# A Bayesian dose-finding design for drug combination clinical trials based on the logistic model.

TL;DR: Under the proposed design, the posterior estimates of the model parameters continuously update to make the decisions of dose assignment and early stopping, and the design is competitive and outperforms some existing designs.

Abstract: In early phase dose-finding cancer studies, the objective is to determine the maximum tolerated dose, defined as the highest dose with an acceptable dose-limiting toxicity rate. Finding this dose for drug-combination trials is complicated because of drug–drug interactions, and many trial designs have been proposed to address this issue. These designs rely on complicated statistical models that typically are not familiar to clinicians, and are rarely used in practice. The aim of this paper is to propose a Bayesian dose-finding design for drug combination trials based on standard logistic regression. Under the proposed design, we continuously update the posterior estimates of the model parameters to make the decisions of dose assignment and early stopping. Simulation studies show that the proposed design is competitive and outperforms some existing designs. We also extend our design to handle delayed toxicities. Copyright © 2014 John Wiley & Sons, Ltd.

## Summary (3 min read)

### 1 Introduction

- For oncologists, the objective of phase I dose-finding studies is to determine the maximum tolerated dose (MTD), defined as the highest dose with a relatively acceptable dose-limiting toxicity (DLT) [1, 2].
- Mandrekar et al. proposed an approach incorporating the toxicity and efficacy of each agent into the identification of an optimal dosing region for the combination by using a continuation ratio model to separate each agent’s toxicity and efficacy curves [8, 9].
- Most of these existing designs rely on complicated statistical models that typically are not familiar to clinicians, which hinder their acceptance and application in practice.
- In addition, the performance of these designs seems comparable and there is no consensus which design should be used [13].

### Dose-combination model

- Let there be a two-drug combination used in a phase I dose-finding clinical trial for which the dose-toxicity relationship is monotonic and increases with the dose levels.
- Before two agents can be combined, each of them typically have been thoroughly investigated individually.
- Therefore, there is often rich prior information on pj’s and qk’s and their values can be readily elicited from physicians.
- Using the prior information to define standardized dose has been widely used in dose-finding trial designs, and the most well known example perhaps is the skeleton of the continuous reassessment method (CRM) [21] with a logistic model.
- Research has shown that this approach improves the estimation stability and trial performance [2, 22].

### Likelihood and posterior inference

- Under the proposed model, the likelihood is simply a product of the Bernoulli density, given by L(β0, β1, β2, β3|data) ∝ (2) We sample this posterior distribution using Gibbs sampler, which sequentially draws each of the parameters from their full conditional distributions .the authors.the authors.
- Dose finding algorithm and determination of the MTD During the trial conduct, the authors use the dose-finding algorithm proposed by Yin and Yuan [11, 10] to determine dose escalation and deescaltion, and propose a different criterion for MTD selection at the end of the trial.
- Shown in dark gray is the AUC for a toxicity probability greater than 0.4, which is equal to the probability of overdosing.

### BCOPULA and BGUMBEL methods

- Yin and Yuan proposed two Bayesian methods that use copula regression for combinations.
- The parameter γ characterizes the drug interactive effect, and α and β the uncertainty of the initial guesses.
- The combination allocation algorithm is the same as that presented in Section 2.1.
- The final MTD is the combination with a toxicity probability closest to the target among the combinations already administered in the trial.

### LOGODDS

- By backsolving this equation, an explicit expression for the probability of toxicity is obtained.
- A normal prior centered in 0 and with variance of 100 was chosen for the interaction parameter.
- The rest of the dose allocation process, estimation and MTD determination was the same as their proposed design in order to compare their method involving a simple interaction logistic model with other logistic models.

### One-dimensional CRM

- In practice, one-dimensional CRM sometimes is used to conduct dose-combination trials [13].
- Under this method, the authors first preselect a subset of combinations, for which the toxicity probability order is known, and then apply the standard CRM to find the MTD.
- The drawback of such an approach is that the authors only investigate a subset of the whole two-dimensional dose space and may miss the true MTD.

### 3 Simulations

- The authors simulated 2000 independent replications of phase I trials that evaluate two agents in drug combinations, with five dose levels for Agent 1 and three for Agent 2, giving 15 possible combinations.
- The authors fixed the toxicity target at 0.3 and used an overall sample size of 60.
- The design parameters of all the designs (e.g., working model) have been calibrated via simulation before used for the comparison.
- The authors selected these features in order to employ typical trial set-ups in their simulation study.
- The authors set the length around the targeted interval, δ, at 0.1.

### 4 Results

- For each scenario, the authors present the correct MTD selection rate, or percentages of correct selection (PCS), in Table 2.
- For these scenarios, all model-based designs gave high PCS, whereas LOGISTIC seemed to perform better than the other methods.
- The addition of the stopping rule for unacceptable toxicity resulted in PCS that were similar to those presented above (Table 5), except in scenario 4 in which the first dose combination, (1, 1), was the MTD.

### 4.1 Sensitivity analysis

- The authors conducted a sensitivity analysis in order to study the performance of their design using different prior distributions and parameters values.
- According to Table 5, the authors can see that the PCS for all scenarios were very similar under these different prior distributions.

### 4.2 Time-to-event outcome

- In practice, a longer follow-up time may be required to assess the toxicity outcome.
- Before combination assignment, the likelihood is defined as L(β0, β1, β2, β3|data) =.
- The authors simulated the time-to-toxicity outcomes using an exponential distribution such that the toxicity probabilities at the end of follow-up matched those given in Table 1.
- Table 6 shows the results of the extended LOGISTIC for all 15 scenarios.
- The authors observe that the performance of the design decreases only slightly by 2%, and the PCS for all scenarios are still very high.

### 5 Discussion

- The authors have proposed a statistical method for clinical trial designs that evaluate various dose combinations for two agents.
- One benefit of their method compared with the other proposed designs is that it is also efficient when the MTDs are not necessarily located on the same diagonal.
- When combining several agents, designs developed for single-agent dose-finding trials cannot be applied to combination studies.
- This approach performs well if the target dose combinations happen to be included in the subset.
- These files are freely available upon request.

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##### References

1,277 citations

### "A Bayesian dose-finding design for ..." refers methods in this paper

...In practice, one-dimensional CRM sometimes is used to conduct dose-combination trials [13]....

[...]

...We also compared the performance of this multidimensional designs with a one-dimensional CRM, CRM anti-diag, as described previously....

[...]

...As the CRM focused on only a subset of doses, it had more patients per dose to find the target doses than the multidimensional designs, given the same total number of patients....

[...]

...Under this method, we first preselect a subset of combinations, for which the toxicity probability order is known, and then apply the standard CRM to find the MTD....

[...]

...Using the prior information to define standardized dose has been widely used in dose-finding trial designs, and the most well-known example perhaps is the skeleton of the continuous reassessment method (CRM) [21] with a logistic model....

[...]

730 citations

### "A Bayesian dose-finding design for ..." refers methods in this paper

...Therefore, as suggested by other authors [10,16,20,26,27], we implement an algorithm-based start-up phase in order to gather enough information to estimate the j,k ....

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654 citations

### "A Bayesian dose-finding design for ..." refers methods in this paper

...These full conditional distributions do not have closed forms, and we use the adaptive rejection Metropolis sampling (ARMS) method [23] to sample them....

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560 citations

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...Our start-up phase shares the spirit of accelerated titration design [28] and can be described as follows: Treat the first cohort of patients at the lowest dose combination ....

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428 citations

### "A Bayesian dose-finding design for ..." refers background or methods in this paper

...For instance, the standard algorithm-based method for phase I dose-finding clinical trials in oncology is the so-called “3+3” design, which is referred to as “memory-less” since allocation to the next dose level for an incoming group of 3 patients depends only upon what has happened to the total of 3 to 6 patients previously treated at the current dose level [14, 15, 16, 17, 18]....

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...As a result, many of the dose-finding clinical trials conducted to evaluate drug combinations still use the conventional “3+3” approach, which was developed for single agents and was shown to be inefficient in terms of dose identification [14, 15, 16, 17, 18]....

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