Two‐stage design for phase I–II cancer clinical trials using continuous dose combinations of cytotoxic agents

Summary

We present a two-stage phase I/II design of a combination of two drugs in cancer clinical trials. The goal is to estimate safe dose combination regions with a desired level of efficacy. In stage I, conditional escalation with overdose control is used to allocate dose combinations to successive cohorts of patients and the maximum tolerated dose curve is estimated as a function of Bayes estimates of the model parameters. In stage II, we propose a Bayesian adaptive design for conducting the phase II trial to determine dose combination regions along the MTD curve with a desired level of efficacy. The methodology is evaluated by extensive simulations and application to a real trial.

1. Introduction

The primary objective of cancer phase I/II clinical trials is to determine a tolerable dose level that maximizes treatment efficacy. For treatments where efficacy is ascertained in a relatively short period of time such as one or two cycles of therapy, sequential designs for updating the joint probability of toxicity and efficacy and estimating the optimal dose have been studied extensively in the literature for single agent trials, see e.g. Murtaugh and Fisher (1990); Thall and Russell (1998); Braun (2002); Ivanova (2003); Thall and Cook (2004); Chen et al. (2015); Sato et al. (2016). These methods are designed to identify a safe dose that maximizes the probability of treatment response or a pre-defined utility function. More recently, these designs have been extended to dose combination trials where the dose levels of two drugs are allowed to vary during the trial and explored for toxicity and efficacy (Yuan and Yin (2011); Wages and Conaway (2014); Cai et al. (2014), Riviere et al. (2015)) or to determine the optimal treatment regimen Wages et al. (2016). These methods accommodate cytotoxic and biologic drugs and are appropriate for short term efficacy endpoints to allow sequential updating of the probabilities of toxicity and efficacy.

For treatments where response evaluation takes few cycles of therapy, it is standard practice to perform a two-stage design where a maximum tolerable dose (MTD) of a new drug or combinations of drugs is first determined, then this recommended phase II dose is studied in stage II and evaluated for treatment efficacy, possibly using a different population of cancer patients from stage I, see Rogatko et al. (2008); Le Tourneau et al. (2009); Chen et al. (2012) for a review of such a paradigm. The two-stage design is routinely used in early phase cancer trials investigating a single agent or combinations of two or more drugs except that in stage II, a single dose level is investigated. For drug combination phase I trials where the dose levels of two or more drugs are allowed to vary during the trial, more than one MTD can be recommended at the conclusion of the trial, e.g. Thall et al. (2003); Wang and Ivanova (2005); Braun and Wang (2010); Tighiouart et al. (2014); Mander and Sweeting (2015); Wages (2016); Tighiouart et al. (2017). Choosing a single MTD combination for efficacy study may result in a failed phase II trial since other MTDs may present higher treatment efficacy. Hence, adaptive or parallel phase II trials may be more suitable for searching an optimal dose combination that is well tolerable with desired level of efficacy as mentioned by Thall et al. (2003).

In this article, we present a two-stage design for drug combination trials with continuous dose levels when treatment efficacy is evaluated after three or more cycles of therapy. Examples of early phase trials that use continuous dose levels can be found in Cheng et al. (2004); Borghaei et al. (2009). In the first stage, the method of Tighiouart et al. (2017) is used to estimated the MTD curve. In stage II, a Bayesian adaptive design is proposed to allocate patients to drug combinations along the MTD curve obtained from stage I with the goal of determining a dose with maximum probability of treatment efficacy. To accommodate various functional forms of the efficacy curve, a flexible family of cubic splines is used to model the dose-response relationship. Adaptive randomization is used sequentially after the efficacy status of each cohort of patients is resolved and the efficacy curve is updated in order to minimize the number of patients treated at sub-therapeutic dose combinations. At the end of the trial, a dose combination with maximum probability of response a posteriori is selected for further phase III studies. The method is illustrated by an application to a phase I/II trial of the combination of cisplatin and cabazitaxel (ciscab) in patients with advanced prostate cancer. A previous phase I trial with dose expansion cohort explored three dose combinations of these agents and declared the combination cabazitaxel/cisplatin of 15/75 mg/m² to be the MTD although no DLT was observed at this dose in the phase I part of the study. Based on safety and efficacy data from the phase I stud and expansion cohort, we hypothesize that a series of tolerable and active dose combinations exist.

The manuscript is organized as follows. In Section 2, we review the stage I design described in Tighiouart et al. (2017) for estimating the MTD curve. Modeling the efficacy curve along the MTD curve and the adaptive design in stage II of the trial are described in Section 3 along with operating characteristics to evaluate the performance of the phase II trial. In Section 4, we apply this methodology to the ciscab trial described above with the goal of finding a tolerable dose combination with maximum benefit rate. Some concluding remarks and a discussion are found in Section 5.

2. Stage I

2.1. Model

Consider the dose-toxicity model of the form

$$P(T=1\mid x,y)=F({\eta }_0+{\eta }_{1}x+{\eta }_{2}y+{\eta }_{3}xy),$$ (1)

where T is the indicator of DLT, T = 1 if a patient given the dose combination (x, y) exhibits DLT within one cycle of therapy, and T = 0 otherwise, x ∈ [X_min,X_max] is the dose level of agent A₁, y ∈ [Y_min, Y_max] is the dose level of agent A₂, and F is a known cumulative distribution function. Here, X_min,X_max and Y_min, Y_max are the lower and upper bounds of the continuous dose levels of agents A₁ and A₂, respectively. Suppose that the doses of agents A₁ and A₂ are continuous and standardized to be in the interval [0, 1] using the transformations h₁(x) = (x − X_min)/(X_max − X_min), h₂(y) = (y − Y_min)/(Y_max − Y_min) and the interaction parameter η₃ > 0.

We will assume that that the probability of DLT increases with the dose of any one of the agents when the other one is held constant. A necessary and sufficient condition for this property to hold is to assume η₁, η₂ > 0. The MTD is defined as any dose combination (x^*, y^*) such that

$$\text{Prob}(T=1\mid x^{\ast },y^{\ast })=\theta .$$ (2)

The target probability of DLT θ is set relatively high when the DLT is a reversible or non-fatal condition, and low when it is life threatening. We reparameterize model (1) in terms of parameters clinicians can easily interpret. One way is to use ρ₁₀, the probability of DLT when the levels of drugs A₁ and A₂ are 1 and 0, respectively, ρ₀₁, the probability of DLT when the levels of drugs A₁ and A₂ are 0 and 1, respectively, and ρ₀₀, the probability of DLT when the levels of drugs A₁ and A₂ are both 0. It can then be shown that the MTD is

$$C=\left\{(x^{\ast },y^{\ast }):y^{\ast }=\frac{(F^{-1}(\theta )-F^{-1}({\rho }_{00}))-(F^{-1}({\rho }_{10})-F^{-1}({\rho }_{00}))x^{\ast }}{(F^{-1}({\rho }_{01})-F^{-1}({\rho }_{00}))+{\eta }_3{x}^{\ast }}\right\}.$$ (3)

Following the work of Tighiouart et al. (2017), we assume that ρ₀₁, ρ₁₀, η₃ are independent a priori with ρ₀₁ ~ beta(a₁, b₁), ρ₁₀ ~ beta(a₂, b₂), and conditional on (ρ₀₁, ρ₁₀), ρ₀₀/min(ρ₀₁, ρ₁₀) ~ beta(a₃, b₃). The prior distribution on the interaction parameter η₃ is a gamma with mean a/b and variance a/b². If D_k = {(x_i, y_i, T_i)} is the data after enrolling k patients to the trial, the posterior distribution of the model parameters is

$$\begin{gathered} \pi ({\rho }_{00},{\rho }_{01},{\rho }_{10},{\eta }_3)\propto \prod _{i=1}^k{(G({\rho }_{00},{\rho }_{01},{\rho }_{10},{\eta }_3;x_i,y_i))}^{T_i}{(1-G({\rho }_{00},{\rho }_{01},{\rho }_{10},{\eta }_3;x_i,y_i))}^{1-T_i} \\ \times \pi ({\rho }_{01})\pi ({\rho }_{10})\pi ({\rho }_{00}\mid {\rho }_{01},{\rho }_{10})\pi ({\eta }_3), \end{gathered}$$

where

$$G({\rho }_{00},{\rho }_{01},{\rho }_{10},{\eta }_3;x_i,y_i)=F(F^{-1}({\rho }_{00})+(F^{-1}({\rho }_{10})-F^{-1}({\rho }_{00}))x_i+(F^{-1}({\rho }_{01})-F^{-1}({\rho }_{00}))y_i+{\eta }_3{x}_i{y}_i).$$ (4)

2.2. Trial Design

Dose escalation/de-escalation proceeds by treating cohorts of two patients simultaneously. It is based on the escalation with overdose control (EWOC) principle where at each stage of the trial, the posterior probability of overdosing a future patient is bounded by a feasibility bound α, see e.g. Babb et al. (1998); Tighiouart et al. (2005); Tighiouart and Rogatko (2010); Tighiouart and Rogatko (2012). For a given cohort, one subject receives a new dose of agent A₁ for a given dose of agent A₂ that was previously assigned and the other patient receives a new dose of agent A₂ for a given dose of agent A₁ that was previously assigned. The algorithm is described in Tighiouart et al. (2017) and is reviewed here for convenience.

1. The first two patients receive the same dose combination (x₁, y₁) = (x₂, y₂) = (0, 0) and let D₂ = {(x₁, y₁, T₁), (x₂, y₂, T₂)}.

2. In the second cohort, patients 3 and 4 receive doses (x₃, y₃) and (x₄, y₄), respectively, where y₃ = y₁, x₄ = x₂, x₃ is the α-th percentile of π(Γ_A1|A2=y1|D₂), and y₄ is the α-th percentile of π(Γ_A2|A1=x2|D₂). Here, π(Γ_A1|A2=y1|D₂) is the posterior distribution of the MTD of drug A₁ given that the level of drug A₂ is y₁, given the data D₂.

3. In the i-th cohort of two patients, if i is even, then patient (2i − 1) receives dose (x_2i−1, y_2i−3), patient 2i receives dose (x_2i−2, y_2i), where $x_{2i-1}={\prod }_{{\mathrm{\Gamma }}_{A_1\mid A_2=y_{2i-3}}}^{-1}(\alpha \mid D_{2i-2}),y_{2i}={\prod }_{{\mathrm{\Gamma }}_{A_2\mid A_1=x_{2i-2}}}^{-1}(\alpha \mid D_{2i-2})$. Similar dose allocations can be derived when i is odd. Here, ${\prod }_{{\mathrm{\Gamma }}_{A_1\mid A_2=y}}^{-1}(\alpha \mid D)$ denotes the inverse cdf of the posterior distribution π(Γ_A1|A2=y|D).

   Repeat step (iii) until N patients are enrolled to the trial subject subject to the following stopping rule.

Stopping Rule

We stop enrollment to the trial if P(P(DLT|(x, y) = (0, 0)) > θ + ξ₁|data) > ξ₂, i.e. if the posterior probability that the probability of DLT at the minimum available dose combination in the trial exceeds the target probability of DLT is high. The design parameters ξ₁ and ξ₂ are chosen to achieve desirable model operating characteristics.

At the end of the trial, we estimate the MTD curve using (3) as

$$C_{\text{est}}=\left\{(x^{\ast },y^{\ast }):y^{\ast }=\frac{(F^{-1}(\theta )-F^{-1}({\widehat{\rho }}_{00}))-(F^{-1}({\widehat{\rho }}_{10})-F^{-1}({\widehat{\rho }}_{00}))x^{\ast }}{(F^{-1}({\widehat{\rho }}_{01})-F^{-1}({\widehat{\rho }}_{00}))+{\widehat{\eta }}_3{x}^{\ast }}\right\},$$ (5)

where ρ̂₀₀, ρ̂₀₁, ρ̂₁₀, η̂ are the posterior medians given the data D_N.

Uncertainty about the estimated MTD curve is evaluated by the pointwise average bias and percent selection described in Tighiouart et al. (2017) when assessing the operating characteristics of the design. Specifically, for i = 1, …, m, let C_i be the estimated MTD curve and C_true be the true MTD curve, where m is the number of simulated trials. For every point (x, y) ∈ C_true, let

$$d_{(x,y)}^{(i)}=\mathit{sign}(y^{\prime }-y)\times {\mathit{min}}_{\{(x^{\ast },y^{\ast }):(x^{\ast },y^{\ast })\in C_i\}}{({(x-x^{\ast })}^2+{(y-y^{\ast })}^2)}^{1/2},$$ (6)

where y′ is such that (x, y′) ∈ C_i. This is the minimum relative distance of the point (x, y) on the true MTD curve to the estimated MTD curve C_i. Let

$$d_{(x,y)}=\frac{1}{m}\sum _{i=1}^m{d}_{(x,y)}^{(i)}.$$ (7)

Equation (7) can be interpreted as the pointwise average bias in estimating the MTD.

Let Δ(x, y) be the Euclidean distance between the minimum dose combination (0, 0) and the point (x, y) on the true MTD curve and 0 < p < 1. Let

$$P_{(x,y)}=\frac{1}{m}\sum _{i=1}^{m}I(\mid d_{(x,y)}^{(i)}\mid \le p\mathrm{\Delta }(x,y)).$$ (8)

This is the pointwise percent of trials for which the minimum distance of the point (x, y) on the true MTD curve to the estimated MTD curve C_i is no more than (100×p)% of the true MTD. This statistic is equivalent to drawing a circle with center (x, y) on the true MTD curve and radius pΔ(x, y) and calculating the percent of trials with MTD curve estimate C_i falling inside the circle. This will give us the percent of trials with MTD recommendation within (100×p)% of the true MTD for a given tolerance p. This is interpreted as the pointwise percent selection for a given tolerance p. In this manuscript, we present the operating characteristics of stage I in the context of the ciscab trial in Section 4.

3. Stage II

Let C_est be the estimated MTD curve obtained from stage I and suppose it is defined for x ∈ [X₁,X₂] and y ∈ [Y₁, Y₂]. Here, [X₁,X₂] ⊂ [X_min,X_max] and [Y₁, Y₂] ⊂ [Y_min, Y_max]. Let E be the indicator of treatment response such as tumor shrinkage, E = 1 if we have a positive response after a pre-defined number of treatment cycles, and E = 0 otherwise. Let p₀ be the probability of efficacy of the standard of care treatment. We propose to carry out a phase II study to identify dose combinations (x, y) ∈ C_est such that P(E = 1|(x, y)) > p₀.

3.1. Model

For every dose combination (x, y) ∈ C_est, let x be the unique vertical projection of (x, y) on the interval [X₁,X₂]. Next, denote by z ∈ [0, 1] the standardized dose of x ∈ [X₁,X₂] using the transformation z = h₃(x) = (x − X₁)/(X₂ − X₁). In the sequel, we will refer to z as dose combination since there is a one-to-one transformation mapping z ∈ [0, 1] to (x, y) ∈ C_est, x ∈ [X₁,X₂], y ∈ [Y₁, Y₂]. This process is further illustrated in the supplement using the CisCab trial example described in Section 4. We model the probability of treatment response given dose combination z in C_est as

$$P(E=1\mid z,\mathit{\psi })=F(f(z;\mathit{\psi })),$$ (9)

where F is a known link function, f(z;ψ) is an unknown function and ψ is an unknown parameter. A flexible way to model the probability of efficacy along the MTD curve is the cubic spline function

$$f(z;\mathit{\psi })={\beta }_0+{\beta }_{1}z+{\beta }_2{z}^2+\sum _{j=3}^k{\beta }_j{(z-{\kappa }_j)}_{+}^3,$$ (10)

where ψ = (β, κ), β = (β₀, …, β_k), κ = (κ₃, …, κ_k) with κ₃ = 0. Let D_m = {(z_i,E_i), i = 1, …, m} be the data after enrolling m patients in the trial, where E_i is the response of the i-th patient treated with dose combination z_i and let π(ψ) be a prior density on the parameter ψ. The posterior distribution is

$$\pi (\mathit{\psi }\mid D_m)\propto \prod _{i=1}^m{[F(f(z_i;\mathit{\psi }))]}^{E_i}{[1-F(f(z_i;\mathit{\psi }))]}^{1-E_i}\pi (\mathit{\psi }).$$ (11)

Let p_z be the probability of treatment efficacy at dose combination z and denote by p₀ the probability of efficacy of a poor treatment or treatment not worthy of further investigation. We propose an adaptive design to conduct a phase II trial in order to test the hypothesis

• H₀: p_z ≤ p₀ for all z versus H₁: p_z > p₀ for some z.

3.2. Trial Design

1. Randomly assign n₁ patients to dose combinations z₁, …, z_n1 equally spaced along the MTD curve C_est so that each combination is assigned to one and only one patient.

2. Obtain a Bayes estimate ψ̂ of ψ given the data D_n1 using (11).

3. Generate n₂ dose combinations from the standardized density F(f(z; ψ̂)) and assign them to the next cohort of n₂ patients.

4. Repeat steps (ii) and (iii) until a total of n patients have been enrolled to the trial subject to pre-specified stopping rules.

This algorithm can be viewed as an extension of a Bayesian adaptive design to select a superior arm among a finite number of arms Berry et al. (2011) to selecting a superior arm from an infinite number of arms.

Decision Rule

At the end of the trial, we accept the alternative hypothesis if

$${\text{Max}}_z[P(F(f(z;\mathit{\psi }))>p_0\mid D_n)]>{\delta }_u,$$ (12)

where δ_u is a design parameter.

Stopping Rules

For ethical considerations and to avoid exposing patients to sub-therapeutic doses, we stop the trial for futility after j patients are evaluable for efficacy if there is strong evidence that none of the dose combinations are promising, i.e. Max_z[P(F(f(z;ψ)) > p₀|D_j)] < δ₀ where δ₀ is a small pre-specified threshold. In cases where the investigator is interested in stopping the trial early for superiority, the trial can be terminated after j patients are evaluable for efficacy if Max_z[P(F(f(z;ψ)) > p₀|D_j)] > δ₁ where δ₁ ≥ δ_u is a pre-specified threshold and the corresponding dose combination z^* = argmax_u{P(F(f(u;ψ)) > p₀|D_j)} is selected for future randomized phase II or III studies.

4. Application to the CisCab Trial

The methodology described in Sections 2 and 3 was used to design a phase I/II trial of the combination cisplatin and cabazitaxel in patients with prostate cancer with visceral metastasis. A recently published phase I trial of this combination by Lockhart et al. (2014) identified the MTD of cabazitaxel/cisplatin as 15/75 mg/m². This trial used a “3+3” design exploring 3 pre-specified dose levels 15/75, 20/75, 25/75. In part 1 of the trial, 9 patients we evaluated for safety and no DLT was observed at 15/75 mg/m². In part 2 of the study, 15 patients were treated at 15/75 mg/m² and 2 DLTs were observed. Based on these results and other preliminary efficacy data, we hypothesize that there exists a series of active dose combinations which are tolerable and active in prostate cancer. Cabazitaxel dose levels will be selected in the interval [10, 25] and cisplatin dose levels were selected in the interval [50, 100] administered intravenously. For stage I, we plan to enroll N = 30 patients and estimate the MTD curve and in stage II, n = 30 patients will be enrolled to identify dose combinations along the MTD curve with maximum clinical benefit rate. Clinical benefit is defined as either a complete response, partial response, or stable disease within three cycles of treatment. The probability of a poor clinical benefit is p₀ = 0.15 and we expect that a tolerable dose combination achieves a clinical benefit rate of p = 0.4.

4.1. Stage I

For stage I, trial design proceeds as described in Section 2.2 with a target probability of DLT θ = 0.33 and a logistic link function for F(·) in (1). DLT is resolved within one cycle (3 weeks) of treatment. The starting dose combination for the first cohort of two patients is 15 mg/m² cabazitaxel and 75 mg/m² cisplatin. The prior distributions were calibrated so that the prior mean probability of DLT at the dose combination 15/75 mg/m² equals the target probability of DLT. Specifically, informative priors were used for the model parameters ρ₀₁, ρ₁₀ ~ beta(1.4, 5.6), and conditional on ρ₀₁, ρ₁₀, ρ₀₀/min(ρ₀₁, ρ₁₀) ~ beta(0.8, 7.2) and a vague prior for η₃ was used as in Tighiouart et al. (2017) so that E(P(DLT|(15; 75))) ≈ 0.33 a priori. We derived the operating characteristics of stage I by simulating 2000 trial replicates as in Tighiouart et al. (2017) under four scenarios for the true MTD curve. Due to space consideration, we only present the estimated MTD curves from two scenarios A and B that are expected by the principal investigator of the trial. Figure 1 shows the true and estimated MTD curve obtained obtained using (3) with the parameters ρ₀₀, ρ₀₁, ρ₁₀, η₃ replaced by their posterior median averaged across all 2000 simulated trials. Scenario A shown on the left panel of Figure 1 is a case where the true MTD curve passes through a point very close to the dose combination (15, 75) identified as the MTD from the previous trial. Scenario B shown on the right panel is a case where the MTD curve is way above this dose combination. In each case, the estimated MTD curves are very close to the true MTD curves. This is also evidenced by the pointwise bias and percent selection (graphs included in the supplement). The trial was also safe since the percent of trials with DLT rate above θ + 0.1 were 3.5% for the scenario on the left and 5.0% for the scenario on the right.

Fig. 1.

True and estimated MTD curve under two different scenarios for the MTD curve. The grey diamonds represent the last dose combination from each simulated trial along with a 90% confidence region.

4.2. Stage II

4.2.1. Simulation Set-up and Scenarios

For stage II, we present simulations based on six scenarios that include three situations favoring the alternative hypothesis and three instances supporting the null hypothesis. A logistic link function F(u) = (1+exp(u))⁻¹ is used in (9) and f(z;ψ) is modeled as a cubic spline function with two knots in (0,1). This is a very flexible class of efficacy curves and accommodates cases of constant probability of efficacy along the MTD curve, high probability of efficacy around the middle of the MTD curve and high probability of efficacy at one or both edges of the MTD curve. Vague priors are placed on the model parameters by assuming that β ~ 𝒩(0, σ²I₆) with σ² = 10⁴ and (κ₄, κ₅) ~ Unif{(u, v): 0 ≤ u < v ≤ 1}. It can be shown that the induced prior mean and variance of the probability of treatment response are E_prior(F(f(z;ψ))) ≈ 0.5 and Var_prior(F(f(z;ψ))) ≈ 0.25 for all dose combinations z ∈ [0, 1]. The trial sample size is n = 30, the initial number of patients enrolled to the trial was set to n₁ = 10, and n₂ = 5 was used in the adaptive randomization phase of the design. The design parameter for the decision rule in (12) was taken as δ_u = 0.8. In each scenario, we simulated M = 2000 trial replicates. The true probability of response curves under scenarios (a,b,c) are shown in blue in Figure 2. The black horizontal lines correspond to the probability of a poor treatment response p₀ = 0.15 and the green horizontal lines represent the target probability of response p = 0.4. Scenario (a) is a case where the probability of efficacy is maximized near the middle of the estimated MTD curve with dose combinations in the interval (0.03, 0.76) having probability of efficacy greater than p₀ = 0.15. The target probability of response is achieved at a single dose combination z = 0.42. Scenario (b) is a case where higher doses of cisplatin and lower doses of cabazitaxel achieve higher efficacy. Specifically, standardized dose combinations in the interval (0.00, 0.49) have probability of efficacy greater or equal to p₀ = 0.15. Scenario (c) is an unusual situation where the probability of efficacy is maximized at the edges of the MTD curve. In this case, dose combinations in the interval (0.00, 0.41) ∪ (0.90, 1.00) have probability of efficacy greater or equal to p₀ = 0.15. Corresponding to these scenarios are situations (d–f) favoring the null hypothesis shown in Figure 2 (d,e,f). The true probability of response curves shown in blue have been shifted downward so that the probability of response equals to p₀ at one dose combination only for scenarios (d), (e), and (f).

Fig. 2.

True and estimated efficacy curve under six scenarios favoring the null and alternative hypotheses.

4.2.2. Operating Characteristics

For each scenario favoring the alternative hypothesis, we estimate the Bayesian power as

$$\text{Power}\approx \frac{1}{M}\sum _{i=1}^{M}I[{\text{Max}}_z\{P(F(f(z;{\mathit{\psi }}_i))>p_0\mid D_{n,i})\}>{\delta }_u],$$ (13)

where P(F(f(z;ψ_i)) > p₀|D_n,i) is estimated using an MCMC sample of ψ_i,

$$P(F(f(z;{\mathit{\psi }}_i))>p_0\mid D_{n,i})\approx \frac{1}{L}\sum _{j=1}^{L}I[F(f(z;{\mathit{\psi }}_{i,j}))>p_0],$$ (14)

where ψ_i,j, j = 1, …, L is an MCMC sample from the i-th trial. For scenarios favoring the null hypothesis, (13) is the estimated Bayesian type I error probability. The optimal or target dose from the i-th trial is

$$z_i^{\ast }={\text{argmax}}_v\{P(F(f(v;{\mathit{\psi }}_i))>p_0\mid D_{n,i})\}.$$ (15)

We also report the estimated efficacy curve by replacing ψ in (9) by the average posterior medians across all simulated trials

$$F(f(z;\overline{\mathit{\psi }})),$$ (16)

where ψ̄ = (β̄, κ̄), ${\overline{\beta }}_l=M^{-1}{\sum }_{i=1}^M{\widehat{\beta }}_{i,l}$, l = 0, …, 5, ${\overline{\kappa }}_k=M^{-1}{\sum }_{i=1}^M{\widehat{\kappa }}_{i,k}$, l = 4,5 and β̂_i,l, κ̂_i,k are the posterior medians from the i-th trial. Finally, we also report the mean posterior probability of declaring the treatment as efficacious for all dose combination z as

$$\frac{1}{M}\sum _{i=1}^{M}P(F(f(z;{\mathit{\psi }}_i))>p_0\mid D_{n,i}).$$ (17)

4.2.3. Results

The estimated efficacy curves shown in black dashed-lines in Figure 2 computed using equation 16 are fairly close to the true probability of efficacy curve in all scenarios except for scenario (c) near the lower edge of the MTD curve. The mean posterior probability of efficacy curve shown in red dashed-line computed using equation 17 is 80% or more at dose combinations where the true probability of efficacy is maximized for scenarios (a,b) adn close to 80% for scenario (c). Similar conclusions can be drawn for scenarios favoring the null hypothesis where the maximum of the mean posterior probability of efficacy is less than 50%. Figure 3 is the estimated density of the target dose z^* defined in equation 15 under scenarios favoring the alternative hypothesis (ac) and the shaded region corresponds to dose combinations with corresponding true probability of efficacy greater than p₀ = 0.15. The mode of these densities are close to the target doses. Moreover, the estimated probabilities of selecting a dose with true probability of efficacy greater than p₀ = 0.15 varies between 0.90 and 0.96 across the three scenarios. The Bayesian power for scenarios (a–c) and type I error probability for scenarios (d–f) estimated using equation 13 using a threshold δ_u = 0.8 are reported in Table 1. Power varies between 0.81 and 0.92 and the type I error probability varies between 0.10 and 0.19. The coverage probability in the last column of Table 1 is the estimated probabilities of selecting a dose with true probability of efficacy greater than p₀ = 0.15.

Fig. 3.

Estimated density of the target dose combination under three scenarios favoring the alternative hypothesis.

Table 1.

Bayesian power, type I error and coverage probabilities.

Scenarios	Power	Scenarios	Prob(Type I error)	Coverage Prob.
(a)	0.896	(d)	0.100	0.964
(b)	0.921	(e)	0.190	0.897
(c)	0.810	(f)	0.143	0.937

We carried out a sensitivity analysis with respect to the initial number of patients enrolled to the trial n₁ and the cohort size used in the adaptive randomization phase of the design n₂. Table 2 gives the power, type I error and coverage probabilities as n₁ varies between 5 and 20 and n₂ varies between 3 and 7. For scenario (a), power is about 2% higher when using n₂ = 3, 7 relative to the other three cases but the type I error (scenario (d)) increases significantly. Under scenario (b), there is a 3% drop in power when n₁ = 20 relative to the case n₁ = 10, n₂ = 5. The type I error under scenario (e) is fairly constant. For scenario (c), an even higher drop in power of 6% is observed when n₁ = 20 relative to the case n₁ = 10, n₂ = 5 with no significant changes in the type I error probabilities under scenario (f). In all cases, the coverage probabilities are not affected by n₁ and n₂. These findings show that the effects of n₁ and n₂ on the power depend on the scenarios under study, at least for a final sample size n = 30. The drop in power of 6% under scenario (c) suggests that in practice, the initial number of patients n₁ to be randomized to dose combinations equally spaced along the estimated MTD curve be selected between 1/4 and 1/3 of the total sample size in stage II. For binary efficacy outcomes, it is well known that outcome adaptive randomization is not suitable for fast accruing trials or for trials with a longer time τ to resolve efficacy outcome, e.g., response at 6 months or one year disease free survival. Otherwise, we recommend that the number of patients n₂ to use for the adaptive phase of the trial to be around the number of patients expected to be accrued within the interval [0, τ]. For this trial, the PI expects to enroll about 2 patients per month. Since the length of a cycle is 3 weeks and clinical benefit is resolved withing 3 cycles of therapy, n₂ should be selected between 4 and 5 patients. It is important to note that these guidelines do depend on p₀, the effect size, and total trial sample size and the investigators have to evaluate the corresponding operating characteristics taking into account the accrual rate and length of the trial.

Table 2.

Sensitivity analysis for n₁ and n₂.

Scenario	n1	n2	Power	Scenario	Prob(Type I error)	Coverage Prob.
(a)	10	5	0.896	(d)	0.100	0.964
	9	3	0.916		0.170	0.946
	9	7	0.915		0.159	0.944
	5	5	0.898		0.091	0.954
	20	5	0.906		0.119	0.979
(b)	10	5	0.921	(e)	0.190	0.897
	9	3	0.931		0.183	0.899
	9	7	0.930		0.183	0.881
	5	5	0.94		0.183	0.892
	20	5	0.887		0.189	0.904
(c)	10	5	0.810	(f)	0.143	0.937
	9	3	0.810		0.150	0.934
	9	7	0.825		0.136	0.929
	5	5	0.861		0.135	0.959
	20	5	0.750		0.144	0.916

Finally, we assessed the uncertainty about the estimated MTD curve in stage II in terms of safety of the phase II trial as determined by first cycle DLT under scenario A shown in Figure 1 (A). To estimate how likely a dose used in the phase II trial is toxic taking into account the pointwise percent selection given in (8), for each dose x_i,j ∈ (0, 1) from the i-th simulated trial allocated to the j-th patient using the algorithm in Section 3.2, i = 1, …, M, j = 1, …, n, let (x_i,j, y_i,j) be the dose combination on the true MTD curve shown in Figure 1 (A). Next, generate a dose combination ( $x_{i,j}^{★},y_{i,j}^{★}$) uniformly inside the circle with center (x_i,j, y_i,j) and radius pΔ(x_i,j, y_i,j), where the tolerance probability p and distance Δ(x, y) are defined in Section 2.2. A first cycle DTL T_i,j is then generated from model (1) with (η₀, η₁, η₂, η₃) replaced by the true values defining scenario A and $(x,y)=(x_{i,j}^{★},y_{i,j}^{★})$. Safety of the phase II trial is summarized by reporting the average DLT rate ${(Mn)}^{-1}{\sum }_{i=1}^M{\sum }_{j=1}^n{T}_{i,j}$ and the percent of trials with an excessive DLT rate, i.e., a DLT rate exceeding θ + 0.1. These statistics are reported in Table 3 under the six scenarios and various values of n₁ and n₂ when the tolerance probability is p = 0.05. As expected, the mean first cycle DLT rate is essentially equal to the target probability of DLT θ = 0.33. The percent of trials with excessive DLT rate varies between 16% and 19% when n₁ = 10, n₂ = 5. These rates are not constant due to the dependence of the radius of the circle on the location of the dose combination on the MTD curve. We conclude that the phase II trial is safe with respect to first cycle DLT rate. Since treatment in the ciscab consists of three cycles of therapy, a separate stopping rule for safety was included in the clinical protocol. Specifically, let θ^* be the true probability that a DLT occurs within three cycles of therapy. The trial will be stopped for safety if P(θ^* > 0.43|data) > 0.9. A Bayesian continuous monitoring was used by evaluating this rule after every 6-th patient is evaluable for DLT with a uniform prior on θ^*. Other simulations testing the same hypotheses showed similar results. We conclude that the design has good operating characteristic in identifying tolerable dose combinations with maximum benefit rate. The trial design has recently been approved by the scientific review committee of Cedars-Sinai Medical Center and is scheduled to open enrollment. Operating characteristics of the phase II trial design under a smaller effect size and different values for the probability of a poor treatment response p₀ can be found in the supplement.

Table 3.

Safety assessment on the uncertainty of the MTD curve.

Scenario	n1	n2	Ave. DLT rate	% Trials: DLT rate > θ + 0.1
(a)	10	5	34.0	19.0
	9	3	34.0	19.0
	9	7	33.0	19.0
	5	5	34.0	18.0
	20	5	34.0	19.0
(b)	10	5	33.0	16.0
	9	3	33.0	17.0
	9	7	33.0	16.0
	5	5	33.0	17.0
	20	5	33.0	18.0
(c)	10	5	34.0	18.0
	9	3	33.0	17.0
	9	7	34.0	18.0
	5	5	34.0	18.0
	20	5	34.0	18.0
(d)	10	5	33.0	17.0
	9	3	33.0	16.0
	9	7	33.0	18.0
	5	5	33.0	18.0
	20	5	33.0	18.0
(e)	10	5	34.0	19.0
	9	3	33.0	19.0
	9	7	33.0	17.0
	5	5	33.0	16.0
	20	5	34.0	17.0
(f)	10	5	34.0	17.0
	9	3	34.0	16.0
	9	7	33.0	17.0
	5	5	33.0	19.0
	20	5	34.0	20.0

5. Discussion

We described a two-stage Bayesian adaptive design for cancer phase I clinical trials using two drugs with continuous dose levels. The goal is to (1) estimate the MTD curve in the two-dimensional Cartesian plane and (2) search for dose combination regions along the MTD curve that yield a desired probability of treatment response. Design of the phase I trial and estimation of the MTD curve in stage I was carried out using EWOC (Tighiouart et al. (2014, 2017)). In the context of the ciscab trial, we showed that good operating characteristics are obtained using informative prior distributions and sample size of 30 patients. In stage II, we modeled treatment efficacy as a binary indicator of treatment response using a cubic spline form of the dose combination-treatment response relationship. This is a very flexible form for the efficacy curve since it accommodates cases of constant probability of efficacy along the MTD curve, high probability of efficacy around the middle of the MTD curve, high probability of efficacy at the edges of the MTD curve. In this stage, a Bayesian adaptive design is proposed to conduct a phase II trial with the goal of identifying dose combination regions that yield a desired probability of treatment response. Initially, a number of patients are treated with dose combinations equally spaced along the estimated MTD curve from stage I and after resolving their treatment response status, the estimated probability of efficacy curve is updated. A small number of patients are then allocated to dose combinations generated from this updated efficacy curve. The trial continues until we reach the final sample size. This design can be viewed as an extension of the Bayesian adaptive design comparing a finite number of arms (Berry et al. (2011)) to comparing an infinite number of arms. In particular, if the dose levels of the two agents are discrete, then methods such as the ones described in Thall et al. (2003); Wang and Ivanova (2005); Wages (2016) can be used to identify a set of MTDs in stage I and the trial in stage II can be done using adaptive randomization to select the most efficacious dose. Unlike phase I/II designs that use toxicity and efficacy data simultaneously and require a short period of time to resolve efficacy status, the use of a two-stage design is sometimes necessary in practice if it takes few cycles of therapy to resolve treatment efficacy or if the populations of patients in phase I and II are different. In fact, for the ciscab trial described in Section 4, efficacy is resolved after three cycles (9 weeks) of treatment and patients in stage I must have metastatic, castration-resistant prostate cancer whereas patients in stage II must have visceral metastasis.

We studied the properties of this design under various scenarios for the true probability of efficacy as a function of dose combinations and we found that the method yields high power and small type I error probability using a sample size of 40 patients with a target effect size of 0.2. For the ciscab trial, a sample size of 30 patients with effect size of 0.25 yield reasonable operating characteristics. Based on these scenarios and proposed models, we conclude that this two-stage design is feasible with sample size ranging between 60 and 80 patients. The uncertainty of the estimated MTD curve in stage I is not taken into account in stage II of the design in the sense that the MTD curve is not updated as a result of observing DLTs in Stage 2. This is a limitation of this approach since patients in stage II may come from a different population and may have different treatment susceptibility relative to patients in stage I. This problem is also inherent to single agent two-stage designs where the MTD from the phase I trial is used in phase II studies and safety is monitored continuously during this phase. An alternative design would account for first, second, and third cycle DLT in addition to efficacy outcome at each cycle. In addition, the nature of DLT (reversible vs. non-reversible) should be taken into account since patients with a reversible DLT are usually treated for that side effect and kept in the trial with dose reduction in subsequent cycles. For single agent trial and a discrete set of doses, Lee et al. (2015) introduced such designs that account for cycles one and two toxicities and efficacy. The authors noted the increase in model complexity if the number of cycles is increased to three and this would ultimately require a more parsimonious model. For drug combinations with continuous dose levels and three cycles of therapy, another layer of model complexity will be introduced and such designs are beyond the scope of this manuscript and are subjects of future research. For the ciscab trial, a separate stopping rule using Bayesian continuous monitoring for excessive toxicity is included in the clinical protocol. We further emphasize the fact that phase I/II designs such as the ones described in Wages and Conaway (2014); Cai et al. (2014) or Wages et al. (2016) require a short period of time to resolve efficacy endpoint and may not be appropriate in the setting of the ciscab trial. Finally, we note that our proposed method is applicable to any combination of cytotoxic and biologic agents since the models used in the phase I part of the study still assume monotonicity of the probability of DLT of either agent for fixed level of the other agent. We plan to extend this work to time to event endpoints such as progression free survival and to accommodate patients baseline characteristics such as biomarker expression that is known to be a target of one or both drugs.