get.boundary.pop {PoPdesign}

Generate the dose escalation and de-escalation boundaries for single-agent trials.

Description

Use this function to generate the dose escalation and deescalation boundaries for single-agent trials.

Usage

get.boundary.pop(target, n, cohortsize, cutoff, K, cutoff_e)

Arguments

target

the target DLT rate

n

total sample size

cohortsize

the cohort size

cutoff

the cutoff for the predictive Bayes Factor (PrBF). Users can specify either a value or a function for cutoff. If PrBF < cutoff, we assign the next cohort of patients to an adjacent dose based on observed DLT. Otherwise, the evidence is in favor of H_{0j} and we need to retain the current dose.

K

number of dose levels. It is required when argument cutoff is a function that requires K.

cutoff_e

the cutoff for the dose exclusion rule. If PrBF_{0,1}<E(n_j), the evidence is in favor of H_{1j}. If \hat{\pi}_j < \phi, the current dose is deemed as subtherapeutic and we exclude the current dose and lower doses; If \hat{\pi}_j > \phi, the current dose is overly toxic and we exclude the current dose and higher doses.

Details

We assume that there are J pre-specified dose levels of the drug of interest. Let d_1,d_2,\ldots,d_J denote these dose levels. The dose-limiting toxicity (DLT) is assessed as a binary outcome, experiencing toxicity or not. The true dose toxicity is monotonically increasing as the dose level increases. Let \phi be the target toxicity rate and \pi_j be the true dose-toxicity of dose level d_j, for j=1,2,\ldots,J.

We formulate our hypothesis as:

H_{0j}: \pi_j=\phi

H_{1j}: \pi_j\ne\phi

H_{0j} indicates that d_j is the desired MTD so that we should stay; H_{1j} reflects the current dose is either below or above the MTD so that we should transit to a lower or upper dose level. Whether escalate or de-escalate the dose is straightforward: if the observed toxicity rate is above the target toxicity rate \phi, we de-escalate the dose; if the observed toxicity rate is below \phi, we escalate the dose.

With the hypothesis, the predictive Bayes factor comparing H_{0j} and H_{1j} is given by

PrBF_{0,1}=\frac{\phi^{y_i}(1-\phi)^{n_j-y_j}B(y_j+1,n_j-y_j+1)^{n_j}exp(1)}{B(y_j+2,n_j-y_j+1)^{y_j}B(y_j+1,n_j-y_j+2)^{n_j-y_j}}

where x_j is the toxicity response of the ith subject among n_j subjects that received dose d_j, for j=1,2,\ldots,J. y_j denotes the sum of toxicity response. We assume that

y_j \sim Bin(n_j,\pi_j)

According to the calibration of the PrBF, a decision rule based on PrBF_{0,1} is: 1. If PrBF_{0,1}>C(n_j), the evidence is in favor of H_{0j} and we need to retain the current dose; 2. Otherwise, we assign the next cohort of patients to an adjacent dose according to the observed DLT \hat{\pi}_j = y_j/n_j, such as:

(a) If \hat{\pi}_j < \phi, we escalate the dose;

(b) If \hat{\pi}_j > \phi, we de-escalate the dose.

For patient safety and trial efficiency, the PoP design employs a dose exclusion rule. On the one hand, if the PrBF based on the observed DLT indicates a dose is above the MTD with a certain evidence, we exclude the current dose and doses above to avoid treating patients at an overly toxic dose; on the other hand, if the PrBF implies that a dose is substantially below the MTD, we eliminate the current dose and doses below to prevent wasting patients at a subtherapeutic dose. Such a dose exclusion rule is as follow:

If PrBF_{0,1}<E(n_j), the evidence is in favor of H_{1j} and:

1. If \hat{\pi}_j < \phi, the current dose is deemed as subtherapeutic and we exclude the current dose and lower doses;

2. If \hat{\pi}_j > \phi, the current dose is overly toxic and we exclude the current dose and higher doses.

Once all the doses are eliminated from further investigation, the trial is terminated early. The selection of the cut-off value for the dose exclusion is critical for the performance of the PoP design, because it ensure the safety of the patients and efficiency of the design by influencing the early termination rule. The exclusion boundaries in the table above were determined using E(n_j)=exp(-1).

Value

get.boundary.pop() returns a list object, including the corresponding decision tables $out.boundary and $out.full.boundary.

Examples

## get the dose escalation and deescalation boundaries for PoP design with
## the target DLT rate of 0.3, maximum sample size of 30, and cohort size of 3
bound <- get.boundary.pop(target=0.5, n = 15, cohortsize = 3,
                          cutoff=2.5,K=4,cutoff_e=5/24)
summary(bound) # get the descriptive summary of the boundary
plot(bound)    # plot the flowchart of the design along with decision boundaries