In oncology phase I/II clinical trials, the primary objective is to identify the maximum tolerable or assessed dose (MTD/MAD) for new therapies and to evaluate their preliminary efficacy. These trials often employ dose escalation/de-escalation algorithms, using dose-limiting toxicity events and available safety and pharmacokinetic information to identify tolerable and therapeutically effective doses. Various methods are available for the optimal dose search, including the Bayesian optimal interval (BOIN) design for Phase I clinical trials [8], and the modified toxicity probability interval (mTPI-2) method [5]. However, these methods primarily focus on safety and may not be able to adequately evaluate the therapeutic efficacy due to the constrained sample sizes at each dose level. Typically, trials employing these designs proceed to the dose-expansion phase once the MTD/MAD dose is identified.

In trials where only one dose is identified for expansion and further efficacy assessment in phase II, Simon’s two-stage design [10] is frequently used. In this design, a dose may be terminated at the first stage if the anti-tumor activity is insufficient or it may proceed to the second stage for further evaluation with more subjects. However, although the identified dose level, often the MTD/MAD, exhibits anti-tumor activities, it may not optimize the benefit-risk ratio, which may hinder successful development in the later phase. Hence, finding a dose level that optimizes the balance between therapeutic benefits and safety risks in the early phases is paramount for the overall success of therapy development. In recognition of this importance, the FDA issued guidance on optimizing the dosage for developing oncology therapy [12]. In the guidance “Expansion Cohorts: Use in First-in-Human Clinical Trials to Expedite Development of Oncology Drugs and Biologics Guidance for Industry” [11], the FDA advises developing statistical analysis plans that encompass not only the justification of the maximum sample size but also stopping rules for lack of activity to minimize subject exposure to ineffective treatments. Furthermore, the FDA’s guidance also encouraged exploring randomized designs for multiple doses/regimens, including justification of sample size in the early phases of therapy development.

In response to the increasing need for new statistical methodologies that explore dose-response relationships in early-phase oncology therapy development, we propose a design method that extends Simon’s single-arm two-stage design. Many modifications of Simon’s design exist in the literature. For example, Chen expanded the two-stage framework into three stages, reducing the expected sample size when the treatment is ineffective [3], while Lin and Shih introduced adaptive extensions that allow modifications based on interim results, improving flexibility in decision-making [7]. Whitehead adapted Simon’s methodology for trials with survival endpoints, broadening its applicability to time-to-event outcomes [13]. Other methods also explored balancing groups to optimize trial efficiency and minimize bias in two-stage designs. For instance, Ye and Shyr propose balanced two-stage designs that maintain equal allocation across groups, improving trial efficiency in oncology Phase II studies [14]. Parashar introduce stratified designs that extend Simon’s two-stage methodology to stratified patient groups, optimizing the assessment of heterogeneity across subgroups [9]. Methods were also developed to enhance Simon’s design for specific trial contexts, such as incorporating sequential monitoring and decision-making processes. Lee proposed sequential dose-finding designs that optimize patient safety by continuously evaluating efficacy and toxicity [6]. Bartroff introduced group-sequential methodologies that integrate dose-escalation strategies with efficacy evaluations, enhancing the applicability of two-stage designs in dose-response contexts [1]. Further advancements include Bayesian methods and meta-analytic methods. For example, Zohar and Chevret presented a Bayesian two-stage design for dose-ranging trials, refining dose selection through adaptive decision rules [15]. Crippa introduced meta-analytic methods for synthesizing dose-response data, which can be integrated into sequential trial designs for more comprehensive analysis [4].

Despite their advancements, existing methods are limited to evaluating a single dose and do not explore the dose-response relationships of multiple doses. In contrast, we propose to extend Simon’s design to evaluate two doses and derive decision rules to identify better doses between two doses with optimal sample sizes. The decision rules allow early termination of the study at Stage I if none of the doses show sufficient anti-tumor potential or if at least one dose exhibits optimal anti-tumor activities. The decision rules also allow the selection of one dose for further evaluation in the second stage if the anti-tumor potential is promising. The paper is organized into the following sections: an overview of Simon’s two-stage design is provided in Section 2; the new method is introduced in Section 3 along with the calculation of the decision probabilities; Section 4 presents an enumeration algorithm for sample size calculation leading to the optimal and minimax design under the constraints of overall type I error and power; and examples of the application of the new method are illustrated in Section 5.

In Simon’s two-stage design, the null and alternative hypotheses are specified as ${H_{0}}:\theta \le {\theta _{0}}$, ${H_{1}}:\theta \ge {\theta _{A}}$, where θ represents a response rate for anti-tumor activities such as objective response rates (ORR). The null hypothesis implies that response rates equal to ${\theta _{0}}$ or lower are considered ineffective whereas the alternative corresponds to the parameter space that has treatment effect. A study design may consider powering certain desired response rates $\theta \ge {\theta _{A}}$. Stage I of Simon’s design uses a small sample size to explore if there is any desired anti-tumor activity. The study may be terminated at Stage I if the activity is lower than expected. Otherwise, additional subjects will be enrolled in Stage II to further evaluate the anti-tumor activity. Denote the observed number of responses at Stage I and Stage II as ${S_{1}}$ and ${S_{2}}$, respectively, and let $S={S_{1}}+{S_{2}}$. The response thresholds for Stage I and Stage II are denoted as a and r, respectively, out of the sample size ${n_{1}}$ and ${n_{2}}$, the number of subjects enrolled at Stages I and II, respectively. At the end of the first stage, if ${S_{1}}\le a$, the therapy is considered ineffective and the study will be terminated in Stage I. Otherwise, the study enters into Stage II. If $S\gt r$ at the end of the second stage, the therapy is considered effective.

Simon’s design determines the efficient sample sizes ${n_{1}}$ and ${n_{2}}$ for the decision rule a and r by controlling type I error, α, and type II error, β, at the desired levels. Let $n={n_{1}}+{n_{2}}$ be the total number of subjects. The expected sample size is expressed as $\text{EN}={n_{1}}+(1-\text{PET}){n_{2}}$, where $\text{PET}$ is the probability of early termination. The optimal design is derived by enumerating n, ${n_{1}}$, a, r with exact binomial probabilities based on the given ${\theta _{0}}$, ${\theta _{A}}$, α, and β. While controlling the type I error and achieving the desired power, Simon introduced two designs for finding sample sizes: the optimal design, which minimizes the expected sample size EN, and the minimax design, which minimizes the maximum sample size n.

To control the maximum type I error in the null space $0\le {\theta _{1}}\le {\theta _{0}}$ and $0\le {\theta _{2}}\le {\theta _{0}}$, it is necessary to search the entire null space as the type I error function in 3.1 is not monotone. To enhance the searching speed for the maximum type I error, we divided the search algorithm into two parts. The first part grids the partial sample size and decision rule, ${n_{1}}$ and ${r_{1}}$, in the null space $0\le {\theta _{1}}\le {\theta _{0}}$ and $0\le {\theta _{2}}\le {\theta _{0}}$ by an interval of 0.02. The partial type I error, $1-Bin({r_{1}}|{\theta _{1}},{n_{1}})Bin({r_{1}}|{\theta _{2}},{n_{1}})$, is calculated for ${n_{1}}$ and ${r_{1}}$. Note that ${n_{1}}$ is the sample size of each dose during the first stage, which is usually small, therefore, ${n_{1}}$ is limited to less than 50 subjects. The choice of ${r_{1}}$ is bound to be less or equal to ${n_{1}}$. In addition, only the combinations of ${n_{1}}$ and ${r_{1}}$ that control the maximum partial type I error in the null space are kept for the next part of search that completes the derivation of the full decision rules and sample size. The first step eliminates a significant portion of the ${n_{1}}$ and ${r_{1}}$ combinations and therefore enhances the efficiency of the enumeration process to identify the decision rules in the second step.

The second step of the algorithm iterates over the selected set of $({n_{1}},{r_{1}})$ from the first step to identify the combinations of $({n_{2}},{a_{1}},r)$ that satisfy the constraints: maximum type I error less than α and power greater than $1-\beta $. To ensure the maximum type I error is less than α for each set of decision rules and sample sizes $({n_{1}},{r_{1}},{n_{2}},{a_{1}},r)$, type I errors are calculated over the null space: ${\theta _{10}}\in (0,{\theta _{0}})$ and ${\theta _{20}}\in (0,{\theta _{0}})$ by a step of 0.01. Decision rules with the overall type I error greater than α are eliminated. The selected decision rules require the control of the maximum individual type I errors to be less than α for each individual dose. As a result, the type I errors are strongly controlled.

Designs are not considered adequate if ${a_{1}}$ and ${r_{1}}$ are too close to each other, for example, only 1 response apart. Such designs terminate the study at Stage I either for futility or efficacy without leave much room for the grey zone (the response is between ${a_{1}}$ and ${r_{1}}$), which would warrants further evaluation in Stage II. To ensure that designs don’t make a black-or-white decision but allow the dose to move on to Stage II when responses are in between the two termination criteria, additional rules, such as ${r_{1}}\gt {a_{1}}+2$, are imposed in design selections. In addition, when the sample size assigned to the first stage ${n_{1}}$ is too small relative to the total number of sample size n, such designs will not be adequate for dose-response evaluation in the first stage. Therefore, further restrictions, such as $2{n_{1}}\ge {n_{2}}\ge {n_{1}}/2$, are also placed in the derivation algorithm. For the decision rule in Stage II, r, it is natural to require $r\gt {r_{1}}$.

Each study design will include sample size ${n_{1}}$ and n as well as decision rules ${a_{1}},{r_{1}},r$ and satisfy both the type I error and power constraints. The expected sample size $\text{EN}$ can be calculated as $\text{EN}=2{n_{1}}+(1-\text{PET}){n_{2}}$. Among all decision rules that satisfy the constraints, the decision rule that yields the minimum total sample size will be selected as the minimax design and the decision rule that yields the minimum EN is the optimal design. Figure 3 illustrates the algorithm of decision rules. Here, $\text{EN}$ can be calculated under null or alternative, denoted as ${\text{EN}_{1}}$ and ${\text{EN}_{2}}$, respectively. In addition, an average ${\text{EN}_{Avg}}$ can also be calculated as the average of ${\text{EN}_{1}}$ and ${\text{EN}_{2}}$.

This section illustrates the results using the proposed algorithm and the selection of efficient designs that are considered optimal. Various design parameters ${\theta _{0}}$ and ${\theta _{A}}$ are used as examples. The type I error is controlled at 1-sided 0.05 and the required power is 80%. For the optimal design, the average expected sample size ${\text{EN}_{Avg}}$ is used to select the design. The optimal designs are selected from all designs that satisfy the constraints of Type I error and the required power. The corresponding design selected from the algorithm for each set of design parameters are listed in Appendix D.

Tables 1 shows the results of the optimal and minimax design when powering at ${\theta _{1}}={\theta _{A}}$ and ${\theta _{2}}={\theta _{A}}$, the same targeting anti-tumor activities for both doses, with relative treatment differences ${\theta _{A}}-{\theta _{0}}$ of either 0.3 or 0.2. Table 1 includes the total sample size, n, the sample size of the first stage ${n_{1}}$, the sample size of the second stage ${n_{2}}$, the termination rules ${a_{1}}$ and ${r_{1}}$ for responses, as well as the total responses for claiming efficacy from Stage II, r, the probability of early termination ${\text{PET}_{Avg}}$ at Stage I, and the expected sample size ${\text{EN}_{Avg}}$. The study designs presented in Table 1 power the alternative points at the minimum of 80% and control the type I error at the level of 1-sided 0.05.

Notice that in Table 1, when ${\theta _{A}}-{\theta _{0}}=0.3$, the best design with a minimal total sample size, the minimax design, and the minimal expected sample size, the optimal design, for most design scenarios are the same. For the scenario where ${\theta _{A}}=0.5$ and ${\theta _{0}}=0.2$, the first stage consists of ${n_{1}}=6$ subjects for both minimax and optimal design. If responses are no more than 1 (${a_{1}}=1$) in both doses, indicating that the compound in both doses does not exhibit sufficient anti-tumor activity, then the trial can be terminated for futility. On the other hand, if there are more than 4 (${r_{1}}=4$) responses in one or both doses, the corresponding doses are considered efficacious and the trial can be terminated after the first stage for efficacy. Otherwise, if the response is in between the two thresholds ${a_{1}}$ and ${r_{1}}$, the dose with a better response rate or the lower dose when both doses have the same response rate is moved to the second stage, and the trial continues to accrual more subjects up to ${n_{2}}=8$ in Stage II. The dose is considered efficacious at the end of Stage II if there are total $r=6$ responses from both stages out of ${n_{1}}+{n_{2}}=14$ subjects. With this design, the total sample size is $2{n_{1}}+{n_{2}}=20$ and the average sample size is ${\text{EN}_{Avg}}=16$ with ${\text{PET}_{Avg}}=0.52$.

It is also observed in Table 1 that as expected, when the response differences ${\theta _{A}}-{\theta _{0}}$ become smaller, larger total and expected sample sizes are needed to meet the desired power and type I error. In design scenarios with a low threshold of ${\theta _{0}}$, for example, ${\theta _{0}}=0.2$, the study will terminate for failure in the first stage with relatively lower ${a_{1}}$ and ${r_{1}}$ as opposed to larger ${\theta _{0}}$. Such design makes sense as lower ${\theta _{0}}$ may indicate diseases that have no good treatments, therefore a treatment with a lower response rate ${r_{1}}$ will be acceptable and the study will be terminated for futility if the treatment response is very low (with a lower threshold ${a_{1}}$).

Table 2 shows the results of the minimax and optimal design when powering the regions ${\theta _{1}}={\theta _{A}}$ or ${\theta _{2}}={\theta _{A}}$. In those designs, the studies are powered when either dose or both doses have response rates ${\theta _{A}}$. As a result of this powering strategy, the overall sample size has increased by close to 50% in comparison to the design that powers the alternative ${\theta _{1}}={\theta _{A}}$ and ${\theta _{2}}={\theta _{A}}$. Similar to Table 1 for treatment difference ${\theta _{A}}-{\theta _{0}}=0.3$, the optimal and minimax designs are the same design.

It is interesting to observe from both Tables 1 and 2 that even though the maximum total sample sizes are similar in both optimal and the minimax designs, the expected sample sizes and the decision rules can be very different between the two. In the optimal design, the sample size allocated in the first stage is smaller than the minimax design, and ${a_{1}}$, the threshold to terminate the study to accept the null, is lower than that of the minimax design in general. Because of the lower sample size in the first stage, the threshold of terminating the study for efficacy, ${r_{1}}$, is also lower than that of the minimax design. Overall, when the two designs are different, the minimax designs invest more subjects in the first stage and has a higher probability to terminate under null and some cases under alternative, relative to the optimal design.

In the evolving landscape of anti-cancer treatments, such as the emergence of high response rate treatments like cell and immuno-oncology therapies, the need for adaptive trial designs that can dynamically explore dose-response relationships is expanding. Therefore, in this paper, our goal is to propose a two-dose two-stage design, which extends on the popular traditional two-stage design proposed by Simon in 1989, that is more suitable for the modern advancements of anti-cancer treatments and more flexible in exploring the dose-response relationship in early-phase oncology treatment development.

In the proposed method, the flexibility of implementing the decision rules enables the design to accommodate the risk-benefit profile evaluation for two doses. Such adaptability is a key strength of our proposed method, providing a framework that aligns with the comprehensive nature of early-phase oncology therapy development.

In the proposed design, we acknowledge the potential high response rates of modern oncology therapy by incorporating early termination for efficacy in the first stage. This approach enables further sample size reduction and is aligned with the treatment capabilities of recent medicine compared to Simon’s original design, which only included early termination for futility. Moreover, we introduce decision rules in the proposed method which enables our design to be flexible under different therapeutic outcomes when identifying the superior dose that optimizes the balance between therapeutic benefits and safety risks. By controlling type I error rigorously in the algorithm, our design also provides high confidence for the anti-tumor activities of the superior dose that the design identifies for the next phases of development. The numerical examples also illustrate the flexibility of our design in its application under varying scenarios.

While our method primarily centers around a single efficacy endpoint, ORR, we acknowledge that the evaluation of anti-tumor activities often encompasses a broader spectrum of metrics. Evidence for anti-tumor activities may involve multiple metrics such as the overall and partial response, response durability, overall survival, and relevant biomarkers. However, at the design stage, it is important to focus on a single parameter, which is different from the totality of evidence for treatment evaluation.

We recognize the limitation that the proposed method only selects one dose from the first stage among the two doses evaluated. In situations where there are no clear differences between the two doses in risk-benefit profiles in all available data, the lower dose is recommended moving forward to the next stage. In such situations, there might be reasons to expect a higher efficacy from the higher doses so that both doses will be moved to the next stage. Further research will be required for such study designs so that the optimal sample size can be derived by controlling the type I error and achieving adequate power.

Moreover, although it is not common to establish quantitative targeting criteria for safety performance except for certain well-characterized safety endpoints such as dose-limiting toxicities at the design stage, the safety profiles such as serious adverse events, the pharmacokinetic parameters, and a comprehensive evaluation of the frequency, severity, duration, and the manageability of adverse events are also important factors to consider for dose selection. The benefit and risk ratio should also assessed to ensure that the gain in efficacy for an increased dose will result in no substantial loss in safety.