Backfilling with the BF-BOIN and BE-BOIN designs
Source:vignettes/simFastBOIN-backfill.Rmd
simFastBOIN-backfill.RmdWhy backfill
In the BOIN design the next cohort is dosed only once the current one has completed the DLT assessment, and patients who become eligible in the meantime are usually turned away. Backfilling treats them at a lower dose that has already been cleared, so that more patients contribute data on safety and activity at doses that may later be carried forward, while the dose escalation goes on as before.
Two designs are provided. BF-BOIN (Zhao et al., 2024) keeps the BOIN decision table and fully staggered cohorts. BE-BOIN (Chen et al., 2026) combines backfilling with the time-to-event BOIN design, so that the escalation itself decides while some patients are still pending. In both, the package selects the MTD from the data of all patients, backfilled ones included. The second stage of Chen et al. (2026), which randomizes patients between doses to choose the optimal biological dose, is not part of the package.
The rules of BF-BOIN
The trial has two parts that run side by side.
- Dose escalation. Cohorts are fully staggered. When every patient of the current cohort has completed the assessment, the next dose follows the BOIN decision table applied to the completed patients at the current dose, backfilled ones included. Elimination is checked at the same moments, at every dose.
-
Backfilling. A patient who arrives while the
escalation waits is treated at the highest dose below the current one
that has not been eliminated, has fewer than
n_cappatients (12 by default) and has a response observed at or below it. A dose is closed when its own DLT rate exceeds the de-escalation boundary and so does the rate pooled with the next higher dose; the doses above it close with it. - Conflicts. When the data of a backfilled dose call for a more cautious decision than those of the current dose (Table 2 of Zhao et al., 2024), the decision is taken from the data pooled from that dose up to the current one.
The trial ends when the escalation has treated n_cohort
cohorts, when n_earlystop patients, backfilled ones
included, have been treated at the current dose and the decision is to
stay, or when the lowest dose is eliminated.
Operating characteristics of BF-BOIN
Scenario 3 of Zhao et al. (2024) has the MTD at dose 3 and responses
that rise with the dose. The design is that of the article: a target of
0.25, ten cohorts of three, n_earlystop = 9, one DLT out of
three treated as a stay, three patients a month and an assessment window
of one month.
zhao_design <- list(target = 0.25, n_cohort = 10, cohort_size = 3,
window = 1, accrual_rate = 3, n_earlystop = 9,
stay_on_1_of_3 = TRUE, accrual = "uniform",
no_slot = "leave", n_trials = 2000, seed = 123)
p_true <- c(0.04, 0.12, 0.25, 0.43, 0.63)
p_resp <- c(0.10, 0.20, 0.30, 0.45, 0.58)
oc_bf <- do.call(sim_bf_boin,
c(zhao_design, list(p_true = p_true, p_resp = p_resp)))
oc_bf
#> BF-BOIN operating characteristics (Zhao et al., 2024)
#> target DLT rate : 0.25
#> trials : 2000
#> cohorts : 10 of size 3
#> escalation size : 30
#> backfill cap : 12 patients per dose
#> lambda_e / _d : 0.1968 / 0.2984
#> window : 1 (DLT), 1 (response)
#> accrual : 3 patients per unit of time (uniform)
#>
#> DL1 DL2 DL3 DL4 DL5 Total / No MTD
#> True DLT rate (%) 4.0 12.0 25.0 43.0 63.0
#> True response rate (%) 10.0 20.0 30.0 45.0 58.0
#> MTD selected (%) 2.3 30.3 57.8 9.6 0.0 0.0
#> Patients treated 6.0 10.8 9.8 3.7 0.5 30.8
#> of which backfilled 1.8 3.4 1.9 0.3 0.0 7.4
#> Patients with DLT 0.2 1.3 2.4 1.6 0.3 5.9
#>
#> Doses above a true DLT rate of 0.25: DL4, DL5
#> Patients treated there : 13.6%
#> Trials treating any patient there : 49.5%
#> Trials selecting an MTD there : 9.6%
#>
#> Backfilling and timing
#> Trials with backfilled patients : 85.2%
#> Average trial duration : 14.2
#> Trials in which the escalation waited: 100%
#> Average time the escalation waited : 6.2
#> Average patients turned away : 11.2Without any response no dose is opened for backfilling and the design is the BOIN design. The table below sets the two runs next to the values that Zhao et al. (2024) report in their Table 4: the percentage of trials selecting the MTD and a dose above it, the average number of patients below, at and above the MTD and in all, and the average duration in months.
oc_boin <- do.call(sim_bf_boin,
c(zhao_design, list(p_true = p_true, p_resp = rep(0, 5))))
zhao_metrics <- function(oc) {
c(oc$sel_percent[[3]], sum(oc$sel_percent[4:5]),
sum(oc$n_pts_dose[1:2]), oc$n_pts_dose[[3]], sum(oc$n_pts_dose[4:5]),
oc$total_n_pts, oc$duration_mean)
}
comparison <- rbind(
"BF-BOIN, Table 4" = c(57.6, 9.7, 16.9, 9.8, 4.2, 30.9, 14.3),
"BF-BOIN, simulated" = zhao_metrics(oc_bf),
"BOIN, Table 4" = c(56.7, 12.3, 11.4, 8.0, 4.6, 23.9, 14.3),
"BOIN, simulated" = zhao_metrics(oc_boin)
)
colnames(comparison) <- c("MTD (%)", "Above (%)", "n below", "n at",
"n above", "n total", "Months")
round(comparison, 1)
#> MTD (%) Above (%) n below n at n above n total Months
#> BF-BOIN, Table 4 57.6 9.7 16.9 9.8 4.2 30.9 14.3
#> BF-BOIN, simulated 57.8 9.6 16.8 9.8 4.2 30.8 14.2
#> BOIN, Table 4 56.7 12.3 11.4 8.0 4.6 23.9 14.3
#> BOIN, simulated 57.1 11.7 11.5 8.1 4.6 24.2 14.5With 2,000 trials the simulated values carry a simulation error of
about one point for the percentages. The tests of the package compare
all twelve scenarios of Table 4 with 10,000 trials each; over 20 seeds
the largest gaps were 2.1 points for the selection, 0.4 patients and
0.17 months. The script
inst/validation/compare-with-Zhao2024.R repeats the
comparison.
The response profile decides which doses are backfilled. Scenarios 2 and 6 of the article share the DLT probabilities but not the response probabilities.
scenarios <- list(
list(name = "Scenario 2", p_true = c(0.12, 0.25, 0.42, 0.49, 0.55),
p_resp = c(0.20, 0.30, 0.40, 0.50, 0.60)),
list(name = "Scenario 6", p_true = c(0.12, 0.25, 0.42, 0.49, 0.55),
p_resp = c(0.30, 0.35, 0.36, 0.36, 0.36))
)
zhao_multi <- zhao_design[names(zhao_design) != "n_trials"]
do.call(sim_bf_boin_multi,
c(zhao_multi, list(scenarios = scenarios, n_trials = 1000)))
#> BF-BOIN operating characteristics across 2 scenarios
#> target DLT rate : 0.25
#> trials each : 1000
#> escalation size : 30
#> backfill cap : 12 patients per dose
#> window : 1 (DLT), 1 (response)
#> accrual : 3 patients per unit of time (uniform)
#>
#> Scenario Item DL1 DL2 DL3 DL4 DL5 Total / No MTD
#> Scenario 2 True DLT rate (%) 12.0 25.0 42.0 49.0 55.0
#> True response rate (%) 20.0 30.0 40.0 50.0 60.0
#> MTD selected (%) 32.4 57.5 9.2 0.6 0.0 0.3
#> Patients treated 10.4 10.0 3.9 0.6 0.0 24.8
#> Backfilled patients 3.0 1.8 0.3 0.0 0.0 5.2
#> Patients with DLT 1.2 2.5 1.6 0.3 0.0 5.7
#> Trial duration (mean) 11.8
#> Scenario 6 True DLT rate (%) 12.0 25.0 42.0 49.0 55.0
#> True response rate (%) 30.0 35.0 36.0 36.0 36.0
#> MTD selected (%) 31.5 57.3 10.4 0.5 0.0 0.3
#> Patients treated 11.3 10.2 3.7 0.6 0.0 25.8
#> Backfilled patients 3.6 1.8 0.4 0.0 0.0 5.9
#> Patients with DLT 1.4 2.6 1.5 0.3 0.0 5.8
#> Trial duration (mean) 12.0Points the articles leave open
The articles do not settle every detail of a simulated trial. The package makes the following choices, and offers an argument where more than one reading is reasonable.
-
A patient with nowhere to go. When the escalation
waits and no dose is open for backfilling,
no_slot = "wait"(the default) lets the patient wait until one can be treated, as insim_tite_boin(), andno_slot = "leave"turns the patient away. Zhao et al. (2024) state a Poisson process of arrivals, but their durations, for BOIN as well as for BF-BOIN, agree with uniform times between arrivals (accrual = "uniform") and patients turned away, which is why the comparison above uses these settings. - The end of backfilling. Backfilling stops when the last patient of the escalation is enrolled, and every enrolled patient is then followed to the end of the assessment.
- Pending backfill patients. The escalation of BF-BOIN uses only the patients who have completed the assessment.
-
Responses. Neither article states how the time to
response is distributed or how a response is related to a DLT. The time
to response is Weibull, with
resp_late_fractionof the responses in the second half ofresp_window, and the DLT and the response are linked by a Gaussian copula with correlationresp_cor, as in Takeda et al. (2025), who set it to 0.2. The default here is 0, independence. -
Which conflicting dose anchors the pooled estimate.
With several conflicting backfilled doses, Zhao et al. (2024) pool from
the highest of them and Chen et al. (2026) from the lowest.
conflict_dosechooses, with the default of each article.
BE-BOIN
BE-BOIN decides the dose for a cohort when its first patient arrives,
with the time-to-event BOIN rule, so the escalation waits only when the
rule asks it to. Two rules of Chen et al. (2025) decide when that is:
rule 1 suspends accrual when fewer than 51 percent of the patients at
the current dose have completed the assessment
(max_pending_ratio = 0.49), and rule 2 allows escalation
only when every pending patient at the current dose has been followed
for at least a quarter of the window
(min_follow_up = 0.25). The decision table with both rules
is Table A1 of the supplementary materials of Chen et al. (2026), which
tite_boin_decision_table() reproduces entry by entry; an
escalation that needs rule 2 is marked MF >= 0.25.
be_table <- tite_boin_decision_table(target = 0.25, max_n = 9,
max_pending_ratio = 0.49,
min_follow_up = 0.25)
print(be_table, cohort_size = 3)
#> TITE-BOIN decision table with target 0.25
#> Method: single mean imputation (Yuan et al., 2018); boundaries refer to STFT
#>
#> Patients DLTs Pending Escalate Stay De-escalate
#> 3 0 0 Y
#> 3 0 1 MF >= 0.25
#> 3 0 >= 2 Suspend Suspend Suspend
#> 3 1, 2 <= 2 Y
#> 3 3 0 Y & Elim
#> 6 0 0 Y
#> 6 0 1-2 MF >= 0.25
#> 6 0 >= 3 Suspend Suspend Suspend
#> 6 1 0 Y
#> 6 1 1 >= 0.22 & MF >= 0.25 < 0.22
#> 6 1 2 >= 1.38 & MF >= 0.25 < 1.38
#> 6 1 >= 3 Suspend Suspend Suspend
#> 6 2, 3 <= 4 Y
#> 6 >= 4 <= 2 Y & Elim
#> 9 0 0 Y
#> 9 0 1-4 MF >= 0.25
#> 9 0 >= 5 Suspend Suspend Suspend
#> 9 1 0 Y
#> 9 1 1-3 MF >= 0.25
#> 9 1 4 >= 0.66 & MF >= 0.25 < 0.66
#> 9 1 >= 5 Suspend Suspend Suspend
#> 9 2 <= 4 Y
#> 9 2 >= 5 Suspend Suspend Suspend
#> 9 3, 4 <= 6 Y
#> 9 >= 5 <= 4 Y & Elim
#>
#> Patients = treated at the current dose, DLTs = DLTs observed, Pending = assessment not yet completed
#> STFT = total follow-up time of the pending patients divided by the length of the assessment window
#> Accrual is suspended when more than 49% of the patients are pending
#> MF = shortest follow-up time of the pending patients divided by the length of the assessment window
#> While patients are pending, escalation requires MF >= 0.25; otherwise accrual is suspended
#> Y & Elim = de-escalate and eliminate this dose and all higher dosesEvery estimate of BE-BOIN, for the escalation, for opening and closing backfill doses and for conflicts, imputes the pending patients with the single mean imputation of TITE-BOIN. Patients who arrive while the escalation is suspended are backfilled with the rules of BF-BOIN. With a window of three months and two patients a month, the two designs can be compared on the same scenario; Chen et al. (2026) report shorter trials for BE-BOIN.
slow_design <- list(target = 0.25, p_true = p_true, p_resp = p_resp,
n_cohort = 10, cohort_size = 3, window = 3,
accrual_rate = 2, n_earlystop = 9, n_trials = 2000,
seed = 123)
oc_be <- do.call(sim_be_boin, slow_design)
oc_be
#> BE-BOIN operating characteristics (Chen et al., 2026)
#> target DLT rate : 0.25
#> trials : 2000
#> cohorts : 10 of size 3
#> escalation size : 30
#> backfill cap : 12 patients per dose
#> lambda_e / _d : 0.1968 / 0.2984
#> window : 3 (DLT), 3 (response)
#> accrual : 2 patients per unit of time (exponential)
#>
#> DL1 DL2 DL3 DL4 DL5 Total / No MTD
#> True DLT rate (%) 4.0 12.0 25.0 43.0 63.0
#> True response rate (%) 10.0 20.0 30.0 45.0 58.0
#> MTD selected (%) 3.5 31.6 54.1 10.4 0.4 0.0
#> Patients treated 7.0 10.6 8.5 3.2 0.4 29.8
#> of which backfilled 1.9 2.8 1.2 0.2 0.0 6.1
#> Patients with DLT 0.3 1.3 2.1 1.4 0.2 5.4
#>
#> Doses above a true DLT rate of 0.25: DL4, DL5
#> Patients treated there : 12.2%
#> Trials treating any patient there : 49.3%
#> Trials selecting an MTD there : 10.8%
#>
#> Backfilling and timing
#> Trials with backfilled patients : 70.7%
#> Average trial duration : 22.3
#> Trials in which the escalation waited: 99.7%
#> Average time the escalation waited : 8.1
oc_bf_slow <- do.call(sim_bf_boin, slow_design)
round(rbind(
"BF-BOIN" = c(MTD = oc_bf_slow$sel_percent[[3]],
patients = oc_bf_slow$total_n_pts,
backfilled = oc_bf_slow$total_n_bf,
months = oc_bf_slow$duration_mean),
"BE-BOIN" = c(MTD = oc_be$sel_percent[[3]],
patients = oc_be$total_n_pts,
backfilled = oc_be$total_n_bf,
months = oc_be$duration_mean)
), 1)
#> MTD patients backfilled months
#> BF-BOIN 54.1 33.2 9.7 31.4
#> BE-BOIN 54.1 29.8 6.1 22.3Agreement with the BOIN and TITE-BOIN designs
The DLT variates of the escalation are drawn from R’s stream exactly
as in sim_boin(), and arrivals, backfill patients and
responses have generators of their own. When no dose is ever opened for
backfilling, the trials are therefore those of sim_boin()
with the same seed, and those of sim_tite_boin() for
BE-BOIN with no_slot = "wait".
no_backfill <- sim_bf_boin(target = 0.25, p_true = p_true,
p_resp = rep(0, 5), n_cohort = 10,
cohort_size = 3, window = 1, accrual_rate = 3,
n_trials = 1000, seed = 7)
plain_boin <- sim_boin(target = 0.25, p_true = p_true, n_cohort = 10,
cohort_size = 3, n_trials = 1000, seed = 7)
all.equal(no_backfill$sel_percent, plain_boin$sel_percent)
#> [1] TRUE
all.equal(no_backfill$n_pts_dose, plain_boin$n_pts_dose)
#> [1] TRUEHow the implementation is checked
- The operating characteristics of Table 4 of Zhao et al. (2024), twelve scenarios of BF-BOIN and five of BOIN, are reproduced within simulation error, durations included.
- The trial example of Figure 1 of Zhao et al. (2024) and that of Section S2 of the supplementary materials of Chen et al. (2026) are replayed patient by patient, with the random numbers scripted to give the outcomes of the articles.
- The decision tables of Chen et al. (2025) and of the supplementary materials of Chen et al. (2026) are reproduced entry by entry.
- The simulated trials agree exactly, trial by trial, with a separate implementation written in Python and run on a replica of R’s random number stream, over 2,700 trials of both designs and several settings.
- Without backfilling the trials are identical to those of the BOIN and TITE-BOIN engines, as shown above.
Chen et al. (2026) give their scenarios only as figures, so their operating characteristics cannot be reproduced number for number.
References
Zhao, Y., Yuan, Y., Korn, E. L. and Freidlin, B. (2024). Backfilling Patients in Phase I Dose-Escalation Trials Using Bayesian Optimal Interval Design (BOIN). Clinical Cancer Research, 30(4), 673-679.
Chen, K., Zhao, Y., Takeda, K. and Yuan, Y. (2026). BE-BOIN: A Dose Optimization Design Accommodating Backfill and Late-Onset Toxicity. Therapeutic Innovation and Regulatory Science.
Chen, K., Chen, T.-Y., Zhang, Y., Lin, R. and Yuan, Y. (2025). Practical Considerations for Using the TITE-BOIN Design to Handle Late-Onset Toxicity or Fast Accrual in Phase I Trials. Clinical Cancer Research, 31(13), 2573-2580.
Takeda, K., Zhu, J. and Hirakawa, A. (2025). BF-BOIN-ET: A Backfill Bayesian Optimal Interval Design Using Efficacy and Toxicity Outcomes for Dose Optimization. Pharmaceutical Statistics, 24(2), e2470.
Liu, S. and Yuan, Y. (2015). Bayesian Optimal Interval Designs for Phase I Clinical Trials. Journal of the Royal Statistical Society: Series C, 64, 507-523.