^{1}

^{2}

The authors have declared that no competing interests exist.

Although many novel phase I designs have been developed in recent years, few studies have discussed how to incorporate external information into dose-finding designs. In this paper, we first propose a new method for developing a phase I design, Bayesian optimal interval design (BOIN)[Liu S et al. (2015), Yuan Y et al. (2016)], for formally incorporating historical information. An algorithm to automatically generate parameters for prior set-up is introduced. Second, we propose a method to relax the fixed boundaries of the BOIN design to be adaptive, such that the accumulative information can be used more appropriately. This modified design is called adaptive BOIN (aBOIN). Simulation studies to examine performances of the aBOIN design in small and large sample sizes revealed comparable performances for the aBOIN and original BOIN designs for small sample sizes. However, aBOIN outperformed BOIN in moderate sample sizes. Simulation results also showed that when historical trials are conducted in settings similar to those for the current trial, their performance can be significantly improved. This approach can be applied directly to pediatric cancer trials, since all phase I trials in children are followed by similar efficient adult trials in the current drug development paradigm. However, when information is weak, operating characteristics are compromised.

In the field of drug development, there is high interest in conducting clinical trials using designs that can enable the incorporation of external information, such as prior or historical information, with trial data to save sample sizes, improve the power, and expedite the trial process. Several studies have focused on developing designs that incorporate external information for phase II or III trials, for example, meta-analytic power prior–based multiple historical sources [

The BOIN design’s escalation/de-escalation decisions are based on two boundaries. Given the DLT target, the two boundaries are fixed (derived by minimizing the overall decision error rate). However, in some situations, we might need to have an unbalanced control of misallocation of patients to under-toxic and over-toxic dose levels. By having accumulative information, we could have a better understanding of the toxicity rate for each dose level tried; fixed boundaries cannot reflect these dynamics. The second goal of this study is to propose flexible boundaries that can change during the trial process. This design is termed as adaptive BOIN (aBOIN).

The rest of the paper gives a brief introduction to the BOIN design, followed by a methodology proposed to incorporate external information based on the BOIN design framework. Next, an approach for extending the BOIN with fixed boundaries to the aBOIN design with non-fixed boundaries is proposed. Empirical findings are shown by comprehensive simulations with derivation of the theoretical properties. The paper ends with a final discussion.

The BOIN design proposed by Liu and Yuan in 2015 [

The BOIN design can be summarized as follows:

Patients in the first cohort are treated with the lowest or a pre-specified dose level.

Let

if

if

otherwise, i.e.

To ensure that dose levels of treatment always remain within the pre-specified dose range, the dose escalation or de-escalation rule needs to be adjusted for the lowest or highest levels of

This process continues until the maximum sample size is reached or the trial is terminated because of excessive toxicity, as described next.

The selection of interval boundaries λ_{1} and λ_{2} is critical, because two parameters essentially determine the operating characteristics of the design. The BOIN design is optimal in the sense that it selects λ_{1} and λ_{2} to minimize incorrect decisions of dose escalation and de-escalation during the trial.

By using _{j} to denote the true toxicity probability of dose level _{1} denotes the highest toxicity probability that is deemed sub-therapeutic (i.e., below the MTD) such that dose escalation is required, and _{2} denotes the lowest toxicity probability that is deemed overly toxic, such that dose de-escalation is required.

Under the Bayesian paradigm, each hypothesis was assigned an equal prior probability, denoted as

Details can be found in [

Viele et al. [

By using the BOIN design framework, prior information can be incorporated naturally via only modifying

Comparing _{0j}, _{1j}, and _{2j}.

By notations, assuming there are _{0,j}, _{1,j} and _{2,j}, _{0j}, _{1j}, _{2j},

Priors | _{1} |
_{2} |
⋯ | _{j} |
⋯ | _{d−1} |
_{J} |
---|---|---|---|---|---|---|---|

_{0} |
_{0,1} |
_{0,2} |
⋯ | _{0,j} |
⋯ | _{0,J − 1} |
_{0,J} |

_{1} |
_{1,1} |
_{1,2} |
⋯ | _{1,j} |
⋯ | _{1,J − 1} |
_{1,J} |

_{2} |
_{2,1} |
_{2,2} |
⋯ | _{2,j} |
⋯ | _{2,J − 1} |
_{2,J} |

Prob | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

On the basis of data in _{j} is closest to the target DLT rate, we assign a larger probability to _{0,j} (e.g., 0.6) and then assign _{1,j}, _{2,j} to be equally half of the rest of the probability (e.g., 0.4/2 = 0.2). Here, we believe a priori with 60% confidence that dose _{j} would be the MTD and 20% confidence that this dose would be under-dosing or over-dosing. We define odds_{j} to be

By eliciting values of the remaining cells in _{0}, that is, (_{0,1}, ⋯, _{0,J}). We emphasize here that pre-specification of the probability vector for _{0} is feasible. For example, if we have strong evidence that one dose is near to the MTD, as in pediatric trials, because MTDs in children and adults correlate strongly and 80% of the adult dose is recommended as the starting dose for children, the investigator can effortlessly select with high confidence the dose that can be the MTD and also other doses. If there is weak prior knowledge, equally likely probabilities can be assigned to this vector.

When eliciting values for two probability vectors of _{1} and _{2}, the two vectors need to be in decreasing and increasing orders, respectively. This is because _{1} refers to the under-dosing hypothesis; therefore, probabilities of believing in _{1} would decrease when dose level increases and vice versa for the probability vector for _{2}. For example, for a trial with five dose levels, if we assign the probability vector to _{0} to be _{0,1} = 0.05, _{0,2} = 0.15, _{0,3} = 0.6, _{0,4} = 0.15, _{0,5} = 0.05, then _{0,1} = 0.05 it means we have very low confidence that the first dose is the MTD. Since the first dose is the lowest dose among the five doses, the _{1,1} should be the highest among (_{0,1}, _{1,1}, _{2,1}), since it is the safest dose level; that is, we have high confidence that the first dose will lie in the interval defined by the hypothesis _{1}, which corresponds to the under-dose interval. As an example, let us assign values (_{0,1} = 0.05, _{1,1} = 0.85, _{2,1} = 0.10) to them by considering the constraint _{0,j} + _{1,j} + _{2,j} = 1, ∀, _{0,1} = 0.15, this again means that we have little confidence that this dose is the MTD and, similarly, _{1,2} should still be the highest dose among (_{0,2}, _{1,2}, _{2,2}). However, the probability of _{1,2} to be in _{1} should now be lower than that for _{1,1}, since dose 2 has a higher toxic rate than dose 1. For example, if we assign the probabilities as (_{0,2} = 0.20, _{1,2} = 0.60, _{2,2} = 0.20), there should be a decreasing trend in the probability vector of _{1} and an increasing trend in the probability vector of _{2}. Similarly, a decreasing trend will be observed for a vector probability of _{2}.

Given the above premise, we propose the following algorithm that can automatically implement the assignment of horizontal probability vectors in

Assign each dose a probability for _{0}, that is, a prior probability vector of (_{0,1}, ⋯, _{0,J}), to best “guess” which of these

This step is not so challenging if clinicians have strong confidence on which dose is closest to the MTD target. For example, in pediatric trials, we can often choose the MTD for adult patients or a starting MTD dose for pediatric patients. In this step, clinicians can also choose a set of skeletons for the CRM.

If the dose _{j} = odds(_{1,j}, _{2,j}) = 1 to assign probabilities to _{1,j} and _{2,j} given _{0,j} in Step 1. Also, let the lowest dose have odds_{1} = odds(_{1,1}, _{2,1}) = 10 and the highest dose have _{1,1}, _{2,1} and _{1,J}, _{2,J}. If the lowest or highest dose levels are believed to be the MTD, then the odds for it is set to be 1.

Use extrapolation method (see details in Appendix) to elicit prior probabilities for the rest of two vectors (_{1,1}, ⋯, _{1,J}) and (_{2,1}, ⋯, _{2,J}) can be easily derived.

The above algorithm is easy to use since it only requires the investigator to provide probability guesses for _{0}s for each investigated dose level. All the other remaining probabilities in

This section discusses the extension of the BOIN to aBOIN design with adaptive boundaries. For interval-based designs, the first step is to specify an indifference interval defined by two _{1} and λ_{2}, which are indirectly linked to the under- and over-dose hypotheses introduced earlier. If we denote the MTD toxicity rate as _{1} = 0.6_{2} = 1.4_{1} and _{2}.

The BOIN design also has useful theoretical properties, such as minimizing the decision-making error, long-term memory coherence, and convergence to the MTD dose. In this section, we first demonstrate that the proposed aBOIN design also inherits theoretical properties from the BOIN design and then conduct simulation studies to see whether this extension could improve the original BOIN design.

Extensive simulation studies have shown that the BOIN design is simple but has excellent operating characteristics comparable with those of the more complicated model-based CRM designs [

Adaptive shrinking boundaries can possibly be used to further control the misallocation of patients to over-toxic doses. In the BOIN design framework, we reconstruct the _{1} and _{2} to be _{j} is the cumulative number of patients treated at a dose level of _{1}, _{2} < 1 are _{1}, Δ_{2} can be interpreted as pre-specified effect sizes to construct the decision intervals in the BOIN design as given above. Obviously, by doing so, the two fixed boundaries of the original BOIN design now depend on the dynamic number _{j}s, which is number of patients treated at the dose level _{1} and _{2} converge to the MTD target _{1} increases to _{1}, _{2}) is bound to converge to the MTD as sample sizes increase. This construction is also very flexible for designing trials. For example, if safety of the design is a very big concern, we can make the upper boundary _{2} to shrink faster than the lower boundary _{1} by using discounting factors _{1} < _{2} to penalize assignment of patients to dose levels beyond the MTD.

Based on the above redefinition of _{1} and _{2}, we have the updated three-point hypotheses of the BOIN design as

The above definition of _{1} and _{2} is reminiscent of boundaries of an optimal symmetric group sequential design by Eales & Jennison (1992) [

In a similar vein of deviations for the BOIN design, the optimal λ_{1j} and λ_{2j} minimize the decision error rate can be derived as λ_{1}(Δ_{1}, _{j}) and λ_{2}(Δ_{2}, _{j}) with

Different from the original BOIN design, λ_{1}(Δ_{1}, _{j}) and λ_{2}(Δ_{2}, _{j}) is presently depend on accumulative sample size _{j} along the trial process instead of constants. We can show that the aBOIN design still enjoys the following theoretical properties

_{1} < _{2} >

See Appendix for proofs.

To use the proposed aBOIN design in practice, we need to specify the values of Δ_{1} and Δ_{2}, which determine the _{1}, _{2} and subsequently the values of λ_{1}, λ_{2}. Since the original BOIN design recommends _{1} = 0.6_{2} = 1.4_{1} = Δ_{2} = 0.4_{j} = 1.

In practice, we also introduce a lead-in process in which we follow the procedure given in the original BOIN design for a pre-specified number of patients, for example, _{1}, and the trial then switches to the aBOIN design with adaptive shrinking boundaries.

Our exploratory simulations (not shown here) with a maximum sample size of 30 show negligible differences in the performance of the trial when _{1} = 6 or _{1} = 9 is used. Hereafter, we will use _{1} = 6 in simulation studies for the lead-in period.

Note that by adopting the accelerating parameter _{1} and _{2}, hypotheses of _{1j} and _{2j} are no longer symmetric. However, including accelerating parameters _{1} and _{2} does not influence the asymptotic properties of the aBOIN design. Furthermore, different _{1} and _{2} may satisfy practical needs; for example, if we want a tighter control of the over toxicities, we can let _{2} > _{1}, which means that the upper boundary would shrink quicker than the below boundary.

Additionally, the aBOIN design that incorporates external information can be derived straightforwardly to have the following form:

In this section, we explore the operating characteristics of the proposed aBOIN design with and without incorporating prior information by comparing it to the original BOIN design. The aims of the simulation study are twofold: (i) to explore the behavior of the aBOIN design that incorporates prior information compared with that of the original BOIN design and the aBOIN design that does not incorporate prior information, and (ii) explore the operating characteristics of the original BOIN and aBOIN designs.

We consider trials with five dose levels and a maximum sample size of 30 patients, with a cohort size of three patients. Twenty different scenarios (one half with dose-limiting toxicity (DLT) rates of 20%, and the other half with DLT rates of 30%) with various locations and DLT rates are shown in

Scenario | Dose Level | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

DLT 20% | DLT 30% | |||||||||

1 | 2 | 3 | 4 | 5 | 1 | 2 | 3 | 4 | 5 | |

1 | 0.22 | 0.23 | 0.25 | 0.27 | 0.33 | 0.34 | 0.35 | 0.36 | ||

2 | 0.18 | 0.22 | 0.23 | 0.25 | 0.27 | 0.33 | 0.34 | 0.35 | ||

3 | 0.17 | 0.18 | 0.22 | 0.23 | 0.26 | 0.27 | 0.33 | 0.34 | ||

4 | 0.1 | 0.15 | 0.18 | 0.22 | 0.15 | 0.2 | 0.27 | 0.33 | ||

5 | 0.08 | 0.1 | 0.15 | 0.18 | 0.1 | 0.15 | 0.2 | 0.27 | ||

6 | 0.3 | 0.35 | 0.4 | 0.45 | 0.4 | 0.45 | 0.5 | 0.55 | ||

7 | 0.1 | 0.3 | 0.35 | 0.4 | 0.2 | 0.4 | 0.45 | 0.5 | ||

8 | 0.05 | 0.1 | 0.3 | 0.35 | 0.1 | 0.2 | 0.4 | 0.45 | ||

9 | 0.01 | 0.05 | 0.1 | 0.3 | 0.05 | 0.1 | 0.2 | 0.4 | ||

10 | 0.01 | 0.05 | 0.08 | 0.1 | 0.05 | 0.1 | 0.15 | 0.2 |

For each scenario, we simulated 10,000 trials. We implemented the BOIN design using the R package BOIN with its default design parameters. For the aBOIN design, we specified the accelerating factors as _{1} = 0.4, _{2} = 0.9, which were derived by trial and error, and we only activated the adaptive shrinking mechanism in at least six patients who had been treated (referred to as the lead-in period). As introduced in [

To incorporate prior information, we first specify a probability vector for _{0} row in _{0} with probabilities (_{0,1}, ⋯, _{0,5}) = (0.2, 0.45, 0.7, 0.45, 0.2) for all scenarios, and the other two probability vectors are derived by using the procedure introduced above to be (_{1,1}, ⋯, _{1,5}) = (0.72, 0.44, 0.15, 0.12, 0.08) and (_{2,1}, ⋯, _{2,5}) = (0.08, 0.11, 0.15, 0.43, 0.72). Considering these specific settings, this means that we have 70% confidence that dose level 3 could be the MTD and 45% confidence that dose level 2 or 4 could be the MTD.

Simulation results of the PCS (%) for the BOIN and the aBOIN design with or without incorporating prior information are shown in ^{1} refers to PCS (%) of the aBOIN design without incorporating prior information. Similarly, for scenarios 1, 3, and 8, the corresponding PCSs (%) are 32.22%, 16.75%, and 43.82% with MTD locations at dose levels 1, 3, and 8, respectively. The third row with aBOIN^{2} refers to the PCS (%) of the aBOIN design incorporating prior information. The corresponding PCS (%) for scenarios 1, 3, and 8 are 26.03%, 22.61%, and 47.65%, respectively. Because the prior has placed high confidence on dose level 3 being the MTD, and scenarios 3 and 8 are scenarios with MTD locations at dose level 3, for scenarios 3 and 8, aBOIN^{2} has highest PCSs (%) among the three designs. For the remaining scenarios, the PCS (%) of aBOIN^{2} is comparable to or lower than that of the BOIN and aBOIN^{1} designs. For example, for scenario 3 of DLT with a 20% toxicity rate, the PCS (%) of the aBOIN^{2} is 22.61%, whereas those of the BOIN and aBOIN^{1} are 16.1% and 16.75%, respectively, because for scenario 3 with the MTD located at dose level 3, the prior guess also has the strongest confidence at dose level 3. However, if we check scenario 1 with the MTD located at dose level 1, we put only 20% confidence into dose level 1 and find that the aBOIN^{2} design has the worst performance in terms of the PCS% (26.03%), whereas BOIN and aBOIN^{1} have a higher PCS% (37.15% and 32.22%, respectively). At a DLT rate of 30%, similar patterns can be observed.

Scenario | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | Average |
---|---|---|---|---|---|---|---|---|---|---|---|

DLT rate 20% | |||||||||||

PCS(%) | |||||||||||

BOIN | 37.15 | 22.81 | 16.1 | 16.38 | 27.3 | 55.11 | 48.96 | 45.41 | 46.27 | 56.69 | 37.22 |

aBOIN^{1} |
32.22 | 20.67 | 16.75 | 18.17 | 34.33 | 51.08 | 46.32 | 43.82 | 46.63 | 60.05 | 37.0 |

aBOIN^{2} |
26.03 | 23.39 | 22.61 | 14.99 | 33.12 | 51.07 | 45.2 | 47.65 | 39.1 | 57.28 | 36.04 |

# of Patients at MTD | |||||||||||

BOIN | 16.81 | 8.1 | 4.78 | 4.02 | 4.98 | 20.4 | 13.62 | 11.22 | 10.11 | 10.93 | 10.49 |

aBOIN | 14.48 | 8.13 | 5.49 | 4.89 | 6.67 | 18.24 | 12.82 | 10.81 | 10.15 | 11.92 | 10.36 |

Risk of overdosing 60% | |||||||||||

BOIN | 31.44 | 16.53 | 6.59 | 3.71 | 0 | 16.84 | 10.69 | 6.77 | 3.67 | 0 | 9.62 |

aBOIN | 40.5 | 23.63 | 9.77 | 2.59 | 0 | 22.85 | 14.81 | 7.46 | 2.14 | 0 | 12.38 |

Risk of underdosing 60% | |||||||||||

BOIN | 0 | 32.67 | 55.45 | 70.38 | 80.93 | 0 | 23.82 | 35.38 | 41.35 | 59.09 | 39.91 |

aBOIN | 0 | 24.86 | 43.54 | 60.42 | 74.36 | 0 | 19.85 | 29.79 | 33.25 | 53.75 | 33.98 |

DLT rate 30% | |||||||||||

PCS(%) | |||||||||||

BOIN | 41.08 | 24.5 | 18.8 | 21.57 | 38.92 | 55.17 | 45.65 | 43.29 | 43.92 | 59.06 | 39.2 |

aBOIN^{1} |
39.08 | 26.3 | 19.7 | 21.16 | 36.14 | 54.1 | 46.52 | 41.51 | 42.37 | 56.85 | 38.37 |

aBOIN^{2} |
31.01 | 27.29 | 25.67 | 19.97 | 39.64 | 53.23 | 46.25 | 45.31 | 38.39 | 58.8 | 38.56 |

# of Patients at MTD | |||||||||||

BOIN | 17.66 | 9 | 5.49 | 5.4 | 7.25 | 20.47 | 13.4 | 11.16 | 10.01 | 11.66 | 11.15 |

aBOIN | 17.58 | 9.19 | 5.59 | 4.97 | 5.97 | 20.32 | 12.58 | 10.17 | 9.21 | 9.68 | 10.53 |

Risk of overdosing 60% | |||||||||||

BOIN | 35.29 | 18.63 | 6.25 | 1.92 | 0 | 23.08 | 15.19 | 8.55 | 2.17 | 0 | 11.11 |

aBOIN | 29.36 | 12.01 | 4.1 | 0.78 | 0 | 16.73 | 8.24 | 4.71 | 0.78 | 0 | 7.67 |

Risk of underdosing 60% | |||||||||||

BOIN | 0 | 33.07 | 54.08 | 62.45 | 71.54 | 0 | 23.77 | 33.04 | 37.28 | 52.5 | 36.77 |

aBOIN | 0 | 29.07 | 56.42 | 70.05 | 82.23 | 0 | 21 | 38.26 | 45.8 | 68.17 | 41.1 |

aBOIN^{1}: adaptive BOIN design without incorporating prior information.

aBOIN^{2}: adaptive BOIN design with incorporating prior information.

From these results, we infer that when the prior guess for the MTD location is close to the truth, the aBOIN version incorporating prior information performs the best in terms of the PCS metric; in other scenarios, its results vary widely and can sometimes even be very inaccurate. Given these observations, we recommend that in actual practice, the aBOIN incorporating prior information should be used only when the investigator has strong confidence or there is prior or historical information on which dose is or approximate to the MTD.

In this subsection, we investigate the aBOIN^{1} design, that is, the aBOIN design without incorporating prior information. However, in this subsection, we still call this version aBOIN for brevity. We closely examine not only the PCS% but also the other three metrics, percentages of patients allocated to a true MTD during the trial (MTD%), and the mean number of observed DLTs throughout the trial (# of DLTs). Results are shown in

For a DLT rate of 20%,

We also examined the convergence rate with the PCS (%) metric for asymptomatic properties for both designs. We present partial results for the first four scenarios for DLT rates of 20% and 30%.

Note that if investigators have vague or less confidence about prior experience or information, we still suggest that they use the BOIN design without prior information.

We have developed two extensions of the BOIN design. The first one develops an accessible approach to allow the incorporation of prior or historical information in the phase I trial. The second extension proposes adaptive shrinking boundaries (aBOIN design), whereas the original BOIN design has fixed boundaries. The aBOIN design uses accelerating factors to control the shrinking speed rates of lower and upper boundaries. Theoretical properties were derived for the aBOIN design.

Performances of the proposed methods were discussed by simulations. When setting up the location for the MTD a priori that was close to the MTD, the aBOIN design incorporating prior information showed better performance than the original BOIN design. However, if the prior deviated from the truth, performance of the aBOIN design was inferior to that of the BOIN design. This is understandable, since there were very few sample sizes and therefore it was hard to dominate the estimation procedure for deciding the dose. Therefore, we caution practitioners to use prior information in real trials unless there is strong confidence. The second extension of the proposed aBOIN design was examined numerically by using a finite sample and a large sample. For finite sample sizes, performances were similar when comparing the aBOIN without incorporating prior information to the BOIN design. Although the proposed aBOIN design outperforms in asymptotic properties, it has limited use in actual phase I trials due to the small sample size. In summary, the original BOIN design can be improved only if very informative historical information is available.

_{1j} < _{2j} >

Thus, the aBOIN is long-term memory coherent.

_{1} and _{2}, we can get λ_{1} → _{2} → _{j} tends to ∞. By the L’hopital’s rule, we get

The proof of λ_{2} → _{j} → ∞ is similar as above.

That is, both λ_{1} and λ_{2} shrink toward the MTD target

_{1} < _{2} > _{1} <

Since we have proved that λ_{1} converges to _{1} is an increasing function of _{1}, then we can prove λ_{1} <

(since 0 < _{1} < 1)

Thus, _{1} = _{1}) is an increasing function of _{1} with limit at

Hence, we have λ_{1} <

Similarly, we can prove λ_{2} >

_{j} is large enough, pr(_{nj → ∞}λ_{1}/_{1} = 1 and lim_{nj → ∞}λ_{2}/_{2} = 1, for _{1}, _{2} < 1 by the CLT, we can get

Then, the proof provided by Oron, Azriel, and Hoff (2011) can be directly used to obtain the result.

Assuming _{0}, that is, guessing which dose would possibly be the MTD, denoted as _{0,1}, ⋯._{0,j}, ⋯, _{0,J}. We can also assume odds of _{1} to _{2} at dose level 1, since at the lowest dose, it would have high confidence that this first dose would be under-dosing than over-dosing, thus, we let _{1} to _{2} at dose level

Based on the above odds_{1} and odds_{J}, we use the definition of the odds to evaluate the prior probabilities for _{1} at dose levels 1 and J as:
_{2} at dose levels 1 and J are:

Now, we have the prior probabilities of the first row for _{0} and first and last (J-th) columns in

To be specific, for computing probabilities of _{1} for dose levels from 2 to

For computing probabilities of _{1} for dose levels from

Thus, based on the above steps, we assign probabilities for the first (_{0}) and second rows (_{1}) in _{2}) are:

We provide a numerical example for showing the above procedure. Assuming there are 5 dose levels and, without losing generality, assuming the 3rd dose level is closest to the MTD prior to the study. For example, the prior probability vector is set to be (_{0,1}, ⋯, _{0,5}) = (0.2,0.45,0.7,0.45,0.2), that is, this is the 1st row in _{3=} 1, so we have _{1,3} = _{2,3} = 0.15 since _{0,3} = 0.7 now. By the algorithm, we also know that odds for the first and last dose levels are as odds_{1} = 10 and

By using the above formula

Then, we can have _{2,1} = 1 − _{1,1} − _{0,1} = 1 − 0.72 − 0.2 = 0.08 and _{2,5} = 1 − _{1,5} − _{0,5} = 1 − 0.08 − 0.2 = 0.72.

Thus, by using the formula _{1}) in _{1,1}, _{1,2}, _{1,3}, _{1,4}, _{1,5}) = (0.72, 0.44, 0.15, 0.12, 0.08).

Then, for the third row (_{2}), we have (_{2,1}, _{2,2}, _{2,3}, _{2,4}, _{2,5}) = 1 − (_{0,1} + _{1,1}, _{0,2} + _{1,2}, _{0,3} + _{1,3}, _{0,4} + _{1,4}, _{0,5} + _{1,5}) = (0.08, 0.11, 0.15, 0.43, 0.72).

PONE-D-20-07779

A Phase I Dose-finding Design with Adaptive Shrinking Boundaries and Incorporation of Historical Information -- Extensions of the Bayesian optimal interval (BOIN) design

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Reviewer #1: Yes

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Reviewer #1: Yes

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Reviewer #1: Bayesian optimal interval design (BOIN) is a recently developed model-assisted method for phase I dose-finding trials. This design has been accepted as a popular design in this field due to its simplicity and desirable operating characteristics. In this paper, the authors extend the original BOIN design from two aspects: incorporate the prior knowledge and relax the fixed decision boundaries. Both of these two extensions are important since the 1st one could possibly improve the performance of the phase I trial if relevant prior information can be used and the 2nd is an interesting theoretical exploration of the original BOIN design.

For the 1st extension, the authors proposed an automatic algorithm for its pre-specification of the parameter setups, which makes the proposed method feasibly to be used in real practices. The simulating conclusions are expected, which demonstrates the proposed algorithm works well empirically.

For the 2nd extension, though the results are not so exciting, that is, only when sample size are large, the proposed aBOIN design performs better than the original version, this exploration is still worthy from my perspective since it literally gives a solidified support for robustness of the original BOIN design.

My only concern is that it would be better that the paper can be edited by a professional language editor since there are still grammatic and language inappropriateness.

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Reviewer #1: No

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Response letter is provided.

Submitted filename:

A Phase I Dose-finding Design with Incorporation of Historical Information and Adaptive Shrinking Boundaries

PONE-D-20-07779R1

Dear Dr. Pan,

We’re pleased to inform you that your manuscript has been judged scientifically suitable for publication and will be formally accepted for publication once it meets all outstanding technical requirements.

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Jed N. Lampe, Ph.D.

Academic Editor

PLOS ONE

PONE-D-20-07779R1

A Phase I Dose-finding Design with Incorporation of Historical Information and Adaptive Shrinking Boundaries

Dear Dr. Pan:

I'm pleased to inform you that your manuscript has been deemed suitable for publication in PLOS ONE. Congratulations! Your manuscript is now with our production department.

If your institution or institutions have a press office, please let them know about your upcoming paper now to help maximize its impact. If they'll be preparing press materials, please inform our press team within the next 48 hours. Your manuscript will remain under strict press embargo until 2 pm Eastern Time on the date of publication. For more information please contact

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on behalf of

Dr. Jed N. Lampe

Academic Editor

PLOS ONE