Adapting isotonic dose-finding to a dynamic set of drug combinations with application to a phase I leukemia trial

Abstract

Background/aims:

This paper describes the proposed design of a Phase I study evaluating the safety of ceramide nanoliposome and vinblastine among an initial set of nineteen possible dose combinations in patients with relapsed/refractory acute myeloid leukemia and patients with untreated acute myeloid leukemia who are not candidates for intensive induction chemotherapy.

Methods:

Extensive collaboration between statisticians and clinical investigators revealed the need to incorporate several adaptive features into the design, including the flexibility of adding or eliminating certain dose combinations based on safety criteria applied to multiple dose pairs. During the design stage, additional dose levels of vinblastine were added, increasing the dimension of the drug combination space and thus the complexity of the problem. Increased complexity made application of existing drug combination dose-finding methods unsuitable in their current form.

Results:

Our solution to these challenges was to adapt a method based on isotonic regression to meet the research objectives of the study. Application of this adapted method is described herein, and a simulation study of the design’s operating characteristics is conducted.

Conclusions:

The aim of this manuscript is to bring to light examples of novel design applications as a means of augmenting the implementation of innovative designs in the future and to demonstrate the flexibility of adaptive designs in satisfying changing design conditions.

Introduction

The paper describes the proposed design of a Phase I trial evaluating the safety of ceramide nanoliposome and vinblastine in leukemia patients. The primary objective of the study was to determine the maximum tolerated dose combination of ceramide nanoliposome and vinblastine among the possible combinations provided in Figure 1. The maximum tolerated dose combination was defined as the combination with a dose-limiting toxicity rate closest to the pre-specified target rate of 20%, which was chosen based on the expectedness of adverse events according to the toxicity profiles observed in previous studies that involved vinblastine or ceramide nanoliposome in solid tumors. The study included several unique design considerations that made direct application of an existing Phase I drug combination method problematic. The initial set of regimens of interest to investigators included the nineteen combinations in the shaded region of Figure 1, with combination 5 being the starting combination. Investigators wanted the design to allow for exploration of another four possible dose combinations (combination 1 through combination 4) if the lowest combinations in the initial set (combinations 5 and 6) were determined to be too toxic. Therefore, not all dose pairs were admissible at the startup, and some dose pairs, such as (54 mg/m² vinblastine, 0.375 mg/m² ceramide nanoliposome) and (54 mg/m² vinblastine, 0.75 mg/m² ceramide nanoliposome), were never intended to be explored. If at any time in the accrual process combinations 5 (the starting combination) and 6 were deemed too toxic, then future accrual to combination 5 or above was no longer allowed. This required that the design be able to handle the adaptive opening and closing of certain dose pairs, based on safety stopping criteria applied to multiple dose pairs.

Figure 1.

Combination and zone designation for a phase I trial of ceramide nanoliposome and vinblastine in leukemia patients.

The clinical considerations of which dose pairs to explore and when created a high-dimensional, dynamic drug combination space that necessitated adaptations to existing methods for combination dose-finding. While it is the case that some existing dose-finding methods for drug combinations can eliminate dose combination candidates, no current method will move to specific areas of the drug combination matrix based on default elimination rules. All current methods would need to be adapted in some form, and it is unknown what impact these types of modifications have on the operating characteristics of any particular design. There are many methods available^1–2 that could be considered for modification, yet we chose to focus our attention on methods that have been implemented in published drug combination studies. One possibility is the partial order continual reassessment method,³ which has been adapted for implementation into several real studies.^4–6 The method of Ivanova and Wang⁷ was executed in a Phase I study of neratinib in combination with temsirolimus in patients with human epidermal growth factor receptor 2-dependent and other solid tumors.⁸ The method of Conaway, Dunbar, and Peddada⁹ was implemented in a Phase I trial investigating induction therapy with bortezomib and vorinostat in patients with surgically resectable non-small cell lung cancer.¹⁰ The partial order continual reassessment method requires pre-specification of a constant, termed the “skeleton” of the model, that is a representation of the probability of dose-limiting toxicity at each combination. The skeleton values must be adequately spaced to ensure good operating characteristics.¹¹ This proved to be very difficult in the current study due to the large number of combinations being studied, as we will demonstrate in our simulation studies. The latter two methods^7,9 rely on isotonic regression¹² for estimation of dose-limiting toxicity probabilities. The Conaway-Dunbar-Peddada method has been extensively evaluated and has demonstrated favorable operating characteristics in recent simulation studies among a variety of more complex dose-finding problems.^13–17 Based on these results, as well as its ability to balance simplicity with good statistical properties, we decided to adapt the Conaway-Dunbar-Peddada method to meet the challenges of the Phase I leukemia trial. Changes in access to one of the study agents ultimately lead to further design modifications with a more limited scope, but we describe the tailoring of the original design to the dynamic set of combinations herein.

Methods

Estimation in the Conaway-Dunbar-Peddada method accounts for the uncertainty in toxicity ordering among the combinations under investigation when identifying the maximum tolerated dose combination. Toxicities were to be assessed according to the National Cancer Institute Common Toxicity Criteria for Adverse Events, version 5. The primary outcome measure that was to guide accrual decisions was the frequency of treatment related dose-limiting toxicities. For the purposes of dose escalation, dose-limiting toxicities were to be determined by protocol-specific severe adverse events occurring during the first cycle of treatment, and intra-patient dose escalation was not to be allowed.

Participant allocation

Participant allocation was to occur in two stages. The escalation plan for the Stage 1 was based upon following an initial pre-specified escalation path, noted by the dashed arrow in Figure 1. Participants were to be accrued to the study in cohorts of size 2 to the initial pre-specified escalation path beginning with Combination 5. The statistical team for this study originally proposed escalation in cohorts of size 1 in Stage 1 in order to quickly and efficiently move through the many combinations being studied. The clinical team (and review entities) felt more comfortable slowing initial escalation to cohorts of size 2 in order to be a bit more conservative. If no participant in the cohort experienced a dose-limiting toxicity, escalation was to proceed along the pre-specified path. Escalation to the next combination in the path was to occur only when both participants had been followed for the minimum follow-up period, and no dose-limiting toxicity had been observed. This allocation strategy was to be followed for accrual along the pre-specified path until the first dose-limiting toxicity was observed, or a stopping rule was triggered. Once a dose-limiting toxicity had been observed in any participant or the highest combination had been reached, Stage 2 based on the estimation procedure of Conaway-Dunbar-Peddada⁷ was to begin. The Stage 1 cohort size and the initial escalation path had little impact on the overall operating characteristics of the design (Supplemental Figures 1 and 2).

Stage 2 was to allocate eligible participants in cohorts of size 1 based upon estimates of the dose-limiting toxicity probabilities at each of the combinations. Estimates are based on a modification of the Conaway-Dunbar-Peddada method using a selected set of possible orderings for the dose-limiting toxicity probabilities (Table 1). Six orderings provides an appropriate balance between choosing enough orderings so that we include adequate information to account for the uncertainty surrounding partially ordered dose–toxicity curves, without increasing the dimension of the problem so much so that we diminish performance.¹⁴ At any point in the trial, the dose-limiting toxicity data for dose combination d_i is of the form Ω_i = {(y_i,n_i):i = 1,…,I} with y_i equal to the number of observed dose-limiting toxicities and n_i equal to the number of participants who have been evaluated for toxicity at combination d_i. To model the probability of dose-limiting toxicity π_i at each combination, we assumed a beta-binomial model

$$y_i\mid {\pi }_i~\text{ Binomial }\left({\pi }_i\right);{\pi }_i~\text{Beta}\left({\alpha }_i,{\beta }_i\right),$$

where Beta(α_i,β_i) is a beta distribution with parameters α_i and β_i. Based on the accumulated toxicity data Ω_i=(y_i,n_i) at d_i, the posterior distribution of π_i follows a beta distribution

$${\pi }_i\mid {\Omega }_i~\text{Beta}\left({\alpha }_i+y_i,{\beta }_i+n_i-y_i\right).$$

Based on this distribution, the updated dose-limiting toxicity probabilities, are given by the following posterior mean.

$${\widehat{\pi }}_i=\frac{y_i+{\alpha }_i}{n_i+{\alpha }_i+{\beta }_i}$$

For each of the six assumed orders in Table 1, isotonic regression estimates of the dose-limiting toxicity probabilities were to be computed using the pool-adjacent-violators algorithm.¹⁸ Under ordering m, the dose-limiting toxicity probability for combination i is estimated using isotonic regression as

$${\tilde{\pi }}_{mi}=\underset{i\le s\le I}{\text{min}}\underset{1\le r\le i}{\text{max}}\sum _{i=r}^s{\widehat{\pi }}_i$$

This algorithm replaces adjacent estimates that violate the monotonicity assumption with their weighted average, where the weights are the current sample size at each dose level. To satisfy the known ordering relationships among combinations, the set of m = 6 isotonic regression estimates were to be averaged to produce a single set of estimates for the dose-limiting toxicity probabilities across the combinations so that

$${\overline{\pi }}_i=\frac{{\sum }_{m=1}^M{\tilde{\pi }}_{mi}}{m}$$

This simplifies the estimation procedure described in Conaway et al⁹ in that it does not rely on the more complex estimation procedure of Hwang and Peddada.¹⁹

Table 1.

Selected set of possible orderings for the dose-limiting toxicity probabilities among the combinations.

Set	Possible orderings
1	1-2-3-4-6-5-7-8-9-13-12-11-10-14-15-16-17-20-19-18-21-22-23
2	1-2-3-4-5-6-9-8-7-10-11-12-13-17-16-15-14-18-19-20-22-21-23
3	1-2-3-4-5-6-7-8-9-10-11-12-13-14-15-16-17-18-19-20-21-22-23
4	1-2-3-4-6-5-9-8-7-13-12-11-10-17-16-15-14-20-19-18-22-21-23
5	1-2-3-4-6-9-13-5-8-12-17-7-11-16-20-10-15-19-22-14-18-21-23
6	1-2-3-4-5-7-10-14-6-8-11-15-18-9-12-16-19-21-13-17-20-22-23

Updating the recommended combination

Let A_i denote a set of admissible combinations to consider for assignment after a dose-limiting toxicity has been observed in the most recent patient treated at d_i. Let B_i denote set of admissible combinations to consider for assignment after a non-dose-limiting toxicity has been observed in the most recent patient treated at d_i. When the most recent participant experiences a dose-limiting toxicity (or non-dose-limiting toxicity) on d_i, we want the design to either stay at d_i, move to a combination adjacent to d_i that is a known de-escalation (or escalation), or move to a combination whose toxicity relationship is unknown relative to d_i because of the partial order. These restrictions ensure that the design is coherent²⁰ in that it does not move to a combination that we know to be an escalation (de-escalation) decision in the presence (absence) of a dose-limiting toxicity in the most recent participant. For instance, for d₈, we know that d₅ and d₆ are de-escalations, and the toxicity relationship between d₈, d₇ and d₉ is unknown. Therefore, the set A₈ = {d₅, d₆, d₇, d₈, d₉}. With these restrictions, the algorithm does not allow for escalation of both agents simultaneously, or a change of more than one dose level of a single agent.

1. If the most recent participant experiences a dose-limiting toxicity, for all d_i ∈ A_i, compute the loss $L_i\left({\overline{\pi }}_i,{\pi }^{*}\right)=\left|{\overline{\pi }}_i-{\pi }^{*}\right|$ associated with each combination based on the target dose-limiting toxicity rate π*. If the most recent participant experiences a non-dose-limiting toxicity, compute $L_i\left({\overline{\pi }}_i,{\pi }^{*}\right)$ for all d_i ∈ B_i.

2. Let $l_{min}=\text{min}{L}_i\left({\overline{\pi }}_i,{\pi }^{*}\right)$, and let T_i be the set of combinations with losses equal to the minimum observed loss so that $T_i=\left\{d_i:L_i\left({\overline{\pi }}_i,{\pi }^{*}\right)=l_{min}\right\}$.

3. If T_i contains a single combination with minimum observed loss, then the suggested combination is the element contained in T_i. If T_i contains more than one combination, then we choose from among them according to the following rules:

   1. If ${\overline{\pi }}_t>{\pi }^{*}\forall t\in T_i$, the suggested combination is randomly chosen from among the combinations in T_i with the lowest estimated dose-limiting toxicity probability.
   2. If ${\overline{\pi }}_t\le {\pi }^{*}$ for at least one t ∈ T_i, the suggested combination is randomly chosen from among the combinations in T_i with the highest estimated dose-limiting toxicity probability.

If at any time in the accrual process, Combination 5 (the starting combination) and Combination 6 were deemed too toxic and the model recommends accrual to Combination 1 then future accrual to Combination 5 or above was no longer to be allowed.

Based on the posterior distributions π₅|Ω₅ and the pre-specified target dose-limiting toxicity rate π*, we can calculate the posterior probability that combination d₅ is too toxic

$$\text{Pr}\left({\pi }_5>{\pi }^{*}\mid {\Omega }_5\right)=1-\mathcal{B}\left({\pi }^{*};{\alpha }_5+y_5,{\beta }_5+n_5-y_5\right)$$

where $\mathcal{B}\left({\pi }^{*};{\alpha }_5+y_5,{\beta }_5+n_5-y_5\right)$ is the cumulative density function of the beta distribution with parameters α₅ + y₅ and β₅ + n₅ – y₅. Similarly, we can calculate Pr (π₆ > π*|Ω₆) We compare these probabilities to an upper probability cutoff p_T. If Pr (π₅ > π*|Ω₅) > P_T and pr (π₆ > π*|Ω₆) > p_T, we say that combinations d₅ and d₆ are not safe combinations. Combinations 5 and 6 were considered the gate keepers for moving to Combination 1 and are deemed too toxic when the posterior probability is at least p_T = 70% that the risk of dose-limiting toxicity at Combinations 5 and 6 exceed the target rate. The impact of various values of p_T are explored in Supplemental Figure 3, with 70% yielding the most desirable operating characteristics on average across all scenarios considered. Under these circumstances, future accrual was to be limited to those combinations in the first column (Combinations 1 to 4) in Figure 1. Allocation of future participants was to be based on the Conaway-Dunbar-Peddada method using isotonic regression estimates, but in this case, there is only a single ordering: {i.e., 1-2-3-4}. Otherwise, accrual to the study continues until either (1) N participants have been evaluated for toxicity or (2) one of the following accrual stopping rules is triggered.

Accrual stopping rules

If, for the lowest combination the posterior probability was at least 70% that the dose-limiting toxicity rate at Combination 1 exceeded the target rate, then Combination 1 was to be deemed too toxic, the trial was to stop for safety, and no combination was to be recommended as the maximum tolerated dose combination. Otherwise, accrual to the study was to continue until the model recommended accrual to a combination that had already accrued 12 eligible participants or the maximum accrual target of 60 eligible participants had been reached. The target of 12 participants was based upon having preliminary information to estimate complete remission rate.

Sample size and estimated accrual

Maximum target sample size was based upon acquiring enough information to assess the goal of establishing the maximum tolerated dose combination and preliminary assessment of response to treatment at the maximum tolerated dose combination. Simulation results provided below indicated that a maximum target accrual of 60 eligible participants performed well and based upon the simulation results the study goals may be achieved with less than 45 participants. Accrual was expected to be about 1 patient per month, and the dose-limiting toxicity evaluation window was 1 cycle of treatment (i.e. 1 month). Therefore, we did not anticipate logistical difficulties associated with partially observed dose-limiting toxicity outcomes of previously treated patients at the time of a new dose assignment, and thus we did not account for this possibility in our simulation study. Yuan, Nguyen, and Thall²¹ quantify the severity of this potential problem by defining the logistical difficulty index

$$\zeta =Q\times \alpha$$

Where Q is the length of the observation window and α is the accrual rate. In our study, Q = 1 month and α = 1 patient per month for an overall logistical difficulty index of ζ = 1. Yuan et al²¹ state that when ζ ≤ 1, there is no logistical difficulty applying a standard adaptive decision rule.

Results

Application to the phase I leukemia trial

The primary objective of the trial was to identify the maximum tolerated dose combination, defined by the dose combination with a dose-limiting toxicity rate closest to the target rate of 0.20. Prior information for the dose-limiting toxicity probabilities π_i at each combination is expressed through a distribution of the form π_i~Beta(α_i,β_i). Based on the expected value of π_i and a 95% upper limit u_i on the dose-limiting toxicity probability, the equations

$$\mathbb{E}\left({\pi }_i\right)=\frac{{\alpha }_i}{{\alpha }_i+{\beta }_i}\text{ and Pr}\left({\pi }_i\le u_i\right)=0.95,$$

are solved to obtain prior specifications for α_i and β_i. In the absence of prior information, a practical prior specification can be acquired by setting the prior mean equal to the target dose-limiting toxicity rate π* and setting the 95% upper limit u_i equal to 2×π* at each dose level. This prior specification is recommended by Wages and Conaway²² to avoid the problem of rigidity²³ in which allocation can become confined to a sub-optimal combination regardless of the ensuing observed data. Based on a target dose-limiting toxicity rate of 0.20, we used a Beta (2.6, 10.4) prior for π at each combination. The maximum target accrual was N = 60 participants who will be evaluated for toxicity. The trial will begin on combination 5, and the upper probability cutoff used to define the safety stopping rule were p_T = 0.70.

We illustrate the behavior of the described design under a set of hypothesized dose-limiting toxicity probabilities. The assumed probabilities for combinations 1–23 are (0.08, 0.09, 0.11, 0.13, 0.10, 0.10, 0.12, 0.13, 0.13, 0.15, 0.16, 0.16, 0.15, 0.18, 0.19, 0.20, 0.20, 0.23, 0.25, 0.25, 0.30, 0.31, 0.38), indicating combinations 16 and 17 to be the true maximum tolerated dose combinations since they have the dose-limiting toxicity probabilities closest to the target rate of 0.20. The data from the entire simulation trial are provided in Table 2. The first two eligible participants are assigned to combination 5 and escalation proceeds in cohorts of size two according to the dashed arrow in Figure 1 without a dose-limiting toxicity event until participant 8 experiences a dose-limiting toxicity on combination 15. At this point in the trial, Stage 1 ends and Stage 2, based on the estimation procedure described above, begins. After the dose-limiting toxicity in the most recent participant on d₁₅, the set A₁₅ of admissible combinations consists of combinations d₁₀, d₁₁, d₁₄, d₁₅, and d₁₆. Based on the currently observed dose-limiting toxicity data at the admissible combinations, we update ${\widehat{\pi }}_{10}=0.20$, ${\widehat{\pi }}_{11}=0.173$, ${\widehat{\pi }}_{14}=0.20$, ${\widehat{\pi }}_{15}=0.24$, ${\widehat{\pi }}_{16}=0.20$. Based on the chosen possible orderings, the averaged isotonic estimates $\overline{\pi }$ for the dose-limiting toxicity probabilities for each of the admissible combinations are ${\tilde{\pi }}_{10}=0.187$, ${\tilde{\pi }}_{11}=0.173$, ${\tilde{\pi }}_{14}=0.216$, ${\tilde{\pi }}_{15}=0.24$, ${\tilde{\pi }}_{16}=0.22$. Following Step 3 in the above algorithm for updating the recommended combination, we compute the loss for each combination in A₁₅. In this case, the set T₁₅ will only consist of combination 10 since it is the single combination with estimated probability $\left({\tilde{\pi }}_{10}=0.187\right)$ closest to the target. Therefore, Step 3 of the assignment algorithm indicates that combination 10 is the suggested combination. Working now in cohorts of size 1, the next participant is administered combination 10 on which he/she does not experience dose-limiting toxicity. Ultimately, the design settles on combination 16 with a true dose-limiting probability of 0.20 as the maximum tolerated dose combination after the accrual of N = 48 participants to the study. At study conclusion, the total observed dose-limiting toxicity data at combination 16 are 2/12, with an estimated dose-limiting toxicity probability of ${\overline{\pi }}_{16}=0.182$.

Table 2.

A simulated sequential trial using the described two-stage Phase I design.

Pt	Combo	DLT	Stage	Pt	Combo	DLT	Stage
1	5	0	1	25	17	1	2
2	5	0	1	26	16	0	2
3	7	0	1	27	15	0	2
4	7	0	1	28	18	0	2
5	11	0	1	29	18	0	2
6	11	0	1	30	19	1	2
7	15	0	1	31	18	1	2
8	15	1	2	32	15	0	2
9	10	0	2	33	16	0	2
10	14	0	2	34	16	0	2
11	14	0	2	35	16	0	2
12	15	1	2	36	17	0	2
13	15	0	2	37	17	0	2
14	15	1	2	38	17	0	2
15	18	0	2	39	17	0	2
16	21	0	2	40	17	0	2
17	23	1	2	41	17	0	2
18	22	0	2	42	16	0	2
19	22	0	2	43	17	0	2
20	21	1	2	44	16	1	2
21	22	1	2	45	16	0	2
22	20	1	2	46	16	0	2
23	17	0	2	47	16	0	2
24	16	0	2	48	16	0	2

Estimated maximum tolerated dose combination=16

DLT = dose-limiting toxicity.

Operating characteristics

Simulations were run to display the performance of the design’s operating characteristics under a wide range of possible dose-toxicity scenarios. The scenarios are provided in Figure 2, with shaded combinations indicating true probabilities within 5% of the target rate. For each scenario, 5000 simulated trials were run. Figure 3 reports the percentage of trials in which a shaded combination was recommended as the maximum tolerated dose combination, the average number of participants treated at shaded combinations, the average sample size, and the accuracy index²⁴ of maximum tolerated dose combination selection. The maximum value of the accuracy index is 1, with larger values (close to 1) indicating that the method possesses high accuracy. Supplemental Tables 1–4 provide the entire distribution of maximum tolerated dose selection and participant allocation for the proposed design and a modified partial order continual reassessment method.

Figure 2:

True dose-limiting probability scenarios for the motivating Phase I leukemia trial. Shaded combinations indicate true dose-limiting toxicity probabilities within 5% of the target dose-limiting toxicity rate of 20%.

Figure 3:

Operating characteristics for the proposed design and a modified partial order continual reassessment method (POCRM). The target dose-limiting toxicity rate is 20%. Results are based on 5000 simulated trials, with each trial starting at combination 5 and accruing a maximum sample size of N=60 participants. The stage 1 cohort size is 2 participants.

The modified partial order continual reassessment method used the same Stage 1 escalation scheme as the proposed design until observance of the first dose-limiting toxicity. Stage 2 used an empiric working model $x_i^{\text{exp}(a)}$, where x_i is the skeleton of the model obtained using the function getprior(0.05, 0.20, 9, 23) in the R package dfcrm, and the set of possible orderings in Table 1. Safety guidelines used to guide the trial to combinations 1 through 4 and to stop the trial early were based on Agresti-Coull binomial confidence interval estimation. If the lower bound of an 80% confidence interval for the observed dose-limiting toxicity rate at combinations 5 and 6 exceeds the target rate of 20%, then the trial restricts assignments for future participants to combinations 1 through 4. If the lower bound of an 80% confidence interval for the observed dose-limiting toxicity rate at combination 1 exceeds 20%, then the trial terminates early for safety.

The results displayed in Figure 3 were based upon a maximum target accrual of 60 eligible participants. In Scenario 1, the true dose-limiting toxicity probabilities indicate that all available combinations are safe (i.e., ≤ 20%). The design selects, as the maximum tolerated dose combination, a combination with a true dose-limiting toxicity probability between 0.15 and 0.20 in 75.9% of simulated trials (accuracy index 66%), while treating 16.2 of 39.8 participants (40.7%) on average at these combinations. The modified partial order continual reassessment method selected a combination with a true dose-limiting toxicity probability between 0.15 and 0.20 in 43.8% of simulated trials (accuracy index 39.8%), while treating 9.3 of 33.8 participants (27.5%) on average at these combinations. The diminished performance of the partial order continual reassessment method in this case relative to the proposed design can likely be attributed to the lack of spacing between adjacent skeleton values due to the number of combinations being studied. In Scenario 2, the true dose-limiting toxicity probabilities indicate that all available combinations are safe (i.e., ≤ 20%), except for combination 23 with a true dose-limiting toxicity rate of 0.32. The proposed design selected, as the maximum tolerated dose combination, a combination with a true dose-limiting toxicity probability between 0.15 and 0.23 in 58.7% of simulated trials (accuracy index 57%), while treating 14.6 of 41.7 participants (35.0%) on average at these combinations. The modified partial order continual reassessment method selected a combination with a true dose-limiting toxicity probability between 0.15 and 0.23 in 37.1% of simulated trials (accuracy index 37.4%), while treating 8.7 of 36 participants (24.2%) on average at these combinations. In Scenario 3, the true dose-limiting toxicity probabilities indicate that most available combinations are safe (i.e., ≤ 20%), with the exception of combinations 19 through 23. The proposed design selected, as the maximum tolerated dose combination, a combination with a true dose-limiting toxicity probability between 0.15 and 0.25 in 47.2% of simulated trials (accuracy index 46.7%), while treating 15.5 of 43.0 participants (36.0%) on average at these combinations. The modified partial order continual reassessment method selected a combination with a true dose-limiting toxicity probability between 0.15 and 0.25 in 40.3% of simulated trials (accuracy index 35.5%), while treating 10.9 of 34.3 participants (31.8%) on average at these combinations.

Scenarios 4–6 represent more toxic scenarios in which there exist multiple combinations with dose-limiting toxicity probabilities exceeding 20%. In Scenario 4, there are 11 of 23 combinations that have true dose-limiting toxicity probabilities between 0.15 and 0.25. The proposed design selects one of these combinations as the maximum tolerated dose combination in 71.3% of simulated trials (accuracy index 28.2%) while allocating 28.4 of 43.7 participants (65.0%) on average to these combinations. The partial order continual reassessment method selects one of these combinations as the maximum tolerated dose combination in 71.2% of simulated trials (accuracy index 30.7%) while allocating 19 of 31 participants (61.3%) on average to these combinations. In Scenario 5, there are 6 of 23 combinations that have true dose-limiting toxicity probabilities between 0.15 and 0.25. The proposed design selects one of these combinations as the maximum tolerated dose combination in 50.2% of simulated trials (accuracy index 34.2%) while allocating 18.5 of 48.02 participants (38.5%) on average to these combinations. The partial order continual reassessment method selects one of these combinations as the maximum tolerated dose combination in 45.1% of simulated trials (accuracy index 32.7%) while allocating 10.4 of 33 participants (32.7%) on average to these combinations. In Scenario 6, there are 6 of 23 combinations that have true dose-limiting toxicity probabilities between 0.15 and 0.25. The proposed design selects one of these combinations as the maximum tolerated dose combination in 75.7% of simulated trials (accuracy index 64.7%) while allocating 25.6 of 37.5 participants (68.3%) on average to these combinations. The partial order continual reassessment method selects one of these combinations as the maximum tolerated dose combination in 70.3% of simulated trials (accuracy index 62.1%) while allocating 19.4 of 27.6 participants (70.3%) on average to these combinations. The final two scenarios represent situations in which (1) the maximum tolerated dose combination is only located in combinations 1–4, and combinations 5 and 6 are too toxic (Scenario 7); and (2) all of the combinations are overly toxic (Scenario 8). In Scenario 7, the proposed design selects the single maximum tolerated dose combination in 40.6% of simulated trials (accuracy index 76.8%) while allocating 6.1 of 28.6 participants (21.3%) on average to this combination. The partial order continual reassessment method selects the single maximum tolerated dose combination in 37.6% of simulated trials (accuracy index 76.7%) while allocating 5.8 of 26.6 participants (21.8%) on average to this combination. In Scenario 8, both methods stop early for safety in a large percentage of simulated trials (proposed 83.9%, partial order continual reassessment method 86.9%). The simulation results indicate that the proposed design and stopping rules have good statistical properties under a wide range of possible dose toxicity relationships and that on average the trial will achieve the study goal with accrual of fewer than 60 participants. It also performs well when compared to an alternative modified method.

Conclusions

The advance of novel methods for the design of clinical trials in early oncology development has grown in recent years, yet the use of innovative designs remains uncommon in real studies. In this article, we have outlined an adaptive design for a proposed Phase I study evaluating the safety of the combination of ceramide nanoliposome and vinblastine among an initial set of nineteen possible dose combinations in patients with relapsed/refractory acute myeloid leukemia and patients with untreated acute myeloid leukemia who are not candidates for intensive induction chemotherapy. The isotonic regression method presented offers an alternative to rigid rule-based methods that lack the capacity to manage the complexity presented by studies involving a large and dynamic set of drug combinations. The execution of more contemporary approaches that are tailored to the research questions that are currently being asked in early development trials are being supported by the FDA and review entities.^25–28 Simulation studies were performed to justify and evaluate the statistical properties of the design. The results in Figure 3 establish the method’s ability to efficiently recommend desirable combinations, defined by acceptable toxicity, in a high percentage of simulated trials with practicable sample sizes.

Software in the form of R²⁹ code for both simulation of design operating characteristics and direct protocol implementation of the method is available upon request from the first author. The numerical results presented in the simulation studies such as the distribution of sample size and frequency of early trial termination is the type of simulation information that improve understanding, acceptance, and approval of adaptive designs such as the one described in this manuscript.^30,31 Design specifics often are not provided on sites such as clinicaltrials.gov, and current clinical trials lack the transparency needed to support the timely implementation of novel designs. The objective of this current work is to highlight design strategies that tackle modern study objectives as a means of enhancing the use of novel approaches in future trials and to illustrate the flexibility of adaptive designs in handling dynamic design considerations. Presentation of novel design use in current clinical trials are needed to mitigate the obstacles that prevent more frequent execution of contemporary design strategies in early development. Thus, this paper can support the acceptance of novel design implementation in early-phase trials. In addition, given the extensive amount of time between study conception and protocol submission, it is important to offer design solutions that have wide applicability. Moreover, journals do not require complete protocols as supplemental material for completed dose-finding trials, and final clinical trial papers do not have the space to adequately present design details. The proposed study design complements a growing number of studies that have endorsed adaptive design strategies to answer complex research questions posed in current early-phase cancer trials. This endorsement for adaptive designs will strengthen early-phase trial design in drug combination studies.³² Early-phase designs with good statistical properties can have a significant influence on the drug development process.³³