Dose-finding is an important part of the clinical development of a new drug. The purpose of dose-finding studies is to determine a suitable dose for future development based on both efficacy and safety. Optimal experimental designs have already been used to determine the design of this kind of studies, however, often that design is focused on efficacy only. We consider an efficacy-safety model, which is a simplified version of the bivariate Emax model. We use here the clinical utility index concept, which provides the desirable balance between efficacy and safety. By maximizing the utility of the patients, we get the estimated dose. This desire leads us to locally c-optimal designs. An algebraic solution for c-optimal designs is determined for arbitrary c vectors using a multivariate version of Elfving's method. The solution shows that the expected therapeutic index of the drug is a key quantity determining both the number of doses, the doses itself, and their weights in the optimal design. A sequential design is proposed to solve the complication of parameter dependency, and it is illustrated in a simulation study.

The knowledge of how good or optimal designs look like is essential for dose-finding trials. In many cases, dose-finding trials consider both efficacy and safety. We analyse, therefore, a bivariate model for these two outcomes. In contrast to earlier research, we consider a model also having placebo effects to see the impact on the optimal design. We calculate D-optimal designs algebraically and numerically. We see that one more design point is necessary, but that otherwise, the optimal design has a similar structure compared to the model without placebo effects. We confirm that the drug's therapeutic index has a significant impact on the shape of the optimal design.

Optimal experimental designs are an essential part of clinical development of a drug and are used to determine the design of dose-finding studies. However, often, these designs for dose-finding aim on one drug. We consider an efficacy Emax model for the combination of two drugs. The interaction is characterized as synergy if it is positive, antagonism when it is negative and additive when there is no interaction between the two drugs. We calculate D-optimal designs algebraically and numerically. The solutions show that the number of doses, the doses itself, and their weights in the D-optimal design depend on the parameter values in the model, ED₅₀^x, ED₅₀^y and the interaction term γ.

Much of traditional optimal design theory relies on specifying a model with only a small number of parameters. In many applications, such models will give reasonable approximations. However, they will often be found not to be entirely correct when enough data are at hand. We consider a low-dimensional model with a distortion term. Our objective is to estimate the combined model, including the distortion. In our situation, the low-dimensional model can be viewed as a fixed effect and the distortion term as a random effect in a mixed-effects model. Since we are interested in estimating the combination of fixed and random effects, our aim is to predict within the mixed model. We describe how we minimize the prediction error using an optimal design by constructing the Best Linear Unbiased Estimator and Predictor in our model. Many algorithms can be used in order to construct an optimal design. We apply here the Fedorov algorithm, which exchanges observations between the design points. By performing the algorithm built on the distorted model, we present the optimal design in different cases. The results indicate that the optimal design depends strongly on the sample size. In low-information situations, optimal designs are sufficient, while distorted terms produce better designs in data-rich cases.