Assume a total of D ascending doses of a new therapeutics is under investigation. At a given moment of the trial, suppose n d patients have been allocated to dose d among whom y d experience the DLT, d = 1 , … , D. Let p d denote the toxicity probability at dose d, and we assume y d | n d , p d ∼ B i n ( n d , p d ) . Also assume the efficacy outcome of each patient to be binary which is usually observed 10–12 weeks after the treatment for solid tumors in oncology trials. Letting the number of patients experiencing efficacy be v d and the response rate q d at dose d, we assume v d | n d , q d ∼ B i n ( n d , q d ) d = 1 , … , D .

The proposed Bi3+3 design extends the algorithm in the i3+3 design [9] given in Table 1. Let p T denote the target toxicity probability of the MTD, which is the highest toxicity rate that can be tolerated in the trial. Define the equivalence interval (EI) as [ p T − ϵ 1 , p T + ϵ 2 ], which describes a range of toxicity probabilities within which doses are deemed equivalent to the MTD. For example, p T = 0.25 and EI = [ 0.2 , 0.3 ]. Then the dose-finding algorithm of the i3+3 design can be summarised in Table 1. Here, a value “below” the EI means that the value is less than ( p T − ϵ 1 ), the lower bound of the EI. A value “inside” the EI means that the value is larger than or equal to ( p T − ϵ 1 ) but less than or equal to ( p T + ϵ 2 ), the upper bound of the EI. A value “above” the EI means that the value is larger than ( p T + ϵ 2 ). Decisions E, S, and D represent escalation to the next higher dose, stay at the current dose, and de-escalation to the next lower dose, respectively.

Phase I oncology trials enroll patients in cohorts, say a group of three patients per cohort. After a cohort is enrolled and assigned to a dose level for treatment, the patients are followed for three to four weeks to evaluate drug safety and record any DLT outcomes. While the patients are being followed, new patients may be eligible for trial enrollment. The proposed Bi3+3 design allocates these patients, as backfill cohorts, to lower doses, which have already been tested and exhibit sufficient safety.

Assume dose d is currently used for treating patients in the trial. We denote “main cohort” (mc) the cohort of patients for dose escalation, and “backfill cohort” (bc) the cohort of patients for backfill at lower doses. At a given moment of the trial, assume an mc of patients is allocated to dose d. After the mc is allocated, Bi3+3 allocates bc to one or multiple doses that are lower than the current dose d. Denote the set of doses assigned to bc by B d = { k 0 , k 0 + 1 , … , d − 1 } where k 0 in B d is the lowest dose available for backfill. The proposed Bi3+3 design continuously and adaptively determine k 0 throughout the trial. We will describe a method to determine k 0 later in Section 2.3. But first, we describe the main algorithm for the Bi3+3 design next.

The following algorithm (also summarized in Figure 2) 1.

Enroll an mc of patients at dose d in the main cohort. The default cohort size is 3.

In the Bi3+3 algorithm described above, when the mc is allocated to dose d, in principle all doses lower than d are available for backfilling, i.e., B d = { k 0 = 1 , 2 , … , d − 1 }. Due to the up-and-down nature of the decisions in Table 1, all doses lower than d must have been tested and their safety has been established by the DLT data observed from the patients treated at these doses. Otherwise, it is impossible for the design to reach dose d for mc if any lower dose exhibits excessive toxicity, in which case the dosing decision would not have been E, to escalate. Therefore, all the doses in B d are deemed safe based on observed data. To this end, to determine B d , we consider efficacy data from patients and exclude low doses with insufficient efficacy. These doses are considered having little chance to induce beneficial efficacy response from cancer patients and therefore are not worth further exploration.

We follow the idea in [3]. We slightly abuse the notation d hereinafter to either denote the current dose in the trial or a general index of any doses when needed. Occasionally, we also use k or i to index doses when d is used to denote the current doses. Given context, these notations should be clear to readers. We start with B d = { k 0 = 1 , 2 , … , d − 1 } assuming the current dose is d for the mc. When a new efficacy outcome is observed from either mc or bc, we refresh the lowest dose k 0 in B d based on the following procedure. Recall the efficacy is assumed to be a binary outcome, e.g., objective response (OR) as in Response Evaluation Criteria in Solid Tumors (RECIST).

To update k 0 , we first define for a dose k ∈ B d , q k + = ∑ i > k D n i q i ∑ i > k D n i , where q i is the true efficacy probability for dose i, and n i denotes the number of patients with available efficacy outcomes at dose i. By convention, 0 / 0 = 0. The quantity q k + represents the expected response rate at doses higher than dose k. Under a Bayesian model that will be presented later, we compute the posterior probability ξ k = Pr { q k + > q k ∣ D }, k ∈ B d , where D = { ( v d , n d ) , d = 1 , … , D }, the observed efficacy data. If ξ k > ξ 0 , say ξ 0 = 0.8, dose k is deemed to be less efficacious than doses higher than k. The first dose k 0 in B d is set to dose ( k + 1 ), i.e., k 0 = k + 1.

Throughout the trial, posterior probabilities ξ k for k ∈ B d are monitored continuously to determine k 0 and update B d .

Following the i3+3 design [9], two safety rules are added as ethical constraints to avoid excessive toxicity. Both rules are applied in step 5 of the Bi3+3 design (Section 2.2).

In safety rules 1 and 2, P r { p d > p T ∣ H } is under the posterior distribution B e t a ( α 0 + y d , β 0 + n d − y d ) which corresponds to an independent B e t a ( α 0 , β 0 ) for p d , and α 0 = β 0 = 1 is used. As a special case of the safety rule 1, if the decision for the current dose d is “E” according to the decision rule in Section 2.2, and if the next higher dose ( d + 1 ) level has been excluded due to Rule 1, the decision for the current dose is changed to “S” Stay for the current dose because there is no available dose to escalate to.

After a prespecified maximum sample size of the trial is reached, all the DLT and efficacy data will be collected for all patients enrolled in the trial. We apply the same MTD selection procedure as in many recent designs based on the isotonic regression [7, 5, 9]. In particular, an independent Beta(0.005, 0.005) prior is assumed for p d , d = 1 , … , D. And the pooled adjacent violators algorithm (PAVA) [13] is applied to calculate the posterior mean toxicity probabilities for all doses, p ˆ d , d = 1 , … , D, subject to the order constraints p ˆ k > p ˆ d for ∀ k > d. Among all tried doses, select as the estimated MTD d ˜ the dose with the smallest difference p ˆ d − p T and p ˆ d ≤ p T + ϵ 2 , i.e., d ˜ = argmin { d : n d ≠ 0 , p ˆ d ≤ p T + ϵ 2 } p ˆ d − p T .

To select the OBD, we modify the change-point model in [3] and construct a four-parameter model for the dose-efficacy response. Specifically, recall that q d is the probability of efficacy at dose d. Then assume logit ( q d ) = β 0 + β 1 x d I ( x d ≤ h ) + ( β 2 + h ) I ( x d > h ) . Here x d ∈ { 1 , … , D } denotes the D discrete dose levels, β 0 is the intercept, β 1 is the slope, and h ∈ { 1 , … , D } is the change point of the model after which efficacy is assumed to plateau. Under this model, when the dose level is less than or equal to h, by assuming β 1 > 0, there is a positive monotonic relationship between dose level and efficacy response rate; when the dose level is higher than h, efficacy response rate no longer increases with dose level and stays as constant β 0 + β 1 ( β 2 + h ) at the logit scale. Quantity β 1 β 2 represents the increment in comparison to the response rate β 0 + β 1 h at the change point h. This increment is needed to describe the response rate of the dose immediately after the change point h due to discrete dosing.

The prior of β 1 is set as log-normal distribution with mean 0 and a large variance. Parameter β 2 describes a jump after the change point in response rate, and its prior distribution follows another log-normal. We use a normal prior with mean −2 for β 0 so that the prior probability of response when there is no treatment is small. As for parameter h, we set a discrete prior where h takes the value of each dose level from the starting dose 1 to the highest dose D with a probability. Specifically, we set the prior probability to be a small value e, say e = 0.05, for any dose with no enrolled patients, and divide the remaining prior probability evenly across doses with enrolled patients. We use D ′ ≤ D to denote the number of doses with enrolled patients. The prior distributions are given as follows. β 0 ∼ N ( − 2 , 10 ) , β 1 ∼ L N ( 0 , 10 ) , β 2 ∼ L N ( 0 , 10 ) , and h ∼ C a t ( r 1 , … , r D ) where r d = e ∗ I ( n d = 0 ) + 1 − ( D − D ′ ) ∗ e D ′ I ( n d > 0 ) , where C a t ( r 1 , … , r D ) is a discrete distribution taking values in { 1 , … , D } with corresponding probabilities ( r 1 , … , r D ), respectively. Note r d is simply the mathematical expression that gives us the aforementioned prior probability for h.

Given available efficacy data D = { ( v d , n d ) , d = 1 , … , D }, the posterior probability ϕ d = Pr { h = d ∣ D }, d = 1 , … , D, can be calculated and used to select OBD. The change point h ∗ is estimated as h ∗ = D ′ ∧ ( argmax d ϕ d ) . Considering both toxicity and efficacy effects of these doses, we finally select the OBD d ∗ based on the selected MTD d ˜ and the estimated change point h ∗ as d ∗ = d ˜ ∧ ( h ∗ + 1 ) .

We first compare Bi3+3 with the Backfill CRM design in [3]. We briefly summarize the Backfill CRM design below, which provides instructions for allocating patients in the main cohort and backfill cohort.

We implement the Backfill CRM design and benchmark our implementation against the simulation results in [3]. Our results, shown in Appendix B, are very close to those presented in [3], thus assuring our implementation. We then compare the Bi3+3 design with Backfill CRM on a variety of scenarios. See Figure 3 for the five scenarios used in our simulation. In the subsequent simulation, the target probability of toxicity for MTD is set to p T = 0.3, and the equivalence interval for Bi3+3 is E I = [ 0.25 , 0.35 ]. As the backfill strategy is different in Bi3+3 design and Backfill CRM design, we try to match the overall sample size (main cohorts + backfill cohorts) for both designs for fair comparison. For the Bi3+3 design, 30 patients are enrolled in the main cohorts which result in an average sample size of 42.8, with the additional 12.8 patients coming from the backfill cohorts. As for the Backfill CRM design, we set 24 as the main cohort sample size which produces an average of 45.0 patients for the entire trial. We use the skeleton in [8] for Backfill CRM, a popular choice in practice. The operating characteristics of both designs are presented in Table 2.

For scenarios 1 and 2, the true MTD is the fifth dose and fourth dose, respectively. The efficacy response rates of doses reach a plateau at the third dose, which is the true OBD. In scenarios 1 and 2, Bi3+3 selects the true MTD in 41.7% and 40.8% of the simulated trials, highest among all the doses, and it selects the true OBD, dose level three, 34.3% and 39.2% of the times. The Backfill CRM design has slightly higher percentages in OBD selection in these two scenarios although it also tends to select the next higher dose with a high frequency. On average, Bi3+3 assigns 10.3 patients to dose levels three and four in scenario 1, respectively. Compared with Backfill CRM, the total sample size of Bi3+3 is 2.2 patients smaller, although there is not much difference in the patient allocation to the OBDs. In scenarios 3 and 4 both toxicity and efficacy increase with dose level and the MTD and OBD are the same, which is dose level five. The Bi3+3 model identifies dose five as the MTD, with frequencies of 36.1% and 42.3%, and as the Optimal Biological Dose (OBD), with frequencies of 30.5% and 40.1% in scenarios 3 and 4, respectively. In contrast, the Backfill CRM design selects dose three as OBD with probabilities 39.4% and 43.9% in the two scenarios, which is less desirable. For patient allocation, Bi3+3 allocates on average 6.0 and 7.7 patients to dose five compared to 4.9 and 6.4 by Backfill CRM in scenarios 3 and 4, respectively. In the last scenario 5, efficacy plateaus at the true MTD, indicating that the true MTD is the same as the true OBD, which is dose four. However, dose three has a toxicity rate close to the MTD, which makes it difficult for both designs to differentiate the two doses. This can be seen in the selection percentages of the MTD and OBD for both designs. Across all the scenarios, Backfill CRM is slightly more conservative than Bi3+3 as it allocates more patients to lower doses. However, it tends to select doses in the middle. This could be desirable if the true OBD is near the middle like in scenarios 1 and 2. In contrast, in cases where the true OBD is at a high dose, like scenarios 3 and 4, Backfill CRM may have less chance finding them. In summary, both designs exhibit desirable features in these five scenarios.

Apart from these features, we also calculate the mean of efficacy using the regression model in each scenario. The results are reported in Table 2 corresponding to the line “Efficacy estimation”. The estimated efficacy response rates in general do not appear to deviate much from the true efficacy probabilities.

Next, we apply mTPI-2 [5] to illustrate some features of Bi3+3 as mTPI-2 is a design that does not allow patient backfill. For fair comparison, we first enroll 30 patients for mTPI-2, select an MTD, and allow thirteen more patients to be enrolled at the selected MTD (to adjust for the fact that Bi3+3 would enroll more patients in the backfill cohorts). After the additional thirteen patients are enrolled and their DLT data are observed, we re-select the MTD based on the additional data. We report the frequency of the re-selected MTD in the simulation results for mTPI-2.

The results comparing Bi3+3 and mTPI-2 are presented in Table 3. Bi3+3 has an advantage in many scenarios as it allows patients to be backfilled. For example, it does not increase trial duration while allowing more patients to be assigned to lower doses, resulting in better OBD selection. See Table 3 for a summary.

As a sensitive analysis, we vary the target toxicity probability from p T = 0.3 to p T = 0.25 and change the EI from [0.25, 0.35] to [0.2, 0.3] (Table 6). We also construct four more scenarios to assess the trial duration and patients enrollment. Bi3+3 shows desirable results as expected, thanks to the ability to backfill patients and time-to-event modeling (Table 4).