Optimized adaptive enrichment designs

Abstract

Based on a Bayesian decision theoretic approach, we optimize frequentist single- and adaptive two-stage trial designs for the development of targeted therapies, where in addition to an overall population, a pre-defined subgroup is investigated. In such settings, the losses and gains of decisions can be quantified by utility functions that account for the preferences of different stakeholders. In particular, we optimize expected utilities from the perspectives both of a commercial sponsor, maximizing the net present value, and also of the society, maximizing cost-adjusted expected health benefits of a new treatment for a specific population. We consider single-stage and adaptive two-stage designs with partial enrichment, where the proportion of patients recruited from the subgroup is a design parameter. For the adaptive designs, we use a dynamic programming approach to derive optimal adaptation rules. The proposed designs are compared to trials which are non-enriched (i.e. the proportion of patients in the subgroup corresponds to the prevalence in the underlying population). We show that partial enrichment designs can substantially improve the expected utilities. Furthermore, adaptive partial enrichment designs are more robust than single-stage designs and retain high expected utilities even if the expected utilities are evaluated under a different prior than the one used in the optimization. In addition, we find that trials optimized for the sponsor utility function have smaller sample sizes compared to trials optimized under the societal view and may include the overall population (with patients from the complement of the subgroup) even if there is substantial evidence that the therapy is only effective in the subgroup.

Keywords: Adaptive design; enrichment design; optimal design; precision medicine; subgroup analysis.

Figure 1.

Case 1. Optimized utilities as function of the population prevalence for adaptive enrichment trials (solid lines), single-stage designs (dashed lines) and single-stage designs restricted to a fixed prevalence (FP) $\lambda =\gamma$ (dotted lines). Red lines show the sponsor utility and blue lines the societal utility.

Figure 2.

Case 1. Optimized trial designs for the single-stage designs. For single-stage designs the sample sizes stratified by subgroup S and $S\prime$ are shown. For the single-stage designs with restricted trial prevalence ($\gamma =\lambda$, dotted lines) the total sample size is shown. Red lines correspond to the sponsor view, blue lines to the societal view.

Figure 3.

Case 1. Optimized trial designs for adaptive enrichment designs. The first column show the optimal first stage sample sizes, the second column the average second stage sample sizes conditional on the event that they are larger than zero. The third column shows the probability that the second stage is conducted in S only/in F.

Figure 4.

Optimized interim decisions under the weak and the strong biomarker prior for Case 1 and $\lambda =0.5$ as function of the first stage test statistics $Z_S^{(1)},Z_{S\prime }^{(1)}$. Grey (red) areas correspond to regions where patients from S and $S\prime$ (only S) are recruited in the second stage of the trial, respectively. White areas correspond to regions where a futility stop is optimal. Note that the first stage sample sizes have been optimized as well and therefore differ between the considered scenarios.

Figure 5.

Case 1. $\rho ({\pi }_0,{\pi \text{'}}_0,\mathcal{D})$ is the ratio of the expected utilities under the prior π₀ if designs $d_{{\pi \text{'}}_0}^{*}$ (the optimal design from family $\mathcal{D}$ under prior ${\pi \text{'}}_0$) and $d_{{\pi }_0}^{*}$ (the optimal design from family $\mathcal{D}$ under π₀) are applied. The proportion ρ is shown for several prior distributions and families of designs $\mathcal{D}$, for both the sponsor and the societal view.