Bayesian Two-Stage Biomarker-Based Adaptive Design for Targeted Therapy Development

Abstract

We propose a Bayesian two-stage biomarker-based adaptive randomization (AR) design for the development of targeted agents. The design has three main goals: (1) to test the treatment efficacy, (2) to identify prognostic and predictive markers for the targeted agents, and (3) to provide better treatment for patients enrolled in the trial. To treat patients better, both stages are guided by the Bayesian AR based on the individual patient’s biomarker profiles. The AR in the first stage is based on a known marker. A Go/No-Go decision can be made in the first stage by testing the overall treatment effects. If a Go decision is made at the end of the first stage, a two-step Bayesian lasso strategy will be implemented to select additional prognostic or predictive biomarkers to refine the AR in the second stage. We use simulations to demonstrate the good operating characteristics of the design, including the control of per-comparison type I and type II errors, high probability in selecting important markers, and treating more patients with more effective treatments. Bayesian adaptive designs allow for continuous learning. The designs are particularly suitable for the development of multiple targeted agents in the quest of personalized medicine. By estimating treatment effects and identifying relevant biomarkers, the information acquired from the interim data can be used to guide the choice of treatment for each individual patient enrolled in the trial in real time to achieve a better outcome. The design is being implemented in the BATTLE-2 trial in lung cancer at the MD Anderson Cancer Center.

Appendices

Appendix 1

The prior for $\eta _k$ in (8) can be obtained by integrating out $\tau _k^2$:

$$\begin{aligned}&\int _{0}^{\infty }{\left( {\frac{{\tau _{k}^{2} }}{{2\pi }}} \right) ^{{\frac{{m_{k} }}{2}}} } \exp \left[ { - \frac{{\left\| {\eta _{k} } \right\| ^{2} }}{2}\tau _{k}^{2} } \right] \frac{{\left( {\frac{{\lambda _{k}^{2}}}{2}} \right) ^{{\frac{{m_{k} + 1}}{2}}} }}{{\varGamma \left( {\frac{{m_{k} + 1}}{2}} \right) }}(\tau _{k}^{2} )^{{ - \frac{{m_{k} + 1}}{2} - 1}} \exp \left[ { - \frac{{\lambda _{k}^{2} }}{{2\tau _{k}^{2} }}} \right] \mathrm{d}\tau _{k}^{2}\\&\quad = \int _{0}^{\infty } {\left( \! {\frac{1}{{2\pi }}} \!\right) ^{{\frac{{m_{k} }}{2}}} } \frac{{\left( \! {\frac{{\lambda _{k}^{2} }}{2}} \!\right) ^{{\frac{{m_{k} + 1}}{2}}} }}{{\varGamma \left( \! {\frac{{m_{k} + 1}}{2}} \!\right) }}\left( \! {\frac{1}{{(\tau _{k}^{2} )^{3} }}} \!\right) ^{{\frac{1}{2}}} \exp \left[ { - \frac{{\left\| {\eta _{k} } \right\| ^{2} \left( \! {\tau _{k}^{2} - \frac{{\lambda _{k} }}{{\left\| {\eta _{k} } \right\| }}} \!\right) ^{2} }}{{2\tau _{k}^{2} }} - \lambda _{k} \left\| {\eta _{k} } \right\| } \!\right] \mathrm{d}\tau _{k}^{2}\\&\quad = \int _{0}^{\infty } {\!\left( \! {\frac{1}{{2\pi }}} \!\right) ^{{\frac{{m_{k} }}{2}}} } \frac{{\left( \! {\frac{{\lambda _{k}^{2} }}{2}} \!\right) ^{{\frac{{m_{k} + 1}}{2}}} }}{{\varGamma \left( \! {\frac{{m_{k} + 1}}{2}} \!\right) }}\left( \! {\frac{{\lambda _{k}^{2} }}{{(\tau _{k}^{2} )^{3} }}} \!\right) ^{{\frac{1}{2}}} \!\exp \left[ \! { - \frac{{\lambda _{k}^{2} \left( \! {\tau _{k}^{2} - \frac{{\lambda _{k} }}{{\left\| {\eta _{k} } \right\| }}} \!\right) ^{2} }}{{2\frac{{\lambda _{k}^{2} }}{{\left\| {\eta _{k} } \right\| ^{2} }}\tau _{k}^{2} }}} \right] \exp [ - \lambda _{k} \left\| {\eta _{k} } \right\| ]\mathrm{d}\tau _{k}^{2}\\&\quad = \left( {\frac{1}{{2\pi }}} \right) ^{{\frac{{m_{k} - 1}}{2}}} \frac{{\left( {\frac{1}{2}} \right) ^{{\frac{{m_{k} + 1}}{2}}} }}{{\varGamma \left( {\frac{{m_{k} + 1}}{2}} \right) }}\lambda _{k}^{{m_{k} }} \exp [ - \lambda _{k} \left\| {\eta _{k} } \right\| ]\int _{0}^{\infty } {\left( {\frac{{\lambda _{k}^{2} }}{{2\pi (\tau _{k}^{2} )^{3} }}} \right) ^{{\frac{1}{2}}} }\\ {}&\qquad \ \times \exp \left[ { - \frac{{\lambda _{k}^{2} \left( {\tau _{k}^{2} - \frac{{\lambda _{k} }}{{\left\| {\eta _{k} } \right\| }}} \right) ^{2} }}{{2\frac{{\lambda _{k}^{2} }}{{\left\| {\eta _{k} } \right\| ^{2} }}\tau _{k}^{2} }}} \right] \mathrm{d}\tau _{k}^{2} \end{aligned}$$

The pdf of inverted Gaussian distribution is

$$\begin{aligned} \left( {\frac{\gamma }{2\pi x^3}}\right) ^{\frac{1}{2}}\exp \left[ {-\frac{\gamma ( {x-\mu })^2}{2\mu x}} \right] . \end{aligned}$$

When letting $\tau _k^2 =x$, $\frac{\lambda _k }{\left\| {\eta _k } \right\| }=\mu$, $\lambda _k^2 =\gamma$, and integrating out $\tau _k^2$, we get the prior in (8).

Appendix 2

We used the same true scenario in Table 2 to simulate true patient response. However, instead of randomizing patients based on individual biomarker information (BATTLE-2 design strategy), patients were randomized based on biomarker group information (BATTLE-1 design strategy). Following BATTLE-1 design strategy, 3 biomarkers with decreasing priorities were used to group patients into 4 groups. A patient will be in group 1 if the first marker (highest priority) is positive; otherwise, the patient will be in group 2 if the second marker is positive; otherwise, the patient will be in group 3 if the third marker is positive; and the patient will be in group 4 if all 3 biomarkers are negative.

For the null case with the BATTLE-1 design strategy, the 3 markers used to group patients were marker 3, 4, and 5. Three different alternative cases were simulated using BATTLE-1 design strategy to represent 3 different situations. In Alternative 1 case, we assumed that the 3 true predictive markers were correctly identified, and marker 3, 4 and 5 were used to group patients. Unless specified explicitly, we always assumed that the marker with lower index has higher priority. In Alternative 2 case, we used marker 3, 6, and 7 to group patients, which means that one true predictive marker has been assigned the highest priority by chance. In Alternative 3 case, we used 3 randomly chosen markers to group patients, and this is equivalent to grouping patients with no credible marker information. The same hypothesis testings for the Go/No-Go decisions as in the simulation for BATTLE-2 design were conducted at the end of stage 1, and the same interim futility analyses were implemented. To make the results comparable to BATTLE-2 design, KRAS mutation and prior erlotinib history, in additional to the marker group information, were always included in the adaptive randomization. The same models with KRAS mutation and prior erlotinib history in (1) and (2) were used in hypothesis testing, and the marker group information was only used in adaptive randomization. Therefore, the differences in simulation results of BATTLE-1 and 2 design strategies were due to differences in randomization strategies and the added variable selection in BATTLE-2:

• (1) BATTLE-1 design strategy: Markers were used to group patients, and group information was used in adaptive randomization in BATTLE-1 design;

• (2) BATTLE-2 design strategy: Individual biomarkers were subjected to variable selection and, upon being selected, markers were directly used in adaptive randomization in BATTLE-2 design.

For comparison purpose, we simulated the clinical trials by following the design strategy of BATTLE-1 trial, and the results for hypothesis testing at the end of stage 1 are shown in Supplemental Table 1. The patient responses were simulated from the same true scenarios in Table 2. The same decision criteria and modeling approaches of BATTLE-2 design were also used for BATTLE-1 simulation. The error rates were controlled at the same rate of 10 % for each experimental treatment. Therefore, the differences in operating characteristics between BATTLE-1 and BATTLE-2 simulations can be attributed to the different strategies of incorporating biomarker information in adaptive randomization. The simulation results for Alternative 1 case in Supplemental Table 1 show that, in the best scenario where the predictive markers were correctly identified before the start of the clinical trial, BATTLE-1 design has higher power on the treatment (arm 2) associated with the marker having the highest priority, and has lower power on the remaining 2 experimental treatments (arm 3 and 4). In the worst scenario where none of the true predictive markers were correctly identified before the start of the trial (Alternative 3 case), BATTLE-2 trial design has higher power on all experimental treatments. In practice, Alternative 3 case is more likely to happen than Alternative 1 case. Because the purpose of clinical trial is to study experimental treatment and the biomarkers associated the treatment. It is unlikely that we are going to have perfect biomarker information at the beginning of clinical trial.

The patient allocation results for BATTLE-1 design strategy are shown in Supplemental Table 2. As compared with the results in Table 5 for BATTLE-2 design, the predictive marker with the highest priority can allocate more patients to the beneficial treatment arm in the most optimal situation (Alternative 1 and 2 cases) where the predictive marker has been correctly identified before the trial. In all other cases, the BATTLE-1 design strategy failed to learn from ongoing trial to treat patients better.

If credible informative biomarkers are available, using pre-selected biomarkers will be more powerful [49]. Our simulation comparison with BATTLE-1 design strategy also demonstrated that. However, in the exploratory stage of study when no credible biomarker information is available, our simulations showed that combining biomarkers into biomarker groups may lead to higher error rate in biomarker selection.

We also simulated the first stage of the trial using an equal randomization. The results are shown in the Supplemental Table 3. Under the null hypothesis, when the overall error rate for the trial is still controlled at 20 %, the average error rate for each experimental treatment arm is about 1–2 % lower (0.077–0.091 under ER and 0.088–0.100 for AR). When the alternative hypothesis is true, the increase in average power for each experimental treatment arm can be as high as 15 % (0.97 to 0.99 for ER and 0.82 to 0.95 for AR). However, the increase of the overall power of the trial is only 2 % (0.999 for ER and 0.978 for AR). With AR and early stopping rule, it can result in a sample size saving of 60 patients under the null hypothesis. AR also resulted in more patients to be treated with more effective treatments.

Appendix 3

The likelihood function of logistic model can be expressed as:

$$\begin{aligned} \mathop \prod \limits _{i=1}^n \left[ {\left( {\frac{\exp ( {X_i \theta })}{1+\exp ({X_i \theta })}}\right) ^{y_i } \left( {\frac{1}{1+\exp ( {X_i \theta })}}\right) ^{1-y_i }} \right] =\frac{\exp ( {\mathop \sum \nolimits _{i\in \Omega _R } X_i \theta })}{( {1+\exp ( {X_i \theta })})^n}, \end{aligned}$$

where $i$ is the index for individual patient, $n$ is the total number of patients, ${{\varvec{\Omega }}}_R$ is the set of index for all responders, $X_i$ is a generic representation of covariate, and $\theta$ is a generic representation of model parameter. Without loss of generality, we assume that both $X_i$ and $\theta$ are scales.

For Bayesian adaptive lasso, the prior for $\theta$ given $\lambda$ is

$$\begin{aligned} \pi (\theta \vert \lambda )\propto \text{ exp }\left( -\lambda \frac{\vert \theta \vert }{\hat{\theta }}\right) , \end{aligned}$$

where $\hat{\theta }$ is a square root n consistent initial estimate of $\theta$. The square of $\lambda$ follows a gamma distribution:

$$\begin{aligned} \pi ( {\lambda ^2})=\frac{\lambda ^{2(a-1)}e^{-\frac{\lambda ^2}{b}}}{b^a\varGamma (a)}, \end{aligned}$$

where $a$ and $b$ are hyper parameters.

The conditional posterior distributions of $\theta$ and $\lambda$ are

$$\begin{aligned} \theta \vert \lambda \propto \frac{\exp ( {\mathop \sum \nolimits _{i\in {{\varvec{\Omega }}}_R } X_i \theta })}{( {1+\exp ( {X_i \theta })})^n}{\cdot }\text{ exp } \left( {-\lambda \frac{\left| \theta \right| }{\hat{\theta }}}\right) \end{aligned}$$

$$\begin{aligned} \lambda \vert \theta \propto \lambda ^{2( {a-1})}e^{-\frac{\lambda ^2}{b}}{\cdot }\exp \left( {-\lambda \frac{\left| \theta \right| }{\hat{\theta }}}\right) . \end{aligned}$$

It is easy to show that all above conditional distributions are log-concave. Therefore, the adaptive rejection sampling implemented in OpenBugs can be conveniently utilized to generate the posterior distribution of model parameters for Bayesian adaptive lasso.

Appendix 4

For group parameter ${\varvec{\eta }}$ in Bayesian adaptive group lasso, the prior distribution is

$$\begin{aligned} \pi (\eta \mid \lambda )\propto \text{ exp }(-\lambda \vert \left| \eta \right| \vert /\vert |{\hat{{\varvec{\eta }}}}|\vert ), \end{aligned}$$

where $\hat{{\varvec{\eta }}}$ is an initial estimate of $\eta$. In our simulation, we used the gamma mixture of normals in Eq. (9). After $\tau _k$ in Eq. (9) is integrated out, Appendix 1 shows that these two formulations are equivalent while conditional on $\lambda$.

The same gamma prior distribution is assumed for the square of $\lambda$:

$$\begin{aligned} \pi ( {\lambda ^2})=\frac{\lambda ^{2(a-1)}e^{-\frac{\lambda ^2}{b}}}{b^a \varGamma (a)} \end{aligned}$$

The likelihood function of logistic model can be expressed as:

$$\begin{aligned} \mathop \prod \limits _{i=1}^n \left[ {\left( {\frac{\exp ( {X_i^\mathrm{T} \eta })}{1+\exp ( {X_i^\mathrm{T} \eta })}}\right) ^{y_i } \left( {\frac{1}{1+\exp ( {X_i^\mathrm{T} \eta })}}\right) ^{1-y_i }} \right] =\frac{\exp ( {\mathop \sum \nolimits _{i\in {{\varvec{\Omega }}}_R } X_i^\mathrm{T} \eta })}{( {1+\exp ({X_i^\mathrm{T} \eta })})^n}. \end{aligned}$$

Both $X_i$ and $\eta$ are column vectors. Without loss of generality, we assume that there is only one group of variables in the model. Therefore, the posterior distribution of model parameters can be sampled from

$$\begin{aligned} \begin{array}{l} \eta ,\tau ,\lambda \vert X,Y \\ \propto \frac{\exp ( {\mathop \sum \nolimits _{i\in {{\varvec{\Omega }}}_R } X_i^\mathrm{T} \eta })}{( {1+\exp ( {X_i^\mathrm{T} \eta })})^n}\tau ^m\text{ exp }\left[ {-\frac{\left\| \eta \right\| ^2}{2}\tau ^2} \right] \lambda ^{m+1}\tau ^{-m-3}\text{ exp }\left[ {-\frac{\lambda ^2}{2 \Vert {\hat{{\varvec{\eta }}}}\Vert ^2\tau ^2}} \right] \lambda ^{2(a-1)}\exp \left[ {-\frac{\lambda ^2}{b}} \right] , \\ \end{array} \end{aligned}$$

where Y is a vector of observed response outcomes, $m$ is the dimension of $\eta$, and $\left\| \eta \right\| ^2=\eta _1^2 +\cdots +\eta _m^2$. The full conditional distributions are

$$\begin{aligned} \eta \vert \tau ,\lambda ,X,Y\propto \frac{\exp ( {\mathop \sum \nolimits _{i\in {{\varvec{\Omega }}}_R } X_i^\mathrm{T} \eta })}{( {1+\exp ( {X_i^\mathrm{T} \eta })})^n}\text{ exp }\left[ {-\frac{\left\| \eta \right\| ^2}{2}\tau ^2} \right] \end{aligned}$$

$$\begin{aligned} \tau \vert \eta ,\lambda ,X,Y\propto \tau ^{-3}\text{ exp }\left[ {-\frac{\lambda ^2}{2 \Vert {\hat{{\varvec{\eta }}}} \Vert ^2\tau ^2}} \right] \text{ exp }\left[ {-\frac{\left\| \eta \right\| ^2}{2}\tau ^2} \right] \end{aligned}$$

$$\begin{aligned} \lambda \vert \eta ,\tau ,X,Y\propto \lambda ^{2a+m-1}\text{ exp }\left[ {-\lambda ^2\left( {\frac{1}{b}+\frac{1}{2 \Vert {\hat{{\varvec{\eta }}}} \Vert ^2\tau ^2}}\right) } \right] . \end{aligned}$$