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.