Methods
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. 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.
Instead of the Bayes factor, we use the predictive Bayes factor
(PrBF) (Zhou, 2011) to conduct the hypothesis testing. Denote \(M_k (\mathbf{y}),k=0,1\) the posterior
model under hypotheses \(H_k\) with the
parameter updated in the presence of observed data \(\mathbf{y}=\{y_1,y_2,\cdots,y_n\}\). Denote
\(\theta_k\) the parameters considered
in the model \(M_k\) and specify a
prior distribution as \(\pi(\theta_k)\). The PrBF comparing \(H_0\) to \(H_1\) is defined as:
\[
\text{PrBF}_{1,2}=\frac{\prod_i P(y_i|M_1(\mathbf{y}))}{\prod_i
P(y_i|M_2(\mathbf{y}))}\frac{\exp{(n\cdot
\hat{b}_1})}{\exp{(n\cdot\hat{b}_2)}}
\] where the estimator \(\hat{b}_k\) corrects the over-estimation
bias in the empirical log posterior predictive distribution, owing to
the ‘double use’ of observed data in both model fitting and
evaluation.
The PrBF has attractive features going beyond Bayes factors by
employing the posterior predictive distribution in model assessment. It
significantly reduces the sensitivity to the choice of the prior
distribution and avoids the degeneration of the integrated likelihood
underlying Lindley’s paradox.
We then use the PrBF to perform the hypothesis testing of the PoP
design, We denote \(y_j\) the sum of
DLTs among \(n_j\) subjects that
received dose \(d_j\), for \(j=1,2,\dots,J\), following a binomial
distribution:
\[
y_j\sim\text{Bin}(n_j,\pi_j)
\]
We apply the predictive Bayes factor (prBF) for dose transition.
Dose transition
If \(\text{PrBF}_{0,1}<C(\phi,n_j)\), the
evidence is in favor of \(H_{1j}\). We
are going to transit the current dose. We assign the next cohort of
patients to an adjacent dose according to \(y_j\), such as
If \(y_j<\phi \cdot n_j\), we
escalate the dose;
If \(y_j>\phi \cdot n_j\), we
deescalate the dose.
Otherwise, we retain the current dose.
The threshold \(C(\phi, n_j)\)
describes the tolerance in strength of evidence for dose transition.
Mathematically, it can be intrinsically determined by the loss function
in making incorrect decisions.
Dose elimination
For patient safety and trial efficiency, the PoP design employs a
dose elimination rule. If the PrBF based on the observed \(y_j\) indicates a dose is above the MTD
with a certain evidence, we exclude the current dose and doses above to
avoid treating patients at overly toxic doses; alternatively, if the
PrBF implies that a dose is substantially below the MTD, we eliminate
the current dose and doses below to prevent patients from receiving
subtherapeutic doses. Such a dose elimination rule is as follow:
If \(\text{PrBF}_{0,1}<E(\phi,
n_j)\), the evidence is \(\textit{strongly}\) in favor of \(H_{1j}\) and:
If \(y_j<\phi \cdot n_j\),
the current dose is deemed as subtherapeutic and we exclude the current
dose and lower doses;
If \(y_j>\phi \cdot n_j\),
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 threshold \(E(\phi, n_j)\) is critical for the PoP
design, because it ensures the safety of the patients and efficiency of
the design by influencing the early termination rule. The plot() output
of the get.boundary.pop results demonstrates the process of the PoP
design to conduct a trial.
Theoretical properties
Compared to other model-assisted design, PoPdesign possesses these
three theoretical properties:
\(\textit{Theorem 1: Global
optimality}\):
The proposed dose transition rules for PoP design minimizes the risk
of incorrect decision of dose assignment under the Bayesian decision
framework.
\(\textit{Theorem 2: Long-term
coherence}\):
The probability of dose escalation (or de-escalation) is zero when
the observed toxicity rate \(\hat{\pi}_j\) at the current dose is higher
(or lower) than the target toxicity rate \(\phi\).
\(\textit{Theorem 3:
Consistency}\):
Dose allocation based on the escalation and de-escalation boundaries
in the PoP design converges almost surely to dose level \(j^*\) if \(\exists j^*\), s.t. \(\pi_{j^*}=\phi\).
\(\textit{Global optimality}\): and
\(\textit{Consistency}\) are two unique
properties of the PoP design.
Obtaining dose escalation and de-escalation boundaries
By specifying the number of cohorts (n.cohort), cohort size and the
target DLT rate (target), we can obtain the dose escalation and
de-escalation boundaries through the get.boundary.pop() function. The
more completed version of the decision boundaries gives a flexible
choice when the patient enrollment cannot strictly stick to the
cohortsize.
bd <- get.boundary.pop(n = 15, cohortsize = 3, target=0.3,
cutoff=2.5, K=3,cutoff_e=5/24)
summary(bd)
#> The decision boundaries are
#> [,1] [,2] [,3] [,4] [,5]
#> Number of patients treated 3 6 9 12 15
#> Escalation if # of DLT (U1) <= 0 1 1 2 3
#> Deescalation if # of DLT (U2) >= 2 3 4 5 6
#> Subtherapeutic exclusion if # of DLT (V1) <= NA NA NA 0 0
#> Overly toxic exclusion if # of DLT (V2) >= 3 5 7 8 9
#>
#> A more completed version of the decision boundaries is given by
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7]
#> Number of patients treated 1 2 3 4 5 6 7
#> Escalation if # of DLT (U1) <= NA 0 0 0 0 1 1
#> Deescalation if # of DLT (U2) >= 1 1 2 2 2 3 3
#> Subtherapeutic exclusion if # of DLT (V1) <= NA NA NA NA NA NA NA
#> Overly toxic exclusion if # of DLT (V2) >= NA NA 3 4 4 5 6
#> [,8] [,9] [,10] [,11] [,12] [,13]
#> Number of patients treated 8 9 10 11 12 13
#> Escalation if # of DLT (U1) <= 1 1 2 2 2 3
#> Deescalation if # of DLT (U2) >= 3 4 4 4 5 5
#> Subtherapeutic exclusion if # of DLT (V1) <= NA NA 0 0 0 0
#> Overly toxic exclusion if # of DLT (V2) >= 6 7 7 7 8 8
#> [,14] [,15]
#> Number of patients treated 14 15
#> Escalation if # of DLT (U1) <= 3 3
#> Deescalation if # of DLT (U2) >= 5 6
#> Subtherapeutic exclusion if # of DLT (V1) <= 0 0
#> Overly toxic exclusion if # of DLT (V2) >= 9 9
The plot() output includes one flowchart along with the dose
escalation/de-escalation table. The flowchart provides detailed
information on how to conduct the PoP design. To open the flowchart,
please extend the image Viewer window.
plot(bd)
#> format width height colorspace matte filesize density
#> 1 PNG 1339 954 sRGB TRUE 146482 72x72
#> [,1] [,2] [,3] [,4] [,5]
#> Number of patients treated 3 6 9 12 15
#> Escalation if # of DLT (U1) <= 0 1 1 2 3
#> Deescalation if # of DLT (U2) >= 2 3 4 5 6
#> Subtherapeutic exclusion if # of DLT (V1) <= NA NA NA 0 0
#> Overly toxic exclusion if # of DLT (V2) >= 3 5 7 8 9
