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A clinical phase I dosefinding design with adaptive shrinking boundaries for drug combination trials
BMC Medical Research Methodology volume 23, Article number: 57 (2023)
Abstract
Background
Combinations of drugs are becoming increasingly common in oncology treatment. In some cases, patients can benefit from the interaction between two drugs, although there is usually a higher risk of developing toxicity. Due to drug–drug interactions, multidrug combinations often exhibit different toxicity profiles than those of single drugs, leading to a complex trial scenario. Numerous methods have been proposed for the design of phase I drug combination trials. For example, the twodimensional Bayesian optimal interval design for combination drug (BOINcomb) is simple to implement and has desirable performance. However, in scenarios where the lowest and starting dose is close to being toxic, the BOINcomb design may tend to allocate more patients to overly toxic doses, and select an overly toxic dose combination as the maximum tolerated dose combination.
Method
To improve the performance of BOINcomb in the above extreme scenarios, we widen the range of variation of the boundaries by setting the selfshrinking dose escalation and deescalation boundaries. We refer to the new design as adaptive shrinking Bayesian optimal interval design for combination drug (asBOINcomb). We conduct a simulation study to evaluate the performance of the proposed design using a real clinical trial example.
Results
Our simulation results show that asBOINcomb is more accurate and stable than BOINcomb, especially in some extreme scenarios. Specifically, in all ten scenarios, the percentage of correct selection is higher than the BOINcomb design within 30 to 60 patients.
Conclusion
The proposed asBOINcomb design is transparent and simple to implement and can reduce the trial sample size while maintaining accuracy compared with the BOINcomb design.
Introduction
Drug combination therapy provides an important method for the treatment of difficult diseases such as cancer. The purpose of drug combination therapy is to induce synergistic therapeutic effects, increase the combined dose intensity, and achieve better therapeutic effects without crosstoxicity. The purpose of a drug combination dose escalation trial is to identify the maximum tolerated dose combination (MTDC) based on a prespecified target toxicity rate. In a singledrug dose escalation trial, toxicity is typically assumed to increase with an increasing dose. However, the order of toxicity between dose combinations in twodrug combination trials is only partially known. If the dose of one drug in the combination increases while that of the other drug decreases, it is unknown whether the toxicity increases or decreases (Fig. 1). This partial ordering assumption impacts interim dose allocation decisions. On the other hand, the MTDC may differ from the combination formed by the maximum tolerated dose (MTD) of each drug alone due to unknown potential interaction between the drugs (synergy, antagonism, or no interaction). In fact, multiple dose combinations may have the same target doselimiting toxicity (DLT) probability, thus yielding an MTDC contour (Fig. 2). Therefore, the traditional singledrug dosefinding design cannot be used directly for combination drug dosefinding trials. The most common method for twodrug combination trials is to fix the dose of one drug and conduct a dosefinding trial on the other drug, i.e., reducing a twodimensional dose combinationfinding problem to a onedimensional dosefinding one, which largely limits the ability to evaluate potential synergy between drugs. In addition, an increasing number of methods applicable to twodrug combination dosefinding trials have been proposed. These methods can be broadly classified into three categories: algorithmbased design, modelbased design, and modelassisted design. Algorithmbased designs rely on several prespecified rules to determine when doses are escalated, deescalated, or selected as MTDC. Examples include the upanddown design [1], 2 + 1 + 3 design [2], and 3 + 3 + 3 and its derivative designs [3]. Modelbased designs simulate the relationship between dose and toxicity probabilities via parametric functions. During the trial, parameter estimates are continuously updated to better describe this relationship. Thall et al. [4] proposed an adaptive twostage Bayesian design by considering a sixparameter joint toxicity rate model. Wang and Ivanova developed a method based on a threeparameter model that uses Bayesian inference [5] to estimate parameters. Yin and Yuan [6] used copulatype regression models to relate the toxicity rates of two drugs based on several feasible conditions, which can be viewed as a generalized or twodimensional version of the continuous reassessment method (CRM) [7]. Wages et al. [8] developed a partially ranked CRM by listing the possible order of the combined toxicity rates. Yin et al. [9] developed a Bayesian adaptive design for dose discovery based on a potential 2 × 2 table. Thomas et al. [10] proposed a hierarchical model for the probability of DLT for dose combinations and applied this model to a Bayesian adaptive trial design. Riviere et al. [11] developed a Bayesian dose discovery design for clinical trials combining cytotoxic drugs with molecularly targeted drugs. Manjrekar et al. [12, 13] proposed using a continuation ratio model to separate the toxicity and efficacy profiles of each drug, and combining them into the optimal dosing region to determine the combination. Li et al. [14] proposed a Bayesian hierarchical model that jointly models the unordered probabilities of toxicity and efficacy in the design of a dose/schedulefinding trial, and applied a Bayesian isotonic transformation to the posterior samples of the toxicity probabilities to impose a partial ordering constraint. Guo et al. [15] developed a doseschedulefinding algorithm to allocate patients sequentially to a desirable dose–schedule combination and to select an optimal combination at the end of the trial. Modelassisted designs do not prespecify any relationship between dose and toxicity and therefore do not rely on any parametric assumptions when finding the MTDC. However, unlike in algorithmbased designs, the decision process for dose escalation and deescalation is aided by statistical models. Lin et al. [16] proposed a twodimensional Bayesian optimal interval design, Zhang et al. [17] proposed a Bayesian optimal intervalbased design for exploring multiple MTDCs, and Pan et al. [18] extended the keyboard design to a twodrug combination dosefinding trial. Among them, the twodimensional Bayesian optimal interval (BOINcomb) design has overall good operating characteristics [16]. The BOINcomb design is also comprehensible and easily implementable as an extension of the onedimensional Bayesian optimal interval (BOIN) design to the twodrug combination case. However, due to the complexity of twodrug combination trial scenarios, the twodimensional Bayesian optimal interval design may not perform well in some extreme scenarios and may tend to allocate more patients at overly toxic doses [19]. In practice, extreme scenarios may not be common, but they do arise in real trials [20,21,22]. For example, in a phase I study of Ganetespib and ZivAflibercept in patients with advanced carcinomas and sarcomas, the lowest and starting dose combination may be adjacent to toxic dose combinations in the dosecombination grid being investigated [22]. To improve the performance, specifically, flexibility and stability, of the BOINcomb design in extreme scenarios, we modify BOINcomb by proposing adaptivelyshrinking dose escalation and deescalation boundaries, the idea of which has been successfully implemented with improvements in design performance demonstrated in onedimensional phase I dosefinding trials [23, 24] and twodimensional doseschedulefinding trials.
The rest of this paper is organized as follows. In Section 2, the BOIN design, the BOINcomb design, and the improved BOINcomb design, denoted as asBOINcomb design, are presented. In Section 3, a simulation study is used to evaluate our proposed design. The results of the simulation study are also analyzed. Section 4 uses a trial example to illustrate our proposed design. Section 5 provides some brief discussions. Finally, a summary is presented in Section 6.
Methods
BOIN design
The BOIN (Bayesian optimal interval) design is a Bayesian modelassisted phase I dosefinding design proposed by Liu and Yuan Y (2015). The design is simple and flexible and performs comparably to modelbased designs [25]. Dose escalation and deescalation in this design are determined by comparing the observed DLT rate at the current dose with a fixed pair of dose escalation and deescalation boundaries. The specific rules are as follows: assume that \({\hat{p}}_j={y}_j/{n}_j\) is the observed DLT probability at the current dose, λ_{e} and λ_{d} denote the predetermined dose escalation and deescalation boundaries, respectively, and j is the current dose level. The BOIN design determines the next dose as follows:

if \({\hat{p}}_j\le {\lambda}_e\), then escalate the dose level to j + 1,

if \({\hat{p}}_j>{\lambda}_d\), then deescalate the dose level to j – 1,

otherwise, i.e., λ_{e} < \({\hat{p}}_j\le {\lambda}_d\), retain the same dose level j.
The trial continues until the prespecified sample size is exhausted or the trial is stopped due to excessive toxicity.
The boundary designation algorithm for the BOIN design aims to minimize incorrect decisions on dose assignment. It makes three assumptions:
where p_{j} denotes the true toxicity probability at dose level j (1, 2..., j), where ϕ_{1} denotes the highest toxicity probability that is deemed subtherapeutic (i.e., below the MTD) such that dose escalation should be made, and ϕ_{2} denotes the lowest toxicity probability that is deemed overly toxic such that dose deescalation is required.
Under the Bayesian model, the three hypotheses are given equal prior probabilities, denoted as π_{kj} = Pr(H_{kj}), k = 0, 1, 2, and the escalation and deescalation boundaries are:
The BOIN design is easy to implement and has comparable performance to that of existing onedimensional phase I dosefinding design [25].
BOINcomb design
The twodimensional BOIN design for drug combination trials is based on an extension of the onedimensional design. The details are as follows. Assume that \({\hat{p}}_{jk}\) denotes the probability of toxicity of two drugs at dose combination (j, k), 1 ≤ j ≤ J,1 ≤ k ≤ K. Assume that the current dose combination is (j, k), and let \({\hat{p}}_{jk}\) denote the estimated probability of toxicity based on the cumulative information of the dose combination (j, k). \({\hat{p}}_{jk}={y}_{jk}/{n}_{jk}\), where y_{jk} and n_{jk} denote the number of patients with toxicity and patients treated at that dose combination, respectively. The acceptable dose escalation set is defined as A_{E} = {(j + 1, k), (j, k + 1)}, and the allowable dose deescalation set is A_{D} = {(j − 1, k), (j, k − 1)}.
The rules of BOINcomb are as follows:

(1)
Treat the first cohort of subjects with the lowest dose combination (1, 1).

(2)
Assuming that the current cohort is treated with dose combination (j, k), then for the next cohort of patients:

(a)
If \({\hat{p}}_{jk}\le {\lambda}_e\), escalate the dose to the dose combination with the largest value of \(\mathit{\Pr}\left\{{p}_{j^{\prime }{k}^{\prime }}\in \left({\lambda}_e,{\lambda}_d\right){y}_{j^{\prime }{k}^{\prime }}\right\}\) in A_{E}.

(b)
If \({\hat{p}}_{jk}\ge {\lambda}_d\), deescalate the dose to the dose combination with the largest value of \(\mathit{\Pr}\left\{{p}_{j^{\prime }{k}^{\prime }}\in \left({\lambda}_e,{\lambda}_d\right){y}_{j^{\prime }{k}^{\prime }}\right\}\) in A_{D}.

(c)
Otherwise, if \({\lambda}_e<{\hat{p}}_{jk}<{\lambda}_d\), the doses remain the same for the combination (j, k).

(a)

(3)
This process continues until the total sample size is exhausted.
During dose escalation and deescalation, if there are multiple optimal dose combinations in sets A_{E} and A_{D}, we randomly choose one with equal probability. If no dose combination exists in sets A_{E} and A_{D}, we retain the current dose combination. Regarding the encountered boundary case, if j = 1 and \({\hat{p}}_{jk}\ge {\lambda}_d\), the next dose combination is (j, k  1), and if (j, k) = (1, 1), the current dose is kept. If j = J and \({\hat{p}}_{jk}\le {\lambda}_e\), then the next dose combination is (j, k + 1), and if (j, k) = (J, K), then the current dose is maintained. Due to the symmetry between j and k, the same rules apply to k. Each p_{jk} follows the (noninformative) Jeffreys prior Beta (0.5, 0.5) distribution [26].
asBOINcomb design
To improve the flexibility and stability of BOINcomb in extreme scenarios, we adopt nonfixed boundaries and introduce sample size n_{j} to dynamically adjust the boundary such that it shrinks with increasing sample size. However, if only this aspect is considered, both dose escalation and deescalation would be more likely to occur in the later stages of the trial, potentially increasing the number of people treated at high doses. Therefore, we also use parameter t to further control the change trend of the boundary and ensure that the initial boundary is the same as in the BOIN design (when n_{j} is 1, ϕ_{1} and ϕ_{2} are the same as in the BOIN design). In the BOIN design framework, we reconstruct ϕ_{1} and ϕ_{2} as
where n_{j} is the cumulative number of patients treated at dose level j during the trial and t_{1},t_{2} > 0 are two acceleration factors to control the contraction rate of the two boundaries. Parameters ∆_{1} and ∆_{2} are the initial value of the prespecified boundaries, i.e., the initial boundary value when the first patient is enrolled in the trial. Clearly, the two fixed boundaries of the original BOIN design now depend on the dynamic number n_{j}, i.e., the number of patients treated at dose level j. This method of construction would clearly make ϕ_{1} and ϕ_{2} converge to the MTDC target ϕ as n_{j} increases. This construction is also highly flexible for designing clinical trials. For example, to reduce toxicity, we can penalize the practice of assigning patients to dose levels above the MTDC by using discount parameters t_{1} > t_{2} so that the deescalation boundary shrinks more rapidly than the escalation boundary. Similar to the BOIN design, the optimal λ_{1j} and λ_{2j} minimize the decision error rate and can be derived as λ_{e}(Δ_{1}, n_{j}) and λ_{d}(Δ_{2}, n_{j}) with the redefined ϕ_{1} and ϕ_{2} plugged into (1) and (2), respectively:
where we set a parameter set controlling the boundary shrinkage with thousands of parameter combinations to ensure that the optimal design performance can be achieved in various clinical trial scenarios. The rest of the design rules are the same as in BOINcomb. In contrast to the BOINcomb design, since a dynamic boundary value is adopted in the asBOINcomb design, the dose decision is related not only to the number of patients treated at the current dose but also the total number of patients enrolled in the trial. Moreover, like the BOINcomb design, the asBOINcomb design also has the advantages of transparency and convenience, and the dose decision table can still be generated in advance according to the trial settings.
The parameter t_{1}and t_{2}, which control the boundaries, play an important role in a clinical trial. Its value can be determined through simulation and will influence the rate of dose escalation in the trial. When conducting a clinical trial, it is important to carefully select the value of t_{1}and t_{2} in order to ensure the safety and performance. Therefore, when conducting simulation studies to assess the risks of a trial, biostatisticians must collaborate more closely with clinical investigators to assess the risks of trials. In general, when setting the t parameters, to further enhance safety of the design, especially in oncology trials, we recommend to choose t_{1} > t_{2}, the trial will be more conservative in terms of dose escalation, and conversely, a fast rate of dose escalation may increase the toxicity of the trial.
To clarify our design, Table 1 provides examples of the values of (λ_{e} (n_{j}), λ_{d}(n_{j})) for target ϕ = 0.3.
Simulation studies
Determination of the boundary range
Due to the dynamic properties of the asBOINcomb boundaries, there may be a better choice than the original BOINcomb recommended boundaries. We conducted a simulation study for the specified values of the default boundaries. In addition to the default values recommended by the original BOINcomb design (∆_{1} = 0.6ϕ, ∆_{2} = 1.4ϕ), we considered eight different sets of values (Table 2).
We set DLT = 0.3 and assigned ten different dosetoxicity scenarios (Table 3), setting 5 × 3 dose levels for the trial with a maximum sample size of 60 and an enrollment cohort size of 3. Each scenario was simulated 1000 times. The following metrics were assessed:

(1)
The percentage of simulated clinical trials in which the correct dose combination was selected as the MTDC (PCS).

(2)
The percentages of patients at the MTDC (PNMTDC).

(3)
Average number of DLTs (NDLTS).
According to Fig. 3, the first 7 groups of boundaries tend to outperform the last two groups of boundaries in most scenarios when evaluated using the PCS metric. In particular, the first group of boundaries in scenario 4 exhibits a 7.3% higher PCS value compared to the original recommended boundary. Overall, the PCS values of the first 7 groups of boundaries are comparable.
According to Fig. 4, the first 7 groups of boundaries tend to be better than the last two groups of boundaries in most scenarios when evaluated using the PNMTDC metric. Specifically, the first group of boundaries in scenario 4 exhibits a 13% higher PNMTDC value compared to the original recommended boundaries. Among the first five groups of boundaries, the boundary of group 3 performs particularly well.
Figure 5 indicates that the latter two groups of boundaries often have smaller DLTs, but this is due to a significant decrease in PCS and PNMTD. As a result, these groups are not selected. In general, the first three boundary groups perform better in terms of DLT control and outperform the original recommended boundary group (group 6) in most scenarios. Overall, the design performance under the first three groups of boundaries is better. These boundaries provide more adjustment space for dose escalation and deescalation, allowing for greater flexibility in dosing and the potential to explore the most suitable method for the current scenario. Additionally, all three indicators PCS, PNMTDC and DLTs are more stable under the boundary of group 3. Therefore, ∆_{1} = 0.3ϕ and ∆_{2} = 1.7ϕ are recommended.
Simulation studies
After determining the optimal bounding range (∆_{1} = 0.3ϕ, ∆_{2} = 1.7ϕ), we conducted a simulation study of the performance of asBOINcomb and BOINcomb at multiple sample sizes (from 30 to 60 step by 3) using the settings in Section 3.1. The BOINcomb design was implemented using the R package BOIN and with the same bounding range. Each trial continued until the sample was exhausted, unless early termination occurred due to excessive toxicity at the lowest dose combination.
According to Fig. 6, asBOINcomb outperforms BOINcomb when the sample size is 30–60 for all 10 scenarios and converges faster, especially in some extreme scenarios (scenarios 4, 5, 9). For example, in scenario 4, the PCS of asBOINcomb with a sample size of 39 is comparable to that of BOINcomb with a sample size of 51. This demonstrates that asBOINcomb can achieve comparable accuracy to BOINcomb with a smaller sample size, which is beneficial in a clinical context.
According to Fig. 7, asBOINcomb is generally comparable to BOINcomb in terms of PNMTDC when the sample size is between 30 and 60. However, in certain extreme scenarios (scenarios 4, 5, and 9), asBOINcomb performs better. For example, in scenario 4, asBOINcomb exhibits a 8–10% higher PNMTDC than BOINcomb at sample sizes between 33 and 39.
According to Fig. 8, asBOINcomb is generally comparable to BOINcomb in terms of DLTs when the sample size is between 30 and 60. In scenario 2, asBOINcomb exhibits improved DLTs at the cost of a partial decrease in PNMTDC.
Combining the three metrics of PCS, PNMTDC, and DLTs, asBOINcomb has better accuracy than BOINcomb, especially in some extreme scenarios, or equivalently, asBOINcomb can achieve performance comparable to BOINcomb with a smaller sample size.
Trial example
To further evaluate the proposed strategy, we applied the proposed design to a clinical trial of PF03084014 combined with doxorubicin for the treatment of advanced triplenegative breast cancer (TNBC). PF03084014 is a reversible, noncompetitive, and selective secretase inhibitor that blocks the NOTCH signaling pathway. In preclinical studies, this combination has demonstrated anticancer efficacy in solid tumor models, such as advanced thyroid cancer and sclerofibrosarcoma, and in Tcell acute lymphoblastic leukemia. In TNBC patientderived and cell lineage xenograft models, the combination of PF03084014 with doxorubicin greatly improved the inhibition of tumor growth [27,28,29,30,31].
In this phase I study (A8641016), patients with advanced TNBC were evaluated for safety, tolerability, pharmacokinetics, and antitumor efficacy. The goal of the study was to determine the MTDC of the drug PF03084014 when combined with the chemotherapy drug doxorubicin. The dose exploration component used a modified probability interval technique based on a 2 × 3 matrix design to evaluate toxicity. On day 1 of each 21day cycle, oral PF03084014 was coadministered with intravenous doxorubicin twice a day.
To redesign the trial, we fitted logistic regression models to the data observed in this PF03084014 with doxorubicin combination trial. Table 4 illustrates the DLT generated using the estimated probability of toxicity for various combinations of PF03084014 with doxorubicin. This probability reflects a clinician’s estimation of the toxicity of the drug combination.
The target toxicity probability was set to 0.33, the maximum sample size was 30 patients, the cohort size was 3, and the first group of patients was treated with the lowest dose combination (1, 1). After the parameter set simulation, the optimal parameters t_{1} = 300 and t_{2} = 1 were selected as the boundary control conditions for the trial. Figure 9 shows the dose allocation path for the subsequent cohort.
As shown in Fig. 9, asBOINcomb can quickly locate the MTDC and treat most patients with the correct dose combination. The estimated toxicity probability matrix at the end of the trial is shown in Fig. 10.
“” represents a dose combination that was not administered in the trial. The dose combination (2, 2) is selected as the MTDC. As shown in Figs. 9 and 10, the dose levels administered converged to the MTDC starting with 6 sequence of enrollment patients, indicating that the asBOINcomb design can rapidly target the MTDC and treat most patients (60%) with the correct dose combination, while also preventing any patients from being treated at doses above the MTDC. This minimizes the risk of DLT and ensures the safety of the trial.
Discussion
From statistical and clinical viewpoints, the proposed combination designs are simple and easy to understand. But there are some practical or ethical issues that have not been considered in actual trials. The above design aims to find individual MTDC in the drug combination trial and is not suitable for finding MTDC profiles. This is a topic of our future research.
Conclusion
We improve the twodimensional Bayesian optimal interval design by proposing adaptivelyshrinking dose escalation/deescalation boundaries, which converge with increasing sample size. In addition, we introduce the parameter t_{1} and t_{2} to control the variation trend of the dose escalation/deescalation boundaries and establish a parameter set to achieve optimal performance. Our simulation results demonstrate that the improved BOINcomb design has better accuracy and stability than the original design, especially in some extreme scenarios. This is largely due to the controllable dynamic boundary which allows for different rates of shrinking for the escalation and deescalation boundaries. For instance, in highly toxic scenarios, the deescalation boundary can be set to shrink at a slightly faster rate than the escalation boundary, as stricter or smaller boundaries reduce the risk of exposing subjects to overtoxic doses. The improved accuracy of asBOINcomb means that it can achieve comparable performance to BOINcomb with a smaller sample size, which is highly beneficial in clinical trials. Additionally, the asBOINcomb design allows clinical trial investigators to make decisions based on decision tables, while maintaining the same flexibility and transparency as the BOINcomb design. In conclusion, the proposed design provides a new reference method for Phase I dose finding in subsequent drug combination clinical trials.
Availability of data and materials
All data generated or analyzed during this study are included in this published article.
Abbreviations
 asBOINcomb:

Adaptive shrinking Bayesian optimal interval design for combination drugs
 BOINcomb:

Bayesian optimal interval design for combination drugs
 CRM:

Continuous reassessment method
 DLT:

Doselimiting toxicity
 MTD:

Maximum tolerated dose
 MTDC:

Maximum tolerated dose combination
 NDLTS:

Average number of DLTs
 PCS:

Percentage of correct selections
 PNMTDC:

Percentages of patients at the MTDC
 TNBC:

Triplenegative breast cancer
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Acknowledgments
The authors appreciate Dr. Zuojing Li from Shenyang Pharmaceutical University for scientific editing of this manuscript.
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This work was supported in part by the National Natural Science Foundation of China (No.41310164).
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LZH was involved in the conception and design of the article, LZJ was involved in the supervision and critical review and editing of the manuscript, and ZAJ, XZ, and QGP were involved in discussions about the content of the manuscript. The author(s) read and approved the final manuscript.
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Li, Z., Xu, Z., Zhang, A. et al. A clinical phase I dosefinding design with adaptive shrinking boundaries for drug combination trials. BMC Med Res Methodol 23, 57 (2023). https://doi.org/10.1186/s1287402301867y
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DOI: https://doi.org/10.1186/s1287402301867y
Keywords
 Dose escalation
 Modelassisted design
 Drug combination
 Bayesian adaptive dosefinding design
 Phase I clinical trial design