CRM

The above example uses the 3+3 design but other methods are available. In fact, early phase statisticians would much prefer that you use a model-based method (Wheeler et al. 2019). The escalation package is intentionally written so that all dose selectors look the same, regardless the implementation-level details. In computing terminology, dose selectors in escalation support a common interface. This makes it trivial to calculate dose-paths for all dose-selectors - we just use the get_dose_paths function again.

Let us investigate a continual reassessment method (CRM) design now, as described by O’Quigley, Pepe, and Fisher (1990) and implemented by the dfcrm package (Cheung 2013). We must specify a dose-toxicity skeleton, and a target toxicity level:

skeleton <- c(0.05, 0.1, 0.25, 0.4, 0.6)
target <- 0.25

We can then specify a CRM model and calculate dose-paths as before:

paths <- get_dfcrm(skeleton = skeleton, target = target) %>%
  get_dose_paths(cohort_sizes = c(3, 3))

When we graph the paths this time (and adopt a different colour palette for fun):

if(Sys.getenv("RSTUDIO") == "1") {
  graph_paths(paths, viridis_palette = 'magma')
}

we see that the Stop node is absent - all paths recommend a dose, even those seeing substaantial toxicity. Stopping behaviours must be specified for the CRM method. Fortunately, escalation makes this simple.

An intuitive method for stopping in the Bayesian setting is to test the posterior distribution for the probability of toxicity. In our example, we will stop if there is at least a 90% probability that the toxicity rate at the lowest dose is 35% or greater:

paths <- get_dfcrm(skeleton = skeleton, target = target) %>%
  stop_when_too_toxic(dose = 1, tox_threshold = 0.35, confidence = 0.9) %>% 
  get_dose_paths(cohort_sizes = c(3, 3))

When we visualise the paths from the updated model:

if(Sys.getenv("RSTUDIO") == "1") {
  graph_paths(paths, viridis_palette = 'inferno')
}

we see that some paths once again recommend stopping. The paths that recommend a dose are otherwise unchanged.

BOIN

Another dose-escalation model implemented in escalation is the Bayesian optimal interval (BOIN) method by Liu and Yuan (2015), implemented in the BOIN package (Yuan and Liu 2018).

To spice things up, we will visualise how this model behaves over four cohorts of two patients:

paths <- get_boin(num_doses = 4, target = target) %>% 
  get_dose_paths(cohort_sizes = rep(2, 4))
#> You have requested 121 model evaluations. Be patient.

if(Sys.getenv("RSTUDIO") == "1") {
  graph_paths(paths, RColorBrewer_palette = 'YlOrRd')
}

The first thing to note about the graph above is that it is much more complex than the previous graphs. The number of nodes in dose-paths increases faster than linearly as more cohorts are considered. Consideration of this must be given when visualising dose-paths.

Secondly, the visual method makes it simple to discern messages about future model behaviour. For instance, we can easily see that dose 4, the darkest red node, is only reached in the first four cohorts if the first cohort sees no toxicity.

In contrast to CRM, BOIN does have a stopping rule for excess toxicity built in. We see that TT in the first cohort is not enough to advocate stopping but that any further toxicity in the second cohort will be sufficient to warrant stopping.

Non-uniform cohorts

There is no reason that the cohort sizes should be uniform. Specify whatever cohort_sizes you like:

paths <- get_boin(num_doses = 4, target = target) %>% 
  get_dose_paths(cohort_sizes = c(3, 1, 2))

if(Sys.getenv("RSTUDIO") == "1") {
  graph_paths(paths, RColorBrewer_palette = 'Blues')
}

You can even evaluate the dose advice after each patient using cohort sizes of 1:

paths <- get_boin(num_doses = 4, target = target) %>% 
  get_dose_paths(cohort_sizes = rep(1, 4))

if(Sys.getenv("RSTUDIO") == "1") {
  graph_paths(paths, RColorBrewer_palette = 'RdPu')
}

Partially completed trials

It is possible to calculate dose-paths from trials that are partially completed. For instance, let us continue with our BOIN model and assume that we have seen outcomes 1NNN 2TNT so far in our trial. Thus, we are reasonably sure that dose 1 is safe but wary that dose 2 might be too toxic. However, these beliefs are tempered by the tiny sample size. From this starting point, how might the next two cohorts of three proceed? We just specify the previous outcomes using the previous_outcomes parameter:

paths <- get_boin(num_doses = 4, target = target) %>% 
  get_dose_paths(cohort_sizes = rep(3, 2), previous_outcomes = '1NNN 2TNT')

if(Sys.getenv("RSTUDIO") == "1") {
  graph_paths(paths, viridis_palette = 'viridis')
}

Notice how this is different to the advice we get at the start of the trial (i.e. when omitting the previous_outcomes):

paths <- get_boin(num_doses = 4, target = target) %>% 
  get_dose_paths(cohort_sizes = rep(3, 2))

if(Sys.getenv("RSTUDIO") == "1") {
  graph_paths(paths, viridis_palette = 'viridis')
}

This is a feature of model-based methods like CRM and BOIN - they use all information at all doses when making dose decisions. They are not memoryless like the 3+3.

Next dose

The dose at which dose-paths commence is inferred from the model. It can be specified manually, however:

paths <- get_three_plus_three(num_doses = 5, allow_deescalate = TRUE) %>% 
  get_dose_paths(cohort_sizes = c(3, 3), next_dose = 3)

if(Sys.getenv("RSTUDIO") == "1") {
  graph_paths(paths, viridis_palette = 'plasma')
}

A 3+3 trial with de-escalation enabled will de-escalate through the doses when toxicity is seen, as expected.

Dose-paths vs simulation

We saw above that probabilistic inference was possible with dose-paths. Researchers have typically used simulation to achieve this task. escalation supports simulation as well through the simulate_trials function. However, this does raise the question, when should you use each method, and what are their relative merits?

The answer to the first question comes down to the expected number of model fits required. Fitting dose-finding models takes computer time. In dose-paths, the model is fit once at each node. We have seen examples above of how the number and size of cohorts affects the number of nodes in dose-paths. In fact, escalation provides a function to calculate the number of nodes.

In a phase 1 dose-finding trial, each patient experiences exactly one of two outcomes: T or N. Let us calculate how many nodes there are in a graph of dose-paths using five cohorts of three patients. We run:

num_dose_path_nodes(num_patient_outcomes = 2, cohort_sizes = rep(3, 5))
#> [1]    1    4   16   64  256 1024

The num_patient_outcomes = 2 parameter reflects that patients may experience T or N. The returned vector of integers is the number of nodes at each depth. There is one starting node. That node is connected to four children via outcomes NNN, NNT, NTT, and TTT. The number of nodes at greater depths proceeds multiplicatively thereafter. The total number of nodes is:

num_dose_path_nodes(num_patient_outcomes = 2, cohort_sizes = rep(3, 5)) %>% 
  sum
#> [1] 1365

Thus it requires exactly 1,365 model fits to calculate the exact operating characteristics in this \(n = 5 \times 3 = 15\) patient scenario. To compare this to simulations, consider that each simulated trial iteration will fit the model up to five times, once at the end of each cohort. The total number of model fits in a simulation study is bounded by this number multiplied by the number of simulated iterations. Generally simulation studies use thousands of replicates, so it is easy to see that exact inference via crystallised dose-paths will be much less computaionally burdensome, and therefore faster here.

In contrast, now consider a trial of eight cohorts of three. The total number of model fits to enumerate the complete graph of dose-paths is

num_dose_path_nodes(num_patient_outcomes = 2, cohort_sizes = rep(3, 8)) %>% 
  sum
#> [1] 87381

Ten thousand simulated iterations of 8 cohorts each would only require up to 80,000 model fits. Thus, a reasonably accurate simulation study would be expected to be faster here.

However, speed is not the only concern: there is also precision to consider. Simulations have the disadvantage of suffering from Monte Carlo error, that is the uncertainty about the estimated statistics arising from the use of a finite number of simulated iterations. In contrast, exact inference via dose-paths has the great advantage of being exact. That is, there is no uncertainty in the calculated probabilities. (Note: there is still uncertainty about which path will be taken because that is determined by random patient outcomes). Thus, there are scenarios when dose-paths may still be preferable to simulations, even when they are expected to take longer.

It is likely that in practice, simulation is used when dose-paths would be a better option. If true, that would likely be linked to provision of software that performs the two methods. escalation plugs that gap.