Setup

To execute the code in this vignette, you first need to install and load the {ppseq} package. You will also need the {future} package to parallelize your code to improve speed.

install.packages("ppseq")
install.packages("future")

library(ppseq)

library(future)

Introduction

While traditional phase I clinical trials in oncology were focused on identification of the maximum tolerated dose, typically using the rule-based 3+3 design or at times an alternative model-based design, the focus is now shifting to interest in obtaining additional safety data across a range of doses or a variety of disease subtypes, and characterizing preliminary efficacy. As a result, one-sample expansion cohorts are becoming increasingly common in phase I clinical trials in oncology, particularly in the context of immunotherapies or other non-cytotoxic treatments, where the maximum tolerated dose may not exist. An expansion cohort is a group of additional patients being treated at the recommended phase 2 dose (RP2D), or possibly several contending doses, that was identified in the dose finding portion of the phase I study, and is typically used to further characterize the toxicity and/or efficacy of the RP2D. While expansion cohorts have been used extensively in phase I clinical trials in oncology, they have often not been planned in advance or with statistical properties in mind, allowing the sample size at times to expand to very large numbers of treated patients. While having data on more patients can sometimes seem desirable, the goal in phase I trials should still be focused on identifying promising treatments for further study in phase 2 and beyond while limiting the number of patients treated with possibly inefficacious drugs. We propose a design for phase I expansion cohorts based on sequential predictive probability monitoring that allows an investigator to identify an optimal clinical trial design within constraints of traditional type I error control and power, and with futility stopping rules that preserve valuable human and financial resources (Zabor et al. 2022). The {ppseq} package provides functions to implement the proposed design. This vignette will demonstrate how to use functions from the {ppseq} package to design a phase I expansion cohort, in the context of a case study based on the study of atezolizumab in metastatic urothelial carcinoma.

Case study background

Atezolizumab is an anti-PD-L1 treatment that was originally tested in the phase I setting in a basket trial across a variety of cancer sites harboring PD-L1 mutations. The atezolizumab expansion study in metastatic urothelial carcinoma had the primary aim of further evaluating safety, pharmacodynamics and pharmacokinetics and therefore was not designed to meet any specific criteria for type I error or power. An expansion cohort in metastatic urothelial carcinoma (mUC) was not part of the original protocol design, but was rather added later. The expansion cohort in mUC ultimately enrolled a total of 95 participants (Petrylak et al. 2018). Other expansion cohorts that were included in the original protocol, including in renal-cell carcinoma, non-small-cell lung cancer, and melanoma, were planned to have a sample size of 40. These pre-planned expansion cohorts were designed with a single interim analysis for futility that would stop the trial if 0 responses were seen in the first 14 patients enrolled. According to the trial protocol, this futility rule is associated with at most a 4.4% chance of observing no responses in 14 patients if the true response rate is 20% or higher. The protocol also states the widths of the 90% confidence intervals for a sample size of 40 if the observed response rate is 30%. There was no stated decision rule for efficacy since efficacy was not an explicit aim of the expansion cohorts.

Re-design of case study

We will demonstrate the use of the functions in {ppseq} to re-design the one-sample expansion cohort for the study of atezolizumab in mUC using sequential predictive probability monitoring. We assume a null, or unacceptable, response rate of 0.1 and an alternative, or acceptable, response rate of 0.2. We plan a study with up to a total of 95 participants. In our sequential predictive probability design we will check for futility after every 5 patients are enrolled. To design the study, we need to calibrate the design over a range of posterior probability thresholds and predictive probability thresholds.

The posterior probability is calculated from the posterior distribution based on the specified priors and the data observed so far, and represents the probability of success based only on the data accrued so far. A posterior probability threshold would be set during the design stage and if, at the end of the trial, the posterior probability exceeded the pre-specified threshold, the trial would be declared a success. The posterior predictive probability is the probability that, at a given interim monitoring point, the treatment will be declared efficacious at the end of the trial when full enrollment is reached, conditional on the currently observed data and the specified priors. Predictive probability provides an intuitive monitoring framework that tells an investigator what the chances are of declaring the treatment efficacious at the end of the trial if we were to continue enrolling to the maximum planned sample size, given the data observed so far in the trial. If this probability drops below a certain threshold, which is pre-specified during the design stage, the trial would be stopped early. Predictive probability thresholds closer to 0 lead to less frequent stopping for futility, whereas thresholds near 1 lead to frequent stopping unless there is almost certain probability of success.

We consider a grid of thresholds so that we have a range of possible designs from which to select a design with optimal operating characteristics such as type I error, power, and sample size under the null and alternative. In this example we will consider posterior thresholds of 0.7, 0.74, 0.78, 0.82, 0.86, 0.9, 0.92, 0.93, 0.94, 0.95, 0.96, 0.97, 0.98, and 0.99, and predictive thresholds of 0.05, 0.1, 0.15, and 0.2.

Using calibrate_thresholds() to obtain design options

To conduct the case study re-design, we use the calibrate_thresholds() function from the {ppseq} package. This function is written using the future and furrr packages, but the user will have to set up a call to future::plan that is appropriate for their operating environment and their simulation setup prior to running the function. In this example, we used the following code on a Unix server with 192 cores, with the goal of utilizing 56 cores since our grid of thresholds to consider was 14 posterior thresholds by 4 predictive thresholds.

The calibrate_thresholds() function will simulate nsim datasets under the null hypothesis and nsim datasets under the alternative hypothesis. For each simulated dataset, every combination of posterior and predictive thresholds will be considered and the final sample size and whether or not the trial was positive will be saved. Then across all simulated datasets, calibrate_thresholds() will return the average sample size under the null, the average sample size under the alternative, the proportion of positive trials under the null (i.e. the type I error), and the proportion of positive trials under the alternative (i.e. the power).

Because calibrate_thresholds() randomly generates simulated datasets, you will want to set a seed before running the function in order to ensure your results are reproducible.

The inputs to calibrate_thresholds() for our case study re-design include p_null = 0.1 as the null response rate, p_alt = 0.2 as the alternative response rate, n = seq(5, 95, 5) indicates interim looks after every 5 patients up to a total of 95, direction = "greater" specifies that interest is in whether the alternative response rate exceeds the null response rate, delta = NULL because this argument specifies the clinically meaningful difference between groups, which is not relevant in the one-sample case, prior = c(0.5, 0.5) specifies that both hyperparameters of the prior beta distribution be set to 0.5, S = 5000 specifies that 5000 posterior samples will be drawn to calculate the posterior and predictive probabilities, N = 95 indicates the final total sample size of 95, nsim = 1000 specifies that we will generate 1000 simulated datasets under both the null and the alternative, pp_threshold is a vector of the posterior thresholds of interest, and ppp_threshold is a vector of predictive thresholds of interest. Note that due to the computational time involved, the object produced from the below example code one_sample_cal_tbl is available as a dataset in the {ppseq} package.

set.seed(123)

future::plan(future::multicore(workers = 56))

one_sample_cal_tbl <- 
  calibrate_thresholds(p_null = 0.1, 
                       p_alt = 0.2, 
                       n = seq(5, 95, 5),
                       N = 95, 
                       pp_threshold = c(0.7, 0.74, 0.78, 0.82, 0.86, 0.9, 
                                        0.92, 0.93, 0.94, 0.95, 0.96, 0.97, 
                                        0.98, 0.99),
                       ppp_threshold = seq(0.05, 0.2, 0.05),
                       direction = "greater", 
                       delta = NULL, 
                       prior = c(0.5, 0.5), 
                       S = 5000, 
                       nsim = 1000
                       )

Results

We will limit our consideration to designs with type I error between 0.05 and 0.1, and a minimum power of 0.7.

print(one_sample_cal_tbl, 
      type1_range = c(0.05, 0.1), 
      minimum_power = 0.7)

optimize_design(one_sample_cal_tbl, 
                type1_range = c(0.05, 0.1), 
                minimum_power = 0.7)
#> $`Optimal accuracy design:`
#> # A tibble: 1 × 6
#>   pp_threshold ppp_threshold `Type I error` Power `Average N under the null`
#>          <dbl>         <dbl>          <dbl> <dbl>                      <dbl>
#> 1         0.92          0.05          0.081 0.874                       50.7
#>   `Average N under the alternative`
#>                               <dbl>
#> 1                              89.9
#> 
#> $`Optimal efficiency design:`
#> # A tibble: 1 × 6
#>   pp_threshold ppp_threshold `Type I error` Power `Average N under the null`
#>          <dbl>         <dbl>          <dbl> <dbl>                      <dbl>
#> 1         0.92           0.1          0.073 0.793                       38.8
#>   `Average N under the alternative`
#>                               <dbl>
#> 1                              81.8

set.seed(123)

one_sample_decision_tbl <- 
  calc_decision_rules(
    n = seq(5, 95, 5), 
    N = 95, 
    theta = 0.92, 
    ppp = 0.1, 
    p0 = 0.1, 
    direction = "greater", 
    delta = NULL, 
    prior = c(0.5, 0.5), 
    S = 5000
    )

one_sample_decision_tbl

plot(one_sample_cal_tbl, 
     type1_range = c(0.05, 0.1), 
     minimum_power = 0.7,
     plotly = TRUE)

plot(one_sample_decision_tbl)