Escalation with Overdose Control Using Ordinal Toxicity Grades for Cancer Phase I Clinical Trials

Abstract

We extend a Bayesian adaptive phase I clinical trial design known as escalation with overdose control (EWOC) by introducing an intermediate grade 2 toxicity when assessing dose-limiting toxicity (DLT). Under the proportional odds model assumption of dose-toxicity relationship, we prove that in the absence of DLT, the dose allocated to the next patient given that the previously treated patient had a maximum of grade 2 toxicity is lower than the dose given to the next patient had the previously treated patient exhibited a grade 0 or 1 toxicity at the most. Further, we prove that the coherence properties of EWOC are preserved. Simulation results show that the safety of the trial is not compromised and the efficiency of the estimate of the maximum tolerated dose (MTD) is maintained relative to EWOC treating DLT as a binary outcome and that fewer patients are overdosed using this design when the true MTD is close to the minimum dose.

1. Introduction

Cancer phase I clinical trials are sequential designs enrolling late stage cancer patients who have exhausted standard treatment therapies [1]. For cytotoxic agents or combinations of biologic with cytotoxic drugs, the main objectives of these trials are to characterize treatment-related toxicities and estimate a dose level that is associated with a predetermined level of dose limiting toxicity (DLT). Such a dose is called maximum tolerated dose (MTD) or phase II dose. Specifically, the MTD, 𝛾, is defined as the dose that is expected to produce DLT after one cycle of therapy in a specified proportion 𝜃 of patients: 𝑃DLT∣Dose=𝛾=𝜃. (1.1) Although the definition of DLT depends on the cancer type and the agent under study, it is typically defined as a grade 3 or 4 nonhematologic and grade 4 hematologic toxicity for cytotoxic agents, see the National Cancer Institute (NCI) common toxicity criteria [2]. The value chosen for the target probability of DLT 𝜃 depends on the nature and severity of the DLT; it is set relatively high when the DLT is a transient, reversible, and nonfatal condition and low when it is lethal or life threatening.

Model-based designs for cancer phase I clinical trials have been studied extensively in the last two decades; see O’Quigley et al. [3], Gatsonis and Greenhouse [4], Durham and Flournoy [5], Korn et al. [6], Whitehead and Brunier [7], Whitehead [8], Babb et al. [9], Gasparini and Eisele [10], Mukhopadhyay [11], and Haines et al. [12]. In particular, the continual reassessment method (CRM) first proposed by O’Quigley et al. [3] is Bayesian outcome adaptive, and has been modified and extended by many authors; see, for example, Faries [13], Moller [14], Goodman et al. [15], O’Quigley and Shen [16], Piantadosi et al. [17], Cheung and Chappell [18], Storer [19], and Leung and Wang [20]. The attribute of the CRM is that at each stage of the trial, we seek a dose for which a Bayes estimate of the probability of DLT is closest to the target probability of DLT 𝜃. Escalation with overdose control (EWOC) originally proposed by Babb et al. [9] is another alternative Bayesian outcome adaptive design for dose finding in early phase cancer trials. Unlike CRM, its main feature is that at each stage of the trial, we seek a dose for which the posterior probability of exceeding the MTD 𝛾 is bounded by a feasibility bound 𝛼. Statistical properties of EWOC have been studied by Zacks et al. [21], Tighiouart et al. [22], and Tighiouart and Rogatko [23], and examples of phase I trials using EWOC can be found in [24–33]. Further literature review of statistical methods for dose finding in cancer phase I trials can be found in Ting [34] and Chevret [35].

The above methods allocate future doses based on a binary outcome of DLT of previously treated patients. Such designs may not be efficient in the sense that the dose recommended for the next patient is the same regardless whether the previously treated patient had no toxicity or had an intermediate grade 2 toxicity. The work we present in this manuscript is motivated by the ethical concern raised by clinical colleagues regarding dose escalation in the absence of DLT. Specifically, if the current patient experiences drug related grade 2 toxicity at the most, then the dose to be allocated to the next patient should not be as high as the one had the current patient experienced a maximum of grade 0 or 1 toxicity. We present a Bayesian outcome adaptive design which is an extension of EWOC by accommodating an intermediate grade 2 toxicity to the model. We use a proportional odds model to describe the dose-toxicities relationship and the design is termed EWOC proportional odds model, written as EWOC-POM. We show that the design satisfies the above ethical consideration without compromising the safety and efficiency of the trial. Furthermore, we show that the design maintains the coherence properties of EWOC.

Cancer phase I clinical trials designs taking into account all grades and types of toxicities have been proposed by many authors with the goal of improving the safety and efficiency of the trial, see Gordon and Willson [36], Wang et al. [37], Bekele and Thall [38], Yuan et al. [39], Potthoff and George [40], Bekele et al. [41], Van Meter et al. [42, 43], Iasonos et al. [44], Lee et al. [45], and Chen et al. [46]. Some of these methods use multivariate models for characterizing the different grades of toxicities as a function of dose while others summarize the information from all toxicity grades into a continuous score. Depending on the underlying scenarios, in general, adding multiple toxicity grades information to the model has little effect on the safety of the trial with a modest gain in the precision of the estimate of the MTD. Our contribution in this manuscript is motivated by the ethical constraint that dose escalation in the absence of DLT should be mitigated by the occurrence of an intermediate grade 2 toxicity. The model and design we propose is constructed in such a way as to maintain the main properties of EWOC while at the same time, satisfying those ethical considerations raised by our clinical colleagues.

The manuscript is organized as follows. In Section 2, we give a detailed description of the methodology and describe the trial design. In Section 3, we state and prove the ethical considerations and coherence properties of EWOC-POM. The simulation results of the design operating characteristics and comparison with EWOC design are included in Section 4. Section 5 contains some final remarks and discussion of practical implementations.

2. Method

2.1. Model

Let 𝐺=0,1,…,4be the maximum grade of toxicity experienced by a patient by the end of one cycle of therapy and define DLT as a maximum of grade 3 or 4 toxicity. Let 𝑌=0if𝐺=0or11if𝐺=22if𝐺=3or4. (2.1) We model the dose-toxicities relationship by assuming that 𝑃𝑌≥𝑗∣𝑥=𝐹𝛼_𝑗+𝛽𝑥,𝑗=1,2, (2.2) where 𝐹(⋅) is a known strictly increasing cdf. This implies that 𝛼₂≤𝛼₁. We assume that 𝛽>0 so that the probability of DLT is an increasing function of dose. The MTD, 𝛾, is defined as the dose that is expected to produce DLT in a specified proportion 𝜃 of patients: 𝑃𝑌=2∣𝑥=𝛾=𝐹𝛼₂+𝛽𝛾=𝜃. (2.3) The value chosen for the target probability 𝜃 depends on the nature and clinical manageability of the DLT; it is set relatively high when the DLT is a transient, correctable, or nonfatal condition and low when it is lethal or life threatening. Suppose that dose levels in the trial are selected in the interval [𝑋_min,𝑋_max].

2.1.1. Likelihood

Let 𝐷_𝑛={(𝑥_𝑖,𝑌_𝑖),𝑖=1,…,𝑛} be the data after enrolling 𝑛 patients to the trial. The likelihood function for the parameters 𝛼₁, 𝛼₂, and 𝛽 is 𝐿𝛼₁,𝛼₂,𝛽∣𝐷_𝑛=^𝑛_𝑖=11−𝐹𝛼₁+𝛽𝑥_𝑖^{𝐼(𝑌𝑖=0)}𝐹𝛼₁+𝛽𝑥_𝑖−𝐹𝛼₂+𝛽𝑥_𝑖^{𝐼(𝑌𝑖=1)}×𝐹(𝛼₂+𝛽𝑥_𝑖)^{𝐼(𝑌𝑖=2)}, (2.4) where 𝐼(⋅) is the indicator function.

We reparameterize model (2.2) in terms of 𝜌₀=𝑃(𝑌=2∣𝑥=𝑋_min), the probability that a DLT manifests within the first cycle of therapy for a patient given dose 𝑥=𝑋_min,  𝜌₁=𝑃(𝑌≥1∣𝑥=𝑋_min), the probability that a grade 2 or more toxicity manifests within the first cycle of therapy for a patient given dose 𝑥=𝑋_min, and the MTD 𝛾. This reparameterization is convenient to clinicians since 𝛾 is the parameter of interest. Assuming that the dose is standardized to be in the interval [0,1], it can be shown that 𝛼₁=𝐹⁻¹𝜌₁,𝛼₂=𝐹⁻¹𝜌₀,1𝛽=𝛾𝐹⁻¹(𝜃)−𝐹⁻¹𝜌₀. (2.5) The conditions 𝛼₂≤𝛼₁, 𝛽>0, and (2.2) imply that 0≤𝜌₀≤𝜌₁ and 0≤𝜌₀≤𝜃. Define 𝐹₁𝜌₀,𝜌₁,𝛾;𝑥=𝐹𝐹⁻¹𝜌₁+𝐹⁻¹(𝜃)−𝐹⁻¹𝜌₀𝑥𝛾𝐹₂𝜌₀,𝜌₁,𝛾;𝑥=𝐹𝐹⁻¹𝜌₀+𝐹⁻¹(𝜃)−𝐹⁻¹𝜌₀𝑥𝛾. (2.6) Using (2.4), (2.5), and (2.6), the likelihood of the reparameterized model is 𝐿𝜌₀,𝜌₁,𝛾∣𝐷_𝑛=^𝑛_𝑖=11−𝐹₁𝜌₀,𝜌₁,𝛾;𝑥_𝑖^{𝐼(𝑌𝑖=0)}𝐹₁𝜌₀,𝜌₁,𝛾;𝑥_𝑖−𝐹₂𝜌₀,𝜌₁,𝛾;𝑥_𝑖^{𝐼(𝑌𝑖=1)}×𝐹₂𝜌₀,𝜌₁,𝛾;𝑥_𝑖^{𝐼(𝑌𝑖=2)}. (2.7)

2.1.2. Prior and Posterior Distributions

Let 𝑔(𝜌₀,𝜌₁,𝛾) be the prior distribution on Ω, where Ω={(𝑥,𝑦,𝑧)∶0≤𝑥≤𝜃,𝑥≤𝑦≤1,𝑋_min≤𝑧≤𝑋_max}. Using Bayes rule, the posterior distribution of the model parameters is proportional to the product of the likelihood and prior distribution 𝜋𝜌₀,𝜌₁,𝑦∣𝐷_𝑛∝𝐿𝜌₀,𝜌₁,𝛾∣𝐷_𝑛×𝑔𝜌₀,𝜌₁,𝛾. (2.8) We designed an MCMC sampler based on the Metropolis-Hastings algorithm [47, 48] to obtain model operating characteristics. We also used WinBUGS [49] to estimate features of the posterior distribution of the MTD and design a trial. The WinBUGS code is included in the Appendix section. In the absence of prior information about the MTD and probability of DLT at 𝑋_min, we specify vague priors for the model parameters as follows: 𝛾∼Unif𝑋_min,𝑋_max𝜌₀[]𝜌∼Unif0,𝜃₁∣𝜌₀∼Unif𝜌₀,1. (2.9)

2.1.3. Trial Design

Dose levels in the trial are selected in the interval [𝑋_min,𝑋_max]. The adaptive design proceeds as follows. The first patient receives a dose 𝑥₁>𝑋_min that is deemed to be safe by the clinician. Denote by Π𝑘(𝛾)=Π(𝛾∣𝐷_𝑘) the marginal posterior cdf of the MTD, 𝑘=1,…,𝑛−1. The (𝑘+1)−𝑠𝑡 patient receives the dose 𝑥_𝑘+1=Π_𝑘⁻¹(𝛼) so that the posterior probability of exceeding the MTD is equal to the feasibility bound 𝛼. This is the overdose protection property of EWOC, where at each stage of the design, we seek a dose to allocate to the next patient while controlling the posterior probability of exposing patients to toxic dose levels. The trial proceeds until a predetermined number of 𝑛 patients are enrolled to the trial. At the end of the trial, we estimate the MTD as 𝛾=Π_𝑛⁻¹(𝛼).

3. Properties of EWOC-POM

3.1. Characteristics of EWOC-POM

The proposed design EWOC-POM assigns dose levels to future patients by taking into account the maximum observed toxicity grade during the first cycle of therapy according to the following properties.(i) At each stage of the design, we seek a dose to allocate to the next patient while controlling the posterior probability of exposing patients to toxic dose levels.(ii) If the maximum grade of toxicity experienced by patient 𝑘−1 within one cycle of therapy is grade 2, then the dose allocated to patient 𝑘 is lower than the dose that would have been given to patient 𝑘 had the maximum grade of toxicity experienced by patient 𝑘−1 been grade 0 or 1.Property (i) is the overdose protection defining characteristic of EWOC which is also satisfied by EWOC-POM. Property (ii) is naturally appealing because when a patient experiences grade 2 dose-related toxicity, then it is ethical not to increase the dose by the same amount had this patient exhibited grade 0 or 1 toxicity at the most. Characteristic (ii) is summarized in the following theorem.

Theorem 3.1. Let 𝐷_𝑘={(𝑌₁,𝑥₁),…,(𝑌_𝑘,𝑥_𝑘)} be the data on the first k patients generated by the design described in Section 2.1.3, and, Π_𝑘(𝛾;𝑌_𝑘) be the cdf of γ given the data 𝐷_𝑘. Let 𝑥_𝑘+1=Π_𝑘⁻¹(𝛼;𝑌_𝑘) and 𝑥^′_𝑘+1=Π_𝑘⁻¹(𝛼;𝑌^′_𝑘). Suppose that for all 𝑥∈[𝑋_min,𝑋_max] and all (𝜌₀,𝜌₁) such that 0≤𝜌₀≤𝜌₁≤1 and 𝜌₀≤𝜃, (𝐹₁−𝐹₂)/(1−𝐹₁) is a monotonically decreasing function in 𝛾. Then, 𝑥^′_𝑘+1≥𝑥_𝑘+1 whenever 𝑌_𝑘^′=0 and 𝑌_𝑘=1.

The proof of Theorem 3.1 is given in the Appendix section. It is easy to verify that the monotonicity condition of Theorem 3.1 holds for the logistic function 𝐹(𝑤)=1/(1+𝑒^−𝑤). Using this link function and the uniform priors given in (2.9) with 𝜃=0.33, Figure 1 gives all possible dose assignments for patients 1 and 2 and selected situations for patients 3 and 4 using the trial design described in Section 2.1.3. The dose has been standardized so that 𝑋_min=0and  𝑋_max=1, and the first patient is given dose 0.10. If this patient experiences grade 0 or 1 toxicity at the most, the dose recommended for patient 2 is 0.36. On the other hand, if patient 1 experiences grade 2 toxicity at the most, the dose recommended for patient 2 is 0.22, much lower than 0.36. This figure also illustrates some of the coherence properties stated in Theorem 3.2 below.

Figure 1 

Tree of possible dose allocations for patients 1 and 2 and selected situations for patients 3 and 4. 𝐺=0,1 corresponds to 𝑌=0, 𝐺=2 corresponds to 𝑌=1, and 𝐺=3,4 corresponds to 𝑌=2 or DLT.

3.2. Coherence of EWOC-POM

In cancer phase I clinical trials, it is ethical not to increase a dose of a cytotoxic agent for the next patient if the previously treated patient exhibited DLT when given the same dose level. Furthermore, it is desirable not to decrease the dose of an agent for the next patient if the previously treated patient did not experience DLT when given that same dose level. These two properties are known as coherence in escalation and de-escalation, respectively, see Cheung [50]. A design that satisfies both of these properties is said to be coherent. Tighiouart and Rogatko [23] show that EWOC is coherent. The next theorem states some of the coherence properties of EWOC-POM.

Theorem 3.2. Suppose that for all 𝑥∈[𝑋_min,𝑋_max] and all (𝜌₀,𝜌₁) such that 0≤𝜌₀≤𝜌₁≤1 and 𝜌₀≤𝜃, 𝐹₁ and 𝐹₂ are monotonically decreasing in 𝛾. Then the design EWOC-POM described in 2.1.3 is coherent in deescalation. Furthermore, if the toxicity response for patient 𝑘 is 𝑌_𝑘=0, then the dose allocated to patient 𝑘+1 satisfies 𝑥_𝑘+1≥𝑥_𝑘.

The proof of Theorem 3.2 is given in the Appendix section.

4. Simulation Studies

We compare the design operating characteristics of EWOC-POM with the original EWOC by simulating a large number of trials under several scenarios. We used the logistic function 𝐹(𝑤)=1/(1+𝑒^−𝑤) to model the dose-toxicities relationship in (2.2). EWOC was implemented as in Tighiouart et al. [22] using the same logistic function to model the dose-toxicity relationship. For all scenarios, we standardize the dose to be in the interval [0,1], 𝜃=0.33, the feasibility bound 𝛼=0.25, and the trial sample size is 𝑛=30. The priors in (2.9) were adopted for EWOC-POM. We also carried out design operating characteristics for 𝜃=0.20, and the conclusions regarding the performance of EWOC-POM relative to EWOC were essentially the same.

4.1. Algorithm

For a given scenario determined by 𝜌₀, 𝜌₁, and 𝛾, the first patient receives dose 0, and the next dose 𝑥₂ is determined according to the trial design described in 2.1.3. The second response 𝑦₂ is then generated from model (2.2) reparameterized in terms of 𝜌₀, 𝜌₁, and 𝛾 with 𝑥=𝑥₂. This process is repeated until all 𝑛 patients have been enrolled to the trial. We considered 9 scenarios corresponding to a fixed value for 𝜌₀=0.05, three values of 𝜌₁, 0.2, 0.5, and 0.8, and three values of the MTD 𝛾, 0.1, 0.5, and 0.7. The corresponding dose-toxicity relationships for these nine scenarios are illustrated in Figure 2. For each model and each scenario, we simulated 𝑀=1000 trials. EWOC and EWOC-POM were compared in terms of the proportion of patients exhibiting DLT, the average bias, bias_ave=𝑀⁻¹(∑^𝑀_𝑖=1𝛾_𝑖−𝛾_true), and the estimated mean square error MSE=𝑀⁻¹(∑^𝑀_𝑖=1(𝛾_𝑖−𝛾_true)²), where 𝛾_𝑖is the Bayes estimate of the posterior distribution of the MTD at the end of the 𝑖th trial with respect to the asymmetric loss function: 𝐿=𝛼(𝑥,𝛾𝛾−𝑥if𝑥≤𝛾1−𝛼)𝑥−𝛾if𝑥>𝛾. (4.1) We also compared the models with respect to the proportion of patients that were overdosed. Here, a dose 𝑥 is defined as an overdose if 𝑥>𝑥^∗, where 𝑥^∗ is defined as the dose for which 𝑃(DLT∣𝑥^∗)=𝜃+0.05. This probability is calculated using the parameter values from the corresponding scenario. These models are further compared with respect to the proportion of patients that were overdosed given that the previously treated patient exhibited grade 2 toxicity. Finally, we compared EWOC-POM to EWOC in terms of the proportion of trials for which the probability of DLT exceeds 0.4. This gives us an estimate of the probability that a prospective trial will result in an excessively high DLT rate. As for the proportion of trials with correct MTD recommendation, we presented percent of trials with estimated MTD within 10% and 20% of the dose range of the true MTD for EWOC-POM and EWOC.

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Figure 2 

Dose-toxicity relationship for the nine scenarios considered in the design operations characteristics. The solid curves correspond to 𝑃(𝑌=2∣𝑥)=𝑃(𝐷𝐿𝑇∣𝑥) and the dashed curves in bold correspond to 𝑃(𝑌>1∣𝑥). The horizontal dashed lines represent the target probability of DLT 𝜃=0.33 and the vertical lines correspond to the true values of the MTD 𝛾.

4.2. Results

Figure 3 shows that the proportion of patients exhibiting DLT is always less than 34% for both EWOC and EWOC-POM under all scenarios and 4% fewer patients experiencing DLT under EWOC-POM when the MTD is small (𝛾=0.1) and 𝜌₁=0.8. The same figure shows that the proportion of patients that are overdosed using EWOC is uniformly higher relative to EWOC-POM when the MTD is small. The same trend is observed when 𝛾=0.5 except when 𝜌₁=0.2. The difference in the proportion of patients being overdosed when 𝛾=0.7 is negligible. The last panel of Figure 3 shows that the proportion of patients that are overdosed given that the previously treated patient exhibited grade 2 toxicity using EWOC is uniformly higher relative to EWOC-POM when 𝛾=0.1, 0.5 except when 𝜌₁=0.2. The difference in these proportions when 𝛾=0.7 is negligible. The last two columns of Table 1 show that the percent of trials with DLT rate of 0.4 or more is 7.5% at the most for EWOC and 6.6% for EWOC-POM. A more detailed comparison is shown in Figure 4, where side-by-side box plots of the distributions of the proportion of DLTs for EWOC-POM and EWOC under the nine scenarios are displayed. These results show that EWOC-POM maintains the safety of the trial relative to EWOC and is much safer when the true MTD is close to the minimum dose by reducing the number of patients that are exposed to toxic doses.

Table 1 

Percent of trials with estimated MTD within 5% of the dose range and 10% of the dose range of the true MTD 𝛾 and percent of trials for which the rate of DLT exceeds 40% for EWOC and EWOC-POM under the nine scenarios.

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Figure 3 

Summary statistics for trial safety for EWOC and EWOC-POM under all scenarios. Each graph represents mean proportion obtained from all patients from all 1000 simulated trials.

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Figure 4 

Box plots for the proportion of DLTs for EWOC-POM and EWOC under the nine scenarios. Each box plot was constructed from the DLT rates of the 1000 simulated trials. The dashed horizontal line corresponds to the target probability of DLT 𝜃=0.33.

Figure 5 shows that the estimated MTDs using EWOC and EWOC-POM are very close in general, with the highest difference observed when 𝜌₁=0.8. This is reflected by the estimated bias and RMSE shown in Figure 5. This is expected since EWOC-POM is characterized by a conservative dose escalation when a patient experiences grade 2 toxicity. The highest absolute value of the bias is 0.04 and is achieved when 𝛾=0.5, 0.7 and 𝜌₁=0.8. This constitutes 4% of the range of the dose and is practically not significant. The percent of trials with estimated MTD within 5% of the dose range and 10% of the dose range of the true MTD 𝛾 under the nine scenarios are shown in columns 2–5 of Table 1. These results further confirm that the precision of the estimate of the MTD is similar between the two models, with a higher precision for EWOC achieved when 𝛾=0.5 and 𝜌₁=0.8. Values other than 5% and 10% of the dose range were also used, and the conclusions were essentially the same. We conclude that EWOC-POM maintains the efficiency of the trial relative to EWOC for all practical purpose.

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Figure 5 

Summary statistics for trial efficiency for EWOC and EWOC-POM under all scenarios. Each graph represents a mean value obtained from all patients from all 1000 trials.

These simulation results suggest that EWOC-POM is a good alternative design for cancer phase I clinical trials which takes into account the ethical consideration that dose escalation in the absence of DLT is mitigated by the occurrence of a grade 2 toxicity.

4.3. Model Robustness

Model robustness with respect to assumption of proportional odds model in (2.2) was assessed by simulating the toxicity responses from a nonproportional odds model: 𝑃𝑌≥𝑗∣𝑥=𝐹𝛼_𝑗+𝛽_𝑗𝑥,𝑗=1,2. (4.2) The same logistic link function 𝐹(𝑤)=1/(1+𝑒^−𝑤) was used. We considered two different models M₁ and M₂ corresponding to the set of parameters 𝛼₂=−3.94, 𝛽₁=22.36, 𝛽₂=32.36 for model M₁ and 𝛼₂=−1.94, 𝛽₁=22.36, 𝛽₂=12.36 for M₂. For each model 𝑀_𝑖, 𝑖=1,2, we considered three different values for the intercept term 𝛼₁,  𝛼₁=−1.38,0.00, 1.38 which correspond to 𝜌₁=0.2,0.5,0.8. These parameters have been selected so that 𝜌₀=0.020 for model M₁, 𝜌₀=0.126 for model M₂, and the true MTD is 𝛾=0.1. Figure 6 shows the graphs of the probabilities of DLT and probability of grade 2 or more toxicity as a function of dose for the proportional odds model POM and models M₁ and M₂ when 𝜌₁=0.2. For each scenario, we simulated 1000 trials with 𝑛=30 patients where at each stage of the trial, the next dose is calculated using the trial design described in 2.1.3 as in Section 4.1 but the toxicity response is generated using the nonproportional odds model (4.2). The simulation results are summarized in Table 2. The maximum difference in proportion of patients exhibiting DLT (averaged across the simulated trials) between model 𝑀_𝑖, 𝑖=1,2 and EWOC-POM is 3%. Under model M₂, the proportion of patients that are overdosed is higher than EWOC-POM, and this proportion is 13% higher when 𝜌₁=0.2.

Table 2 

Summary statistics for trial safety and efficiency under model misspecification when the true MTD𝛾=0.1. Rows 2–6 give the summary statistics based on all patients from all 1000 trials.

Figure 6 

Dose-toxicity relationship under the proportional (EWOC-POM) and nonproportional odds models (M1 and M2) when the true MTD 𝛾=0.1. The dashed horizontal line corresponds to the target probability of DLT 𝜃=0.33.

The percent of trials with DLT rate exceeding 0.4 is less than 15% in all cases. This percent is highest for model M₂; however, this is still relatively small compared to the results obtained in [42]. The percent of trials with estimated MTD within 10% of the dose range of the true MTD is 100% between the three models and across all scenarios and very good within 5% of the dose range of the true MTD.

In summary, EWOC-POM seems to be robust to model misspecification when the true MTD is near the initial dose. On the other hand, the model is sensitive to model misspecification when the true MTD is high but the safety of the trial is not compromised. We also conducted similar simulations (results not shown) when the true MTD is 𝛾=0.5,0.7. We found that under all scenarios, the proportion of patients exhibiting DLT is always less than 33% but the bias tends to be higher for high values of 𝜌₁ and 𝛾. This is the case when the probability of DLT curve increases very slowly as a function of dose which results in a very slow dose escalation scheme.

5. Discussion

In this paper, we proposed a Bayesian adaptive design for dose-finding studies in cancer phase I clinical trials. The method addresses the ethical concern regarding dose escalation in the absence of DLT. Specifically, if the current patient experiences drug-related grade 2 toxicity at the most, then it is ethical not to escalate the dose for the next patient by the same amount as the one had the current patient experienced a maximum of grade 0 or 1 toxicity. The method termed EWOC-POM is an extension of EWOC by accommodating an intermediate grade 2 toxicity to the model. We used a proportional odds model to describe the dose-toxicities relationship for simplicity. We proved that if the maximum grade of toxicity experienced by patient 𝑘−1 within one cycle of therapy is grade 2, then the dose allocated to patient 𝑘 is lower than the dose that would have been given to patient 𝑘 had the maximum grade of toxicity experienced by patient 𝑘−1 been grade 0 or 1. Furthermore, we also showed that the coherence properties of EWOC are maintained.

We studied design operating characteristics by simulating a large number of trials under different scenarios of the dose-toxicity relationships. EWOC-POM was compared to EWOC with respect to the primary goals of cancer phase I trials, safety and efficiency of the estimate of the MTD. We found that in general, the safety of the trial is not compromised when we account for an intermediate grade 2 toxicity. In particular, when the unknown MTD is close to the initial dose, a substantial number of patients are overdosed when using EWOC relative to EWOC-POM, and if the current patient experiences grade 2 toxicity, then the next patient is more likely to be overdosed using EWOC compared to EWOC-POM. The loss in efficiency of the estimate of the MTD by introducing an extra parameter to the model is very marginal as was shown by the simulation results of the various scenarios. We also showed that the proportional odds assumption is robust to model misspecification when the true MTD is close to the minimum dose. However, the bias of the estimate of the MTD increases as a function of the MTD under model misspecification. In any case, safety of the trial as assessed by the number of patients exhibiting DLT and number of patients that are overdosed was never compromised.

We have shown that Theorem 3.1 holds under the proportional odds assumption using EWOC scheme with link functions satisfying monotonicity conditions as a function of the MTD. One can easily prove that a similar ethical constraint stated in Theorem 3.1 can be established using the Bayesian CRM originally proposed in O’Quigley et al. [3]. In fact, a more general theorem can be established for a Bayes estimate using a general convex loss function. Extensions of EWOC-POM to accommodate patients’ baseline characteristics ([30]) and time to event DLT for late onset toxicities are being investigated. In conclusion, EWOC-POM is a good alternative design to EWOC if clinicians expect to see a substantial number of grade 2 toxicities induced by the agent or drug combinations under study.

Appendix

Proof of Theorem 3.1. To simplify notation and presentation of the proof, we assume that 𝑋_min=0, 𝜌₀≤𝜌₁ are fixed, and let us drop the nuisance parameters 𝜌₀ and 𝜌₁ from 𝐹₁(𝜌₀,𝜌₁,𝛾;𝑥), and 𝐹₂(𝜌₀,𝜌₁,𝛾;𝑥). Let 𝐿_𝑘(𝛾)=𝐿_𝑘(𝜌₀,𝜌₁,𝛾∣𝐷_𝑘), and 𝜋(𝛾) be a proper prior density for 𝛾. If 𝑌_𝑘=0, 𝐿_𝑘(𝛾)=𝐿_𝑘−1(𝛾)[1−𝐹₁(𝛾;𝑥_𝑘)], and if 𝑌_𝑘=1, 𝐿_𝑘(𝛾)=𝐿_𝑘−1(𝛾)[𝐹₁(𝛾;𝑥_𝑘)−𝐹₂(𝛾;𝑥_𝑘)]. Using Bayes’ rule, _𝑘(𝑡;0)−_𝑘()=∫𝑡;1_𝑡0𝐿_𝑘−1𝛾1−𝐹₁𝛾;𝑥_𝑘𝜋𝛾𝑑𝛾∫¹₀𝐿_𝑘−1𝛾1−𝐹₁𝛾;𝑥_𝑘𝜋𝛾−∫𝑑𝛾^𝑡0𝐿_𝑘−1𝛾𝐹₁𝛾;𝑥_𝑘−𝐹₂𝛾;𝑥_𝑘𝜋𝛾𝑑𝛾∫¹₀𝐿_𝑘−1𝛾𝐹₁𝛾;𝑥_𝑘−𝐹₂𝛾;𝑥_𝑘𝜋𝛾𝑑𝛾=𝐴^−1𝑡₀¹₀𝐿_𝑘−1𝛾𝐿_𝑘−1𝛾^′𝜋𝛾𝜋𝛾^′1−𝐹₁𝛾;𝑥_𝑘𝐹₁𝛾^′;𝑥_𝑘−𝐹₂𝛾^′;𝑥_𝑘𝑑𝛾^′𝑑𝛾−𝐴^−1𝑡₀¹₀𝐿_𝑘−1𝛾𝐿_𝑘−1𝛾^′𝜋𝛾𝜋𝛾^′𝐹₁𝛾;𝑥_𝑘−𝐹₂𝛾;𝑥_𝑘1−𝐹₁𝛾^′;𝑥_𝑘𝑑𝛾^′𝑑𝛾=𝐴^−1𝑡₀¹_𝑡𝐿_𝑘−1𝛾𝐿_𝑘−1𝛾^′𝜋𝛾𝜋𝛾^′1−𝐹₁𝛾;𝑥_𝑘𝐹₁𝛾^′;𝑥_𝑘−𝐹₂𝛾^′;𝑥_𝑘𝑑𝛾^′𝑑𝛾−𝐴^−1𝑡₀¹_𝑡𝐿_𝑘−1𝛾𝐿_𝑘−1𝛾^′𝜋𝛾𝜋𝛾^′𝐹₁𝛾;𝑥_𝑘−𝐹₂𝛾;𝑥_𝑘1−𝐹₁𝛾^′;𝑥_𝑘𝑑𝛾^′𝑑𝛾=𝐴^−1𝑡₀¹_𝑡𝐿_𝑘−1𝛾𝐿_𝑘−1𝛾^′𝜋𝛾𝜋𝛾^′1−𝐹₁𝛾;𝑥_𝑘𝐹₁𝛾^′;𝑥_𝑘−𝐹₂𝛾^′;𝑥_𝑘−𝐹₁𝛾;𝑥_𝑘−𝐹₂𝛾;𝑥_𝑘1−𝐹₁𝛾^′;𝑥_𝑘𝑑𝛾^′𝑑𝛾, (A.1) where 𝐴=¹₀𝐿_𝑘−1𝛾𝐿_𝑘−1𝛾^′1−𝐹₁𝛾;𝑥_𝑘𝐹₁𝛾^′;𝑥_𝑘−𝐹₂𝛾^′;𝑥_𝑘𝜋𝛾𝜋𝛾^′𝑑𝛾^′𝑑𝛾. (A.2) Since 𝛾≤𝛾^′ and (𝐹₁−𝐹₂)/(1−𝐹₁) is monotonically decreasing in 𝛾, then [𝐹₁(𝛾^′;𝑥_𝑘)−𝐹₂(𝛾^′;𝑥_𝑘)]/[1−𝐹₁(𝛾^′;𝑥_𝑘)]≤[𝐹₁(𝛾;𝑥_𝑘)−𝐹₂(𝛾;𝑥_𝑘)]/[1−𝐹₁(𝛾;𝑥_𝑘)], which implies that [1−𝐹₁(𝛾;𝑥_𝑘)][𝐹₁(𝛾^′;𝑥_𝑘)−𝐹₂(𝛾^′;𝑥_𝑘)]≤[𝐹₁(𝛾;𝑥_𝑘)−𝐹₂(𝛾;𝑥_𝑘)][1−𝐹₁(𝛾^′;𝑥_𝑘)].Hence, Π_𝑘(𝑡;0)≤Π_𝑘(𝑡;1), which implies that Π_𝑘⁻¹(𝛼;0)≥Π_𝑘⁻¹(𝛼;1), that is, 𝑥_𝑘+1≥𝑥^′_𝑘+1. This completes the proof of Theorem 3.1.

Proof of Theorem 3.2. Using the notation in the proof of Theorem 3.1, if 𝑌_𝑘=0, then 𝐿_𝑘(𝛾)=𝐿_𝑘-1(𝛾)[1−𝐹₁(𝛾;𝑥_𝑘)], and if 𝑌_𝑘=2, then 𝐿_𝑘(𝛾)=𝐿_𝑘−1(𝛾)𝐹₂(𝛾;𝑥_𝑘). Since both 𝐹₁ and 𝐹₂ are monotonically decreasing in 𝛾, then the result of the theorem follows from the proof of Theorem 1 in Tighiouart and Rogatko [23].

WinBUGS Code
 # Assume that the toxicity response 𝑌 has 3 categories: # 𝑌=1 if patient has grade 0 or 1 toxicity # 𝑌=2 if patient has a grade 2 toxicity # 𝑌=3 if patient has grade 3 or 4 toxicity, that is, DLT # Dose is standardized between 0 and 1. model { for (i in 1 : N) { # Likelihood logit(eta[i,1])<- -(logit(rho1)) − (1/mtd)*(logit(theta) − logit(rho0))*X[i] logit(eta[i,3])<- logit(rho0) + (1/mtd)*(logit(theta) − logit(rho0))*X[i]  # rho1 = P(grade 2 or more toxicity at initial dose) # rho0 = P(DLT or grade 3 or 4 toxicity at initial dose) # MTD = dose producing DLT in theta fraction of the population          } for (i in 1 : N) { p[i,1] <-eta[i,1] p[i,3]<-eta[i,3] p[i,2]<-1-(eta[i,1]+eta[i,3]) Y[i]~dcat(p[i,])  }  # Prior Distributions rho0 ~ dunif(0, theta) rho1 ~ dunif(rho0, 1) mtd ~ dunif(0,1)  }  # Data list(𝑌=𝑐(1,2,2,2,3,1,2,3,3,1,2,2,1,3),dose=  c(0.1,0.3262,0.3873,0.4390,0.4892,0.3810,0.4298,0.4681,0.3980, 0.3339,0.3650,0.3788,0.3986,0.4308),theta=0.33333333, N=14) # Initial values list(rho0=0.05, rho1=0.15, mtd=0.3).

Acknowledgments

This paper supported in part by the National Center for Research Resources, Grant UL1RR033176, and is now at the National Center for Advancing Translational Sciences, Grant UL1TR000124 (M. Tighiouart and A. Rogatko), Grant 5P01CA098912-05 (A. Rogatko) and P01 DK046763 (A. Rogatko). The content is solely the responsibility of the authors and does not necessarily represent the official views of the NIH.