Introduction of gBOIN design
Assume there are
J specified doses
d1<⋯<
dJ under investigation. Let
y denote the toxicity outcome which is either binary or quasi-binary (e.g., DLT or ETS) or continuous (e.g., TTB, TBS or TTP). For the motivating trial, after an appropriate transformation, we take the AUC as a continuous end point and model it by a normal distribution. [
14] adopted the binomial and the normal distributions for binary (or quasi-binary) and continuous endpoints, respectively. Define
μ=E(
y) and
μj=E(
y|
dj). Given the dose
dj, the distribution of
y belongs to the exponential family,
$$ f(y|d_{j}) = h(y)\exp\left\{\eta(\theta_{j})T(y)-A(\theta_{j})\right\}, $$
(1)
where,
-
θj=μj, η(θj)= log{μj/(1−μj)}, A(θj)=− log(1−μj), T(y)=y, and h(y)=1, if y follows a binomial distribution;
-
θj=(μj,σ2), η(θj)=μj/σ2, \(A(\theta _{j}) = \mu _{j}^{2}/(2\sigma ^{2})\), T(y)=y, and \(h(y) = \frac {1}{\sqrt {\pi }\sigma }\exp \left \{-y^{2}/(2\sigma ^{2})\right \}\), if y follows a normal distribution.
Let
ϕ0 denote the target value of
μ for dose finding. Specifically, for binary or quasi-binary toxicity endpoints,
ϕ0 is the target DLT probability; for continuous endpoints,
ϕ0 is the targeted value of the TTB, TBS or TTP. Assume there are
nj patients treated at dose level
dj and let
\({\mathcal {D}}_{j} = (y_{1}, \cdots, y_{n_{j}})\) denote the observed toxicity data. Based on
\({\mathcal {D}}_{j}\), the sample mean can be obtained as
\(\hat {\mu }_{j} = \sum _{i=1}^{n_{j}} y_{i} /n_{j}\). For the interval-based design, dose transition decisions are made by comparing
\(\hat {\mu }_{j}\) with the decision boundaries,
λe(
dj,
nj,
ϕ0) and
λd(
dj,
nj,
ϕ0). Specifically, if
\(\hat {\mu }_{j} < \lambda _{e}(d_{j}, n_{j}, \phi _{0})\), escalate to the higher dose level
j+1, and if
\(\hat {\mu }_{j} > \lambda _{d}(d_{j}, n_{j}, \phi _{0})\), de-escalate to the lower dose level
j−1, otherwise retain the same dose level
j. The selection of the decision boundaries
λe(
dj,
nj,
ϕ0) and
λd(
dj,
nj,
ϕ0) is critical because these two parameters essentially determine operating characteristics of a design. Let the decisions retainment, escalation and de-escalation (each based on the current dose level), denoted as
\({\mathcal {R}}\),
\({\mathcal {E}}\) and
\({\mathcal {D}}\), respectively and let
\(\overline {\mathcal {R}}\) denote the decisions that are complementary to
\({\mathcal {R}}\) (i.e.,
\(\overline {\mathcal {R}}\) includes
\({\mathcal {E}}\) and
\({\mathcal {D}}\)), and
\(\overline {\mathcal {E}}\) and
\(\overline {\mathcal {D}}\) denote the decisions that are complementary to
\({\mathcal {E}}\) and
\({\mathcal {D}}\), respectively. Following the same rule of [
9], to obtain optimal decision boundaries under some criteria, the gBOIN [
14] considers three point hypotheses
H0:
μj=
ϕ0,
H1:
μj=
ϕ1,
H2:
μj=
ϕ2 and minimize an incorrect decision probability
α,
$$ {}\alpha = \mathrm{P}(H_{0}) \mathrm{P}(\overline{\mathcal{R}} |H_{0}) + {\mathrm{P}}\left(H_{1}\right){\mathrm{P}}\left({\overline{\mathcal{E}} \left| {{H_{1}}} \right.} \right) + {\mathrm{P}}\left(H_{2}\right){\mathrm{P}}\left(\overline {\mathcal{D}}|H_{2} \right), $$
(2)
where
ϕ1 is a value deemed subtherapeutic such that dose escalation is warranted, and
ϕ2 is a value deemed overly toxic such that dose de-escalation is required. Note that,
H0 indicates that the current dose is the MTD and we should retain the current dose for the next cohort of patient;
H1 indicates that the current dose is below the MTD and we should escalate the dose; and
H2 indicates that the current dose is overly toxic and we should deescalate the dose. Thus, the correct decisions under hypotheses
H0,
H1 and
H2 are retainment, escalation and de-escalation. Correspondingly, the incorrect decisions under
H0,
H1 and
H2 are
\(\overline {\mathcal {R}}\),
\(\overline {\mathcal {E}}\) and
\(\overline {\mathcal {D}}\), respectively. For example, under
H0 (i.e., the current dose is the target), the correct decision is to retain the current dose (i.e.,
\({\mathcal {R}}\)), and incorrect decisions are dose escalation and de-escalation (i.e.,
\({\mathcal {E}}\) and
\({\mathcal {D}}\)). Taking a noninformative prior, i.e., P(
H0)=P(
H1)=P(
H2)=1/3, and minimizing the incorrect decision probability
α in Eq. (
2), the decision boundaries can be obtained as (details can be found in [
14]),
$$ \lambda_{e}^{*} = \frac{A(\vartheta_{1})-A(\vartheta_{0})}{\eta(\vartheta_{1}) - \eta(\vartheta_{0})},\quad \lambda_{d}^{*} = \frac{A(\vartheta_{2})-A(\vartheta_{0})}{\eta(\vartheta_{2}) - \eta(\vartheta_{0})}. $$
(3)
Specifically, when
y follows a Bernoulli or quasi-Bernoulli distribution, we have
𝜗k=
ϕk,
A(
𝜗k)=− log(1−
ϕk),
η(
𝜗k)= log{
ϕk/(1−
ϕk)}. Then,
$$\begin{array}{*{20}l} {\lambda_{e}^{*}} = \frac{\log \frac{ 1-\phi_{1}}{1-\phi_{0}}}{\log\frac{\phi_{0}(1-\phi_{1})}{(1-\phi_{0})\phi_{1}}}, \qquad {\lambda_{d}^{*}} =\frac{\log\frac{1-\phi_{0}}{1-\phi_{2}}}{\log\frac{\phi_{2}(1-\phi_{0})}{(1-\phi_{2})\phi_{0}}}, \end{array} $$
(4)
which are exactly the same as boundaries provided by the original BOIN design [
9]. When
y follows a normal distribution, we have
\(\vartheta _{k}=\left (\phi _{k}, \sigma _{j}^{2}\right)\),
\(A\left (\vartheta _{k}\right) = \phi _{k}^{2}/\left (2\sigma _{j}^{2}\right)\),
\(\eta \left (\vartheta _{k}\right) = \phi _{k}/\sigma _{j}^{2}\). Then,
$$\begin{array}{*{20}l} {\lambda_{e}^{*}} =\frac{\phi_{0}+\phi_{1}}{2 },\qquad {\lambda_{d}^{*}} =\frac{\phi_{0}+\phi_{2}}{2 }. \end{array} $$
(5)
Based on the above decision boundaries, the gBOIN design is summarized as follows:
(a)
Patients in the first cohort are treated at the lowest dose level or at a prespecified dose level.
(b)
At the current dose level
j, assign a dose to the next cohort of patients,
-
if \({\hat \mu _{j}} \le {\lambda _{e}^{*}}\), escalate the dose level to j+1,
-
if \({\hat \mu _{j}} \ge {\lambda _{d}^{*}}\), de-escalate the dose level to j−1, and
-
otherwise, i.e., \({\lambda _{e}^{*}} < {\hat \mu _{j}} < {\lambda _{d}^{*}}\), retain the same dose level, j.
(c)
This process is continued until the maximum sample size is reached or the trial is terminated because of excessive toxicities.
It is remarkable that the optimal decision boundaries \(\left ({\lambda _{e}^{*}}, {\lambda _{d}^{*}}\right)\) are free of dj and nj, which means that the same pair of boundaries are used throughout the trial no matter which dose is the current dose, nor how many patients have been treated at the current dose.
Adaptive gBOIN design
Extensive simulation studies have shown that the gBOIN is transparent and simple to implement, and it yields good performance that is comparable or superior to more complicated model-based designs. As we described in the “
Introduction” section, the un-fixed boundaries may allow a flexibility to penalize mis-allocation rate of patients at over-toxic doses. To account for un-fixed boundaries, firstly, we reformulate the above three hypotheses as follows,
$$H_{0}: \mu_{j} = \phi_{0} \quad versus \quad H_{1}: \mu_{j} = \phi_{1}, $$
and
$$H_{0}: \mu_{j} = \phi_{0} \quad versus \quad H_{2}: \mu_{j} = \phi_{2}. $$
In the Bayesian paradigm, the Bayes factor in favor of the alternative hypothesis
H1 against a fixed null hypothesis
H0 is defined as,
$$ \text{BF}_{10}(D_{j}) = \frac{\mathrm{P}(H_{1}|D_{j})/\mathrm{P}(H_{0}|D_{j})}{\mathrm{P}(H_{1})/\mathrm{P}(H_{0})}, $$
(6)
and the null hypothesis
H0 is rejected if BF
10(
Dj) exceeds a prespecified threshold
γ1. Similarly, the Bayes factor in favor of the alternative hypothesis
H2 against a fixed null hypothesis
H0 is defined as,
$$ \text{BF}_{20}(D_{j}) = \frac{\mathrm{P}(H_{2}|D_{j})/\mathrm{P}(H_{0}|D_{j})}{\mathrm{P}(H_{2})/\mathrm{P}(H_{0})}, $$
(7)
and the null hypothesis
H0 is rejected if BF
20(
Dj) exceeds a prespecified threshold
γ2. Note that, if we want to put more penalties on over-toxic allocation, values of
γ1 and
γ2 would be different and presumably
γ1 should be greater than
γ2 since smaller
γ2 means decisions of de-escalation are easier made if over-toxicities occur. Given the prior odds P(
Hk)/P(
H0)=1 and the threshold
γk,(
k=1,2), we can determine an alternative hypothesis that maximize the probability that the Bayes factor forms a test exceed the specified threshold
γk. In other words, here we can choose the value of
\(\phi _{k}^{*}, (k=1,2)\) (this notation has been introduced in the “
Introduction” section) to maximize P(BF
k0(
Dj)>
γk).
By the Lemma 1 of [
21],
\(\phi _{1}^{*}\) and
\(\phi _{2}^{*}\) can be obtained by,
$$\begin{array}{*{20}l} {}\phi_{1}^{*} = \underset{\mu_{j}<\phi_{0}}{\arg\max}~g_{\gamma_{1}}(\mu_{j}, \phi_{0}) \quad and \quad \phi_{2}^{*} = \underset{\mu_{j}>\phi_{0}}{\arg\min}~g_{\gamma_{2}}(\mu_{j}, \phi_{0}) \end{array} $$
(8)
respectively, where \(g_{\gamma _{k}}(\mu _{j}, \phi _{0}) = \frac {\log (\gamma _{k})+n_{j}\{A(\theta _{j})-A(\theta _{0})\}}{\eta (\theta _{j})-\eta (\theta _{0})}\), k=1,2.
Specifically, for binomial distribution,
\(\phi _{1}^{*}\) and
\(\phi _{2}^{*}\) can be given as,
$$\begin{array}{*{20}l} {}&\phi_{1}^{*} = \underset{\mu_{j}<\phi_{0}}{\arg\max}~\frac{\log(\gamma_{1})-n_{j}\left\{\log(1-\mu_{j})-\log(1-\phi_{0})\right\}}{\log\left\{\mu_{j}/(1-\mu_{j})\right\} - \log\left\{\phi_{0}/(1-\phi_{0})\right\}}, \\ {}&\phi_{2}^{*} = \underset{\mu_{j}>\phi_{0}}{\arg\min}~\frac{\log(\gamma_{2})-n_{j}\left\{\log(1-\mu_{j})-\log(1-\phi_{0})\right\}}{\log\left\{\mu_{j}/(1-\mu_{j})\right\} - \log\left\{\phi_{0}/(1-\phi_{0})\right\}}. \end{array} $$
(9)
Obviously, the values of
\(\phi _{1}^{*}\) and
\(\phi _{2}^{*}\) depend on the target
ϕ0, the sample size
nj and the threshold
γk,
k=1,2. Although their close forms cannot be obtained, they can be solved via numerical optimization methods. For normal distribution,
\(\phi _{1}^{*}\) and
\(\phi _{2}^{*}\) can be given as,
$$\begin{array}{*{20}l} & \phi_{1}^{*} = \phi_{0} - \sigma\sqrt{\frac{2\log\gamma_{1}}{n_{j}}}, \\ & \phi_{2}^{*} = \phi_{0} + \sigma\sqrt{\frac{2\log\gamma_{2}}{n_{j}}}. \end{array} $$
(10)
Note that, for the normal distribution, values of \(\phi _{1}^{*}\) and \(\phi _{2}^{*}\) depend on the value of σ. So, if σ is unknown, we can replace it with its sample estimation \(\hat {\sigma } = \sqrt {\left \{\sum _{i=1}^{n_{j}} \left (y_{i} - \hat {\mu }_{j}\right)^{2}\right \}/n_{j}}\), or alternatively, we can take an Inverse Gamma distribution with shape parameter α0 and rate parameter β0 as its prior, then σ can be replaced by using its posterior mean \(\left (2\beta _{0} + \sum _{i=1}^{n_{j}} \left (y_{i} - {\mu }_{j}\right)^{2}\right)/ \left (n_{j} + \alpha _{0}\right)\) with μ replaced by \(\hat {\mu }_{j}\).
Replacing
ϕk in
\(\lambda _{e}^{*}\) and
\(\lambda _{d}^{*}\) with
\(\phi _{k}^{*}\),
k=1,2, we can get the adaptive shrinkage decision boundaries
\(\lambda _{e}^{*}(n_{j})\) and
\(\lambda _{d}^{*}(n_{j})\). Note that, for a standard binary toxicity endpoint, if we take the same values for
γk,
k=1,2, the boundaries are the same as the UMPBI design [
22]. Based on Lemma 2 in [
21], we have the following double-shrinkage property theorem about the shrinkage boundaries
\(\lambda _{e}^{*}(n_{j})\) and
\(\lambda _{d}^{*}(n_{j})\).
Theorem 1 introduces a double-shrinkage property for the proposed adaptive gBOIN design: The optimal values \(\phi _{k}^{*}\) shrink toward the target toxicity probability ϕ0, and the optimal boundaries \(\lambda _{e}^{*}(n_{j})\) and \(\lambda _{d}^{*}(n_{j})\) based on each combination of \(\phi _{1}^{*}\) and \(\phi _{2}^{*}\) shrinkage toward the target value ϕ0.
Now we give the procedure of the proposed gBOINS design as follows.
(a)
Patients in the first cohort are treated at the lowest dose level or at a prespecified dose level.
(b)
At the current dose level
j, to assign a dose to the next cohort of patients,
-
if \({\hat \mu _{j}} \le {\lambda _{e}^{*}(n_{j})}\), escalate the dose level to j+1,
-
if \({\hat \mu _{j}} \ge {\lambda _{d}^{*}(n_{j})}\), de-escalate the dose level to j−1, and
-
otherwise, i.e., \({\lambda _{e}^{*}(n_{j})} < {\hat \mu _{j}} < {\lambda _{d}^{*}(n_{j})}\), retain the same dose level, j.
(c)
This process is continued until the maximum sample size is reached or the trial is terminated because of excessive toxicities.
After the trial has been completed, we use the pooled adjacent violators algorithm [
23] to select a dose level as the MTD. Denote the isotonically transformed values of the observed value
\(\{\hat \mu _{j}\}\) by
\(\{\tilde \mu _{j}\}\), to be specific, for finding the MTD, we select dose
j∗, for which the isotonic estimate of the toxicity rate
\(\tilde \mu _{j^{*}}\) is closest to
ϕ0; if there are ties for
\(\tilde \mu _{j^{*}}\), we select from the ties the highest dose level when
\(\tilde \mu _{j^{*}} < \phi _{0}\) or the lowest dose level when
\(\tilde \mu _{j^{*}} >\phi _{0}\).
For patient safety, we impose the following overdose control rule when using the gBOIN design.
If \({\mathrm {P}\left ({{\mu _{j}} > \phi _{0} \left | {\mathcal {D}}_{j} \right.} \right) > 0.95}\) and nj≥3, dose levels j and higher are eliminated from the trial, and the trial is terminated if the first dose level is eliminated.
Posterior probability \({\mathrm {P}\left ({{\mu _{j}} > \phi _{0} \left | {\mathcal {D}}_{j} \right.} \right) > 0.95}\) can be evaluated on the basis of a beta-binomial model for the binary or quasi-binary endpoint, assuming μj follows a vague beta prior, e.g., μj∼beta(1,1). For normal endpoint y with mean μj and variance \(\sigma _{j}^{2}\), assuming noninformative prior \((\mu, \sigma _{j}^{2}) \propto \sigma ^{-2}\), the posterior distribution of μj follows a t distribution with nj−1 degrees of freedom, mean \(\hat \mu _{j}\) and scale \(n_{j}^{-1}\sum _{i=1}^{n_{j}}\left (y_{i} - \hat \mu _{j}\right)^{2}\).
Design properties
From a practical viewpoint, a natural requirement for dose-finding trials is that dose escalation should be not allowed if the observed toxicity rate or mean toxicity score at the current dose is higher than the target, and dose de-escalation should not be allowed if the observed toxicity rate or mean toxicity score at the current dose is lower than the target. [
9] referred to this finite sample property as “long-term memory coherent”, which is an extension of a similar concept originally proposed by [
24]. That original definition of design coherence requires the prohibition of dose escalation (or de-escalation) when the observed toxicity rate in the most recently treated cohort is more (or less) than the target toxicity rate. Because that definition is based on the response from only the most recently treated cohort without considering responses from patients who were previously enrolled and treated, [
9] refers this definition as “short-term memory coherence”. Clearly, short-term memory coherence is a stronger counterpart than long-term memory coherence.
As shown in the Appendix, the gBOINS design has the following desirable finite-sample property.
To further enhance safety of the design, we let the upper boundary \(\phi _{2}^{*}\) have a little bit faster shrinking rate than that of the lower boundary \(\phi _{1}^{*}\), since more strict or smaller \(\phi _{2}^{*}\) has less risk of exposing participated patients to over-toxic doses. We propose to take γk as \(\gamma _{k} = \exp (c_{k}n_{j}^{\varepsilon _{k}})\), k=1,2, 1>ε1≥ε2>0 and \(0< c_{1} < n_{j}^{1-\epsilon _{1}}\log (1/(1-\phi _{0}))\) and \(c_{1}< c_{2} < n_{j}^{1-\epsilon _{2}}\log (1/\phi _{0})\). It can be shown that the proposed adaptive gBOIN design has the following desirable large-sample property.
According to Theorem 1, the condition
\(\gamma _{k} = \exp (c_{k}n_{j}^{\varepsilon _{k}})\),
ck>0,
k=1,2, imposed here to leverage the converge rate of
\(\lambda _{e}^{*}{(n_{j})}\) and
\(\lambda _{d}^{*}{(n_{j})}\), yielding
\(\mathrm {P}\{\hat {\mu }_{j} \in (\lambda _{e}^{*}{(n_{j})}, \lambda _{d}^{*}{(n_{j})})\} =1 \), because
\(\hat {\mu }_{j}\) converges in probability to
μj at the
\(\sqrt {n}\) rate. Following the proof of Theorem 1 of [
25], the result can be directly obtained and is omitted here.
Practical implementation
To implement the proposed gBOINS design in practice, we need to specify the values of
εk and
ck,
k=1,2. We recommend the
εk=0.5,
k=1,2. The values of
ck,
k=1,2 need to be calibrated by extensive simulation studies, and even there are no uniform values for different type of endpoints with the same target. For the normal endpoints, the shrinkage boundaries depend on the estimate of
σ, this will influence the pre-tabulation and the simplicity of gBOINS. For practical applications, we suggest to replace it with 1.1
ϕ0. Note that a big (small) value of
\(\lambda _{e}^{*}(n_{j})\) (or
\(\lambda _{d}^{*}(n_{j})\)) will make dose escalation (de-escalation) rapidly, this may lead serious safety problems and reduce the efficiency of the design when the sample size is small, since the smaller value of the sample size the bigger variance of
\(\hat {\mu }_{j}\). To avoid this adverse event problem and improve the design’s efficiency, in practice, we introduce a lead-in process in a trial to follow the original gBOIN design for a pre-specified number of patients (denoted as
N0). After
nj>
N0, the trial is then switched to the gBOINS design. For our simulations,
N0=6 is recommended. Table
1 shows examples of the values of
\((\lambda _{e}^{*}(n_{j}), \lambda _{d}^{*}(n_{j}))\) for target
ϕ0=0.2 and
ϕ0=0.3.
Table 1
Dose escalation and de-escalation boundaries for Bernoulli and continuous toxicity endpoint, with ϕ0=0.2, ϕ0=0.3, εk=0.5, k=1,2 and N0=6
Bernoulli | ϕ0=0.2 | \(\lambda _{e}^{*}{(n_{j})}\) | 0.16 | 0.16 | 0.16 | 0.17 | 0.17 | 0.17 | 0.17 | 0.17 | 0.17 | 0.17 |
or | | \(\lambda _{d}^{*}{(n_{j})}\) | 0.24 | 0.24 | 0.22 | 0.22 | 0.22 | 0.22 | 0.22 | 0.22 | 0.22 | 0.22 |
quasi- | ϕ0=0.3 | \(\lambda _{e}^{*}{(n_{j})}\) | 0.24 | 0.24 | 0.24 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.26 | 0.26 |
Bernoulli | | \(\lambda _{d}^{*}{(n_{j})}\) | 0.36 | 0.36 | 0.33 | 0.33 | 0.33 | 0.33 | 0.33 | 0.33 | 0.33 | 0.32 |
Continuous | ϕ0=0.2 | \(\lambda _{e}^{*}{(n_{j})}\) | 0.16 | 0.16 | 0.17 | 0.17 | 0.18 | 0.18 | 0.18 | 0.18 | 0.18 | 0.18 |
| | \(\lambda _{d}^{*}{(n_{j})}\) | 0.24 | 0.24 | 0.22 | 0.21 | 0.21 | 0.21 | 0.21 | 0.21 | 0.21 | 0.21 |
| ϕ0=0.3 | \(\lambda _{e}^{*}{(n_{j})}\) | 0.24 | 0.24 | 0.26 | 0.26 | 0.27 | 0.27 | 0.27 | 0.27 | 0.27 | 0.27 |
| | \(\lambda _{d}^{*}{(n_{j})}\) | 0.36 | 0.36 | 0.32 | 0.32 | 0.32 | 0.32 | 0.32 | 0.32 | 0.32 | 0.32 |