Incorporating a Patient Dichotomous Characteristic in Cancer Phase I Clinical Trials Using Escalation with Overdose Control

We describe a design for cancer phase I clinical trials that takes into account patients heterogeneity thought to be related to treatment susceptibility. The goal is to estimate the maximum tolerated dose MTD given patient’s specific dichotomous covariate value. The design is Bayesian adaptive and is an extension of escalation with overdose control EWOC . We will assess the performance of this method by comparing the following designs via extensive simulations: 1 design using a covariate; patients are accrued to the trial sequentially and the dose given to a patient depends on his/her baseline covariate value, 2 design ignoring the covariate; patients are accrued to the trial sequentially and the dose given to a patient does not depend on his/her baseline covariate value, and 3 design using separate trials; in each group, patients are accrued to the trial sequentially and EWOC is implemented in each group. These designs are compared with respect to safety of the trial and efficiency of the estimates of the MTDs via extensive simulations. We found that ignoring a significant baseline binary covariate in the model results in a substantial number of patients being overdosed. On the other hand, accounting for a nonsignificant covariate in the model has practically no effect on the safety of the trial and efficiency of the estimates of the MTDs.


Introduction
The main objective of cancer phase I clinical trials is to determine a maximum tolerated dose MTD of a new experimental drug or combination of known drugs for use in a phase II trial.These trials enroll advanced stage cancer patients who have exhausted all standard therapies sequentially in cohorts of size one or more patients and dose level assignment to a given cohort of patients is dependent upon the dose levels and toxicity outcomes of the previously treated cohorts of patients.Adaptive statistical designs for cancer phase I clinical trials have been studied extensively in the last two decades, see for example, O'Quigley et al.
the minimum and maximum dose levels available for use in the trial.Clinicians choose these levels in the belief that X min is safe when administered to humans and X max is too toxic, see 18 how these levels were selected for a real prospective trial.Denote by Z the observable baseline binary covariate taking values 0 or 1 and let

P z x
Prob DLT | Dose x, Z z , 2.1 be the probability of dose limiting toxicity DLT for a patient with baseline covariate z and treated with dose level x.For simplicity, we consider the logistic model to describe the dosetoxicity relation We assume that β 1 ≥ 0 so that P z x is an increasing function of dose x.This is a reasonable assumption for cytotoxic agents.Model 2.2 implies a constant odds ratio of toxicity between the two groups of patients in the sense that this odds ratio does not depend on the dose level.
The MTD for a patient with covariate value z is defined as the dose γ z that results in a probability equal to θ z that a DLT will manifest within one cycle of therapy.The value chosen for the target probability of DLT θ z would depend on the nature and consequences of the dose-limiting toxicity; it would be set relatively high when the DLT is a transient, correctable or nonfatal condition, and low when it is life threatening or lethal 5 .We will assume that θ z is constant in z although the methodology can be adapted to different target probabilities of toxicities.In practice, clinicians use a constant target probability of DLT θ since we do not know a priori how the treatment under study affects the different groups of patients defined by their baseline covariate value.It follows from the dose-toxicity model 2.2 that the MTD is Let ρ 0,z be the probability of DLT at the initial dose given to a patient with covariate value z.In the statistical design of a phase I clinical trial, it is convenient to specify the prior distribution on parameters the clinicians can easily interpret.For instance, Babb and Rogatko 17 reparameterized model 2.2 in terms of the MTD associated with the maximumanticipated plasma concentration of anti-SEA antibodies and the probabilities of DLT when the minimum allowable dose is administered to patients with pretreatment anti-SEA concentrations selected to span the range of this covariate.Here, we reparameterize model 2.2 in terms of γ 0 γ 0 , γ 1 γ 1 , and ρ 0,0 .We chose this reparameterization because the MTDs for each group are the parameters of interest.However, other parameterizations such as difference between the MTDs in both groups are possible.Using the definition of the MTDs and probability of toxicity at the initial dose x 1 , one can show that

2.4
We note that the probability of DLT does not depend on the parameter γ 1 when z 0. Denote by p ρ 0,0 , γ 0 , γ 1 , x 1 and p ρ 0,0 , γ 0 , x 0 the probabilities of DLT for a patient with covariate value 1 and 0, respectively.These probabilities are obtained using the dose-toxicity model 2.2 with β 0 , β 1 , η given by 2.4 .

Likelihood
Suppose that after the lth patient with baseline covariate value z is treated with dose x z,l , there are m l patients with covariate value z 0 and k l patients with covariate value z 1.
Let y z,l be the toxicity outcome 1 for DLT and 0 for no DLT for the ith patient with covariate value z.The likelihood of the data is p ρ 0,0 , γ 0 , x 0,i y 0,i 1 − p ρ 0,0 , γ 0 , x 0,i where D l { x 0,1 , y 0,1 , . . ., x 0,m l , y 0,m l , x 1,1 , y 1,1 , . . ., x 1,k l , y 0,k l } and m l k l l.Let h ρ 0,0 , γ 0 , γ 1 be a prior distribution on the parameters ρ 0,0 , γ 0 , and γ 1 .The posterior distribution is where c D l is a normalizing constant.This joint posterior is clearly intractable and WinBugs and a Markov chain Monte Carlo sampler will be devised to estimate features of this joint posterior distribution as in Tighiouart et al. 18 .

Prior Distributions
Another advantage of the reparameterization in 2.4 is the natural specification of vague but proper prior densities for the model parameters.Indeed, under the assumption that γ 0 , γ 1 belong to X min , X max with prior probability 1 and no prior assumptions on whether one group can tolerate higher doses better than the other, we can take γ 0 , γ 1 ∼ Uniform X min , X max 2 and γ 0 independent of γ 1 .If on the other hand, we have a priori belief that one group can tolerate higher doses better than the other group for example, then γ 0 , γ 1 can be taken to be uniform on the triangle X min < γ 0 < γ 1 < X max .Design operating characteristics should be performed when designing prospective trials when considering informative priors.
The prior distribution for ρ 0,0 is taken as a uniform in 0, θ , which reflects a lack of prior knowledge regarding the probability of DLT at the initial dose.

Trial Design
Denote by A and B the two groups of patients corresponding to covariate values 0 and 1, respectively.We assume that the support of the MTDs γ 0 and γ 1 are contained in X min , X max .
That is, we assume that dose levels X min and X max are identified a priori such that γ 0 , γ 1 belong to X min , X max with prior and hence posterior probability 1.We note that if the prior distribution π γ 1 is independent of the joint prior distribution of ρ 0,0 , γ 0 , then π γ 1 is never updated unless a patient in group B is enrolled in the trial.In the case of such priors, the trial proceeds as follows.
The first patient in either group receives the dose x 1 X min .Let Π z,1 be the marginal posterior cdf of the MTD γ z , z 0, 1. Suppose that the first patient belongs to group A. If the second patient belongs to group A, then he or she will receive the dose x 0,2 Π −1 0,1 α so that the posterior probability of exceeding the MTD γ 0 is equal to the feasibility bound α.If the second patient belongs to group B, then he or she will receive the dose x 1 X min .In general, the first time a patient is assigned to a given group always receives x 1 X min no matter how many patients have been enrolled in the other group.Once l patients have been enrolled in the trial with at least one patient treated in each group, the l 1-st patient with covariate value z receives the dose x z,l 1 Π −1 z,l α .The trial proceeds until a total of npatients have been accrued.At the end of the trial, we estimate the MTD as γ z Π −1 z,n α , z 0, 1.

Comparison of Three Designs
In order to assess the operating characteristics of this design when designing a prospective trial, we explored the behavior of this method when we adjust for a significant covariate.We also evaluated the performance of this design when adjusting for a nonsignificant baseline covariate.Finally, its performance was also explored when two parallel trials are used instead of adjusting for a binary baseline covariate.Therefore, we study design operating characteristics by comparing the following designs.
i Design using a covariate; patients are accrued to the trial sequentially and the dose given to a patient is calculated assuming model 2.2 .
ii Design ignoring the covariate; patients are accrued to the trial sequentially and the dose given to a patient is calculating assuming model 2.2 without the covariate, that is, as in the original EWOC.
iii Design using separate trials; in each group, patients are accrued to the trial sequentially and EWOC is implemented in each group.
Comparisons will be carried out under several scenarios for the true values of the MTDs γ 0 and γ 1 .Since the main goal of cancer phase I clinical trials is to efficiently estimate the MTD 6 Journal of Probability and Statistics while protecting patients from potentially toxic side effects, we will assess the safety of the trial and efficiency of the estimate of the MTDs by simulating a large number of trials M under each model and compare the proportion of patients exhibiting DLT, the average bias bias ave M −1 M i 1 γ z,i − γ z,true and the estimated mean square error MSE M −1 M i 1 γ z,i − γ z,true 2 , where γ z,i is the MCMC estimate of the Bayes estimate of the marginal posterior distribution of the MTD at the end of the ith trial, z 0, 1.In addition, the models are further compared with respect to the proportion of patients that were overdosed.Here, a patient with baseline covariate z is overdosed if this patient has been given a dose x such that x > x * , where x * is defined as the dose for which P DLT | x * , z θ 0.05.This probability is calculated using the parameter values from the corresponding scenario.

Simulation Setup
The simulation results presented below all assume that the feasibility bound α 0.25 and that the dose levels are standardized so that the starting dose x 1 equals to the minimum dose for each trial X min 0 and all subsequent dose levels are selected from the unit interval.The target probability of DLT is fixed at θ 0.33, ρ 0,0 0.05, and the total sample size is n 42.We consider several scenarios corresponding to combinations of four possible values of γ 0 , γ 1 , 0.2, 0.4, 0.6, and 0.8.In all simulations, the prior distributions for ρ 0,0 , γ 0 , γ 1 were taken as uniform in 0, θ × X min , X max 2 with ρ 0,0 , γ 0 , γ 1 independent a priori.
For design 1 , a patient is randomly selected from either group A or B with equal probability so that the total number of patients in group A, m, equals to the total number of patients in group B, k.For each pair γ 0 , γ 1 in {0.2, 0.4, 0.6, 0.8}×{0.2,0.4, 0.6, 0.8}, we simulate 1000 trials and calculate the proportion of patients that were overdosed, the proportion of patients exhibiting DLT, the average bias, and the estimated MSE.For design 2 , the covariate for each patient is recorded but it is not taken into account when calculating the dose level for the next patient.Again, we simulate 1000 trials and calculate the proportion of patients that were overdosed, the proportion of patients exhibiting DLT, the average bias, and the estimated MSE.For the third design, separate trials are simulated in each group and the summary statistics are calculated based on 1000 simulated trials in each group.In all cases, the responses y z,i are generated from model 2.2 .

Results
Table 1 gives the overall proportion of patients exhibiting DLT, the proportion exhibiting DLT in each group, the proportion of patients in each group that are overdosed, the bias, and MSE of the estimates of the MTDs when design i in Section 3.1 is used.Table 2 gives the summary statistics corresponding to the safety of the trial when design ii is used.The overall proportion of patients exhibiting DLT is always less than θ 0.33 under all entertained scenarios and it is uniformly lower for a design which accounts for the baseline covariate relative to the design ignoring this covariate.The same conclusion holds when comparison of these two designs is carried out within each group.On the other hand, the proportion of patients being overdosed in group A is much higher when the two groups of patients differ in their susceptibility to treatment and this difference is not taken into account.This proportion can be as high as 16% in the case where γ 0 , γ 1 0.4, 0.8 .This is not surprising because when a difference in the MTDs is not taken into account in the model, then the sequence of doses generated by the design tends to cluster around a weighted average of the two "true" MTDs, where the weights depends on the number of patients in each group.If on the other hand, the models accounts for the difference in the MTDs, then the distribution of the sequence of doses is bimodal clustering around the two "true" MTDs, as displayed in Figure 1, which shows the histogram of all doses with fitted density dashed line when γ 0 , γ 1 0.3, 0.6 .The difference in the proportion of patients being overdosed in group B between the two designs is practically negligible.When the two MTDs are equal and the design does account for the baseline covariate, Tables 3 and 4 show that the proportions of DLTs overall and within each group is no more than the target probability of DLT θ, and the differences in these proportions between the two designs are practically not important.No design is uniformly better than the other in terms of the proportion of patients being overdosed.For instance, when γ 0 γ 1 0.2, the proportion of patients in group A is 0.336 when a covariate is used and this proportion is 0.275 when the covariate is not taken into account.On the other hand, when γ 0 γ 1 0.4, the proportion of patients in group A is 0.179 when a covariate is used but this proportion is 0.21 when the covariate is not taken into account.In fact, these proportions are equal on the average across the four scenarios for the true value of the MTD γ γ 0 γ 1 .Tables 3 and 4 also show that the bias and MSE of the estimates of the MTD is higher when a nonsignificant covariate is used in the model, with the higher values obtained the true MTD is high, γ 0.6, 0.8.
Table 5 gives the summary statistics when separate trials enrolling n 21 patients are used.As before, the proportion of patients exhibiting DLT does not exceed the target probability of DLT θ.When the true MTDs are the same, the overall proportion of patients that are overdosed using a model with a baseline covariate Table 3 is lower than the corresponding proportion if parallel trials were used Table 5 when γ 0.2, 0.4.The differences in these proportions for γ 0.6, 0.8 are negligible.The bias of the estimate of the MTD is about the same for both designs.In the case where the MTDs γ 0 and γ 1 are  different, more patients are overdosed using a model with a baseline covariate compared to using parallel trials and the differences in the bias and MSE are negligible, see Tables 1 and 5.
Based on these results, we recommend adjusting for a baseline covariate thought to be related to treatment susceptibility when designing a cancer phase I trials whenever possible.We stand to lose little if we were to use a design with a covariate when in fact there is no difference between the MTDs of the two groups.

Discussion
We have presented design operating characteristics of a Bayesian adaptive design which accounts for a patient dichotomous baseline covariate using EWOC scheme.The design is suitable for cancer phase I clinical trials where the goal is to estimate the conditional MTD given patients' covariate value.
We have found that if the two MTDs are different and the design does not adjust for this heterogeneity, then the trial will result in more patients being overdosed.If the two MTDs are the same and the design adjusts for patients' heterogeneity, then slightly more patients can be overdosed if the true MTD is low relative to a design with no covariate but these proportions are equal on the average across the four scenarios for the true value of the MTD.Thus, we stand to lose little if we do include a statistically nonsignificant covariate in the model.Incidentally, this conclusion is in agreement with the findings in O'Quigley et al. 15 .We carried out other simulations results not shown for various sample sizes, allocation ratios, probability of DLT at the initial dose ρ 0 .The results and conclusions were essentially the same.Ratain et al. 13 showed the importance of including patient's plasma concentration of anti-SEA antibodies in order to determine the MTD of the agent PNU-214565 as a function of this continuous baseline covariate.In a similar trial, Tighiouart and Rogatko 16 showed how more patients were overdosed when a baseline covariate, cancer type, was not accounted for in the model.Indeed, a retrospective analysis of a cancer phase I trial using a baseline continuous covariate showed that nonsmall cell cancer patients and pancreatic patients were treated at suboptimal doses whereas renal cell carcinoma patients were overdosed, with 36.4% experiencing DLT; the target probability of DLT was θ 0.2.This last example is in agreement with the simulation results we obtained in this paper.We are in the process of determining model operating characteristics in the presence of a continuous covariate, more than one covariate, and interaction term.An important question is to decide whether or not to include patients' covariate values during the trial.Although the previous results seem to indicate that we stand to lose little in terms of the proportion of patients being overdoses and the efficiency of the estimate of the MTDs when covariate information is taken into account in the model when in fact, this covariate is not predictive of DLT, determining the value of this covariate may involve a monetary cost.This is the case when patients need to be genotyped and certain biomarker expressions need to be determined.

Table 1 :
EWOC with Covariate.Design operating characteristic with respect to safety and efficiency of the trial.

Table 2 :
EWOC with no Covariate.Design operating characteristic with respect to safety of the trial.

Table 3 :
EWOC with Covariate.Design operating characteristic with respect to safety and efficiency of the trial.

Table 4 :
EWOC with no Covariate.Design operating characteristic with respect to safety and efficiency of the trial.

Table 5 :
EWOC.Design operating characteristic with respect to safety and efficiency of the trial.