Neuenschwander, Branson, and Gsponer (2008) (NBG) introduced a derivative of the CRM for dose-escalation clinical trials using the model:
\[ \text{logit} p_i = \alpha + \exp{(\beta)} \log{(x_i / d^*)}, \]
where \(p_i\) is the probability of
toxicity at the \(i\)th dose, \(x_i\), and \(d^*\) is a reference dose. Here \(\alpha\) and \(\beta\) are model parameters on which the
authors place a bivariate normal prior. This model is very similar to
the two-parameter logistic CRM, implemented with
stan_crm(model = 'logistic2'). However, a notable
difference is that the dose, \(x_i\),
enters the model as a covariate. This dispenses with the toxicity
skeleton that is used in the CRM.
The authors introduce their approach in a paper that argues for greater parameterisation in dose-escalation models so that they might more accurately estimate the entire dose-toxicity curve than simplistic one-parameter models.
Let’s run some examples.
To access the Stan implementations of the NBG model, we must load
trialr:
library(trialr)For illustration, let us reproduce the model fit in the lower right pane in Figure 1 of Neuenschwander, Branson, and Gsponer (2008). The authors fit their model to a partially-completed historic trial investigating 15 doses, using the highest dose as the reference dose:
dose <- c(1, 2.5, 5, 10, 15, 20, 25, 30, 40, 50, 75, 100, 150, 200, 250)
d_star <- 250The original investigators sought a dose associated with 30% toxicity:
target <- 0.30and the analysis concerns which dose to give after observing the following outcomes:
outcomes <- '1NNN 2NNNN 3NNNN 4NNNN 7TT'We see that the original investigators escalated through doses 1 to 3, but after having observed no toxicity, seem to have thrown caution to the wind and escalated straight to dose 7. Unfortunately, that move seems to have be imprudent because two toxicities were seen in two patients.
Neuenschwander et al. introduce normal priors for the
parameters \(\alpha\) and \(\beta\) which we specify when we call
stan_nbg to fit the model:
fit <- stan_nbg(outcome_str = outcomes, real_doses = dose, d_star = d_star,
target = target, alpha_mean = 2.15, alpha_sd = 0.84,
beta_mean = 0.52, beta_sd = 0.8, seed = 2020, refresh = 0)Note that presently, the implementation in trialr uses
independent normal priors on \(\alpha\)
and \(\beta\), and not the bivariate
normal that the authors used. This will hopefully be addressed in a
future release of trialr. Nevertheless, we see that the
small difference of prior apparently makes little difference to the
posterior because the mean estimates of the probability of toxicity are
very close to that shown in Figure 1 of NBG’s manuscript:
fit
#> Patient Dose Toxicity Weight
#> 1 1 1 0 1
#> 2 2 1 0 1
#> 3 3 1 0 1
#> 4 4 2 0 1
#> 5 5 2 0 1
#> 6 6 2 0 1
#> 7 7 2 0 1
#> 8 8 3 0 1
#> 9 9 3 0 1
#> 10 10 3 0 1
#> 11 11 3 0 1
#> 12 12 4 0 1
#> 13 13 4 0 1
#> 14 14 4 0 1
#> 15 15 4 0 1
#> 16 16 7 1 1
#> 17 17 7 1 1
#>
#> Dose N Tox ProbTox MedianProbTox ProbMTD
#> 1 1 3 0 0.0126 0.00563 0.00025
#> 2 2 4 0 0.0325 0.01967 0.00050
#> 3 3 4 0 0.0675 0.04934 0.02250
#> 4 4 4 0 0.1385 0.11863 0.10450
#> 5 5 0 0 0.2064 0.18937 0.16775
#> 6 6 0 0 0.2692 0.25641 0.17100
#> 7 7 2 2 0.3264 0.31878 0.14425
#> 8 8 0 0 0.3780 0.37409 0.16850
#> 9 9 0 0 0.4661 0.46884 0.12475
#> 10 10 0 0 0.5371 0.54698 0.07125
#> 11 11 0 0 0.6616 0.67700 0.02000
#> 12 12 0 0 0.7390 0.75753 0.00375
#> 13 13 0 0 0.8259 0.84691 0.00075
#> 14 14 0 0 0.8717 0.89297 0.00025
#> 15 15 0 0 0.8993 0.91986 0.00000
#>
#> The model targets a toxicity level of 0.3.
#> The dose with estimated toxicity probability closest to target is 7.
#> The dose most likely to be the MTD is 6.
#> Model entropy: 2.06We see that the design advocates selecting dose 7 for the next patients. This can be verified in the manuscript.
To illustrate the researchers’ motivation for developing their method, the original one-parameter CRM model using the original investigators’ skeleton was:
skeleton = c(0.01, 0.015, 0.02, 0.025, 0.03, 0.04, 0.05, 0.10, 0.17, 0.30)
fit2 <- stan_crm(outcomes, skeleton = skeleton, target = target,
model = 'empiric', beta_sd = 1.34, seed = 2020, refresh = 0)
fit2
#> Patient Dose Toxicity Weight
#> 1 1 1 0 1
#> 2 2 1 0 1
#> 3 3 1 0 1
#> 4 4 2 0 1
#> 5 5 2 0 1
#> 6 6 2 0 1
#> 7 7 2 0 1
#> 8 8 3 0 1
#> 9 9 3 0 1
#> 10 10 3 0 1
#> 11 11 3 0 1
#> 12 12 4 0 1
#> 13 13 4 0 1
#> 14 14 4 0 1
#> 15 15 4 0 1
#> 16 16 7 1 1
#> 17 17 7 1 1
#>
#> Dose Skeleton N Tox ProbTox MedianProbTox ProbMTD
#> 1 1 0.010 3 0 0.0754 0.0592 0.00725
#> 2 2 0.015 4 0 0.0924 0.0759 0.00600
#> 3 3 0.020 4 0 0.1071 0.0906 0.01125
#> 4 4 0.025 4 0 0.1201 0.1038 0.01150
#> 5 5 0.030 0 0 0.1321 0.1161 0.02425
#> 6 6 0.040 0 0 0.1538 0.1386 0.02650
#> 7 7 0.050 2 2 0.1733 0.1589 0.10250
#> 8 8 0.100 0 0 0.2533 0.2432 0.27850
#> 9 9 0.170 0 0 0.3419 0.3369 0.36500
#> 10 10 0.300 0 0 0.4763 0.4775 0.16725
#>
#> The model targets a toxicity level of 0.3.
#> The dose with estimated toxicity probability closest to target is 9.
#> The dose most likely to be the MTD is 9.
#> Model entropy: 1.61Note that the above fit uses just the lowest ten doses, to be
consistent with Table 1 in Neuenschwander,
Branson, and Gsponer (2008). Incredibly, this design advocates
escalating two more doses to dose 9, despite the outcomes
7TT in the previous cohort.
To be fair to the CRM, this scenario has become a didactic example for how not to choose a skeleton. The prior probabilities of toxicity are far too close together. Choosing a perhaps more sensible skeleton, the model advocates more defensible behaviour:
skeleton = c(0.03, 0.06, 0.12, 0.20, 0.30, 0.40, 0.50, 0.59, 0.67, 0.74)
# Obtained using dfcrm::getprior(0.05, 0.3, 5, 10)
fit3 <- stan_crm(outcomes, skeleton = skeleton, target = target,
model = 'empiric', beta_sd = 1.34, seed = 2020, refresh = 0)
fit3
#> Patient Dose Toxicity Weight
#> 1 1 1 0 1
#> 2 2 1 0 1
#> 3 3 1 0 1
#> 4 4 2 0 1
#> 5 5 2 0 1
#> 6 6 2 0 1
#> 7 7 2 0 1
#> 8 8 3 0 1
#> 9 9 3 0 1
#> 10 10 3 0 1
#> 11 11 3 0 1
#> 12 12 4 0 1
#> 13 13 4 0 1
#> 14 14 4 0 1
#> 15 15 4 0 1
#> 16 16 7 1 1
#> 17 17 7 1 1
#>
#> Dose Skeleton N Tox ProbTox MedianProbTox ProbMTD
#> 1 1 0.03 3 0 0.0177 0.00744 0.00025
#> 2 2 0.06 4 0 0.0344 0.01959 0.00250
#> 3 3 0.12 4 0 0.0700 0.05163 0.01625
#> 4 4 0.20 4 0 0.1229 0.10544 0.08625
#> 5 5 0.30 0 0 0.1977 0.18583 0.19575
#> 6 6 0.40 0 0 0.2820 0.27782 0.26925
#> 7 7 0.50 2 2 0.3759 0.37951 0.24000
#> 8 8 0.59 0 0 0.4689 0.47830 0.13325
#> 9 9 0.67 0 0 0.5584 0.57133 0.04650
#> 10 10 0.74 0 0 0.6421 0.65647 0.01000
#>
#> The model targets a toxicity level of 0.3.
#> The dose with estimated toxicity probability closest to target is 6.
#> The dose most likely to be the MTD is 6.
#> Model entropy: 1.77With this particular skeleton that places the prior guess of MTD at dose 5 and spaces out the prior probabilities of toxicity, de-escalation to dose 6 is suggested.
trialr and the escalation packageescalation
is an R package that provides a grammar for specifying dose-finding
clinical trials. For instance, it is common for trialists to say
something like ‘I want to use this published design… but I want it to
stop once \(n\) patients have been
treated at the recommended dose’ or ‘…but I want to prevent dose
skipping’ or ‘…but I want to select dose using a more risk-averse metric
than merely closest-to-target’.
trialr and escalation work together to
achieve these goals. trialr provides model-fitting
capabilities to escalation, including the NBG method
described here. escalation then provides additional classes
to achieve all of the above custom behaviours, and more.
escalation also provides methods for running simulations
and calculating dose-paths. Simulations are regularly used to appraise
the operating characteristics of adaptive clinical trial designs.
Dose-paths are a tool for analysing and visualising all possible future
trial behaviours. Both are provided for a wide array of dose-finding
designs, with or without custom behaviours like those identified above.
There are many examples in the escalation vignettes at https://cran.r-project.org/package=escalation.
trialr is available at https://github.com/brockk/trialr and https://CRAN.R-project.org/package=trialr