Dose-Finding and Phase I/II Designs
Plans the sample size for a seamless early-phase program by combining a fixed Phase I dose-escalation allowance with a Simon two-stage Phase II design. The Phase II component can be optimized for minimum expected sample size under the null or for minimum maximum sample size.
A seamless Phase I/II design combines the objectives normally addressed in separate early-development studies. In the planning framework used here, Phase I contributes a prespecified dose-escalation patient allowance, while the Phase II component uses Simon's two-stage single-arm design to determine the efficacy-evaluation sample size. Seamless early-phase designs can combine dose escalation with subsequent dose expansion or randomized dose evaluation, but the exact statistical operating characteristics depend on the specific dose-selection and expansion rules. The calculation here therefore makes those assumptions explicit rather than treating "seamless Phase I/II" as one universal formula.
The Phase I planning allowance is calculated as:
where J is the number of planned dose levels and c is the planned cohort size per dose level. This is a fixed planning allowance. Actual Phase I enrollment can differ when the dose-escalation design stops early, expands a cohort, skips doses, or otherwise changes the allocation.
Simon's method tests a null response probability H0: p ≤ p0 against an alternative p ≥ p1, with prespecified Type I error α and Type II error β. For candidate Stage 1 size n1, Stage 2 size n2, Stage 1 boundary r1, and final boundary r, the probability of rejecting an insufficiently active treatment is evaluated using binomial probabilities.
The optimal design searches the admissible integer designs and minimizes the expected Phase II sample size when p = p0. The minimax design instead minimizes the maximum Phase II sample size. Both criteria must satisfy the requested α and β constraints.
The displayed expected total therefore assumes that the Phase I dose-escalation allowance is fixed and that the Phase II component contributes the Simon expected sample size. It should not be interpreted as a universal operating-characteristic calculation for every possible seamless Phase I/II design. Designs that formally reuse Phase I patients in the Phase II inferential analysis, use Bayesian dose selection, incorporate randomized controls, or simultaneously model toxicity and efficacy require their own prespecified statistical model and usually simulation-based operating-characteristic evaluation.
With 4 Phase I dose levels and cohorts of 3 patients, the Phase I allowance is 12 patients. For p0 = 0.05, p1 = 0.20, α = 0.10, and β = 0.10, exhaustive enumeration of Simon designs gives the optimal Phase II design n1 = 12, n2 = 25, with r1 = 0 and r = 3. Its maximum Phase II sample size is 37 and its expected Phase II sample size under p0 is approximately 23.49. Consequently, the corresponding seamless planning values are a maximum total of 49 patients and an expected total of approximately 35.49 patients under p0.
Simon, R. (1989). Optimal two-stage designs for phase II clinical trials. Controlled Clinical Trials, 10(1), 1–10. DOI: 10.1016/0197-2456(89)90015-9.
Pan, H., Xie, F., Liu, P., Xia, J., & Ji, Y. (2014). A phase I/II seamless dose escalation/expansion with adaptive randomization scheme (SEARS). Clinical Trials, 11(1), 49–59. DOI: 10.1177/1740774513500081.
Hobbs, B. P., et al. (2019). Seamless Designs: Current Practice and Considerations for Early-Phase Drug Development in Oncology. Journal of the National Cancer Institute, 111(2), 118–127.