• Adaptive design;
  • Biologically optimal dose combination;
  • Change point model;
  • Dose finding;
  • Drug combination;
  • Non-monotonic pattern


Treating patients with novel biological agents is becoming a leading trend in oncology. Unlike cytotoxic agents, for which efficacy and toxicity monotonically increase with dose, biological agents may exhibit non-monotonic patterns in their dose–response relationships. Using a trial with two biological agents as an example, we propose a dose finding design to identify the biologically optimal dose combination, which is defined as the dose combination of the two agents with the highest efficacy and tolerable toxicity. A change point model is used to reflect the fact that the dose–toxicity surface of the combinational agents may plateau at higher dose levels, and a flexible logistic model is proposed to accommodate the possible non-monotonic pattern for the dose–efficacy relationship. During the trial, we continuously update the posterior estimates of toxicity and efficacy and assign patients to the most appropriate dose combination. We propose a novel dose finding algorithm to encourage sufficient exploration of untried dose combinations in the two-dimensional space. Extensive simulation studies show that the design proposed has desirable operating characteristics in identifying the biologically optimal dose combination under various patterns of dose–toxicity and dose–efficacy relationships.