A Bayesian model-free approach to combination therapy phase I trials using censored time-to-toxicity data

Summary. Theproductofindependentbetaprobabilitiesescalationdesignfordualagentphase I dose escalation trials is a Bayesian model-free approach for identifying multiple maximum tolerated dose combinations of novel combination therapies.Despite only being published in 2015, the design has been implemented in at least two oncology trials. However, these trials require patients to have completed follow-up before clinicians can make dose escalation decisions. For trials of radiotherapy or advanced therapeutics, this may lead to impractically long trial durations due to late-onset treatment-related toxicities.We extend the product of independent probabilities escalation design to use censored time-to-event toxicity outcomes for making dose escalation decisions. We show via comprehensive simulation studies and sensitivity analyses that trial duration can be reduced by up to 35%, particularly when recruitment is faster than expected, without compromising on other operating characteristics.


Introduction
The majority of anticancer therapeutic strategies consist of giving patients two or more treatments together to provide improved treatment effects relative to individual therapies given alone.In phase I trials of novel combination therapies, the aim is to identify one or more maximum tolerated dose combinations (MTDCs), i.e. one or more dose combinations with an expected probability of causing a severe drug-related adverse event equal to or close to a target of interest, known as the target toxicity level (TTL).Once identified, larger comparative trials are conducted to compare the efficacy of the MTDC(s) of the new combined treatment regimen with standard care, in the hope of improving patient response and survival outcomes.
Similarly to phase I trials for monotherapies, trials of two treatments combined are conducted as dose escalation studies.In these, patients are allocated to a dose combination and observed over time, with the number and severity of adverse events recorded.On the basis of these data, the next cohort of patients may receive the same combination or a different combination; if the current dose combination is considered tolerable (e.g.no adverse events reported), the next cohort may receive a combination with only one drug increased, or both drugs increased simultaneously.By gradually exploring the dose-toxicity grid, we aim to identify MTDCs while minimizing the chance of overdosing patients in the trial.
Several approaches for designing phase I dose escalation studies on drug combinations have been proposed (Harrington et al., 2013); these include rule-based and model-based trials.Mander and Sweeting (2015) proposed a probabilistic model-free approach: the product of independent probabilities escalation (PIPE) design.The PIPE design does not assume a model for the dose-toxicity surface; each dose combination is assigned a beta prior distribution, independent of all other combinations, which is updated with trial data.The tail probabilities that each combination are above the TTL are calculated and used to estimate the chance that a particular contour dividing the dose-toxicity surface into two areas of tolerable and intolerable combinations is the true maximum tolerated contour (MTC).This approach gives sensible estimation of toxicity risks when the dose-toxicity surface cannot be well approximated by a model (e.g. if the surface is asymmetric) and, unlike many rule-based designs, allows borrowing of information across dose combinations.The PIPE design has been implemented in practice in at least two trials: these include a study of the monoclonal antibody emactuzumab and RO7009789 in combination in participants with advanced solid tumours (ClinicalTrials.govidentifier NCT02760797), and ORCA-2, a phase I study of olaparib in addition to cisplatin-based chemoradiotherapy for patients with advanced squamous cell head and neck cancer (ClinicalTrials.govidentifier NCT02308072).Since radiotherapy is used in the ORCA-2 trial, late-onset treatment-related toxicities are expected and the dose limiting toxicity (DLT) observation window is 14 weeks from the first administration of olaparib.
One requirement of many phase I trials is that the end point (often a binary outcome denoting DLT or no DLT) must be observed before making dose escalation decisions.Trials with a long DLT observation window, e.g.radiotherapy trials such as ORCA-2, may have a long planned duration and therefore a large cost.Furthermore, the underlying patient population may change over time if the trial is planned to last a long time; patient drift may affect which combinations are considered to be tolerable and intolerable (Villar et al., 2015).In addition, rapid recruitment may mean that we cannot assign suitable patients to trial treatments because we must wait for outcomes to be obtained from current patients.Patients waiting may have their treatment delayed, or in the worst case may never receive treatment because of death.To overcome these problems, time-to-event (TITE) outcomes have been proposed for use in single-agent dose escalation studies (Cheung and Chappell, 2000;Braun, 2006;Ivanova et al., 2016).Rather than an outcome of DLT or no DLT, a patient's outcome measured at any point during their DLT observation window is a function of the time for which they have yet to be observed, conditional on the patient's not yet experiencing a DLT.Wages et al. (2013) applied the time-to-event-continual reassessment method (TITE-CRM) design (Cheung and Chappell, 2000) to dual agent trials by exploiting partial orders of dose-toxicity probabilities per dose combination.They could shorten trial durations by 84-89% in several scenarios, in which only one combination was the MTDC and patients were treated when they arrived.However, multiple MTDCs may exist when combining treatments and a one-parameter CRM model is insufficient to identify multiple MTDCs.Furthermore, the number of possible partial orders increases exponentially as the number of dose levels per agent is increased.
We propose an approach for extending the model-free PIPE design by using a TITE outcome measure.We use the proportion of a patient's DLT observation window that has passed since the time of treatment administration, with DLT unobserved, as a form of censored toxicity response.Even if some patients have not completed follow-up, dose escalation decisions can still be made and trial duration can be reduced.We call this proposed method the TITE-PIPE method.Section 2 outlines the TITE-PIPE methodology and in Section 3 we describe and present comparative simulation studies to compare the performance of the TITE-PIPE against the PIPE method with respect to experimentation, recommendation and trial duration.In Section 5 we conclude with a discussion of our findings and provide practical considerations as well as areas for future research.
The programs that were used to analyse the data can be obtained from https://rss.onlinelibrary.wiley.com/hub/journal/14679876/seriesc-datasets

Methodology
Consider a dual agent phase I trial of J dose levels of agent A, {a 1 , : : : , a J }, and K dose levels of agent B, {b 1 , : : : , b K }.Let Y i,t be a binary random variable that takes value 1 if at time t t patient i has experienced a DLT, and 0 otherwise.If Y i,t = 1, then Y i,t = 1 ∀ t t.Let t i0 be the time at which patient i begins treatment, and let U i 0 be a random variable denoting the time from t i0 at which patient i experiences a DLT.We assume that each patient's DLT observation window is T time units.Then, for DLT observation period [t i0 , t i0 + T ], the random variable Y i,t i0 +T |U i , i.e. the indicator of whether patient i has had a DLT by the end of their DLT observation period conditionally on their time to DLT being equal to U i , is An alternative weight function that was proposed by Cheung and Chappell (2000) adapted weighted outcomes dependent on whether previous DLTs had been observed towards the start or end of the observation window.If more DLTs are observed towards the end of the interval, then a higher weight is given to a patient who has yet to have a DLT and not reached the latter part of the observation window (relative to the uniform case in equation ( 2)).Similarly, if the majority of DLTs are observed at the start of the observation window, a patient's weighted outcome towards the end of the DLT observation window will be lower relative to the uniform case.Specifically, for the set of z ordered DLT times U z = {u .1/, u .2/, : : : , u .z/} that occur in the DLT observation window such that 0 ≡ u .0/< u .1/: : : u .z/< u .z+1/≡ T , where κ = max 0 h z {h : t − t i0 u .h/}.Fig. 1 shows examples of both the uniformly decreasing and adaptive weight functions based on different time-to-toxicity distributions during the DLT observation window.

Posterior dose limiting toxicity probabilities
Regardless of the choice of weighting function, we may use the following approach to incorporate censored DLT outcomes into posterior estimates for the probability of DLT per dose combination.Assume that our prior uncertainty of the probability of DLT π jk for dose combination .aj , b k / can be represented by a beta(r jk,0 , s jk,0 ) distribution.As per Mander and Sweeting (2015), we specify r jk,0 and s jk,0 such that the prior median is equal to a prespecified value πjk (i.Since the beta prior is conjugate for the binomial likelihood function, the posterior probability distribution of π jk at time t is also beta distributed, i.e. π jk ∼ beta.rjk,0 + R jk,t , s jk,0 + S jk,t /.This provides a working model for each π jk , which are independent.Therefore no assumptions are made regarding the marginal dose-toxicity relationship of each agent, specifically that the marginal probabilities are monotonically increasing, i.e. π jk π .j+1/kand π jk π j.k+1/ .Monotonicity is introduced through the MTC.

Posterior contour probabilities
As proposed by Mander and Sweeting (2015), we may use the posterior distributions of π jk to construct an MTC, which satisfies the assumption of monotonicity.A monotonic contour C can be defined by a Let F t be the set of all trial data available at time t, with F jk,t ⊆ F t being the set of all data available at dose combination .aj , b k / at time t (i.e.F jk,t = {∪ i∈I jk,t .wi,t , y i,t /}; F t = ∪ j,k F jk,t ).Let p.F jk,t / be the posterior probability that the toxicity risk at dose combination .aj , b k / at time t is less than TTL θ, i.e.
J + K J /}: the set of all contours that do not violate the marginal assumptions of monotonicity.Assuming that the MTC must be one of these monotonic contours, the q l .F t / are rescaled to form a distribution function, i.e.
ql .F t / = q l .F t / J + K J g=1 q g .F t /, .6/and we may, for simplicity, select the MTC C Å t to be the modal contour at time t, i.e.C Å t = arg max C l ∈C { ql .F t /}.

Dose allocation
Patient allocation to a dose combination can follow similar procedures as specified by Mander and Sweeting (2015).However, with the TITE-PIPE method we do not necessarily need to wait for a patient's follow-up to be complete before allocating doses for the next patient(s).We shall assume the following set-up.The study starts at time t = 0, when the first patient enters.A minimum of c patients must be dosed at a combination before experimentation elsewhere can take place, and patients are followed up for a maximum of T time units (e.g. a week or a month).Under the TITE-PIPE design, though patients enter the study on their arrival time, regardless of whether previous patient have completed follow-up or not, we employ a constraint that the first c patients in the trial (who are given the lowest dose combination .a 1 , b 1 /) must have completed follow-up before new patients can enter the study.For each variant of the TITE-PIPE method, we specify two allocation criteria that may be used.
(a) Completed dose: at least c patients must have completed their follow-up on a particular dose combination before any more patients can be assigned to other dose combinations.
We denote this the TITE-PIPE-C criterion.(b) On dose: at least c patients must have been assigned to a dose combination before any more patients can be assigned to other dose combinations.We denote this the TITE-PIPE-O criterion.
Under the TITE-PIPE-O design, c patients may be assigned to a dose combination and then, immediately after, a decision can be made on where to dose the next patient; this may not be ethical, but it is worthy of investigation (Senn et al., 2007;Bird et al., 2017).Under the TITE-PIPE-C design, though we cannot make a dose escalation decision for the next patient until c patients have completed their follow-up, if another patient is recommended to the same combination (i.e.c + 1 patients are now on the current dose combination), we may make decisions on where to dose future patients midway through the follow-up of patient c + 1.This satisfies ethical constraints (i.e.we have at least c patients' worth of complete follow-up data on the combination), but the trial duration may be reduced relative to the PIPE design where all patients must have completed follow-up before escalation decisions are made.Under both the TITE-PIPE-C and TITE-PIPE-O designs, if two or more patients arrive at the same time, then admissible dose assignments are calculated after each individual patient has been assigned to treatment; this was also undertaken in a trial of clofarabine in combination with fractionated gemtuzumab ozogamicin in patients with refractory or relapsed acute myeloid leukaemia (Foster et al., 2012;Ivanova et al., 2016).Fig. 2 illustrates how the trial entry and When we are ready to allocate patients to a combination, dose combinations are selected on the basis of the following rules (Mander and Sweeting, 2015).
(a) Neighbouring dose combinations; escalation can be to doses that are within at most one dose level of the previously experimented combination.(b) Diagonal dose escalation (increasing doses of both drugs simultaneously) is permitted.
(c) Closest dose combinations to the estimated MTC; admissible dose combinations for the next cohort are those that are closest to the MTC (see Mander and Sweeting (2015) for further details).(d) Minimum sample size for ties; if multiple dose combinations are permitted, then choose the combination with the fewest patients assigned to it.If there are still multiple combinations to choose from after these rules, the next patient is randomly assigned to one such combination with equal probability.

Early termination
The trial is terminated either when the maximum sample size is reached, or if one of the following stopping rules is satisfied.
Let G jk,t be the set of all available trial data collected from patients who have completed their follow-up (either had a DLT, or not had a DLT during their entire DLT observation window) at dose combination .aj , b k /, i.e.G jk,t = {∪ i∈I jk, t .wi,t , y i,t / : .wi,t , y i,t / = .0,0/ ∪ y i,t = 1}.Let G t = ∪ j,k G jk,t and qunsafe .G t / be the probability that the monotonic contour defining all dose combinations as unsafe, denoted C unsafe , is the MTC (computed by using equation ( 6), conditionally on complete follow-up data only).The trial is terminated early if qunsafe .G t / , where is some threshold to be calibrated before the trial.In words, the trial is terminated early if, using completed follow-up data only, the probability that the lowest dose combination is above the MTC is at least .Only data from patients who have completed follow-up are used to mitigate stopping early when several patients are still in observation; if they were not to experience DLT, the trial could have continued.
Furthermore, we adopt a safety rule which is similar to that of Ivanova et al. (2016); if the trial is deemed safe to continue by the aforementioned stopping rule ( qunsafe .G t / < ), but the inclusion of data from patients who are currently in observation means that all combinations are inadmissible for the next patient, we wait until all patients who are currently under observation have completed their follow-up.Mathematically, we say that dose combination .a ∀ j, k, recruitment is suspended until all patients on trial have completed follow-up.We may then reassess whether to terminate the trial or not by using qunsafe .G t / and, if the trial is to continue, decide what combination the next patient should receive.This prevents stopping the trial on the basis of incomplete data and enables a thorough assessment of the safety of the dose combinations before enrolling future patients.

Maximum tolerated dose combinations
At the end of the trial, say time t = t Å , the modal MTC C Å t Å = arg max C l ∈C { ql .G t Å /} is estimated.All dose combinations that lie closest to C Å t Å from below that have been experimented on are declared as MTDCs (Mander and Sweeting, 2015).

Simulation study
We now compare the original PIPE design, where new patients are admitted either once a DLT has been observed in the previous patient, or the previous patient has completed their followup period, with the TITE-PIPE-C and TITE-PIPE-O designs via simulation.In our studies patients will be observed for T = 1 time unit; if patient i enters the study at time 13:2 they will be followed up until time 14:2, or until a DLT is observed, whichever occurs first.
For our simulations we use the seven dose-toxicity scenarios that were presented as simulation 2 in Mander and Sweeting (2015) (4 × 4 dose-toxicity grids also used by Braun and Jia (2013)).We assume a priori median dose-toxicity values that are shown in scenario A (Table 1).The TTL θ is 0:20.The threshold at which doses are considered inadmissible (and also for early termination of the trial) is calibrated to be 0:80, such that the trial is terminated if the first two patients, who are both treated at combination .a 1 , b 1 /, experience DLTs.

Dose-toxicity models
We consider three models for dose-toxicity, which simulate onset of toxicity as uniformly distributed, early onset or late onset.Similarly to work by Cheung and Chappell (2000) and Braun (2006), these are as follows.combination so that the cumulative distribution function at time T is that dose combination's toxicity probability.Fig. 3 shows the probability distribution functions and cumulative distribution functions for the time-to-toxicity distributions under the uniform, Pareto and Weibull models that are specified above.
Arrival times of patients are generated from a Poisson process with rate λ (the average number of patients expected per time unit), with λ ∈ {0:5, 1, 2}.With three arrival rates and three toxicity generation models, we have nine simulation environments per dose-toxicity scenario.For each simulation environment, we investigate the performance of the PIPE, TITE-PIPE-C and TITE-PIPE-O methods; for each TITE-PIPE approach we compare the uniform weighting function (equation ( 2)) and the adaptive weighting function (equation ( 3)).We simulate 2000 trials per simulation environment, using a maximum sample size of 40 patients.

Operating characteristics to compare
We compare the three designs and allocation rules by using the following operating characteristics: (a) experimentation percentages, i.e. the percentage of patients who are assigned to dose combinations within a specific DLT probability range; (b) MTDC recommendation percentages, i.e. the probability that, at the end of the trial, dose combinations within a specific DLT probability range are identified as MTDCs (multiple combinations may be recommended); (c) trial duration (the time until trial is terminated or the last patient completes follow-up); (d) sample size (since some trials may terminate early).

Computational set-up
Simulations were undertaken in R by using code adapted from the pipe.designpackage (Sweeting, 2016) (functions incorporated in the latest version of the package).We use the same sequence of random seeds between each simulation environment to provide comparable operating characteristics.

Illustrative single trial
Before presenting the results from the simulation study, we provide a single illustrative trial of 40 patients (Fig. 4).16 dose combinations formed from four dose levels each of drug A and drug B were under investigation, with the aim of targeting one or more MTDCs with θ = 0:20.Patient arrival times were simulated as a Poisson process with rate λ = 2 (i.e. an average arrival rate of two patients per time unit), and patients were followed up for T = 1 time units.DLTs and their times were simulated from scenario A under a conditional uniform model, with uniform weighting assigned to unobserved DLTs.We conducted this trial by using the TITE-PIPE-C method and required two patients to have completed treatment on a dose combination before another combination may be considered for exploration.
No DLTs were observed in the first four patients (two on .a 1 , b 1 /; two on .a 2 , b 2 /).Combination .a 3 , b 2 / was given to two new patients, one of whom experienced DLT.The modal MTC and safety constraint were updated and patients 7 and 8 were given .a 4 , b 1 /; this combination is closest to the MTC and admissible.At t = 4:93, patients 7 and 8 completed their follow-up (with no DLTs occurring), and patients 9 and 10 are allocated to .a 4 , b 1 / and .a 3 , b 1 / respectively.Since two patients are required at each dose level in our example trial, patient 11 also received .a 3 , b 1 /.After these three patients completed their DLT follow-up at t = 5:93 (n = 11), patient 12 was allocated to .a 3 , b 2 /; this combination was below the safety contour after 11 patients and closest to the modal MTC.

Experimentation and recommendation percentages
Table 2 shows the experimentation percentages, for each simulation, assuming uniformly distributed toxicity times and a uniform weight function.Experimentation percentages are dependent on λ, with increasing λ (more rapid arrival of patients) associated with more experimentation at combinations with lower true probabilities of DLT.This is due to the conservative assumption that an unobserved portion of a patient's observation window is attributed to a DLT outcome.Under the TITE-PIPE-O design, a new patient is treated at a dose under the assumption that the previous patient, who has been followed up for a short amount of time, contributes more to the number of DLTs rather than non-DLTs.However, under the TITE-PIPE-C design, a new patient may have to wait until the minimum number of patients have completed treatment at a particular dose.As a result, the average observed DLT rates per scenario decrease both with increasing λ and as we move from the PIPE to the TITE-PIPE-C to the TITE-PIPE-O design; the TITE-PIPE-C method generally offers experimentation operating characteristics that sit between the PIPE approach and the TITE-PIPE-O approach.
Table 3 shows the chance of recommending combinations with true DLT probabilities within different intervals per design and scenario, as well as the mean number of MTDCs recommended, the percentage of trials that did not recommend any MTDCs and the percentage of trials that were terminated early.The TITE-PIPE approaches and the PIPE design have similar recommendation percentages within the 15-24% DLT risk interval (where the TTL lies), with any differences between designs becoming more pronounced as λ increases.The performance can be slightly improved, or slightly reduced by using the TITE-PIPE over the PIPE design; in scenario A with λ = 2, the chance of choosing combinations with true DLT risk of 15-24% increases from 73% to 76% under the TITE-PIPE design, whereas in scenario B it decreases from 27% to 18%.
The percentage of trials that terminate early increases as we move from the PIPE to the TITE-PIPE-C design (the largest relative increase of 1:4% (scenario C)) to the TITE-PIPE-O design (the largest relative increase of 2:3% (scenario C)).This is because TITE-PIPE approaches are slightly more likely to assign patients to lower dose combinations (since partially followed-up patients contribute a partial DLT response; see Table 2), and any subsequent DLTs at lower combinations will increase the chance of terminating the trial earlier than if those DLTs were observed at higher combinations.Overall, the differences in early stopping probabilities between approaches are very small.The percentage of trials that do not recommend any MTDCs includes those trials that terminate early and trials where all 40 patients are treated, but no MTDCs can be identified (see Section 2.6).
When using the adaptive weighting function instead of the uniform weighting function, no major differences are observed in experimentation or recommendation results (see the on-line supplementary material, Table S1 and Table S2).This is not surprising; given the simulationgenerated uniformly distributed toxicity times, the adaptive weight function will yield similar weighted DLT outcomes to those of the uniform weighting function.

Trial duration and delays in treatment
Table 4 gives the mean trial durations (and the percentage change from the PIPE design) per design and scenario, under both uniform and adaptive weight functions.Trial durations do not differ with the choice of weight function when assuming uniformly distributed time to toxicity.Looking at the results from using the uniform weight function, the trial duration is shortened under the TITE-PIPE approaches by 0.4-1.2units (λ = 0:5), 0.8-2.2units (λ = 1) and 3.6-10.3units (λ = 2).When the average arrival rate is two patients per time unit (λ = 2), the trial  duration can be shortened by 24-33%, without substantially compromising on experimentation and MTDC recommendation performance.
As well as reductions in trial duration, we also consider the mean delay between recruitment of the last patient onto a trial and the administration of their treatment.Fig. 5 shows that, under the PIPE design, the delay between recruitment of the last patient and treatment administration increases substantially as the recruitment rate λ increases.For λ = 0:5, this delay is approximately 0.40 time units (0.16 for scenario D, where the average sample size was around 19 patients).When λ = 1, although the mean trial duration between the PIPE and TITE-PIPE methods is very similar, there is a noticeable difference in the mean delay in treatment for the last recruited patient.When λ = 2, this delay increases dramatically to around 10 time units (3.8 time units for scenario D).Under the TITE-PIPE design, the delay is virtually 0 (a maximum of 0.35 under the TITE-PIPE-C the design (scenario D) and 0.28 under the TITE-PIPE-O design (scenario D)).So, as well as shortening the trial, the delay between recruitment and receiving treatment (as measured in the last patient) is almost completely removed.

Early onset and late onset toxicities
We now assume that the time to toxicity is distributed as a Pareto random variable (Section 3.1) so that DLTs are more likely to occur earlier in a patient's observation period than under a uniform distribution.Tables S3 and S4 in the on-line supplementary material show similar experimentation and MTDC recommendation results to that under the uniform time-to-toxicity distribution.Under the adaptive weight function (Tables S5 and S6), changes of 1-2% are observed when comparing the TITE-PIPE MTDC recommendations with those by using uniform   weightings, but otherwise very little difference is observed.For trial durations (Table S7), we again see similar time savings under different recruitment rates to those observed under uniform time to toxicity.This is probably because we observe few toxicities on average per trial (between four and 12 out of 40 patients approximately), and the mean time to DLT is similar under the uniform and Pareto distributions used, even though the DLT times under the Pareto distribution are skewed towards earlier times.Essentially, the DLTs that do occur are clustered towards the start of a patient's DLT observation window, but there are not enough of them to cause a noticeable change in model performance on the adaptive weighting function.
When the time to DLT follows a Weibull distribution, so that DLT times are skewed towards the end of the DLT observation period (Section 3.1), we again see minimal changes in each design's operating characteristics.This applies to both the uniform weighting function (Tables S8 and S9 in the on-line supplementary material) and the adaptive weighting function (Tables S10 and S11).The trial duration (Table S12) is reduced under the TITE-PIPE approaches regardless of which weighting function is used, and to a similar extent to that of the simulations for uniform-and Pareto-distributed time to toxicity.The greatest reductions in duration are slightly more than observed under a uniform time-to-toxicity model (34-35% reduction).

Discussion
We have proposed an approach for incorporating censored patient toxicity outcomes into the PIPE dose escalation design for dual agent phase I trials.We have shown that substantial reductions in trial duration can be achieved, and delays in starting treatment can be avoided even when recruitment is at a rate that is similar to the outcome follow-up time.These can be done without compromising on the experimentation and MTDC recommendation performance.We found that the TITE-PIPE-C approach, where two patients had to be completely followed up per combination experimented on before censored time-to-toxicity data could be used for dose escalation decisions, gave slightly better experimentation performance than that of the TITE-PIPE-O method.Furthermore, the performance of both the TITE-PIPE-C and the TITE-PIPE-O methods was generally invariant to the underlying time-to-toxicity distribution of the combination therapy and was not affected by the choice of weight function (uniform or adaptive).
We implemented a safety rule that if the probability that the lowest dose combination is above the MTC is at least 0:80, given data from patients who had completed their follow-up, then the trial is terminated.We also implemented a probabilistic rule (Bekele et al., 2008;Polley, 2011;Ivanova et al., 2016) whereby if, on the basis of partial data, the probability that the lowest dose combination is above the MTC is greater than = 0:80, then recruitment is suspended until those patients who are enrolled in the trial have completed their follow-up.Then we use the new set of completely observed data to assess whether the trial should be terminated or not.The threshold should be carefully considered before the design is implemented in practice; we adopted a threshold of = 0:80 so that the trial would be terminated if the first two patients experienced DLTs, but higher thresholds (e.g.= 0:95) have been considered (Ivanova et al., 2016) and may prove more favourable for reducing the chances of terminating a trial when there truly is at least one MTDC.Calibration of this stopping rule will need to be performed per trial and will vary with chosen TTLs, clinical needs and prior set-ups (median DLT risks and prior sample sizes).
The simulation studies that were conducted show that trial duration can be reduced by 5-6% when patients arrive at a rate of one per DLT observation window (in this study, one time unit) and reduced up to 35% when patients arrive twice as fast as expected.

Fig. 2 .
Fig. 2. Example arrival and trial entry times from one simulated trial (patients arriving as a Poisson process with mean 1.25; same random-number seed used across designs) (}, recruitment time; , start time; , follow-up; , DLT): for the PIPE design ((a)) each patient must have completed follow-up or have a confirmed DLT reported before the next patient (if one has been recruited) can enter the study; for both the TITE-PIPE-C ((b)) and TITE-PIPE-O ((c)) designs, the first patient on study is required to have completed follow-up before more patients are assigned to dose combinations; for the TITE-PIPE-C design, at least one patient per combination must have completed follow-up before future patients can be assigned to dose combinations; otherwise (and for the TITE-PIPE-O design), patients can enter the trial on arrival

Fig. 4 .
Fig. 4. A single illustrative trial using the TITE-PIPE design simulated over scenario A, with θ D 0.20 (at least two patients were required to have completed treatment on a dose combination before a new combination could be considered for future patients; combinations marked m.n/ C r.s/ indicate that, of m

Fig. 5 .
Fig. 5. Mean delay in the start of treatment from the time of recruitment for the last patient in the trial versus average trial duration (uniform toxicity distribution; uniform weight function) ( , PIPE; , TITE-PIPE-C; , TITE-PIPE-O; , recruitment rate 0.5; 4, recruitment rate 1; +, recruitment rate 2): (a) scenario A; (b) scenario B; (c) scenario C; (d) scenario D; (e) scenario E; (f) scenario F; (g) scenario G e., for beta cumulative distribution function B.v; r jk,0 , s jk,0 /, B. πjk ; r jk,0 , s jk,0 / = 0:50) and the effective sample size at dose combination .aj , b k / is r jk,0 + s jk,0 = 1=.JK/.Therefore, the effective sample size across all dose combinations is equal to one patient.Assume that n jk,t patients have been assigned to dose combination .aj , b k / at time t, and these patients' identifiers are contained in the indexing set I jk,t .Let R jk,t = Σ i∈I jk, t w i,t and S jk,t = Σ i∈I jk,t .1 − w i,t / = n jk,t − R jk,t be the total number of weighted DLTs and weighted non-DLTs observed for patients in I jk,t .The likelihood function arising from these data is binomial, specifically

Table 1 .
Dose-toxicity scenarios for the simulation study † †Combinations marked in italics have a DLT probability of 20% ± 2%.The MTC is shown as a line bisecting the dosetoxicity grid.

Table 2 .
Experimentation percentages by using the uniform weight function (assuming a uniformly distributed DLT model)

Table 3 .
Recommendation percentages by using a uniform weight function (assuming a uniformly distributed DLT model)

Table 4 .
Mean trial durations per design and scenario, with percentage change from the PIPE design (in parentheses) by using uniform and adaptive weight functions (assuming a uniformly distributed DLT model)