M-K Riviere

Ying Yuan
email: yyuan@mdanderson.org

F Dubois

Sarah Zohar
email: sarah.zohar@inserm.fr

A Bayesian Dose-finding Design for Clinical Trials Combining a Cytotoxic Agent with a Molecularly Targeted Agent

Novel molecularly targeted agents (MTAs) have emerged as valuable alternatives or complements to traditional cytotoxic agents in the treatment of cancer. Clinicians are combining cytotoxic agents with MTAs in a single trial to achieve treatment synergism and better patient outcomes. An important feature of such combinational trials is that, unlike the efficacy of the cytotoxic agent, that of the MTA may initially increase at low dose levels and then approximately plateau at higher dose levels as MTA saturation levels are reached. Therefore, the goal of the trial is to find the optimal dose combination that yields the highest efficacy with the lowest toxicity and meanwhile satisfies a certain safety requirement. We propose a Bayesian phase I/II design to find the optimal dose combination. We model toxicity using a logistic regression and propose a novel proportional hazard model for efficacy, which accounts for the plateau in the MTA dose-efficacy curve. We evaluate the operating characteristics of the proposed design through simulation studies under various practical scenarios. The results show that the proposed design performs well and selects the optimal dose combination with high probability.

Introduction

Traditional cytotoxic agents have played important roles in combating cancer. However, after decades of research, it has become difficult to find new cytotoxic agents that are substantially more effective than the existing therapeutic strategies. Recently, novel molecularly targeted agents (MTAs), such as small molecules or monoclonal antibodies, have emerged as alternatives or complements to cytotoxic agents for treating cancer (Le [START_REF] Tourneau | Current challenges for the early clinical development of anticancer drugs in the era of molecularly targeted agents[END_REF][START_REF] Tourneau | Heterogeneity in the definition of dose-limiting toxicity in phase I cancer clinical trials of molecularly targeted agents: a review of the literature[END_REF][START_REF] Tourneau | Efficiency of new dose escalation designs in dose-finding phase I trials of molecularly targeted agents[END_REF][START_REF] Postel-Vinay | Clinical benefit in Phase-I trials of novel molecularly targeted agents: does dose matter?[END_REF]. Unlike cytotoxic agents, MTAs modulate specific aberrant pathways in cancer cells, while sparing normal tissue. To take advantage of both types of treatment agents, clinicians are exploring the possibility of combining traditional cytotoxic agents with novel MTAs to achieve treatment synergism and better patient response.

This new trend of combining cytotoxic agents with MTAs for treating cancer brings new challenges for early-phase dose-finding trial design. These challenges arise from the difference in the dose-efficacy curves between the two types of treatment agents. For cytotoxic agents, more is better (i.e., a higher dose yields a greater response) until a dose-limiting toxicity level is reached. However, the dose-efficacy relationship of the MTA may not follow a monotonic pattern: the efficacy of the MTA often increases at low dose levels and then plateaus (or approximately plateaus) at higher dose levels once a saturation level in the body is reached (Le [START_REF] Tourneau | Current challenges for the early clinical development of anticancer drugs in the era of molecularly targeted agents[END_REF][START_REF] Hoering | Seamless phase I-II trial design for assessing toxicity and efficacy for targeted agents[END_REF]. Although it is possible that efficacy decreases at higher dose levels, here we focus on the case in which efficacy first increases and then plateaus because such a dose-efficacy relationship is much more commonly encountered in practice.

Consequently, the conventional dose-finding paradigm of searching for the maximum tolerated dose (MTD) is not suitable for combinational trials of a cytotoxic agent with an MTA, and it is imperative to consider efficacy and toxicity simultaneously, with the goal of finding the molecularly targeted optimal dose combination (ODC). This is because once the MTA dose-efficacy curve reaches a plateau, further increases in the dose of the targeted agent will not yield any therapeutic benefit, but will potentially result in greater toxicity [START_REF] Postel-Vinay | Clinical benefit in Phase-I trials of novel molecularly targeted agents: does dose matter?[END_REF]. In this article, the ODC is defined as the most efficacious dose combination that yields the lowest toxicity. As the lowest toxicity can still be excessive, we also require the ODC to satisfy a certain safety requirement, e.g., the toxicity probability must be lower than a certain upper bound.

Numerous designs have been proposed to find the MTD for trials combining multiple cytotoxic agents, without considering the efficacy endpoint. For example, [START_REF] Thall | Dose-finding with two agents in Phase I oncology trials[END_REF] developed a Bayesian approach to identify an entire toxicity "contour" of drug combinations. [START_REF] Conaway | Designs for single-or multipleagent phase I trials[END_REF] proposed a dose-finding method based on the simple and partial orders of drug combinations. [START_REF] Yuan | Sequential continual reassessment method for twodimensional dose finding[END_REF] proposed a sequential dosefinding design that allows single-agent dose-finding methods to be used in multiple-agent combination trials. [START_REF] Braun | A hierarchical Bayesian design for phase I trials of novel combinations of cancer therapeutic agents[END_REF] proposed a hierarchical-model-based approach for dose finding. [START_REF] Yin | Bayesian dose finding in oncology for drug combinations by copula regression[END_REF] developed a Bayesian dose-finding method based on a copula-type regression model. [START_REF] Wages | Continual Reassessment Method for Partial Ordering[END_REF] extended the continual reassessment method to two-dimensional dose finding. Recently, several phase I/II drug-combination trial designs have been proposed to account for both toxicity and efficacy. Focusing on a combination of cytotoxic agents, [START_REF] Huang | A parallel phase I/II clinical trial design for combination therapies[END_REF] proposed a phase I/II design based on the "3+3" type dose escalation scheme, and [START_REF] Yuan | Bayesian Phase I/II Adaptively Randomized Oncology Trials With Combined Drugs[END_REF] developed a model-based approach to accommodate toxicity and efficacy for combination trials. [START_REF] Mandrekar | An adaptive phase I design for identifying a biologically optimal dose for dual agent drug combinations[END_REF] proposed a dose-finding design for trials combining two MTAs based on a continuation ratio model for trinary outcomes. [START_REF] Cai | A bayesian phase i/ii design for oncology clinical trials of combining biological agents[END_REF] proposed a Bayesian dose-finding design for trials combining two MTAs, which used a change point model to reflect that the dose-toxicity surface of combinations may plateau. Despite this rich body of literature, no design is available for clinical trials combining a cytotoxic agent with an MTA, which requires simultaneously accounting for the different behaviors of the cytotoxic agent and the MTA. In addition, the existing phase I/II drug-combination designs assume that the efficacy outcome is immediately ascertainable; however, this assumption may not hold in many practical situations because, unlike the toxicity endpoint, the efficacy endpoint often requires a relatively long time to assess.

We propose a Bayesian phase I/II design to find the ODC for trials combining a cytotoxic agent with an MTA. We model efficacy as a time-to-event outcome rather than a binary outcome, thereby eliminating the requirement that the efficacy outcomes of treated patients must be fully evaluated before a new cohort can be enrolled into the trial. To account for the feature of the MTA whereby the dose-efficacy curve may initially increase and then plateau, we incorporate a plateau parameter into the proportional hazard model for time to efficacy. We model the binary toxicity outcome using a logistic regression model. During the trial, we continuously updated the model estimates and use them to assign patients to the ODC.

The remainder of the article is organized as follows. In Section 2, we present a case study that has motivated the proposed methodology. In Section 3, we propose toxicity and efficacy models, and describe a dose-finding algorithm to identify the ODC. In Section 4, we present simulation studies to evaluate the operating characteristics of the proposed design and investigate its sensitivity to model specifications. We conclude with a brief discussion in Section 5. [START_REF] Pishvaian | A Phase I clinical trial of the combination of imatinib and paclitaxel in patients with advanced or metastatic solid tumors refractory to standard therapy[END_REF] reported a phase I dose-finding clinical trial for the combination of imatinib and paclitaxel in patients with advanced solid tumors refractory to standard therapy. Imatinib is a tyrosine-kinase inhibitor used in the treatment of multiple cancers, most notably chronic myelogenous leukemia (CML). Imatinib works by inhibiting the activity of the BCR-Abl tyrosine kinase enzyme that is necessary for cancer development, thus preventing the growth of cancer cells and leading to their death by apoptosis. Because the BCR-Abl tyrosine kinase enzyme exists only in cancer cells and not in healthy cells, imatinib works effectively as an MTA killing only cancer cells through its action. The goal of the trial was to evaluate the safety of combining imatinib with the traditional cytotoxic chemotherapeutic agent paclitaxel, and to determine whether that combination improved the efficacy of imatinib. In the trial, four doses (300, 400, 600, 800 mg) of imatinib and three doses (60, 80, 100 mg/m 2 ) of paclitaxel were investigated. Most of the grade 3 or 4 toxicities related to therapy involved neutropenia, flu-like symptoms, and pain. The treatment response was evaluated using the Response Evaluation Criteria in Solid Tumors (RECIST). This phase I trial adopted the conventional "3+3" design, which unfortunately suffers from several limitations. First, as the "3+3" design requires that the doses under investigation must be monotonically increasing, only a subset of all 12 possible combinations of imatinib and paclitaxel were investigated in the trial. As a result, the trial might not have even examined the most desirable dose in the 4 × 3 dose combination space. Specifically, the trial selectively investigated 6 dose combinations: (paclitaxel, imatinib) = (60, 300), (60, 400), (80, 400), (80, 600), (100, 600) and (100, 800). The original protocol involved an intensive dose schedule, with continuous daily oral administration of imatinib and weekly paclitaxel infusions. However, after treating patients at the first two doses, the regimen resulted in an excessive number of adverse events, and thus the protocol was amended to a less intensive schedule, with intermittent dosing of imatinib. The second limitation of this use of the "3+3" design is that it ignores the important fact that the efficacy of imatinib does not monotonically increase with the dose, and that the MTD may not be the optimal dose for treating patients. [START_REF] Druker | Perspectives on the development of a molecularly targeted agent[END_REF] pointed out that, for treating CML, a dose of 400 to 600 mg of imatinib reached the plateau of the dose-response curve. As a result, 400 mg or 600 mg is the dose of imatinib that is commonly used in clinical practice. This result was confirmed in a meta-analysis [START_REF] Gafter-Gvili | High-dose imatinib for newly diagnosed chronic phase chronic myeloid leukemia patients-systematic review and meta-analysis[END_REF] of phase III randomized trials, in which no treatment difference was found between 400 mg and higher doses of imatinib. This trial example demonstrates the need for a new dose-finding design to handle the clinical trials that combine an MTA with a traditional cytotoxic agent. We apply our design to the trial in Section 4.

A solid tumor clinical trial

Methods

Toxicity model

Consider a trial combining J doses of cytotoxic agent A with K doses of molecularly targeted agent (MTA) B, and denote (j, k) as the combination of the jth dose level of agent A with the kth dose level of agent B. We assume that toxicity (i.e., the dose-limiting toxicity defined by the trial investigator) is quickly ascertainable and monotonically increases with the doses of both agents A and B; this assumption generally holds for cytotoxic agents and is plausible for most MTAs.

Let y i denote the binary toxicity outcome of patient i with y i = 1 indicating a toxicity response, and p jk denote the toxicity probability of combination (j, k) for j = 1, . . . , J and k = 1, . . . , K. We model toxicity using a logistic model as follows,

logit(p jk ) = β 0 + β 1 u j + β 2 v k (1)
where β 0 , β 1 , and β 2 are unknown parameters, and u j and v k are "effective" doses ascribed to the jth dose level of agent A and the kth dose level of agent B based on the prior estimates of the single-agent toxicity probabilities for these two dose levels. The procedure of determining the values of u j and v k will be described in Section 3.4. We require β 1 > 0 and β 2 > 0 so that toxicity monotonically increases with the dose levels of both agents A and B.

Assuming that during the trial conduct, among the n jk patients administered the combination (j, k), m jk patients experienced toxicity, then the likelihood of the toxicity data

D tox = {n jk , m jk } is L(D tox |β 0 , β 1 , β 2 ) ∝ J ∏ j=1 K ∏ k=1 p m jk jk (1 -p jk ) n jk -m jk .
Letting π(β 0 , β 1 , β 2 ) denote the prior distribution of β 0 , β 1 and β 2 , the posterior is then given by

f (β 0 , β 1 , β 2 |D tox ) ∝ π(β 0 , β 1 , β 2 )L(D tox |β 0 , β 1 , β 2 ).
(2)

In model (1), we do not include an interactive effect of the two agents (e.g., an interaction term β 3 u j v k ) because the reliable estimation of such an interaction term requires a large sample size (e.g., a few hundred), which is typically not available in early-phase trials. Our numerical study suggests that including the interaction term does not improve but often impairs the performance of the design (results are not shown). Note that for the purpose of dose finding, our goal is not to accurately model the entire dose-toxicity surface, but to obtain an adequate local fit to facilitate dose escalation and de-escalation. A model may provide a poor global fit for the entire dose-toxicity surface; however as long as the model provides a good local fit around the current combination, it will lead to correct decisions of dose escalation and dose selection [START_REF] O'quigley | Continual reassessment method for ordered groups[END_REF].

Efficacy model

Unlike toxicity, which often can be observed quickly, the efficacy response may require a relatively long follow-up time to be scored. In this circumstance, the conventional approach of treating efficacy as a binary outcome causes a serious logistic issue: when a new patient is enrolled and is waiting for dose assignment, some of patients already treated in the trial might not have finished their evaluation yet, and thus their response outcomes are not available to make the decision of dose assignment for the new patient. To overcome this difficulty, we herein model the response as a time-to-event outcome, in which the data of the incomplete efficacy evaluations are naturally incorporated into the decision making of dose assignment as censored observations.

Let t denote the time to response. In early phase clinical trials, the typical way to evaluate efficacy is to follow each patient for a fixed period of time T , e.g., 3 months, after the initiation of the treatment. Within the assessment window (0, T], if the patient responds favorably to the treatment (i.e., t ≤ T ), it is scored as a response, otherwise nonresponse.

The efficacy of the drug is defined as the response rate at T . Patient's outcomes after T

will not be used to define the efficacy of the drug and make the decision of dose escalation and selection. In other words, the time to response t is always administratively censored at T . Although we cannot observe any t beyond the time point T , it does not cause any issue here because for the purpose of evaluating the efficacy of the drug and finding the optimal dose, by definition, we only concern the response rate at T , i.e., 1 -S(T ), where S(•) denotes the survival function of t. For the same reason, conceptually, we can regard t = ∞ for the patients who do not benefit from the treatment without affecting the dose finding. A special feature of the trial combining an MTA with a cytotoxic agent is that the dose-efficacy curve behaves differently with respect to the two agents: efficacy is expected to monotonically increase with the dose of the cytotoxic agent, but not with the dose of the MTA. Efficacy often initially increases and then plateaus with the dose of the MTA after the MTA reaches a level of saturation. Let λ jk (t) denote the hazard function associated with combination (j, k), and 1(C) denote the indicator function, which takes a value of 1 if C is true. We model the time to efficacy for the combination of an MTA and a cytotoxic agent using a proportional hazard model, augmented with a plateau parameter τ , as follows,

λ jk (t) = λ 0 (t)exp{γ 1 w j + γ 2 (z k 1(k < τ ) + z τ 1(k ≥ τ ))},
where λ 0 (t) is the baseline hazard, and w j and z k are "effective" doses ascribed to the jth dose level of agent A and the kth dose level of agent B based on the prior estimates of the single-agent efficacy probabilities for these two doses, which will be described in the next section. We assume that γ 1 > 0 and γ 2 > 0, and therefore efficacy monotonically increases with the dose of the cytotoxic agent A (i.e., w j ). The plateau parameter τ is an integer between 1 and K and indicates at which dose level of agent B (i.e., the MTA) efficacy reaches a plateau. When the dose level is lower than τ , the efficacy monotonically increases with the dose of the MTA (i.e., z k ) through the covariate effect

γ 2 (z k 1(k < τ ) + z τ 1(k ≥ τ )) = γ 2 z k ;
and when the dose level is equal to or higher than τ , the efficacy plateaus (with respective to the dose level of agent B) with a constant dose effect γ 2 z τ .

Due to the small sample size of early-phase trials, we take a parameter approach and assume an exponential distribution for the time to efficacy with a constant baseline hazard,

i.e., λ 0 (t) = λ 0 , resulting in the following survival function for the time to efficacy

S jk (t) = exp[-λ 0 t exp{γ 1 w j + γ 2 (z k 1(k < τ ) + z τ 1(k ≥ τ ))}].
Then, the response rate at the end of T for patients treated at the combination (j, k), denoted by q jk , is given by q ij = 1 -S jk (T ). In our design, q jk will be used as the measure of efficacy for determining the dose transition and selection.

For patient i, let s i denote the actual follow-up time, t i denote the time to response, and (j i , k i ) denote the combination administered to the patient. Define

x i = min(T, s i , t i )
and censoring indicator δ i = 1(x i = t i ). Given the efficacy data D eff = {x i , δ i } obtained from n patients, the likelihood is given by

L(D eff |λ 0 , γ 1 , γ 2 , τ ) ∝ n ∏ i=1 λ δi jiki (x i )S jiki (x i ),
and the posterior is

f (λ 0 , γ 1 , γ 2 , τ |D eff ) ∝ π(λ 0 , γ 1 , γ 2 , τ )L(D eff |λ 0 , γ 1 , γ 2 , τ ), (3)
where π(λ 0 , γ 1 , γ 2 , τ ) is the prior for the unknown parameters.

Specification of prior and effective doses

We first discuss the specification of priors for the model parameters. For the toxicity model, we adopted a vague normal prior N (0, 100) for the intercept β 0 , and, following [START_REF] Chevret | The continual reassessment method in cancer phase I clinical trials: a simulation study[END_REF], we assigned the slopes β 1 and β 2 independent exponential distributions with a rate parameter of 1, i.e., β 1 , β 2 ∼ Exp(1). For the efficacy model, we took vague priors λ 0 ∼ Exp(0.01) and γ 1 , γ 2 ∼ Exp(0.1), and assigned τ a multinomial distribution with probability parameters π = (π 1 , ..., π K ), where π k is the prior probability that the dose-efficacy curve plateaus at dose level k of the MTA. When there is rich information on the location of τ , e.g., we know the saturation dosage of the MTA from pharmacokinetic and pharmacodynamic studies, we can choose a set of π to reflect the likelihood of each dose level being the plateau point. When there is no good prior information regarding the location of τ , we recommend assigning τ an increasing sequence of prior probabilities (i.e.,

π 1 < π 2 < ... < π K ) rather than a noninformative flat prior π 1 = π 2 = ... = π K . This
recommendation is based on our experience with numerical studies, in which we found that using a noninformative prior often caused the dose selection to remain at low dose levels due to the sparsity of data; whereas the prior with increasing π k 's encourages the dose-finding algorithm to explore higher dose levels of agent B and actively learn the shape of the doseefficacy curve, thereby improving the ODC selection accuracy. In our simulation study, we took π = (0.14, 0.20, 0.28, 0.39), which led to good operating characteristics across various scenarios. A summary of prior distributions is given in Table 1. After specifying the prior distributions, we sampled posterior distributions ( 2) and (3) using the Gibbs sampler.

We next discuss how to specify the effective doses (i.e., u j 's and v k 's in the toxicity model, and w j 's and z k 's in the efficacy model) based on the prior estimates of the singleagent toxicity and efficacy probabilities. In practice, before two agents are to be combined, each of them typically has been studied individually. For example, prior to the solid tumor clinical trial that combines imatinib with paclitaxel [START_REF] Pishvaian | A Phase I clinical trial of the combination of imatinib and paclitaxel in patients with advanced or metastatic solid tumors refractory to standard therapy[END_REF], many phase I and II trials have been conducted to study the single-agent toxicity and efficacy profiles for imatinib [START_REF] Ramanathan | Phase I and pharmacokinetic study of imatinib mesylate in patients with advanced malignancies and varying degrees of liver dysfunction: a study by the National Cancer Institute Organ Dysfunction Working Group[END_REF][START_REF] Gibbons | Phase I and pharmacokinetic study of imatinib mesylate in patients with advanced malignancies and varying degrees of renal dysfunction: a study by the National Cancer Institute Organ Dysfunction Working Group[END_REF][START_REF] Lipton | Phase I trial of zoledronic acid + imatinib mesylate (Gleevec) in patients with bone metastases[END_REF][START_REF] Van Oosterom | Safety and efficacy of imatinib (STI571) in metastatic gastrointestinal stromal tumours: a phase I study[END_REF] and paclitaxel [START_REF] Kato | A phase II study of paclitaxel by weekly 1-h infusion for advanced or recurrent esophageal cancer in patients who had previously received platinum-based chemotherapy[END_REF][START_REF] Tsimberidou | Phase I clinical trial of hepatic arterial infusion of paclitaxel in patients with advanced cancer and dominant liver involvement[END_REF][START_REF] Takano | Phase I and pharmacological study of single paclitaxel administered weekly for heavily pre-treated patients with epithelial ovarian cancer[END_REF][START_REF] Lim | Phase I pharmacokinetic study of a weekly liposomal paclitaxel formulation (Genexol-PM) in patients with solid tumors[END_REF][START_REF] Horiguchi | Phase II study of weekly paclitaxel for advanced or metastatic breast cancer in Japan[END_REF]. Therefore, we often have good prior estimates of the single-agent toxicity and efficacy probabilities for each of the agents. The purpose of defining and using the effective doses is to match the prior estimates of (single-agent) toxicity and efficacy probabilities under our model with those elicited from the prior information. By doing so, we incorporate the available single-agent dose-toxicity and -efficacy information into our model and thus improve the efficiency of the design. This approach has been previously used for dose finding in single-agent trials [START_REF] Chevret | Statistical Methods for dose-Finding Experiments[END_REF][START_REF] Zohar | Using the continual reassessment method to estimate the minimum effective dose in phase II dose-finding studies: a case study[END_REF] and drug-combination trials [START_REF] Liu | A bayesian dose-finding design for drug combination trials with delayed toxicities[END_REF]. Therefore, based on the prior estimates pj0 and p0k , we backsolve the effective doses u j and v k as

u j = {logit(p j0 ) -β0 }/ β1 v k = {logit(p 0k ) -β0 }/ β2 ,
where β0 and β1 are prior means of β 0 and β 1 . Similarly, under efficacy survival model

(2), the single-agent efficacy model is S j0 (t) = exp{-λ 0 .t.exp(γ 1 w j )} for agent A and

S 0k (t) = exp[-λ 0 .t.exp{γ 2 (z k 1(k < τ ) + z τ 1(k ≥ τ ))}] for agent B.
We determine the effective doses

w j = log[-log{1 -qj0 }/( λ0 T )]/γ 1 z k = log[-log{1 -q0k }/( λ0 T )]/γ 2 ,
where λ0 , γ1 , γ2 are prior estimates of the corresponding parameters, and τ is the highest dose levels.

Dose-finding algorithm

At the beginning of the trial, data are very sparse and the estimates of the toxicity and efficacy models are highly unreliable. To improve the reliability of dose finding, we use a start-up phase to collect some preliminary data prior to switching to the formal model-based dose-finding algorithm.

We adopted a start-up phase similar to that proposed by [START_REF] Huang | A parallel phase I/II clinical trial design for combination therapies[END_REF], which divides the dose combination matrix into a sequence of zones along the diagonal from low doses to high doses (see Figure 1), and then conducts a "3+3" type dose escalation across the zones. Specifically, we initiate the start-up phase by treating the first cohort of 3 patients at the lowest zone, i.e., the lowest combination (1, 1), and then continuously escalate the dose to higher dose zones until we first encounter a zone in which all doses are "closed." Given a dose, if more than 1 patient were to experience toxicity out of the 3 or 6 patients who have been administered that dose, we close the dose and require that all higher doses (i.e., any combination having a higher dose level of A or B or A and B) are automatically closed and not eligible for use in treating future patients in the start-up phase. More precisely, if we close dose combination (j, k), we also close higher doses {(j ′ , k ′ ); j ′ ≥ j and k ′ ≥ k}.

The "closed" dose combinations can be reopened later to treat patients in the subsequent model-based dose-finding phase if the accumulating data indicate that they are actually safe. The dose escalation across zones is analogous to the traditional "3+3" dose escalation rule: among three patients, if we observe no toxicity, we escalate the dose; if more than 2 patients experience toxicity, we close the dose; and if 1 patient experiences toxicity, we treat three more patients at the current dose. In the latter case, if 0 or 1 out of the 6 patients experiences toxicity, we escalate the dose; otherwise we close the dose. When we escalate to a higher dose zone, if there are multiple combinations that are not closed in that zone, we simultaneously assign patients to each of the combinations.

After the start-up phase, we switch to the model-based dose-finding phase. Let θ and ξ denote the prespecified toxicity upper bound and efficacy lower bound, respectively. Let N denote the total sample size, and n denote the number of patients treated in the trial. We define that a combination (j, k) is admissible if it satisfies the safety requirement

P (p jk > θ) < C T (4)
and also the efficacy requirement

P (S jk (T ) > ξ) ≥ C E 1(n ≥ N/2), ( 5
)
where C T and C E are the respective probability thresholds for toxicity and efficacy. Note that the efficacy requirement (5) takes effect when only half of the patients have been enrolled, as controlled by the indicator function 1(n ≥ N/2). We found that introducing the efficacy condition too early caused a high frequency of misclassification of the admissible doses as inadmissible and thus resulted in the early termination of the trial. This situation can arise because, compared to the evaluation of the toxicity condition (4), the reliable evaluation of the efficacy condition (5) requires more data, as the efficacy outcome is not immediately observable and the efficacy model is relatively more complicated.

Let (j, k) denote the current dose, A denote the set of combinations that have been previously used to treat patients, and B = {(j ′ , k ′ ); j ′ ≤ j + 1, k ′ ≤ k + 1, and (j ′ , k ′ ) ̸ = (j+1, k+1)} denote the set of combinations for which the doses are not two levels higher than the current dose (j, k). Our model-based dose-finding algorithm can be described as follows:

after the start-up phase, we assign the next cohort of patients to the optimal combination that is admissible and which also has the highest estimate of efficacy, i.e., 1 -Ŝ(T ), selected from the set A ∪ B. If several such optimal combinations exist, e.g., the efficacy has reached a plateau with respect to the dose level of the MTA, we select the one with the lowest toxicity probability (e.g., the optimal combination with the lowest MTA dose level) to treat the new cohort. At any time, if all combinations are not admissible, then we terminate the trial; otherwise, we continue this dose assignment process until the maximum sample size is reached. At the end of the trial, we select the ODC as the admissible combination that has the highest estimate of efficacy along with the lowest estimate of toxicity.

Numerical Studies

Simulation study

We carried out extensive simulations to evaluate the operating characteristics of the proposed phase I/II design. Taking the setting of the aforementioned solid tumor trial, we assumed 3 dose levels for cytotoxic agent A (i.e., paclitaxel) and 4 dose levels for molecularly targeted agent B (i.e., imatinib), resulting in a total of 12 combinations. We took the initial guesses of the single-agent toxicity and efficacy as (0.2, 0.3, 0.4) and (0.3, 0.4, 0.5), respectively, for agent A, and (0.12, 0.2, 0.3, 0.4) and (0.3, 0.4, 0.5, 0.59) for agent B. The maximum sample size was 75 and patients were treated sequentially in cohorts of size 3.

We assumed that the patient accrual followed a Poisson process with the rate of 1/3.5 patients per week. The toxicity upper bound was θ = 0.30 and the efficacy lower bound was ξ = 0.20. We set the toxicity threshold as C T = 0.85 and the efficacy threshold as C E = 0.10, and took the prior probabilities of the dose-efficacy curve reaching a plateau at the different dose levels of agent B as (0.16, 0.21, 0.27, 0.36). We considered 8 different dose-toxicity and efficacy scenarios (see Table 2), representing what we may encounter in practice. We assumed that toxicity was quickly evaluable, while the evaluation of efficacy required 7 weeks, i.e., T = 7 weeks. Under each scenario, we assumed that at each combination, the time to efficacy followed an exponential distribution. The parameter of the exponential distribution was chosen such that at the end of follow-up, the efficacy rate of each dose combination (i.e., 1-S jk (T )) matched those displayed in Table 2. As a result, the parameter of the exponential distribution had to vary across doses. Under each scenario, we conducted 1,000 simulations.

Table 3 shows the simulation results, including the ODC selection percentage, the average number of patients assigned to the ODC, and the average number of dose-limiting toxicities (DLTs). We also reported the "effective dose combination" (EDC) selection percentage, defining the EDCs as the admissible combinations that yield the same (highest) efficacy as the ODC and which also have acceptable toxicity, i.e., the toxicity probability of the EDC is not necessarily the lowest among several equally efficacious combinations. For example, in scenario 2 in are EDCs as they are both safe and efficacious. Although the ODC is optimal, in practice, the EDCs are also of interest due to their high efficacy even though their dose of agent B may be higher than what is actually needed. Note that under our definitions, the ODC is one of the EDCs, but not vice versa. Table 4 provides more detailed simulation results for the selection percentages and the number of patients treated at each dose combination.

In general, the proposed design performed well across 8 scenarios. The ODC and EDC selection percentages were generally greater than 50%, and the design allocated a large number of patients to the ODC and EDCs. Specifically, in scenario 1, the dose-efficacy curve (approximately) plateaus at the lowest dose level of the targeted agent B, and the ODC is the combination (3, 1), which yields the highest efficacy with the lowest dose of the targeted agent. This ODC is also the only EDC in scenario 1. To mimic what may happen in practice, we designed the scenarios to allow for some variation in efficacy, even when it has reached the plateau. The proposed design selected the ODC 75.6% of the time, and allocated on average 35.1 patients to the ODC. As in scenario 1, in scenario 2, the dose-efficacy curve plateaued from the lowest dose level of the targeted agent B with (3, 1)

as the ODC, but with two EDCs, i.e., (3, 1) and (3, 2). Note that combinations (3, 3) and

(3, 4) have high efficacy probabilities that are similar to that of the ODC (3, 1), but they are not EDCs because they are not admissible combinations due to high toxicity. In this case, the proposed design selected the ODC and EDCs 62.0% and 94.0% of the time. In scenario 3, the dose-efficacy curve plateaus after dose level 1 of agent B. The ODC and EDC selection percentages in that scenario were 51.3% and 96.9%, respectively. Scenarios 4 and 5 both have efficacy plateaus after dose level 2 of agent B, but with different locations for the ODC and EDCs. In these two cases, the ODC selection percentages were more than 40%. Scenarios 6 to 8 simulate efficacy monotonically increasing with the dose of agent B (e.g., agent B does not reach a level of saturation within the range of the investigational doses), with different numbers for the ODCs, which is similar to what may happen in conventional combination trials with two cytotoxic agents. The simulations demonstrate that our proposed design performed well and achieved ODC and EDC selection percentages that were all higher than 60%, suggesting that the proposed design can also be applied to the combination of two cytotoxic agents.

Sensitivity analysis

We performed sensitivity analyses to study the robustness of our design. We varied four factors: (1) the prior estimates of the single-agent toxicity and efficacy probabilities;

(2) the distribution of time to efficacy; (3) the variance of the prior distribution for β 0 , β 1 , β 2 , λ 0 , γ 1 , andγ 2 ; and (4) the prior distribution for τ . We assumed that the single-agent toxicity and efficacy probabilities, used to determine the "effective" dose in the toxicity and efficacy model, were (0.06, 0.12, 0.2) and (0.12, 0.2, 0.3) for agent A, and (0.06, 0.12, 0.2, 0.3) and (0.4, 0.5, 0.59, 0.67) for agent B. We simulated the time to efficacy from a Weibull distribution with a fixed shape parameter of 3. We chose the scale parameter of the Weibull distribu-tion such that the efficacy probabilities at the end of follow-up, i.e., 1-S(T ), matched those given in Table 2. We inflated the prior variances of parameters β 0 , β 1 , β 2 , λ 0 , γ 1 , and γ 2 by 2-fold, and used π = (0.11, 0.17, 0.28, 0.44) as the prior probabilities of τ . Table 5 shows the results of the sensitivity analyses. We can see that the ODC and EDC selection percentages and the number of patients treated at the ODC and EDCs are generally similar to those reported in Table 3, which suggests that the proposed design is not sensitive to the aforementioned design factors.

Application

We retrospectively applied our design to the solid tumor trial. As described previously, the trial selectively studied six dose combinations out of 12 possible combinations. Because the dosing schedule used in the original protocol resulted in too many toxicities, the protocol was amended to use a less intensive dose schedule. As a result, five dose combinations were actually used for dose finding under the amended schedule, as shown in Table 6. The window for assessing treatment response was set at T = 13 weeks. The trial did not report the time to response; thus, we assumed that it was uniformly distributed within the assessment window. To be consistent with the "3+3" method used by the trial, we set C T = 0.33 and C E = 0.0, and forbade skipping untried doses during the dose escalation.

The trial started by treating the first cohort of 3 patients at the lowest dose combination (60, 400), at which one response and no dose-limiting toxicity (DLT) was observed. Based on the data, our method identified dose combination (100, 600) as the ODC, with an estimated response rate of 0.54, and thus recommended dose escalation to (80, 400) for treating the second cohort of patients. Among the three patients treated at (80, 400), two responded to the treatment and no DLT was observed. In light of this new information, our method estimated combination (100, 600) as the ODC, with an estimated response rate of 0.56. Accordingly, we escalated the dose to (80, 600) for treating the third cohort of 3 patients, among which 2 patients responded to the treatment and no DLT was observed.

Our method then escalated the dose and assigned the fourth cohort to dose combination (100, 600), at which we observed 2 responses and no DLT. At that moment, the estimated ODC was dose combination (100, 600), with the estimated response rate of 0.45. Based on this result, our method would retain the current dose and assign the remaining 6 patients to (100, 600); whereas the "3+3" design dictated a dose escalation to (100, 800). At the end of the trial, our design selected (100, 600) as the ODC, while the "3+3" design picked (100, 800). According to the literature, a dose of 600mg of imatinib reaches the plateau of the dose-response curve and actually is the dosage that has been widely administered to cancer patients in practice. It seems that our design successfully identified that, while the "3+3" design might have resulted in overdosing of patients by selecting a dose of 800mg.

Conclusions

We have proposed a Bayesian phase I/II design for trials that combine a cytotoxic agent with a molecularly targeted agent. We assumed that toxicity is quickly evaluable and used a logistic regression model to evaluate toxicity as a binary outcome. In contrast, we assumed that efficacy takes a relatively long time to evaluate, and correspondingly used a proportional hazard model to evaluate efficacy as a time-to-event outcome. To account for the characteristic dose-efficacy curve for MTAs, which initially increases and then plateaus, we incorporated a plateau point into the time-to-efficacy model. During the trial conduct, we continuously updated the model estimates and used them to assign patients to the ODC.

We evaluated our design through a simulation study under various practical scenarios. Our design performed well by selecting the optimal dose combination a high percentage of the time.

The proposed design assumes that the treatment response can be observed anytime during the followup period. For some clinical trials, the response however can only be ascertained at the end of followup, for example, when the response is defined as a certain percentage of tumor shrinkage at time T . In these cases, rather than modeling the time to response, we can model the time to disease progression (i.e., time to nonresponse), which typically is observable in real time based on patient's symptoms. The proposed model and design can still be used. We just need to treat t as the time to nonresponse, and accordingly estimate the response rate q jk at T by S jk (T ), rather than 1 -S jk (T ).

There are several possible extensions of the proposed design to further improve its performance or flexibility in order to accommodate different clinical applications. For example, rather than modeling toxicity as a binary outcome, we can use measurements of the various grades of toxicity and model it as an ordinary outcome. This approach uses more refined information and can potentially improve the efficiency of the trial design. In addition, when late-onset toxicity is of concern, we can model toxicity as a time-to-event outcome, as we have done for efficacy. Lastly, our dose assignment and selection criteria focus on efficacy while controlling for toxicity. In some applications, it may be more appropriate to consider the tradeoff between toxicity and efficacy. To accommodate these cases, we can define a utility function for the tradeoff between toxicity and efficacy and then use the utility as the criterion for dose assignment and selection.

  Specifically, let pj0 and p0k denote the estimates of the single-agent toxicity probabilities for the jth level of agent A and the kth level of agent B, respectively, and qj0 ≡ 1 -Ŝj0 (T ) and q0k ≡ 1 -Ŝ0k (T ) denote the estimates of the single-agent efficacy probabilities for the jth level of agent A and the kth level of agent B (at the end of follow-up). Under toxicity model (1), by setting the dosage of agent B (or A) as zero, we obtain the single-agent toxicity model logit(p j0 ) = β 0 + β 1 u j for agent A and logit(p 0k ) = β 0 + β 2 v k for agent B.

Figure 1 .

 1 Figure 1. An illustration of combination zones for the start-up phase.

Table 2

 2

	, both (3, 1) and (3, 2) have the same high efficacy rate of
	55% and acceptable toxicity rates not higher than 30%. The ODC is (3, 1) as it has a lower
	toxicity probability or lower MTA dose level (i.e., agent B); whereas both (3, 1) and (3, 2)

Table 1 .

 1 Prior distributions for model parameters.

	Parameter	Prior distribution
	β0	N (0, 100)
	β1, β2	Exp(1)
	λ0	Exp(0.01)
	γ1, γ2	Exp(0.1)
	τ	M ultinomial(π1, ..., πK ) with π1 < ... < πK

Table 2 .

 2 Eight toxicity and efficacy scenarios for the combination of a cytotoxic agent (agent A) with a molecularly targeted agent (agent B). The optimal dose combinations (ODCs) are in bold and the effective dose combinations (EDCs) are underlined. The dashed lines indicate the dose level of the MTA at which the efficacy plateaus.

					Agent A					Agent A
	Agent	1	2	3	1	2	3	1	2	3	1	2	3
	B		Toxicity			Efficacy			Toxicity			Efficacy
				Scenario 1					Scenario 2
	4	0.45	0.50	0.65	0.27	0.42	0.56	0.30	0.45	0.55	0.28	0.43	0.57
	3	0.30	0.45	0.55	0.26	0.41	0.56	0.15	0.30	0.45	0.27	0.42	0.56
	2	0.15	0.30	0.45	0.25	0.41	0.55	0.12	0.15	0.30	0.26	0.40	0.55
	1	0.10	0.15 0.30	0.25	0.40 0.55	0.08	0.10 0.15	0.25	0.40 0.55
				Scenario 3					Scenario 4
	4	0.10	0.15	0.30	0.37	0.52	0.67	0.10	0.15	0.30	0.32	0.46	0.61
	3	0.07	0.10	0.15	0.36	0.51	0.66	0.08	0.10 0.15	0.30	0.45 0.60
	2	0.05	0.08 0.10	0.35	0.50 0.65	0.04	0.05	0.10	0.20	0.30	0.40
	1	0.01	0.05	0.07	0.15	0.25	0.35	0.01	0.03	0.07	0.05	0.10	0.20
				Scenario 5					Scenario 6
	4	0.30	0.55	0.65	0.41	0.56	0.67	0.10	0.15 0.30	0.40	0.55 0.70
	3	0.15 0.45	0.55	0.40 0.55	0.65	0.07	0.10	0.15	0.25	0.40	0.55
	2	0.10	0.30	0.45	0.15	0.20	0.25	0.05	0.08	0.10	0.15	0.30	0.40
	1	0.05	0.15	0.30	0.05	0.10	0.15	0.01	0.05	0.07	0.05	0.10	0.30
				Scenario 7					Scenario 8
	4	0.50	0.60	0.65	0.50	0.60	0.70	0.45	0.55	0.65	0.50	0.60	0.70
	3	0.45	0.55	0.60	0.45	0.50	0.60	0.30 0.45	0.60	0.45 0.52	0.63
	2	0.30 0.45	0.50	0.30 0.45	0.50	0.15 0.30 0.50	0.30 0.45 0.50
	1	0.15 0.30 0.45	0.15 0.30 0.45	0.10	0.15 0.30	0.20	0.30 0.45

Table 3 .

 3 Selection percentage of the optimal dose combination (ODC) and effective dose combination (EDC), the average number of patients treated at the ODC and EDC, and the average number of dose-limiting toxicities (DLTs).

					Scenarios
		1	2	3	4	5	6	7	8
	ODC selection percentage	75.6 62.0 51.3 51.3 42.1 66.4 62.8 78.3
	Mean number of patients at ODC 35.1 25.1 17.4 18.1 14.4 23.1 32.5 40.9
	Mean number of DLTs	22.5 16.7	9.4	10.8 21.9 12.0 25.8 23.7
	EDC selection percentage	75.6 94.0 96.9 83.8 68.1 66.4 62.8 78.3
	Mean number of patients at EDC 35.1 43.9 42.2 34.4 28.8 23.1 32.5 40.9

Table 4 .

 4 ODC (in boldface) and EDC (underlined) selection percentages and the average number of patients treated at each combination.

		Selection percentage				Average number of patients
		Scenario 1		Scenario 2			Scenario 1			Scenario 2
	0.4	0.1	0.1	0.2	0.2	0.0	2.0	0.2	0.2	3.0	0.9	1.2
	0.9	1.0	0.0	0.1	1.0	3.6	4.0	1.3	1.3	3.4	2.7	5.2
	0.5	7.4	6.0	0.0	0.8	32.0	3.9	5.8	10.3	3.6	3.5	18.8
	0.1	7.9	75.6	0.0	0.0	62.0	4.0	6.9	35.1	3.7	3.8 25.1
		Scenario 3		Scenario 4			Scenario 3			Scenario 4
	0.0	0.0	16.7	0.0	0.8	32.5	3.5	2.9	10.3	3.5	3.3	16.3
	0.0	0.0	28.9	0.0	0.1	51.3	3.4	3.3	14.5	3.6	3.4 18.1
	0.0	0.0	51.3	0.0	0.0	13.9	3.3	3.5	17.4	3.4	3.3	9.4
	0.0	0.0	3.1	0.0	0.0	1.4	3.1	3.4	6.5	3.1	3.2	4.5
		Scenario 5		Scenario 6			Scenario 5			Scenario 6
	26.0	0.8	0.0	0.0	1.2	66.4	14.4	3.3	0.9	3.4	3.2 23.1
	42.1 16.5	1.3	0.0	0.0	19.8	14.4 10.0	2.6	3.5	3.2	13.1
	0.1	2.9	3.9	0.0	0.0	9.1	4.4	5.5	5.1	3.4	3.4	7.1
	0.0	0.1	5.1	0.0	0.0	3.5	3.4	4.2	6.6	3.1	3.4	5.1
		Scenario 7		Scenario 8			Scenario 7			Scenario 8
	1.8	0.0	0.0	4.8	0.3	0.0	2.6	0.2	0.1	4.8	0.7	0.3
	8.0	0.1	0.0	16.3	3.1	0.0	7.8	0.7	0.3	9.4 3.3	1.4
	27.7	3.6	1.1	3.4	19.5	7.2	14.9	4.1	2.3	5.3 9.5	8.6
	2.7	35.1 13.0	0.0	2.8	42.5	7.1	17.6 14.9	3.8	5.7 22.0

Table 5 .

 5 Results of the sensitivity analysis. Mean number of patients at EDC 34.8 43.9 41.9 34.4 29.8 23.0 33.9 41.8 Time to efficacy following a Weibull distribution ODC selection percentage 78.3 72.0 64.0 47.1 38.1 49.4 64.1 78.5 Mean number of patients at ODC 78.3 96.2 92.6 65.0 56.5 49.4 64.1 78.5

					Scenarios
		1	2	3	4	5	6	7	8
	Different prior estimates for single agents
	ODC selection percentage	74.7 63.4 50.6 56.9 44.2 65.4 62.2 79.7
	Mean number of patients at ODC 74.7 93.6 96.4 85.0 70.0 65.4 62.2 79.7
	Mean number of DLTs	0.4	0.0	0.0	0.0	1.8	0.0	7.4	0.3
	EDC selection percentage	34.8 25.6 17.1 19.1 15.0 23.0 33.9 41.8
	Mean number of DLTs	0.4	0.2	0.0	0.0	0.0	0.0	2.3	0.2
	EDC selection percentage	35.7 27.7 20.2 17.3 12.8 16.9 32.8 41.0
	Mean number of patients at EDC 35.

Acknowledgements

This work was partially funded by grants from the ANRT (Association Nationale de la Recherche et de la Technologie), "Laboratoires Servier" (CIFRE number 2011/0900), and

Denis Diderot University-Paris 7 ("Aide à la mobilité internationale 2012"). Yuan's research was partially supported by Award Number R01 CA154591 and P50 CA098258 from the National Cancer Institute.


Marie-Karelle Riviere
email: marie-karelle.riviere@crc.jussieu.fr

C Le Tourneau

X Paoletti

F Dubois

Sarah Zohar

M-K Riviere

Ms Marie-Karelle Riviere

Designs of drug-combination phase I trials in oncology: a systematic review of the literature

Keywords: Drug combinations, Phase I trials, Dose-finding

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

RESULTS

Among the 847 references retrieved, 162 papers reported drug-combination phase I trials in which at least two agents were dose-escalated. In 88% of trials, a traditional or modified 3+3 dose escalation design was used. All except one trial used a design developed for single-agent evaluation. Our study suggests that drug-combination phase I trials in oncology are very safe, as revealed by the calculated median DLT rate of 6% at the recommended dose, which is far below the target rate in these trials (33%).

We also examined requirements of phase I clinical trials in oncology with drug combinations and the potential advantages of novel approaches in early phases.

CONCLUSION Introduction

Phase I trials in oncology are dose-finding studies that seek to determine the dose to recommend for further evaluation (recommended phase II dose [RP2D]). These trials are designed to obtain reliable information on the safety, pharmacokinetics, and mechanism of action of a drug. In oncology, dose-finding studies focus on determining the highest dose of a new drug with acceptable toxicity [START_REF] Storer | Design and analysis of phase I clinical trials[END_REF][START_REF] Cheung | Dose Finding by the Continual Reassessment Method[END_REF]. They are subject to the ethical constraint of minimizing the number of subjects treated at unacceptable toxic dose levels. Toxicity is measured as a binary endpoint, denoted as "dose limiting toxicity" (DLT), mainly using National Cancer Institute Common Toxicity Criteria. Most dose-escalation methods were developed for cytotoxic agents with the assumption that toxicity increases with dose in a monotonic fashion.

Therefore, the RP2D has traditionally been the highest safe dose, called the "maximum tolerated dose" (MTD). These methods were specifically designed for the evaluation of single agents. In clinical practice, the traditional "3+3" dose escalation design or a modification thereof are the most frequently used dose-escalation methods in phase I trials [START_REF] Tourneau | Dose escalation methods in phase I cancer clinical trials[END_REF].

Drug combinations have been introduced with the goal of improving treatment efficacy by increasing overall anti-tumor activity and, presumably, survival.

Successful drug combinations include a combination of cytotoxic agents for the treatment of germ-cell tumors and lymphoma, polychemotherapy for the treatment of germ-cell tumors [START_REF] Curigliano | Cisplatin, etoposide and continuous infusion bleomycin in patients with testicular germ cell tumors: efficacy and toxicity data from a retrospective study[END_REF][START_REF] Bamias | Report of the long-term efficacy of two cycles of adjuvant bleomycin/etoposide/cisplatin in patients with stage I testicular nonseminomatous germ-cell tumors (NSGCT): a risk adapted protocol of the Hellenic Cooperative Oncology Group[END_REF], combinations of trastuzumab with a taxane for HER2-positive breast cancer [START_REF] Slamon | Use of chemotherapy plus a monoclonal antibody against HER2 for metastatic breast cancer that overexpresses HER2[END_REF], and a combination of BRAF and MEK inhibitors [START_REF] Flaherty | Combined BRAF and MEK inhibition in melanoma with BRAF V600 mutations[END_REF]. Although it can reasonably be assumed that toxicity increases with dose for a single drug, the determination of the relationship between toxicity and doses of multiple drugs remains elusive. When combining two or more agents, there may be a potential synergistic effect, not only in terms of efficacy, but also in terms of toxicity [START_REF] Soria | From theoretical synergy to clinical supraadditive toxicity[END_REF].

Therefore, when combining several agents, the ordering between combinations according to their DLT rates is important. However, only partial ordering of DLT rates can be anticipated when the dose of only one drug is being escalated, whereas the dose of the other drugs in the combination is kept fixed (Figure 1). That is, referring to Figure 1a, in a row (or column), one agent is fixed while the other is increased. In this case, the DLT rates are increased with the dose of the agent. All these order relations in rows and columns (shown with the symbol inferior "<") lead to "a partial ordering of DLT rates" given in Figure 1a. For example, if 2 agents with 3 dose levels are considered, when a monotonic and increasing relationship is assumed with respect to both agents then a partial toxicity order is known between the 9 combinations. The lowest combination is dose level 1 of agent 1 combined to dose level 1 of agent 2 (1,1) and the highest combination is dose level 3 of agent 3 combined with dose level 3 of agent 2 [START_REF] Tourneau | Dose escalation methods in phase I cancer clinical trials[END_REF][START_REF] Tourneau | Dose escalation methods in phase I cancer clinical trials[END_REF]. Presumably, combination (1,2) is less toxic than (2,2), which is also presumably less toxic than [START_REF] Tourneau | Dose escalation methods in phase I cancer clinical trials[END_REF][START_REF] Cheung | Dose Finding by the Continual Reassessment Method[END_REF], etc. However, on a diagonal, when the dose of one agent is increased while the dose of the other is decreased, it is not known which combination is more toxic. For instance, is the (1,2) combination more or less toxic than the (2,1) combination? Therefore, several toxicity orderings between combinations are possible (two examples are given in Figures 1b and1c).

In practice, drug-combination phase I trials raise several challenging points to be defined prior to the trial onset [START_REF] Mandrekar | Dose-finding trial designs for combination therapies in oncology[END_REF][START_REF] Adjei | What is the right dose? The elusive optimal biologic dose in phase I clinical trials[END_REF][START_REF] Cannistra | Challenges and pitfalls of combining agents in phase I studies[END_REF][START_REF] Hamberg | Dose-escalation models for combination phase I trials in oncology[END_REF][START_REF] Harrington | Adaptive designs for dual-agent phase I dose-escalation studies[END_REF][START_REF] Jain | Phase I oncology studies: evidence that in the era of targeted therapies patients on lower doses do not fare worse[END_REF][START_REF] Lorusso | An overview of the optimal planning, design, and conduct of phase I studies of new therapeutics[END_REF][START_REF] Booth | Endpoints and other considerations in phase I studies of targeted anticancer therapy: recommendations from the task force on Methodology for the Development of Innovative Cancer Therapies (MDICT)[END_REF][START_REF] Seymour | Design and conduct of early clinical studies of two or more targeted anticancer therapies: recommendations from the task force on Methodology for the Development of Innovative Cancer Therapies[END_REF]: 1) starting dose of each agent; 2) choice of the dose range of each agent and the number of combinations to be evaluated; and

3) total sample size that is strongly related to the number of possible combinations. In this study, we aimed at evaluating how drug-combination phase I trials in oncology have been designed in the last three years and what the principal investigator's choices were with regard to the dose range, number of combinations, and statistical design.

Material and methods

All drug-combination phase I trials published between January 1, 2011 and December 31, 2013 were reviewed (Figure 2). We restricted our review to phase I combination trials where at least two drugs were planned to undergo dose escalation.

Trials involving radiation therapy or drugs other than cytotoxic agents and molecularly targeted agents (MTAs) were excluded. MTAs were defined in our review as anticancer agents that selectively target molecular pathways, as opposed to DNA, tubulin or cell division machinery. Hormonal therapy and biological therapeutics, such as immunotherapy, were included.

We performed a MEDLINE® PubMed search using the following terms:

"Clinical Trial, Phase I[ptyp] AND cancer[MeSH] AND "2011/01/01"[PDAT] : "2013/12/31"[PDAT AND (combination OR combine OR combined OR combining)".

Among 847 references retrieved, 162 papers reported on a drug combination phase I trial meeting our inclusion criteria, 381 papers involved drug combinations where only one agent was dose-escalated, while the others were fixed (Figure 2).

The following data were recorded: the number of drugs undergoing dose escalation, the types of drugs (cytotoxic agent versus MTA), the number of dose levels planned for each drug, justification of the starting doses, number, choice and justification of drug combinations, dose-escalation design used, addition of drug combinations during the trial, number of patients included, and target toxicity level.

We also performed a quality control analysis of the reviewed papers.

In this review, the lowest combination is defined as the combination corresponding to the lowest dose levels planned of each agent. A monotonic and increasing dose-toxicity relationship with respect to both agents signifies that when fixing one agent or the other to a certain dose independently, the DLT rate of the combination increases with the dose level of the remaining agent.

Results

Characteristics of the drugs

The 162 phase I trials involved 340 drugs that underwent dose escalation. In the majority of the trials, only 2 drugs underwent dose escalation (Table 1). Trials that involved only cytotoxic agents, only MTAs, and a combination of cytotoxic agents and MTAs were roughly equally distributed.

Dose levels

The median number of patients included per trial was 25 [range: 7-136] (Table 1). In 69% of cases, the starting combination in the trial was the one associated with the lowest dose level of each agent considered in the trial. The starting dose used in the trial was justified (short explanation or only references) in 35% of the trials, respectively (Figure 3). The dose levels of each agent involved in the combinations of the clinical trial were justified in only 47 publications (29%). Results of a quality control analysis are provided in Figure 3.

Dose combinations

The median number of planned combinations in the trial protocol was 5 [range: 2-16], The median ratio of the number of planned combinations to the number of possible combinations (defined as the number of planned combinations divided by product of the number of doses levels of each agent) was 0.67 [range: 0.25-1], 0.24 [range: 0.17-0.63] and 0.13 [range: 0.13-0.13] in trials combining two, three, and four drugs, respectively.

Dose escalation method

In most trials, a traditional 3+3 or a modified 3+3 dose escalation design was used (Table 1). Only one trial used a design developed for combination trials. Most of the selected papers assumed a monotonic and increasing dose-toxicity relationship, in 62% of trials, whereas 38% of papers assumed only a partial monotonic and increasing dose-toxicity relationship.

In 24% of the trials, additional drug combinations were evaluated during the trial for safety reasons.

Safety

The DLT target rate associated with the recommended dose was 33% in most studies (Table 1). However, according to the number of patients and DLTs reported at the RP2D, the calculated median DLT rate at the recommended dose was 6%

[range: 0%-40%]. Nevertheless, in only 4% of trials was the DTL rate estimated by the authors at the recommended combination for further studies.

In 3% of the studies, the trial was stopped at the first dose level due to DLTs.

Five trials were stopped for reasons relating to over-toxicity; that is, the lowest combination evaluated in the trial was considered too toxic and the trial was abruptly halted without finding a tolerable combination. Fifty-six per cent of the trials found the MTD according to its initial definition, and 11% of trials found an MTD without observing any DLTs throughout the trial. In 48% of trials, the progress observed in the trial did not match the initial planned method. The trials that did not match the intended plan were all 3+3 or modified 3+3 statistical designs. The main observed differences from planning were: (1) difference in the planned number of patients per cohort with no justification and (2) a different allocation rule during the trial.

Discussion

Our study suggests that drug-combination phase I trials in oncology are safe. Overall, however, the starting doses of the drugs in the trials reviewed, as well as the dose levels and the dose-escalation steps, were barely justified. In addition, the dose levels explored in the drug-combination phase I trials included in our study did not reflect the entire space of possible drug combinations. In most of cases, dose levels seemed to be arbitrarily decided. It remains to be evaluated whether non-explored drug combinations would have been able to produce increased anti-tumor activity without jeopardizing patient safety.

Only a limited number of combinations were explored and only a sub-set of combinations was evaluated, despite the larger number of possible combinations. In our MEDLINE® PubMed search, the median ratio of the number of combinations considered to the number of possible combinations indicated that approximately onethird of the combinations were not considered for two-drug combinations. This indicates that trial investigators may have selected the combinations to be evaluated prior to the trial, and that some combinations were excluded without documented rationale. Exploring the entire combination space is obviously not feasible in practice.

Nevertheless, the choice of the combinations to explore should not be limited by partial toxicity ordering. The design should have the possibility to explore combination estimated to be the best. In fact, due to possible interactions between drugs, pre-selecting an arbitrary reduced sub-set of combinations induces a risk in selecting a combination with a DLT rate far from target toxicity. Even if the targeted DLT proportion was most often about 33% in the papers, the median DLT rate associated with the RP2D at the end of the trial was much lower. That could be a reason why an intermediate combination was added, in some cases, which induced a non-monotonic dose-toxicity relationship in some trials.

During the review of this paper, the question was raised whether the low DLT rate could be due to MTAs for which the toxicity profile is different. Indeed, for these noncytotoxic agents, very low toxicities are often observed with sometimes cumulative low-grade toxicities that may become dose limiting. The cumulative low-grade toxicities partially explain deviance from the intended plan. An FDA guideline [START_REF]Guidance for Industry, Clinical Considerations for Therapeutic Cancer Vaccines[END_REF] reported: "…cancer vaccine trials have used the "3 + 3 design" and the results show that, except in very rare situations, an MTD for a cancer vaccine may not be identified. In these trials, the dose-toxicity curve may be so flat that the highest dose that can be administered is limited by manufacturing or anatomic issues rather than toxicity. Therefore, this "3+3 design" may not be the most suitable approach to gathering information from early phase trials of cancer vaccines, and alternative trial designs should be considered." They added that: "When no DLT is expected or achieved, optimization of other outcomes, such as the immune response, can be useful to identify doses for subsequent studies".

For this reason, standard dose-finding designs dealing only with toxicity, such as the "3+3" do not seem appropriate for some biological agents [START_REF] Tourneau | Efficiency of new dose escalation designs in dose-finding phase I trials of molecularly targeted agents[END_REF]. First, it is true that the dose determination based on less than 33% DLT on the first cycle of treatment for molecularly targeted agents is problematic. These non-cytotoxic agents have different toxicity profiles than cytotoxic agents. One possible reason for the observed low DLT rate at the RP2D could be due to the DLT evaluation only on the first cycle of treatment. Physicians can observe no DLT on the first cycle but cumulative lowgrade toxicities that become dose limiting with later cycles of treatment. For this reason, they decrease the recommended dose level for phase II (in contrast to the statistical design), rendering a low DLT rate (evaluated only on the first cycle) for this dose. All cumulative toxicity grades on all available cycles should be considered in the statistical analysis for dose recommendations. Furthermore, depending on the biological agent, several dose-efficacy relationships could be observed: (1) monotonic and increasing; (2) monotonic increasing and then reaching a plateau; and (3) monotonic increasing and then decreasing with the dose. In the latter two cases, only studying toxicity in the dose-finding process is not sufficient, and efficacy should also be considered. Therefore, alternative designs should be developed.

Adapting the way of doing early phase clinical trials for these innovative molecules is important, but changing usual practices in oncology is very complex and difficult. If regulatory agencies were to give clear instructions, trial sponsors and investigators would need to apply them. There are published statistical designs proposing alternative methods [START_REF] Thall | Dose-finding based on efficacy-toxicity trade-offs[END_REF][START_REF] Doussau | Dose-finding design using mixed-effect proportional odds model for longitudinal graded toxicity data in phase I oncology clinical trials[END_REF][START_REF] Zhang | An adaptive dose-finding design incorporating both toxicity and efficacy[END_REF], therefore statistics should not be a limited factor. However, in calculating the median DLT rate for trials in which the combination involved cytotoxic agents, we observed a DLT rate of 4%. Therefore, we do not believe that this is due to the type of agent but rather to the use of the "3+3" algorithm, where the dose retained is the dose under 2 DLTs over 3 or 6 patients. Indeed, in the trials studied, either the combination level was associated with no (or very few) DLTs, or the highest dose level in the trial did not even reach the target toxicity. Thus, due to the small number of patients (3 or 6) with the "3+3" design, the estimation is unreliable and often close to 0%. It should be noted that combination trials of MTAs included more patients at the RP2D than combination trials of cytotoxic agents, perhaps due to the uncertainty on overall toxicity discussed above. That could explain the small increased difference in DLT rates despite the toxicity profile of MTAs, as the estimation with a greater number of patients is more reliable.

Most of the drug-combination phase I trial designs used the traditional "3+3" design or a modified version. Recent dose-escalation designs have been developed for drug combinations but never employed in the trials reviewed [START_REF] Huang | A parallel phase I/II clinical trial design for combination therapies[END_REF][START_REF] Wang | Two-dimensional dose finding in discrete dose space[END_REF][START_REF] Mandrekar | An adaptive phase I design for identifying a biologically optimal dose for dual agent drug combinations[END_REF][START_REF] Mandrekar | Model-based phase I designs incorporating toxicity and efficacy for single and dual agent drug combinations: methods and challenges[END_REF][START_REF] Yin | A latent contingency table approach to dose finding for combinations of two agents[END_REF][START_REF] Yin | Bayesian dose finding in oncology for drug combinations by copula regression[END_REF][START_REF] Thall | Dose-finding with two agents in Phase I oncology trials[END_REF][START_REF] Riviere | Competing designs for drug combination in Phase I dose-finding clinical trials[END_REF]. In all but one trial reviewed, the dose-toxicity relationship was considered to be one-dimensional, whereas the reality involved several agents inducing a multi-dimensional issue. Most of the time, the problem was brought back into a one-dimensional space by preselecting combinations with a known toxicity order to be evaluated.

The methods for single agents do not always seem appropriate for combination phase I trials in which the doses of several drugs vary, as they are not designed to take a multi-dimensional space into account. Several alternative designs were proposed for either algorithm-based or design-based combinations that give the possibility to explore any appropriate combination in the entire combination space according to the accumulated data. Ivanova and Wang proposed an "up-and-down algorithm-type" method with isotonic regression [START_REF] Ivanova | A non-parametric approach to the design and analysis of two-dimensional dose-finding trials[END_REF] that was used recently in Gandhi et al. [START_REF] Gandhi | Phase I study of neratinib in combination with temsirolimus in patients with human epidermal growth factor receptor 2-dependent and other solid tumors[END_REF]. Conaway et al. developed a design for multiple agents based on partial orders [START_REF] Conaway | Designs for single-or multiple-agent phase I trials[END_REF] that was used in the publications reviewed in Jones et al. [START_REF] Jones | Phase I trial of induction histone deacetylase and proteasome inhibition followed by surgery in non-small-cell lung cancer[END_REF]. Other authors have proposed model-based designs in which the multi-dimensional feature of the entire combination space is taken into account. These methods allow considering the entire combination space that includes a large number of combinations with non-monotonic relationships. It should be noted that these methods do not permit exploring combinations that are estimated to be too toxic. In a recent comparison, based on simulations, Riviere et al showed that these designs were comparable and had high operational characteristics [START_REF] Riviere | Competing designs for drug combination in Phase I dose-finding clinical trials[END_REF]. However, it is true that these designs have only been shown to be effective in simulation studies (Riviere et al, Stat Med 2014), and they require the involvement of a statistical expert.

In a recent editorial, Mandrekar [START_REF] Mandrekar | Dose-finding trial designs for combination therapies in oncology[END_REF] pointed out the importance of using adequate methods for the evaluation of combinations. Our bibliometric work supports this editorial with a large and detailed study on clinical practice in the phase I settings for combination trials.

Our analysis did not include trials published in abstract form. Although this induces a selection bias, the present analysis still provides useful data that may help improve the design of drug-combination phase I trials.

In our review, we did not stated that both agents must be administered at their single-agent MTD when in combination. As two agents can have a synergistic, antagonistic or independent effect on toxicity, the question of achieving doses (for each agent) that nearly approximate the recommended phase II dose is up for debate. It is a strong assumption that the addition of both agents at their MTD would result in the same toxicity as if administered alone. We believe that considering all

  5.5 [range:[START_REF] Tourneau | Dose escalation methods in phase I cancer clinical trials[END_REF][START_REF] Curigliano | Cisplatin, etoposide and continuous infusion bleomycin in patients with testicular germ cell tumors: efficacy and toxicity data from a retrospective study[END_REF][START_REF] Bamias | Report of the long-term efficacy of two cycles of adjuvant bleomycin/etoposide/cisplatin in patients with stage I testicular nonseminomatous germ-cell tumors (NSGCT): a risk adapted protocol of the Hellenic Cooperative Oncology Group[END_REF][START_REF] Slamon | Use of chemotherapy plus a monoclonal antibody against HER2 for metastatic breast cancer that overexpresses HER2[END_REF][START_REF] Flaherty | Combined BRAF and MEK inhibition in melanoma with BRAF V600 mutations[END_REF][START_REF] Soria | From theoretical synergy to clinical supraadditive toxicity[END_REF][START_REF] Mandrekar | Dose-finding trial designs for combination therapies in oncology[END_REF][START_REF] Adjei | What is the right dose? The elusive optimal biologic dose in phase I clinical trials[END_REF][START_REF] Cannistra | Challenges and pitfalls of combining agents in phase I studies[END_REF][START_REF] Hamberg | Dose-escalation models for combination phase I trials in oncology[END_REF][START_REF] Harrington | Adaptive designs for dual-agent phase I dose-escalation studies[END_REF][START_REF] Jain | Phase I oncology studies: evidence that in the era of targeted therapies patients on lower doses do not fare worse[END_REF][START_REF] Lorusso | An overview of the optimal planning, design, and conduct of phase I studies of new therapeutics[END_REF] and 12 [range: in trials combining two, three, and four drugs, respectively. The median number of actually evaluated combinations was 4 [range: 2-12], 4 [range: 2-9] and 3 [range: 3-3] in trials combining two, three, and four drugs, respectively.

Figure 1 .

 1 Figure 1.

Figure 2 .

 2 Figure 2.

Figure 3 .

 3 Figure 3.

Acknowledgments of research support:

This work was partially funded by grants from the ANRT (Association Nationale de la Recherche et de la Technologie), "Laboratoires Servier"(CIFRE number 2011/0900).

Disclosure:

M-K. Riviere and F. Dubois are employees of IRIS (Institut de Recherches Internationales Servier) Pharmaceutical industry. All remaining authors have declared no conflicts of interest.

combinations of dose levels between the two agents as a possible MTD should be acceptable, under medical restrictions and prior knowledge of such combinations.

The recommended combination at the end of the trial should not be limited to the combination of both single-agent MTDs, but the dose-finding process should be performed similarly to that of a single-agent in order to recommend the combination with a toxicity rate closest to or below a pre-defined target. Indeed, in the same way, combining two agents can also induce a synergistic, antagonistic, or independent effect on overall efficacy. This point should be discussed for each combination of drugs, as the mechanism of action of each agent can differ.

In conclusion, we believe that the design of drug-combination phase I trials in oncology can be improved. We recommend that the starting doses of the drugs, as well as the dose levels and the dose-escalation steps, need to be appropriately justified. These parameters should be determined with the aim to: 1) ensure patient safety; 2) treat as few patients as possible at presumably infra-therapeutic doses; and 3) identify the optimal drug combination for further evaluation. We strongly support the use of innovative designs that are able, at least in theory, to fulfil these requirements.