Optimal designs for efficacy-toxicity response in dose finding studies using the bivariate probit model

Phase I clinical trials are the first-in-human studies that primarily focus on the safety profile of drugs. Traditionally, the primary aim of a phase I clinical trial is to establish the maximum tolerated dose and characterize the toxicity profile of the tested agents. As a secondary aim, some phase I studies also include studies to obtain preliminary efficacy information about the experimental agents. In our research, we consider the optimal design of experiments in extended phase I clinical trials where both efficacy and toxicity are measured and the maximum tolerated dose has been established. We represent the response of both outcomes using a bivariate probit model for correlated responses and propose systematic numerical approaches based on Semidefinite Programming to address the problem. We construct locally optimal experimental designs for the following situations: (i) responses with efficacy and toxicity strongly correlated versus non-correlated, by varying the correlation parameter; (ii) a priori known correlation versus unknown correlation; (iii) unconstrained versus constrained designs, where the constraints represent safety limits, budget constraints and probability bounds; (iv) single versus combined drugs. Additionally, we consider four distinct optimality criteria: D-, A-, E-, and K-optimality. Our methodologies are extensively tested, and we demonstrate the optimality of the designs using equivalence theorems. To enrich our analysis, an equivalence theorem for the K-optimality criterion is derived.