Early phase, personalized dose-finding trials for combination therapies seek to identify patient-specific optimal biological dose (OBD) combinations, which are defined as safe dose combinations that maximize therapeutic benefit for a specific covariate pattern. Given the small sample sizes which are typical of these trials, it is challenging for traditional parametric approaches to identify OBD combinations across multiple dosing agents and covariate patterns. To address these challenges, we propose a Bayesian optimization approach to dose-finding which incorporates efficacy and toxicity information into the sequential search strategy. Independent Gaussian processes are used to model the efficacy and toxicity surfaces, and an acquisition function is utilized to define the dose-finding strategy. Furthermore, we define an adaptive stopping rule using the posterior entropy for the location of the OBD. This work is motivated by a personalized dose-finding trial which considers a dual-agent therapy for obstructive sleep apnea (OSA), where OBD combinations are tailored to OSA severity. Via a simulation study, the approach is first investigated across varying degrees of response heterogeneity for both efficacy and toxicity, and then a collection of final designs for the OSA trial are compared. We demonstrate that the proposed approach toward personalized dose-finding yields good performance under the considered scenarios.

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