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We introduce a Bayesian optimization method for estimating the maximum tolerated dose in this article. A number of parametric model-based methods have been proposed to estimate the maximum tolerated dose; however, parametric model-based methods need an assumption that dose–toxicity relationships follow specific theoretical models. This assumption potentially leads to suboptimal dose selections if the dose–toxicity curve is misspecified. Our proposed method is based on a Bayesian optimization framework for finding a global optimizer of unknown functions that are expensive to evaluate while using very few function evaluations. It models dose–toxicity relationships with a nonparametric model; therefore, a more flexible estimation can be realized compared with existing parametric model-based methods. Also, most existing methods rely on point estimates of dose–toxicity curves in their dose selections. In contrast, our proposed method exploits a probabilistic model for an unknown function to determine the next dose candidate without ignoring the uncertainty of posterior while imposing some dose-escalation limitations. We investigate the operating characteristics of our proposed method by comparing them with those of the Bayesian-based continual reassessment method and two different nonparametric methods. Simulation results suggest that our proposed method works successfully in terms of selections of the maximum tolerated dose correctly and safe dose allocations.

Background: Long-acting reversible contraceptives, including hormonal levonorgestrel-releasing intrauterine systems, are the most effective methods of reversible contraception. However, unfavorable bleeding, particularly during the first months of use, is one of the most important reasons for discontinuation or avoidance. Minimizing this as early as possible would be highly beneficial. Nonsteroidal anti-inflammatory drugs inhibiting prostaglandin synthesis are known to reduce bleeding and pain at time of menses. A levonorgestrel-releasing intrauterine system has been developed with an additional reservoir containing indomethacin, designed to be released during the initial postplacement period., Objective: This proof-of-concept study aimed to establish whether the addition of indomethacin to the currently available levonorgestrel-releasing intrauterine system (average in vivo levonorgestrel release rate of 8 μg/24 h during the first year of use) reduces the number of bleeding and spotting days during the first 90 days of use compared with the unmodified system. The dose-finding analysis included 3 doses of indomethacin-low (6.5 mg), middle (12.5 mg), and high (15.4 mg)-to determine the ideal dose of indomethacin to reduce bleeding and spotting days with minimal side-effects., Study Design: This was a multicenter, single-blinded, randomized, controlled phase II trial conducted between June 2018 and June 2019 at 6 centers in Europe. Three indomethacin dose-ranging treatment groups (low-, middle-, and high-dose indomethacin/levonorgestrel-releasing intrauterine system) were compared with the unmodified levonorgestrel-releasing intrauterine system group, with participants randomized in a 1:1:1:1 ratio. The primary outcome was the number of uterine bleeding and spotting days over a 90-day reference (treatment) period. Secondary outcomes were the number of women showing endometrial histology expected for intrauterine levonorgestrel application and the frequency of treatment-emergent adverse events. Point estimates and 2-sided 90% credible intervals were calculated for mean and median differences between treatment groups and the levonorgestrel-releasing intrauterine system without indomethacin. Point and interval estimates were determined using a Bayesian analysis., Results: A total of 174 healthy, premenopausal women, aged 18 to 45 years, were randomized, with 160 women eligible for the per-protocol analysis set. Fewer bleeding and spotting days were observed in the 90-day reference period for the 3 indomethacin/levonorgestrel-releasing intrauterine system dose groups than for the levonorgestrel-releasing intrauterine system without indomethacin group. The largest reduction in bleeding and spotting days was achieved with low-dose indomethacin/levonorgestrel-releasing intrauterine system, which demonstrated a point estimate difference of -32% (90% credible interval, -45% to -19%) compared with levonorgestrel-releasing intrauterine system without indomethacin. Differences for high- and middle-dose indomethacin/levonorgestrel-releasing intrauterine system groups relative to levonorgestrel-releasing intrauterine system without indomethacin were -19% and -16%, respectively. Overall, 97 women (58.1%) experienced a treatment-emergent adverse event considered related to the study drug, with similar incidence across all treatment groups including the unmodified levonorgestrel-releasing intrauterine system. These were all mild or moderate in intensity, with 6 leading to discontinuation. Endometrial biopsy findings were consistent with effects expected for the levonorgestrel-releasing intrauterine system., Conclusion: All 3 doses of indomethacin substantially reduced the number of bleeding and spotting days in the first 90 days after placement of the levonorgestrel-releasing intrauterine system, thus providing proof of concept. Adding indomethacin to the levonorgestrel-releasing intrauterine system can reduce the number of bleeding and spotting days in the initial 90 days postplacement, without affecting the safety profile, and potentially improving patient acceptability and satisfaction., (Copyright © 2022 Elsevier Inc. All rights reserved.)

We use optimal design theory and construct locally optimal designs based on the maximum quasi-likelihood estimator (MqLE), which is derived under less stringent conditions than those required for the MLE method. We show that the proposed locally optimal designs are asymptotically as efficient as those based on the MLE when the error distribution is from an exponential family, and they perform just as well or better than optimal designs based on any other asymptotically linear unbiased estimators such as the least square estimator (LSE). In addition, we show current algorithms for finding optimal designs can be directly used to find optimal designs based on the MqLE. As an illustrative application, we construct a variety of locally optimal designs based on the MqLE for the 4-parameter logistic (4PL) model and study their robustness properties to misspecifications in the model using asymptotic relative efficiency. The results suggest that optimal designs based on the MqLE can be easily generated and they are quite robust to mis-specification in the probability distribution of the responses.