Methods and Applications: A Novel Adaptive Design Approach for Early-Phase Clinical Trials to Optimize Vaccine Dosage

We assume that the number of individuals experiencing DLT events at dose level $ i $, denoted by $ {x}_{i} $, follows a binomial distribution $ binom\left({n}_{i},\rho \right) $, where $ {n}_{i} $ is the sample size at dose level $ i $ and $ \rho $ is the true, but unknown, toxicity probability. To incorporate prior knowledge, we specify a $ \text{beta}\left(\alpha ,\beta \right) $ prior distribution for $ \rho $ (detailed rationale is provided at Supplementary Material).

Based on the observed data $ {x}_{i} $ and $ {n}_{i} $ from the trial, we construct the likelihood function. Using Bayesian updating, we derive the posterior distribution of $ \rho $ as $ \text{beta}\left(\alpha +{x}_{i},\beta +{n}_{i} -{x}_{i}\right) $, from which the posterior probability $ P\left(\rho > \pi |\mathit{H}\right) $ is calculated, where $ \pi $ represents the target toxicity rate and $ \mathit{H} $ denotes the accumulated trial data. To ensure monotonicity in toxicity, isotonic transformation is applied to the probability. If $ P\left(\rho > \pi |\mathit{H}\right) $ exceeds the prespecified toxicity threshold T, the current dose level $ i $ is deemed excessively toxic. Consequently, this dose level is excluded from further consideration for subsequent participant cohorts, ensuring participant safety and optimizing dose selection.

We assume that $ {y}_{i} $ out of $ {n}_{i} $ individuals experience efficacy at dose level $ i $, where $ {\tau }_{i} $ is the efficacy probability and $ {d}_{i} $ denotes the dosage at dose level $ i $. Additionally, we specify a lower boundary of the efficacy rate $ \phi $ for futility monitoring. For the efficacy endpoint, we employ a logistic regression model specified as follows:

The detailed mathematical derivation and explanation for efficacy monitoring are provided at Supplementary Material.

In the Bayesian framework, we specify the Cauchy distribution for the unknown parameters following Gelman’s recommendation (13):

The likelihood function $ L\left(H|a, b,c\right) $ is proportional to:

By integrating the prior distribution with the likelihood function and employing Markov Chain Monte Carlo (MCMC) sampling, we derive posterior distributions for the parameters: a, b, and c, where convergence diagnostics are conducted using trace plots. Leveraging this model, we predict the dose associated with peak efficacy given the current dataset, thus providing guidance for dose decision-making.

The proposed Bayesian logistic dose-finding method follows a systematic approach:

Initial cohort treatment: Administer the first cohort of participants at the lowest dose or a prespecified dose level $ i $.

Interim data collection: At dose level $ i $, collect interim data where $ {n}_{i} $ patients have been treated (calculated as the cohort size multiplied by the number of recruited cohorts), with $ {x}_{i} $ experiencing toxicity and $ {y}_{i} $ showing an immune response.

Toxicity monitoring: Based on the interim data, identify doses where $ P\left(\rho > \pi |\mathit{H}\right) $<T and consider them as safe doses. If no safe dose is identified, the trial should be terminated, and no optimal biological dose (OBD) should be declared.

Efficacy monitoring: Determine the dose level $ {i}^{*} $ with the highest posterior estimate of efficacy probability τ to treat the next cohort.

Iteration: Steps 2 and 3 are repeated until the maximum sample size is reached. At the end of the trial, the posterior estimate of the efficacy probability τ of all doses is calculated based on the final dataset.

OBD selection: Select the final OBD using the following criteria:

a. If τ > φ, declare the dose with an efficacy probability of τ as the OBD.

b. If τ < φ , declare no OBD due to ineffectiveness.

The dose escalation process is illustrated in a flow chart (Figure 1).

Figure 1. 

Dose escalation decision flow chart.

Note: φ means the lower boundary of efficacy rate.

Abbreviation: OBD=optimal biological dose.

To evaluate the performance of our proposed design, we conducted extensive simulation studies across six distinct scenarios (Figure 2). These scenarios were carefully selected to comprehensively assess the proposed method under typical conditions encountered in phase I/II clinical trials. We based our scenario settings on a vaccine trial example (14) (Supplementary Materials). Across all scenarios, dose-toxicity curves were assumed to increase monotonically, while we considered various dose-response relationships that might occur in vaccine clinical trials. For all scenarios, we specified a target toxicity rate of $ \pi =0.2 $, a prespecified toxicity threshold of $ T=0.8 $, and a lower boundary of the efficacy rate $ \phi =0.2 $. The simulations employed a cohort size of 3 participants with a maximum of 30 participants for the entire trial.

Figure 2. 

Six dose-toxicity and dose-efficacy scenarios considered in the simulation study. A–F represent scenarios 1–6, respectively.

Note: The red solid line represents DLT rate $ \rho $; the blue dashed line represents the immune response probability $ \tau $; the horizontal dashed line represents at $ \pi $= 0.2.

We conducted simulation studies across six scenarios, each with 5,000 replications, to evaluate the performance of our proposed Bayesian logistic dose-finding method compared with the traditional 3+3 design (15) and the EffTox design (5). For each simulation, we report the percentages of different doses selected, the average percentages of participants treated at each dose, and the average percentages of DLTs and immune responses (Supplementary Materials). Overall, while the EffTox design achieved near-perfect metrics in certain scenarios, it demonstrated significant instability and variability across different scenarios. Similarly, the 3+3 design showed inadequate performance in both OBD identification and participant allocation. In contrast, our proposed Bayesian logistic dose-finding method demonstrated superior performance across all scenarios, accurately identifying the OBD, optimizing participant allocation, and maintaining robustness across diverse clinical situations.

To assess the robustness of our proposed design under different conditions, we conducted a sensitivity analysis by varying prior distributions and sample sizes. The consistency of the OBD selection results highlighted the model’s flexibility and reliability in guiding dose-finding decisions.

We tested the impact of different prior distributions by modifying the scale parameters of a, b to values of (1,2,3,4,5) according to Andrew Gelman (13). As shown in Figure 3, regardless of variations in the prior parameters, the final results remained largely consistent across all simulation scenarios. Changes in scale parameter did not significantly affect either the selection of the OBD or the average percentage of participants treated at the OBD. These results indicate that our design is relatively insensitive to variations in the prior distribution within the tested range. The underlying data tables are shown in Supplementary Table S1.

Figure 3. 

Results of sensitivity analysis for different prior settings. A shows the percentage of correct dose selection, B shows the average percentage of participants treated at each dose level under different prior settings.

Note: Different colors represent different prior distributions.

Abbreviation: OBD=optimal biological dose.

We also evaluated the impact of different sample sizes by increasing the sample sizes to ranges of 30 and 120 participants in four representative simulation settings, corresponding to scenarios presented in Figure 2. Supplementary Figure S1 shows that larger sample sizes resulted in a stable or higher average percentage of OBD selections, suggesting that more extensive data collection improves both the precision and stability of dosage decision-making. The underlying data tables are shown in Supplementary Table S2.

The sensitivity analysis demonstrates that our proposed design maintains consistent performance despite variations in prior distributions and sample sizes, highlighting its reliability and stability under different conditions.

Vaccines are fundamental to global health efforts and offer significant clinical benefits. Early-phase clinical trials are crucial for identifying doses that are both safe and immunogenic, establishing the foundation for subsequent efficacy studies. However, these trials face significant challenges, including minimal toxicity within dose ranges and nonmonotonic dose response curves. Existing dose-finding designs often prove inadequate or require substantial modifications to address the unique properties of vaccines. In this study, we proposed a novel Bayesian phase I/II trial design specifically tailored for vaccine dose-finding. Given that current methods for optimizing vaccine doses are predominantly empirical, we compared our approach with the traditional 3+3 design and the EffTox design. Our results demonstrated that our method excels in balancing the toxicity–efficacy trade-off and optimally allocating participants. The proposed design offers simplicity and accuracy, providing a more effective alternative to traditional methods. Additionally, extensive sensitivity analyses across various prior settings confirmed the robustness and efficiency of our approach, underscoring its reliability in different scenarios. By offering a more precise and reliable framework for dose determination, our Bayesian design addresses the inherent challenges of early-phase vaccination trials, promising enhanced efficacy and safety in vaccine dosage determination.