Phase I doseescalation trials with more than one dosing regimen
Burak Kürsad Günhan,^{1}^{1}1Department of Medical Statistics, University Medical Center Göttingen, Göttingen, Germany ^{2}^{2}2Correspondence to: Burak Kürsad Günhan; email: burak.gunhan@med.unigoettingen.de Sebastian Weber,^{3}^{3}3Novartis Pharma AG, Basel, Switzerland Abdelkader Seroutou,3 Tim Friede1
Traditionally, phase I doseescalation oncology
trials are designed to find a dose at which an acceptable event rate
of dose limiting toxicities (DLT) occur. However, nowadays the dosing
regimen, which determines the timing of drug administration, is varied
in addition to the drug amount itself; e.g. a weekly or daily schedule.
Standard methods such as the Bayesian Logistic Regression Model do not
directly allow for more than one dosing regimen to be evaluated, and
hence adhoc approaches like dose rescaling are used to make dosing
regimens comparable. To avoid such adhoc approaches, we propose a
new statistical model that uses pharmacokinetic (PK) principles to
integrate varying dosing regimens. We propose to use a latent pseudoPK,
which uses the preplanned dosing regimen. We complement the
pseudoPK by an effect compartment which admits a delay
between the PK and the actual effect, the occurrence of a DLT.
The effect compartment measure is used as exposure measure and set
directly proportional to the instantaneous hazard of the
timetofirst DLT event process. The
model is formulated using interpretable parameters which facilitates
the specification of priors. Moreover, we derive from the
timetoevent model metrics which enable escalation with overdose
control. In a Monte Carlo simulation study, the proposed model displayed
desirable performance in terms of investigated measures
across a range of dosetoxicity profiles including single and
varying dosing regimens. The proposed model is motivated and
illustrated by a reallife example. The software to implement the
proposed model (R and Stan code) is publicly available.
Keywords:Phase I doseescalation trials, multiple dosing regimens, pharmacokinetic models, Stan
1 Introduction
Phase I doseescalation trials constitute the first step in investigating the safety of potentially promising drugs in humans.^{1} In oncology, such trials traditionally focused on identifying the maximum tolerated dose (MTD) and/or the recommended phase II dose through a series of dose escalation steps. The almost exclusive focus on safety in oncology is driven by the fact that cytotoxic drugs are generally most effective at the MTD. While safety is central to all drug therapy, its role in dose selection for other mechanisms of action is more nuanced.
Doseescalation trials in oncology traditionally enroll small cohorts of patients who are treated in cycles. Typically, the estimation of the MTD is based on the toxicity data of the first cycle data only. The observed toxicities are classified into doselimiting toxicities (DLT) and nonDLTs. Each time a cohort finishes its first cycle at a given dose intensity, the available data is assessed to decide how the trial proceeds. A commonly accepted target for the MTD is to allow for a DLT probability of 33% per cycle of treatment.^{2}
To facilitate effective and safe learning in the dose escalation trials, adaptive model based approaches such as the Continual Reassessment Method (CRM)^{3} or the Bayesian Logistic Regression Model (BLRM)^{4, 2} are commonly used. Both of these methods predict the toxicity profile for all doses planned in a given trial, which includes a prediction for already tested and yet untested doselevels. These model based assessments are used to guide the doseescalation trial in order to warrant the safety of patients by avoiding unacceptable high risks for DLTs. The BLRM is a two parameter version of the CRM which avails of the socalled EscalationWithOverdoseControl (EWOC)^{5} principle; i.e. requires that for a given dose the probability for a DLT must not exceed the maximal admissible DLT probability by a fixed uncertainty margin. The BLRM has been shown to have good performance in extensive simulation studies,^{4} and has been used for many phase I trials including drug combination trials (see e.g. ^{6, 7, 8, 9}). Furthermore, historical data can be incorporated via metaanalytic predictive (MAP) priors^{2} to improve predictive performance.
A key limitation of the BLRM is that it cannot naturally take into account differing dosing regimens (different dosing amounts and frequencies) which are more commonly used in oncology, for example starting with a weekly dosing regimen and then changing to a daily dosing regimen. Changes in regimens mean that scaling by dose alone is no longer necessarily a valid assumption because for a given dose level very different responses may be observed depending on the frequency of administration. A trivial analogy is that it matters if we drink 3 units of alcohol in 1 hour or over 3 hours. Often, ad hoc approaches such as dose rescaling are used in an attempt to account for the regimen effect. Such ad hoc approaches are always context specific approximation, meaning multiple models are necessary for multiple regimens and in addition data from different regimens must be discounted due to the often simplistic nonmodel based bridging of the available data. This means, the underlying dosing regimenDLT relationship needs to be learned repeatedly from a discounted base data set, which increases the required sample sizes and strongly limits the use of existing data to predict the dosetoxicity relationship of untested regimens.
A more rational approach to account for dosing regimen is to draw on the theory and practice underlying pharmacokineticpharmacodynamic (PK/PD) models. In particular, we are suggesting to use PK principles to model the relationship between timetofirst DLTs and the planned dosing regimen in a modelbased approach following the spirit of Cox et al.^{10} The key point here is to account for the overall kinetic properties implied by drug input rate of a regimen, drug elimination from the body of the patient and delays between drug exposure and actual drug effect. Our novel model, a timetoevent pharmacokinetic model henceforth referred as TITEPK, requires data on the exact dosing schedule and adverse event history. TITEPK includes a latent pseudoPK based on the preplanned dosing regimen and is therefore considered like a (timevarying) baseline covariate. Formally, the latent pseudoPK process is used to define a timevarying exposure metric, which we use to construct a timevarying Poisson process describing the DLT event process. TITEPK assumes that a given dose input over time, produces a proportional signal that drives the probability of observing a response subsequent to dosing. This signal decays exponentially similar to drug concentration in a one compartment model. Similar to PK/PD models, the signals associated with repeated dosing are superimposed in a manner that accounts for the loss of signal since administration of the previous dose(s). Systematic delays between the signal and the observed responses are accounted for by a further exponential function akin to a socalled effect compartment that can be used to account for delays in the distribution of drug from plasma to the effect site. The instantaneous hazard of the timetofirst DLT is directly proportional to the pseudo PK signal. Analogous to the monotonicity assumption in the BLRM, the proportionality constant which drives the Poisson process is constrained to be strictly positive. Two parameters are required to account for the decay and the delay. Prior information on all parameters may be included by adopting a Bayesian framework.
The proposed approach builds on the positive experience with the BLRM, but has the added benefit of allowing changes in dose and dosing regimens to be taken into account in a systematic manner that draws on the significant positive experience from PK/PD analyses. Note that in the proposed model, PK analysis and safety analysis are not combined as is done for example by Ursino et al.^{11} This simplification is made to ensure the operational applicability of our approach in a realistic clinical trial setting. In clinical trials it is often not feasible to obtain reliable PK data in time for doseescalation decision meetings. Instead, the model is conditioned on known PK parameters from previous analyses following the work of Cox et al.^{10}
The main contribution of this paper is to introduce a novel approach, TITEPK, for analysing and guiding phase I doseescalation trials with different dosing regimens. Our approach builds on experience with the BLRM which has demonstrated the value of the EWOC principle implemented within a Bayesian framework. The R and Stan code for the implementation of the TITEPK are available from Github (https://github.com/gunhanb/TITEPK_code). In Section 2, we describe a motivating phase I trial example from oncology which investigated daily and weekly dosing regimens. In Section 3, we review the BLRM and its adhoc extensions for combining regimens, and introduce the proposed TITEPK model. In Section 4, the BLRM and TITEPK are applied to the motivating example. Finally, the performance of both approaches are compared in a simulation study in Section 5. Scenarios involving simple and complex dosing regimens are considered. We close with discussion and conclusions.
2 Motivating example: Everolimus trial
Everolimus (RAD001) is an oral inhibitor of mammalian target of rapamycin, that is being developed as an antitumor agent.^{12} Everolimus is approved by the US FDA to treat various conditions including certain types of pancreatic cancer and gastrointestinal cancer,^{12} and certain type of tuberous sclerosis.^{13} Everolimus is also approved to treat transplant rejection.^{12} Everolimus was included in a phase Ib trial in combination with standard of care (etoposide and cisplatin chemotherapy) to identify a feasible dosing regimen in the treatment of small cell lung cancer (ClinicalTrials.gov identifier: NCT00466466).^{14} The trial was openlabel and multicentered. Patients were assigned alternately to either weekly or daily regimens of everolimus in treatment cycles of 21 days. The final data can be obtained from the supplementary material of Besse et al.^{14} The dataset is displayed in Table 1. All DLTs were reported at day 15. The elimination halflife and the absorption rate of everolimus for cancer patients are reported as 30 (hours) and 2.5 (1/hours), respectively.^{15} A Bayesian timetoevent model was used to inform the doseescalation decisions.^{16} A dosing regimen of 2.5 mg/day everolimus was identified as the MTD.
Regimen  Dose  Number  Number 

(mg)  of patients  of DLTs  
Weekly  20.0  5  0 
Weekly  30.0  13  4 
Daily  2.5  4  2 
Daily  5.0  6  3 
We used this trial to compare the performance of the TITEPK and BLRM methods because: (1) the trial evaluated two different regimens (weekly and daily dosing), and (2) the large number of DLTs allow a good assessment on the relative performance of the respective methods.
3 Methods
In this section, we briefly review the BLRM and an extension of the BLRM which is necessary to account for the multiple regimens. Subsequently, we describe the proposed TITEPK model, and its software implementation. Our primary focus is the application of these methods to phase I doseescalation trials with more than one dosing regimen, although we also consider trials with only one dosing regimen. For simplicity, consider there are two dosing regimens, R1 and R2, and the aim is to determine the MTD characterized by the dosing amount and the dosing regimen.
3.1 The BLRM
The BLRM is a logistic regression model in the logarithm of a standardized dose. For dose , the number of patients with a DLT () in a cohort of size are assumed to be binomially distributed
(1) 
with DLT probabilities () and two parameters ( and )
(2) 
where is the reference dose used for standardization of the dose. At the reference dose the odds of the DLT are . Thus, the reference dose is critical in choosing a prior for and is usually defined such that the reference dose is set to the anticipated MTD at which an odds of is used as mean for the prior.
In the absence of relevant historical data, Neuenschwander et al^{2} suggest the use of weakly informative priors (WIPs) for and . Their suggested WIP is a bivariate normal distribution with means , standard deviations , and the correlation .
To inform the doseescalation decisions, the posterior distribution of the DLT probability is used. The DLT probabilities are classified into three categories as follows

Underdosing (UD)

Targeted toxicity (TT)

Overdosing (OD)
Escalation or deescalation decisions are informed using the overdosing probability of dose , . The EWOC criteria is fulfilled, if is smaller than the prespecified feasibility bound, , which was recommended as by Babb et al.^{5} In this paper, we use throughout the manuscript, unless another value is stated explicitly.
The BLRM does not distinguish in which frequency doses are administered to the patients (for example weekly dosing frequency or daily dosing frequency), but only the dosing amounts. Hence, an extension of the BLRM is used to incorporate different dosing regimens in a phase I trial, which we describe now.
3.1.1 The BLRM MAP
The BLRM with a metaanalytic predictive (MAP) approach^{17}, ^{18} can be used for informing doseescalation decisions in a phase I study with more than one dosing regimen. Hereafter we refer this method as the BLRM MAP. In the BLRM MAP approach, the doses from the first dosing regimen (R1) are rescaled so that two sets of doses from different regimens are comparable. For example, if R1 is weekly dosing and R2 is daily dosing, the doses from R1 are divided by 7. This ensures that the respective nominal dose in each regimen results in the same cumulative dose. Then, a metaanalyticpredictive (MAP) prior is derived using the data of the R1 assuming some betweenregimen heterogeneity for the parameters. The idea is that the data coming from R1 is treated as concurrent data (codata).^{19} The degree of heterogeneity depends on the switch between R1 and R2. For instance, one may assume less heterogeneity for the switch between “twice in a day” to daily compared to a switch between weekly to daily regimen. Furthermore, it may be desirable to make the MAP prior more robust for possible unwarranted use of data from R1. To achieve this, the robust MAP prior () can be obtained by mixing the MAP prior () with the WIP () as follows:^{20}, ^{2}
(3) 
where is the weight which can be chosen, for example, from the range of and . Neuenschwander et al^{2} suggested the use of , and in this paper we follow their suggestion. After the robust MAP prior is derived, the BLRM is used to inform doseescalation decisions. Note, that the BLRM MAP approach leads to distinct doseDLT models for each regimen and is difficult to conduct in case data is generated concurrently for R1 and R2 at the same time.
3.2 The proposed method: TITEPK
The basic concept in this method is inspired by Cox et al.^{10} Instead of modelling the number of DLTs as in the BLRM, the timetofirst DLTs are modeled using a timevarying (nonhomogeneous) Poisson process. A timevarying Poisson process can be defined using the instantaneous hazard function () for a DLT occurring at time as^{21}
(4) 
The hazard function corresponds to the probability that a patient experiences a DLT in the time interval given that they did not experience a DLT until time . The hazard is modeled as a timedependent function which is directly proportional to an exposure measure of the drug () as^{10}
(5) 
where is the proportionality parameter to estimate. Note that, here, the exposure measure refers to the drug concentration as in a exposureresponse model,^{22} and the calculation of will be explained in Section 3.2.1. Furthermore, if we integrate both sides of Equation (5), we obtain
(6) 
where is the area under the curve of the exposure measure over time and is the cumulative hazard function, respectively.
From the event history analysis,^{21} we know that the probability density for an event to occur at timepoint is
(7) 
and the survivor function for the event to occur past some timepoint is given by
(8) 
In the following we use to denote the censoring time of patient . By convention we restrict the followup period for all patients to cycle 1 only. Thus, all patients without a DLT up to the end of cycle 1 will be censored at the end of cycle 1, . Longterm DLTs will thus not be considered, which will be discussed later. Furthermore, we denote with an event indicator which is set to for censored events and for DLT events. The overall likelihood can be written as
(9) 
where is the total number of the patients. Now, we discuss the exposure measure of the drug.
3.2.1 PseudoPK model
The proposed exposure model in the TITEPK model does not rely on measured drug concentration data, as this data is not routinely available in a form that it may be used directly in the model to support escalation decisions in a timely manner. For this reason, PK is considered as latent variable which we refer to as pseudoPK. The pseudoPK is used to account for the dosing history and the expected accumulation in exposure over time that ultimately drive pharmacological responses, including safety. The main purpose of the pseudoPK model is to account for the natural “waxing and waning” of exposure observed after dosing of drug. This pseudoPK model has a “central” compartment into which the drug is administered which accounts for drug elimination as a linear first order process; i.e. the elimination rate is proportional to the amount of drug in the compartment ^{23}
(10) 
where is the concentration of drug in the central compartment and is the elimination halflife. As the volume of the central compartment cannot be identified for a latent pseudoPK, we set it by convention to unity. The elimination halflife is assumed to be known from previous analyses and we will evaluate the sensitivity of the approach to misspecification of the elimination halflife.
To account for delays between the instantaneous drug concentration in the central compartment, we use a socalled effect compartment^{23}
(11) 
where is the drug concentration in the effect compartment, and is the PK parameter which governs the delay between the concentration in the central compartment () and the concentration in the effect compartment ().
The ordinary differential equations (ODEs) (10) and (11) account dosing over time through administration into the central compartment. The analytical solution to the ODE system for multiple doses is obtained through the use of the superposition principle which holds for linear ODE systems (see e.g. ^{24}). This model can account for the natural history of any number of dosing regimens over time. In order to simplify the notation, we restrict ourselves here to regular dosing regimens which have a dosing frequency (in units of ), start at time and use the same dose amount for all dosing events. With these simplifications (in notation) the solution to the above ODE system is
(12) 
where denotes the Heaviside step function (or unit step function).
To facilitate meaningful interpretation of the parameter , and hence to help prior specification, the exposure measure is obtained by scaling using a reference regimen including a reference dose () and a reference dosing frequency () at the end of cycle 1 () such that
Note that this is analogous to the usage of a reference dose in the BLRM as discussed in Section 3.1. For illustrative purposes, the calculated , , and of the Everolimus trial for 20 mg/weekly and 5 mg/daily dosing using different values are displayed in Figure 1. Notice that of 5 mg/daily at week 3 is 1, since 5 mg/daily is taken as the reference regimen and the length of cycle 1 is 3 weeks.
3.2.2 Informing doseescalation decisions
To inform escalation decisions during the phase I trial, TITEPK uses an adapted EWOC criteria analogous to the BLRM metric. Hence, the probability that a patient experiences at least one DLT within the first cycle (shortly the endofcycle 1 DLT probability) given the regimen with dose and frequency , , is our measure of interest. By using the relationship between and combining Equation (8) with (6) it follows that
(13) 
Since the cumulative hazard is set proportional, see Equation (6), to the area under curve of the exposure metric this leads to
(14) 
For the reference regimen with dose and dosing frequency the AUC of the exposure measure up to the reference timepoint is unity, , such that holds which highlights the importance of the reference regimen to specify the prior for the parameter .
We consider two models, a oneparameter TITEPK model and a twoparameter TITEPK model. In the oneparameter model, the only parameter to estimate is the regression coefficient , and the PK parameter is assumed to be known. In the twoparameter model, is considered unknown. The elimination halflife is assumed to be known for both models.
3.3 Software implementation
The BLRM and the BLRM MAP can be implemented using Markov chain Monte Carlo (MCMC), for example, via BUGSvariant programs such as WinBUGS^{25} or JAGS.^{26} BUGS code for the BLRM and the Robust MAP are included in the Appendix of Neuenschwander et al^{4} and the supplementary material to the paper by Schmidli et al,^{20} respectively. For our implementations, we use JAGS via the R2jags R package to generate four parallel chains of 100,000 MCMC samples after a 10,000 iteration warmup. To check MCMC convergence, we used the GelmanRubin statistics, trace plots and lag 1 sample autocorrelations.
The proposed model TITEPK is implemented in Stan ^{27} via rstan R package. The corresponding code for the implementation of the TITEPK method is available from Github (https://github.com/gunhanb/TITEPK_code). Four parallel chains of 1,000 MCMC iterations after warmup of 1,000 iterations are generated. Convergence diagnostics are checked using the GelmanRubin statistics and traceplots. There were no divergences reported for the implementation of the application.
4 Revisiting the Everolimus trial
Returning to the data set described in Section 2, consider the Everolimus trial shown in Table 1. Firstly, we analyse the data only from the daily regimen using the BLRM, the oneparameter TITEPK, and the twoparameter TITEPK. Secondly, we analyse the data considering both weekly and daily regimen using the BLRM MAP, oneparameter TITEPK, and the twoparameter TITEPK. Note that although the cohorts of weekly and daily regimen are administered simultaneously in the Everolimus trial, we analyse the dataset as if the trial is conducted sequentially, specifically daily regimen after weekly regimen. This is because, we want to compare the proposed method with the BLRM MAP, which is only applicable for phase I trials involving different regimens conducted sequentially. The proposed method TITEPK can be used for phase I trials in which data is generated concurrently for different regimen at the same time, but we come to this point at the discussion. The reference regimen is determined using dosing amount of 5 mg ( mg) and dosing frequency of 24 hours ( 1/h). For TITEPK both models, the elimination halflife is taken as (hours).
To compare the BLRM and TITEPK models, priors are constructed so that a priori DLT probabilities from BLRM and endofcycle 1 DLT probabilities from TITEPK models are similar. To define a WIP for BLRM, we choose a bivariate normal prior with following parameters . For TITEPK models, a normal WIP is chosen such that . For twoparameter TITEPK, a WIP for the parameter is derived using the cycle length and the absorption rate. Specifically, a lognormal distribution is setup by matching the inverse of cycle length, 1/504 (1/h), and the absorption rate, 2.5 (1/h) as the 0.025 and 0.975 quantiles, respectively. This gives a lognormal distribution with mean parameter 0.41 and the standard deviation parameter 0.27. For the oneparameter TITEPK model, is taken as the mean of the derived WIP for . The summaries of a priori DLT probabilities of the BLRM, the oneparameter TITEPK, and the twoparameter TITEPK models are shown in Figure 2A. Points, thick lines and thin lines correspond to median estimates, the 50% and the 95% equitailed credible intervals, respectively. Vertical dashed lines (0.160.33) are the boundaries of the targeted toxicity interval. If the upper bound of 50% credible interval of a dose is higher than the 0.33 value, then the overdosing probability of the corresponding dose exceeds the prespecified feasibility bound 0.25. Therefore, the corresponding dose is an overdose based on the EWOC criteria. From Figure 2A, we can see that for all three models, a priori, doses of 7.5 mg and 10 mg are overdoses, while doses of 2.5 mg and 5 mg are not.
Figure 2B displays the posterior estimates of DLT probabilities, when we only consider daily regimen data. Results obtained by one parameter TITEPK and two parameter TITEPK models are very similar. To compare those two models, we also calculate approximate leaveoneout cross validation (LOO) values ^{28} using the loo R package. LOO values of one parameter TITEPK and two parameter TITEPK models are 81.1 (standard error 27.6) and 81.5 (standard error 27.7), respectively. The difference of the expected log pointwise predictive density of LOO values between one parameter TITEPK and two parameter TITEPK is 0.2 (standard error 0.1). If we compare BLRM and TITEPK models, all three methods suggest that all doses are overdoses, meaning that the trial should be stopped without any dose declared as the MTD. This seems reasonable, since 2 DLTs observed from 4 patients in 2.5 mg, and 3 DLTs from 6 patients in 5 mg dose.
We continue by combining weekly and daily regimen data. We estimate the DLT probabilities of daily doses but also taking into consideration the data coming from the weekly data. As explained in Section 3.1.1, to implement BLRM MAP, the doses of weekly regimen, namely 20 mg and 30 mg, are rescaled dividing them by 7. The Robust MAP prior is constructed by mixing the MAP prior and the WIP using the weight of 0.8 (). Finally, the BLRM is fitted and posterior estimates of DLT probabilities are obtained. On the other hand, our method TITEPK, naturally, combines information from different regimens without requiring any adhoc method. Figure 2C displays the estimated posterior summaries of DLT probabilities of daily doses obtained by one parameter and two parameter TITEPK and BLRM MAP approaches. As in Figure 2B, two parameter TITEPK model did not show any improvement compared to one parameter TITEPK model in terms of the LOO estimate. Hence, hereafter we only consider one parameter TITEPK model, and use TITEPK to refer one parameter TITEPK model.
For both TITEPK and BLRM MAP, the overdosing probability of dose 2.5 mg/daily is decreased substantially, namely from 0.49 to 0.27 for BLRM MAP, and from 0.28 to 0.01 for TITEPK. The reduction of the overdosing probabilities of 2.5 mg/daily seems reasonable, since in the weekly regimen data, 0 DLTs occurred from 5 patients in 20 mg/weekly and 4 DLTs occurred from 13 patients in 30 mg/weekly. However, the interval estimates obtained by TITEPK are shorter, hence more precise estimates compared to BLRM MAP. Unlike BLRM MAP, TITEPK suggests that 2.5 mg/daily is not an overdose, hence it can be declared as the MTD which was the conclusion of the original phase I trial. Also it must be noted that results of BLRM MAP depends on the choice of the weights which are used for the construction of the Robust MAP prior. For instance, if we choose the weight of 1.0 for instead of 0.8, then the overdosing probability of 2.5 mg/daily reduces to 0.20 which is not an overdose based on the EWOC criteria. It is also worthwhile to point out that conceptually BLRM MAP is different in comparison to TITEPK. The BLRM MAP approach for different regimens is a twostep approach which establishes two distinct models; one per dosing regimen. The TITEPK model instead combines in a model based approach all available information.
As pointed out in Section 3.2.1, by construction of TITEPK, the elimination halflife is treated as known. To investigate the influence of misspecification of the parameter, we fit TITEPK using ranging from 5 to 50 hours. The timing of all DLTs (in total 9 DLTs) were reported at day 15. To examine what would be the influence of the timing of DLTs, we also fit TITEPK to two hypothetical datasets. Early DLTs dataset and late DLTs dataset are created by changing timing of DLTs from day 15 to day 1.5 and to day 20.5, respectively. Posterior estimates of DLT probabilities for different values and for different timing of DLTs are shown in Figure 3. The middle plot corresponds to the original Everolimus trial data. Firstly, the posterior medians and credible intervals obtained by different values look very similar. Note that it is highly probable to not to have a precise estimate of elimination halflife during a phase I trial, hence these results are reassuring for the practicality of TITEPK. Secondly, timing of DLTs have a crucial affect on the posterior estimates, and hence the overdosing probabilities. Having the same number of DLTs, the earlier the DLT happened, the higher the overdosing probability of the corresponding dose estimated. This makes sense, since one would expect the drug is more toxic if DLTs happened earlier than later.
5 Simulation study
In order to assess the performance of the TITEPK and to compare with the BLRM under different true doseDLT profiles and different regimens, various scenarios are investigated in a simulation study.
5.1 Simulation settings
Simulation study follows the clinical scenario evaluation framework introduced by Benda et al.^{29} In all scenarios, by mimicking the Everolimus trial, PK parameters are chosen such that (hours) and . Weekly doses of 8, 16, 32, 64, 115, 230 (mg/weekly) and daily doses of 1, 2, 4, 8, 15, 30 (mg/daily) and are considered. Reference dose and reference dosing frequency are determined using 8 mg ( mg) and 24 hours ( 1/h). Cycle length is taken as 28 days instead of 21 day as in Everolimus trial, since the former is more common. As in the Everolimus application, WIPs are constructed so that a priori DLT probabilities of BLRM and TITEPK are similar. For BLRM, we choose a BVN prior with following parameters: . For TITEPK model, a normal WIP is chosen such that log() . Although the choice of prior can have an influence on posterior estimates for a Bayesian phase I design, we did not consider to vary the priors. This is because our aim is to compare the BLRM and TITEPK given that they have a priori similar DLT probabilities for all doses. However, we recommend varying priors as part of sensitivity analyses when TITEPK is used for an application as is the case with other Bayesian phase I models.
For the first setting of scenarios, six different doseDLT profiles with only daily dosing are considered as shown in Figure 4B. Scenario 3 is the basis and equals to the mean of a priori medians of the TITEPK and BLRM. DLT probabilities of Scenario 1 and 2 are 0.25 and 0.75 of the DLT probabilities of Scenario 3, respectively. Thus Scenario 1 and 2 have lower toxicity compared to the basis (Scenario 3). Similarly, Scenario 4 and 5 are 1.25 and 1.75 of Scenario 5, respectively, hence more toxic scenarios. Finally, Scenario 6 is an extreme scenario that is highly inconsistent with the prior, and it is taken from Babb et al.^{5} In the second and third set of scenarios, we consider switching regimen scenarios. We assumed that after the MTD declaration of the weekly dosing, daily dosing is administered to the patients and then MTD is declared for the daily dosing. For the daily dosing, the scenarios illustrated in Figure 4B are used. Figure 4A displays the true doseDLT profiles used in the weekly dosing scenarios and they are calculated using same procedure with daily dosing scenarios. In the second simulation set of scenarios, we consider the same true doseDLT profiles between different regimens. For example, a trial is started with weekly dosing of Scenario 3, after MTD declaration, it is switched to the daily dosing of Scenario 3. In third set of scenarios, we consider the situations when there is conflict of true doseDLT profiles between weekly and daily regimens. For instance, in the weekly regimen, Scenario 2 is used, and then switched to the daily dosing of Scenario 3.
The data generation processes for BLRM and TITEPK are different, since the former uses a binomial distribution whereas the latter uses a timevarying Poisson process. Hence, we run the simulations separately for two methods. For BLRM, specification of the true doseDLT profiles are enough to simulate data. But for TITEPK, we also need to specify the cycle length as well as the PK parameters and . For both TITEPK and BLRM, data for 1,000 trials were generated. The following criteria for the simulations are taken from Neuenschwander et al.^{2} Sample sizes were randomly chosen cohorts of size 36. The maximum number of patients per trial was set to 60. The trial was stopped when one of the following criteria were met:

At least six patients have been treated at the recommended MTD.

The MTD dose satisfies one of the following conditions:

The probability of targeted toxicity at recommended MTD dose exceeds 50%.

A minimum of 21 patients have already been treated in the trial.

5.2 Results
Measures  Explanation 

I  Percentage of patients receiving an underdose. 
II  Percentage of patients receiving a dose that is in the targeted toxicity interval. 
III  Percentage of patients receiving an overdose. 
IV  Probability that recommended MTD is an underdose. 
V  Probability that recommended MTD is in the targeted toxicity interval. 
VI  Probability that recommended MTD is an overdose. 
Stopped  Probability that trial is stopped without recommending MTD. 
AveN  Average number of patients dosed. 
AveDLT  Average number of DLTs observed. 
To assess the performance of the BLRM and TITEPK in the simulations, we used six measures which are demonstrated in Table 2. Performance measures obtained by TITEPK and BLRM for the first set of scenarios (only daily dosing) are shown in Figure 5. For all six scenarios, the percentage of trials with MTD identified in targeted toxicity (TT) region (correctly identifying MTD) are higher in TITEPK compared to the BLRM. On the other hand, for scenarios 2, 4, and 5, the percentage of trials with an overdose identified MTD is higher in TITEPK compared to BLRM, although only scenario 4 can be seen as problematic (14.3% in TITEPK vs 7.1% in BLRM). However, the percentage of patients receiving an overdose is not extremely higher in TITEPK in comparison to BLRM (10% in TITEPK vs 8.3% in BLRM). Furthermore, to improve these results, we consider TITEPK using a modified EWOC, namely using prespecified feasibility bound instead of . Note that reducing the feasibility bound results the conservative doseescalation decisions, that is reducing the percentage of trials with an overdose identified MTD but also reducing the percentage of trials with correctly identified MTD. The percentage of trials with correctly identified MTD obtained by TITEPK using is decreased to 11.1%, while in all scenarios, TITEPK outperforms BLRM in terms of this metric. Scenario 6 needs special consideration. For this very toxic scenario, while the measure for correctly identifying MTD are higher in both TITEPK methods in comparison to the BLRM, the percentage of patients treated at overdoses is also considerably lower compared to BLRM (26.6% in TITEPK using vs 35.8% in BLRM). What is more, the percentage of patients receiving an overdose obtained by BLRM is higher compared to both TITEPK using and TITEPK using . In terms of average number of DLTs and average number of sample sizes, BLRM and TITEPK methods give similar results, although latter is slightly higher.
Now, we consider the second set of the scenarios, switching regimen scenarios with no conflict of true doseDLT profiles of weekly and daily regimens. Performance measures of TITEPK and BLRM MAP are shown in Figure 6. Firstly, we can compare both methods to their performances for the first set of scenarios as shown in Figure 5. It can be clearly seen that for all measures, TITEPK is improved. For example, the percentage of trials with an overdose identified MTD of Scenario 4 now reduces to 12.1% from 14.1%. And the percentage of trials with correctly identified MTD of Scenario 4 increases to 71.3% from 55%. Although in many measures BLRM is also improved, there are some measures in which it’s performance are getting worse, for example the percentage of trials with correctly identified MTD of Scenario 5 reduces from 29.5% to 21.4%. Most importantly, if we compare BLRM MAP and TITEPK, it can be easily seen that TITEPK outperforms BLRM MAP in many measures especially for correctly identifying MTDs.
Lastly, we consider the third set of the scenarios, switching regimen scenarios with conflict of true doseDLT profiles of weekly and daily regimens. Performance measures of TITEPK and BLRM MAP are shown in Figure 7. Firstly, we can notice that TITEPK outperforms BLRM MAP in all considered scenarios in terms of the percentage of trials with correctly identified MTD. For the fifth scenario (starting with with weekly regimen of Scenario 4, then continue with daily regimen of Scenario 1: Sc. 4 and Sc. 1), the difference is 40%, for example. For the percentage of trials with an overdose identified MTD, we see that TITEPK has lower or equal value compared to BLRM MAP except the third scenario (Sc. 3 and Sc. 4). Note that here we consider the measures of daily regimen of Scenario 4, hence we can compare these results to Scenario 4 of the first set of scenarios to see the influence of the inclusion of the weekly regimen of Scenario 3. In the first set of the scenarios, the percentage of trials with an overdose identified MTD of TITEPK for Scenario 4 was 14.1%, and now it increases to 21.5%. On the other hand, the percentage of trials with an overdose identified MTD of BLRM MAP is only slightly increased when we compare the first set and third set of scenarios for Scenario 4, from 7.1% to 8.5%. However, the percentage of patients receiving an overdose obtained by TITEPK is not unacceptably high in compared to the BLRM (11.2% in TITEPK vs 6.7% in BLRM). Note that the difference of these scenarios compared to previous ones is that the assumption that the endofcycle 1 DLT probability is increasing with higher AUC of the exposure of the drug is violated. This is analogous to the monotonicity assumption of the BLRM. For example, in the third scenario (starting with with weekly regimen of Scenario 3, then continue with daily regimen of Scenario 5: Sc. 3 and Sc. 5), true DLT probability of dose 32 mg/weekly (0.10) is smaller than true DLT probability of dose 4 mg/daily (0.15), while of dose 32 mg/weekly is bigger than of dose 4 mg/daily. This assumption is more relevant for TITEPK than BLRMMAP, since in the latter, data from the weekly regimen is used to construct the prior, but not used as data.
6 Discussion and Conclusion
We propose a Bayesian adaptive model, TITEPK, to support design, analysis and guidance of phase I doseescalation trials, where the drug is administered by different dosing regimens and dosing decisions are determined by dose limiting toxicities. TITEPK preserves the advantages of the BLRM including interpretable parameters, and being able to use the EWOC criteria. It uses PK principles to combine different dosing regimens in a modelbased approach. A real application involving weekly and daily dosing is used to show the usage of TITEPK. Moreover, we have demonstrated by means of simulations that TITEPK shows better performance in terms of the investigated measures compared to BLRM for single regimen scenarios and compared to BLRM MAP for combining different regimens in realistic scenarios.
We considered two models, one parameter TITEPK model and two parameter TITEPK model. The second model has the extra parameter of which governs a possible delay between the putative concentration in the central compartment and manifestation of effect. Note, in this case it also accounts for the delay between the administration and the manifestation of the putative concentration. In the Everolimus trial, posterior estimates were not changed when we used the two parameter TITEPK model, also the estimated leaveoneout cross validation values were not improved. Hence, we suggest the usage of one parameter TITEPK model which is computationally less expensive. An advantage of using timetofirst DLTs as in TITEPK instead of only numbers of DLTs as in BLRM is that censored patients can also be included in the analysis. There are some suggested time to event models for phase I trials, for example TITECRM^{16} which, additionally, models long term DLTs not only DLTs occurring in the first cycle as in TITEPK.^{30} TITEPK can be extended to also incorporate long term DLTs which requires the modelling of the recurrent DLTs, and two parameter TITEPK model could be more useful for such longterm safety analysis.
In the simulation study, we only considered the scenarios when different regimens are administered to the cohort of patients sequentially. That is, the MTD is estimated for the first regimen (R1), then the second regimen (R2) is administered to the patients, finally the MTD for the second regimen is estimated. As an alternative design, the data can be generated concurrently for R1 and R2 at the same time. TITEPK can be used to inform doseescalation decisions for both designs. However, we did not investigate the latter design scenarios, since we want to compare the TITEPK with the BLRM MAP, which is only applicable for the former design.
We have used a linear PK model for the pseudoPK to calculate the exposure measure of the drug. This implies that the constructed exposure metric is proportional to the administered dose which extends to the key metric of the model, the endofcycle 1 DLT probability. When there is a clear conflict in doseDLT profiles between different regimens, this assumption can be seen as a limitation of TITEPK. To relax this assumption, one can consider more complicated PK models including a nonlinear PK model which may not have an analytical solution. Such extensions may be implemented in Stan which has a builtin differential equation solver. However, more complicated modelling approaches always need to be calibrated well given the sparseness of the phase I doseescalation data sets. Alternatively, one can consider an adhoc extension of one parameter TITEPK model. For instance, similar to the idea of power model,^{31} a pseudodose such as can be used instead of dose in the model which may be helpful to relax the linear PK assumption.
When relevant historical information or data from a different study population exists, it is desirable to include such information in the analysis of the phase I trial, for example using a MAP prior. Since TITEPK is parametrized by mimicking the interpretable parameters of the BLRM, it can be also extended to use a MAP approach like the BLRM. A key strength of the TITEPK approach is it’s ability to integrate the data from different dosing regimens in a model based approach. This makes adhoc approaches used for the BLRM obsolete which reduces the need for strong discounting of historical data from different regimens. However, discounting may still be needed to account for other sources of heterogeneity in the data under consideration.
Another crucial aspect of the methods for phase I trials is the ability to analyse the combination of drugs. Although, we only consider the single agent case here, it is possible to extend TITEPK to analyse drug combinations which is complicated by the need to model possible drug interactions. Moreover, we only considered the fixed feasibility bound to use EWOC, that is a dose is considered an overdose if the overdosing probability of a dose exceeds a fixed bound. However, one can also consider varying feasibility bounds as suggested by Wheeler et al.^{32}
In summary, we expect that with TITEPK the available historical data can be used more efficiently and that phase I trials can more flexibly explore dosing regimens in doseescalation trials. This is achieved through the use of pharmacokinetic principles in the TITEPK model.
Acknowledgement
We thank Heinz Schmidli who contributed valuable comments and pointed us to several important references, and Michael Looby for carefully proofreading this, and recommending several changes that lead to an improved presentation of this paper.
Conflict of interest
S.W. and A.S. are employees of Novartis, and T.F. is a consultant to Novartis and has served on data monitoring committees for Novartis. Novartis is the manufacturer of everolimus, an everolimus trial was used to motivate and illustrate the investigations presented here (see Section 2 and Section 4).
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