A constrained optimum adaptive design for dose finding in early phase clinical trials.

Recently, interest has grown in the development of dose-finding methods that consider both toxicity and efficacy as endpoints. Along with responses on these, the incorporation of pharmacokinetic (PK) data can be beneficial in terms of patients' safety and can also increase the efficiency of the design for finding the best dose for the next phase. In this paper, the maximum concentration ($C_{max}$) is used as the PK measure guiding the dose selection. The ethically attractive approach, which is based on the probability of efficacy, is used as a dose optimisation criterion. At each stage of an adaptive trial, that dose is selected for which the criterion is maximised, subject to the constraints imposed on the $C_{max}$ and the probability of toxicity. The inter-patient variability of the PK model parameters is considered, and population $D$-optimal sampling time points for measuring the concentration of a drug in the blood are calculated. The method is illustrated with a one-compartment PK model with first-order absorption, with the parameters being assumed to be random. The Cox model for bivariate binary responses is employed to model the dose-response outcomes. The results of a simulation study for several plausible dose-response scenarios show a significant gain in the efficiency of the design, as well as a reduction in the proportion of toxic responses.