Guiding Oncology Dose-Escalation Trials

Introduction

The OncoBayes2 package provides flexible functions for Bayesian meta-analytic modeling of the incidence of Dose Limiting Toxicities (DLTs) by dose level, under treatment regimes involving any number of combination partners. Such models may be used to support dose-escalation decisions and estimation of the Maximum Tolerated Dose (MTD) in adaptive Bayesian dose-escalation designs.

The package supports incorporation of historical data through a Meta-Analytic-Combined (MAC) framework [1], which stratifies these heterogeneous sources of information through a hierarchical model. Additionally, it allows the use of EXchangeable/Non-EXchangeable (EX/NEX) priors to manage the amount of information-sharing across subgroups of historical and/or concurrent sources of data.

Example use-case

Consider the application described in Section 3.2 of [1], in which the risk of DLT is to be studied as a function of dose for two drugs, drug A and drug B. Historical information on the toxicity profiles of these two drugs is available from single agent trials trial_A and trial_B. The historical data for this example is available in an internal data set.

kable(hist_combo2)

The objective is to aid dosing and dose-escalation decisions in a future trial, trial_AB, in which the drugs will be combined. Additionally, another investigator-initiated trial IIT will study the same combination concurrently. Note that these as-yet-unobserved sources of data are included in the input data as unobserved factor levels. This mechanism allows us to specify a joint meta-analytic prior for all four sources of historical and concurrent data.

levels(hist_combo2$group_id)

## [1] "trial_A"  "trial_B"  "IIT"      "trial_AB"

# Design parameters ---------------------

dref <- c(3, 960)
num_comp <- 2 # two investigational drugs
num_inter <- 1 # one drug-drug interaction needs to be modeled
num_groups <- nlevels(hist_combo2$group_id) # four groups of data
num_strata <- 1 # no stratification needed

# Model fit -----------------------------

blrmfit <- blrm_exnex(
  cbind(Ntox, Npat - Ntox) ~
    1 + I(log(DosesAdm1 / dref[1])) |
    1 + I(log(DosesAdm2 / dref[2])) |
    0 + I(DosesAdm1/dref[1] *DosesAdm2/dref[2]) |
    group_id,
  data = hist_combo2,
  prior_EX_mu_mean_comp = matrix(
    c(logit(0.1), 0, # hyper-mean of (intercept, log-slope) for drug A
      logit(0.1), 0), # hyper-mean of (intercept, log-slope) for drug B
    nrow = num_comp,
    ncol = 2,
    byrow = TRUE
  ),
  prior_EX_mu_sd_comp = matrix(
    c(3.33, 1, # hyper-sd of mean mu for (intercept, log-slope) for drug B
      3.33, 1), # hyper-sd of mean mu for (intercept, log-slope) for drug B
    nrow = num_comp,
    ncol = 2,
    byrow = TRUE
  ),
  prior_EX_tau_mean_comp = matrix(
    c(log(0.25), log(0.125),
      log(0.25), log(0.125)),
    nrow = num_comp,
    ncol = 2,
    byrow = TRUE
  ),
  prior_EX_tau_sd_comp = matrix(
    c(log(4) / 1.96, log(4) / 1.96,
      log(4) / 1.96, log(4) / 1.96),
    nrow = num_comp,
    ncol = 2,
    byrow = TRUE
  ),
  prior_EX_mu_mean_inter = 0,
  prior_EX_mu_sd_inter = 1.121,
  prior_EX_tau_mean_inter = matrix(log(0.125), nrow = num_inter, ncol = num_strata),
  prior_EX_tau_sd_inter = matrix(log(4) / 1.96, nrow = num_inter, ncol = num_strata),
  prior_is_EXNEX_comp = rep(FALSE, num_comp),
  prior_is_EXNEX_inter = rep(FALSE, num_inter),
  prior_EX_prob_comp = matrix(1, nrow = num_groups, ncol = num_comp),
  prior_EX_prob_inter = matrix(1, nrow = num_groups, ncol = num_inter),
  prior_tau_dist = 1
)

prior_summary(blrmfit) # not run here

newdata <- expand.grid(
  group_id = c("trial_AB"),
  DosesAdm1 = c(0, 3, 4.5, 6),
  DosesAdm2 = c(0, 400, 600, 800),
  stringsAsFactors = FALSE
)
newdata$group_id <- factor(newdata$group_id, levels(hist_combo2$group_id))

newdata <- expand.grid(
  group_id = factor(c("trial_AB"), levels(hist_combo2$group_id)),
  DosesAdm1 = c(0, 3, 4.5, 6),
  DosesAdm2 = c(0, 400, 600, 800)
)

summ_stats <- summary(blrmfit, 
                      newdata = newdata,
                      prob = 0.95,
                      interval_prob = c(0, 0.16, 0.33, 1))

kable(cbind(newdata, summ_stats), digits = 3)

# set up two scenarios at the starting dose level
# store them as data frames in a named list
scenarios <- expand.grid(
  group_id  = factor("trial_AB", levels(hist_combo2$group_id)),
  DosesAdm1 = 3,
  DosesAdm2 = 400,
  Npat      = 3,
  Ntox      = 1:2
) %>% split(1:2) %>% setNames(paste(1:2, "DLTs"))

candidate_doses = expand.grid(
  group_id = factor("trial_AB", levels(hist_combo2$group_id)),
  DosesAdm1 = c(3, 4.5),
  DosesAdm2 = 400
)

scenario_inference <- lapply(scenarios, function(scen_row){
  
  # refit the model with each scenario's additional data
  scenario_data <- rbind(hist_combo2, scen_row)
  scenario_fit <- update(blrmfit, data = scenario_data)
  
  # summarize posterior at candidate doses
  scenario_summ <- summary(scenario_fit,
                           newdata = candidate_doses,
                           interval_prob = c(0, 0.16, 0.33, 1))
  
  cbind(candidate_doses, scenario_summ)
  
})

Continuation of example

In the example of [1], at the time of completion of trial_AB, the complete historical and concurrent data are as follows.

kable(codata_combo2)

Numerous toxicities were observed in the concurrent IIT study. Through the MAC framework, these data can influence the model summaries for trial_AB.

final_fit <- update(blrmfit, data = codata_combo2)

summ <- summary(final_fit, newdata, prob = c(0.5, 0.95), interval_prob = c(0,0.33,1))

final_summ_stats <- cbind(newdata, summ) %>%
    mutate(EWOC=1*`(0.33,1]`<=0.25)

ggplot(final_summ_stats,
       aes(x=factor(DosesAdm2), colour=EWOC)) +
    facet_wrap(~DosesAdm1, labeller=label_both) +
    scale_y_continuous(breaks=c(0, 0.16, 0.33, 0.4, 0.6, 0.8, 1.0)) +
    coord_cartesian(ylim=c(0,0.8)) +
    geom_hline(yintercept = c(0.16, 0.33),
               linetype = "dotted") +
    geom_pointrange(aes(y=`50%`, ymin=`2.5%`, ymax=`97.5%`)) +
    geom_linerange(aes(ymin=`25%`, ymax=`75%`), size=1.5) +
    ggtitle("DLT Probability", "Shown is the median (dot), 50% CrI (thick line) and 95% CrI (thin line)") +
    ylab(NULL) + 
    xlab("Dose Drug B [mg]")

It can be insightful to consider the continuous representation of the model as shown below using the modern tidybayes package. The EWOC criterion fails to be fullfilled once the upper end of the 50% CrI crosses the critical 33% threshold probability since then the 75% quantile of the distribution exceeds the 33% thresholds such that more than 25% of probability lies beyond 33%.

library(tidybayes)

expand.grid(group_id=factor("trial_AB", levels=levels(codata_combo2$group_id)),
            DosesAdm2=exp(seq(log(100),log(800),length=100)), DosesAdm1=c(0, 3, 4.5, 6)) %>%
    add_draws(posterior_linpred(final_fit, newdata = ., transform=TRUE)) %>%
    median_qi(.width=c(0.5, 0.95)) %>%
    ggplot(aes(y = .value, x = DosesAdm2)) +
    scale_x_log10(breaks=c(100,200,400,600,800)) +
    facet_wrap(~DosesAdm1, labeller=label_both) +
    scale_y_continuous(breaks=c(0, 0.16, 0.33, 0.4, 0.6, 0.8, 1.0)) +
    geom_lineribbon() +
    scale_fill_brewer() +
    coord_cartesian(ylim=c(0,0.8)) + 
    geom_hline(yintercept = c(0.16, 0.33),
               linetype = "dotted") +
    ggtitle("DLT Probability", "Shown is the median (line), 50% CrI (dark) and 95% CrI (light)") +
    ylab(NULL) + 
    xlab("Dose Drug B [mg]")

References

[1] Neuenschwander, B., Roychoudhury, S., & Schmidli, H. (2016). On the use of co-data in clinical trials. Statistics in Biopharmaceutical Research, 8(3), 345-354.

[2] Neuenschwander, B., Wandel, S., Roychoudhury, S., & Bailey, S. (2016). Robust exchangeability designs for early phase clinical trials with multiple strata. Pharmaceutical statistics, 15(2), 123-134.

[3] Neuenschwander, B., Branson, M., & Gsponer, T. (2008). Critical aspects of the Bayesian approach to phase I cancer trials. Statistics in medicine, 27(13), 2420-2439.

[4] Neuenschwander, B., Matano, A., Tang, Z., Roychoudhury, S., Wandel, S. Bailey, Stuart. (2014). A Bayesian Industry Approach to Phase I Combination Trials in Oncology. In Statistical methods in drug combination studies (Vol. 69). CRC Press.

Session Info

sessionInfo()

## R version 3.5.3 (2019-03-11)
## Platform: x86_64-pc-linux-gnu (64-bit)
## Running under: Ubuntu 16.04.6 LTS
## 
## Matrix products: default
## BLAS: /usr/lib/libblas/libblas.so.3.6.0
## LAPACK: /usr/lib/lapack/liblapack.so.3.6.0
## 
## locale:
## [1] C
## 
## attached base packages:
## [1] stats     graphics  grDevices utils     datasets  methods   base     
## 
## other attached packages:
## [1] tidybayes_1.1.0  dplyr_0.8.3      ggplot2_3.2.0    knitr_1.22      
## [5] RBesT_1.4-0      OncoBayes2_0.4-4 Rcpp_1.0.1      
## 
## loaded via a namespace (and not attached):
##  [1] rstan_2.18.2              ggstance_0.3.2           
##  [3] tidyselect_0.2.5          xfun_0.6                 
##  [5] purrr_0.2.5               lattice_0.20-38          
##  [7] colorspace_1.3-2          htmltools_0.3.6          
##  [9] stats4_3.5.3              loo_2.0.0                
## [11] yaml_2.2.0                rlang_0.4.0              
## [13] pkgbuild_1.0.2            pillar_1.4.2             
## [15] glue_1.3.1                withr_2.1.2              
## [17] RColorBrewer_1.1-2        matrixStats_0.54.0       
## [19] plyr_1.8.4                stringr_1.4.0            
## [21] munsell_0.5.0             gtable_0.2.0             
## [23] mvtnorm_1.0-8             coda_0.19-2              
## [25] codetools_0.2-16          evaluate_0.13            
## [27] forcats_0.3.0             inline_0.3.15            
## [29] callr_3.1.0               ps_1.2.1                 
## [31] parallel_3.5.3            bayesplot_1.6.0          
## [33] highr_0.8                 rstantools_1.5.1         
## [35] arrayhelpers_1.0-20160527 scales_1.0.0             
## [37] backports_1.1.3           checkmate_1.8.5          
## [39] StanHeaders_2.18.0-1      abind_1.4-5              
## [41] gridExtra_2.3             svUnit_0.7-12            
## [43] digest_0.6.18             stringi_1.4.3            
## [45] processx_3.2.1            grid_3.5.3               
## [47] cli_1.0.1                 tools_3.5.3              
## [49] magrittr_1.5              lazyeval_0.2.1           
## [51] tibble_2.1.3              Formula_1.2-3            
## [53] crayon_1.3.4              tidyr_0.8.2              
## [55] pkgconfig_2.0.2           prettyunits_1.0.2        
## [57] ggridges_0.5.1            assertthat_0.2.1         
## [59] rmarkdown_1.11            R6_2.4.0                 
## [61] compiler_3.5.3