Optimal designs for identifying effective doses in drug combination studies

We consider the optimal design problem for identifying effective dose combinations within drug combination studies where the effect of the combination of two drugs is investigated. Drug combination studies are becoming increasingly important as they investigate potential interaction effects rather than the individual impacts of the drugs. In this situation, identifying effective dose combinations that yield a prespecified effect is of special interest. If nonlinear surface models are used to describe the dose combination-response relationship, these effective dose combinations result in specific contour lines of the fitted response model. We propose a novel design criterion that targets the precise estimation of these effective dose combinations. In particular, an optimal design minimizes the width of the confidence band of the contour lines of interest. Optimal design theory is developed for this problem, including equivalence theorems and efficiency bounds. The performance of the optimal design is illustrated in different examples modeling dose combination data by various nonlinear surface models. It is demonstrated that the proposed optimal design for identifying effective dose combinations yields a more precise estimation of the effective dose combinations than ray or factorial designs, which are commonly used in practice. This particularly holds true for a case study motivated by data from an oncological dose combination study.

💡 Research Summary

This paper addresses a critical challenge in pharmaceutical development: designing efficient experiments to identify combinations of two drugs that produce a specific target therapeutic effect. The authors propose a novel optimal design criterion specifically tailored for estimating “Multivariate Effective Doses” (MED) in drug combination studies. Unlike single-drug studies where an Effective Dose (ED) is a single point, in combination studies, multiple dose pairs can yield the same effect, forming a contour line on a dose-response surface model.

The core methodological contribution is the development of an optimal design theory that directly minimizes the imprecision in estimating these MED contour lines. The proposed criterion aims to minimize a function of the prediction variance along the contour of interest, effectively narrowing the confidence band around the estimated MED curve. This is a significant shift from classical optimal design criteria like D-optimality, which optimize for precise parameter estimation globally but not necessarily for precise estimation of a specific functional of the model like the MED contour.

The paper establishes a robust theoretical framework. It considers a general nonlinear regression model for the combined drug effect and formally defines the MED. The authors then derive the corresponding information matrix and asymptotic distribution of the MED estimator. Within this framework, they develop the optimal design theory for the new MED-oriented criterion, providing an equivalence theorem for verifying optimality and a lower bound for the design efficiency, which is valuable for assessing designs when the true optimum is unknown.

The performance of these newly derived “MED-optimal” designs is rigorously evaluated and compared against standard practices. The authors examine two common types of nonlinear surface models used in pharmacology: an additive model combining standard one-dimensional dose-response functions (e.g., Emax) with an interaction term, and the Greco model. In various scenarios using these models, the MED-optimal designs are compared to traditional designs like ray designs (testing fixed mixture ratios) and factorial designs, as well as to the parameter-focused D-optimal designs.

The results consistently demonstrate the superiority of the proposed designs for the specific task of MED estimation. The MED-optimal designs achieve substantially higher efficiency in estimating the target contour line than all alternative designs. This advantage is convincingly illustrated through a case study motivated by real data from an oncological dose combination study, highlighting its practical relevance. Furthermore, the paper explores the robustness of the designs to uncertainty in the assumed strength of the drug interaction, showing that the approach remains effective across a range of plausible scenarios.

In summary, this work provides a rigorous and practical optimal design methodology for a pressing problem in modern drug development. By focusing the design objective directly on the precise estimation of effective dose combinations, it offers a more efficient and targeted alternative to conventional experimental designs in drug combination research.