knowledge about which of the available dose levels will correspond to the maximum tolerated dose (MTD), but we may have quite strong knowledge concerning which of the two groups will have the higher level of MTD, if indeed they do not have the same MTD. The motivation for studying this problem came from an investigation into a new therapy for acute leukemia in children. The background to this study is discussed. There were two groups of patients: one group already received heavy prior therapy while the second group had received relatively much lighter prior therapy. It was therefore anticipated that the second group would have an MTD higher or at least as high as the first. Generally, likelihood methods or, equivalently, the use of noninformative Bayes priors, can be used to model the main aspects of the study, i.e., the MTD for one of the groups, reserving more informative Bayes modeling to be applied to the secondary features of the study. These secondary features may simply be the direction of the difference between the MTD levels for the two groups or, possibly, information on the potential gap between the two MTDs.     

Generalized Verlet Algorithm for Efficient Molecular Dynamics Simulations with Long-range Interactions

For the purpose of molecular dynamics simulations of large biopolymers we have developed a new method to accelerate the calculation of long-range pair interactions (e.g. Coulomb interaction). The algorithm introduces distance classes to schedule updates of non-bonding interactions and to avoid unnecessary computations of interactions between particles which are far apart. To minimize the error caused by the updating schedule, the Verlet integration scheme has been modified. The results of the method are compared to those of other approximation schemes as well as to results obtained by numerical integration without approximation. For simulation of a protein with 12 637 atoms our approximation scheme yields a reduction of computer time by a factor of seven. The approximation suggested can be implemented on sequential as well as on parallel computers. We describe an implementation on a (Transputer-based) MIMD machine with a systolic ring architecture.