Two-Dimensional Maximin Power Designs for Combination Experiments of Drugs

The combined use of two drugs is a major treatment approach for complex diseases such as cancer and HIV due to its potential for efficacy at lower, less toxic doses and the need to reduce developmental time and cost. Experimental designs have been proposed in the literature to test whether there are synergistic or antagonistic actions between the combined drugs. The existing designs for synergy testing are primarily one-dimensional (1D), allocating the doses of one drug while keeping the dose of another, the mixing proportion, or the total dose of the two drugs fixed. This paper considers two-dimensional (2D) designs in which the doses of two drugs can be varied simultaneously over the entire dose region. Based on the premise that prior information about the single-drug experiments is already available, we propose a succinct dose-response model that encompasses a wide class of potential synergistic/antagonistic actions deviated from additivity. We show that the uniform design measure over the 2D dose region is optimal under the proposed model in the sense that it maximizes the minimum power in the F-test to detect drug synergy. Methods for sample size calculation and design generation for our 2D optimal design are given. We illustrate the use of the proposed design and demonstrate its advantages over the 1D optimal design via a combination study of two anticancer drugs.