Combination Dose-Finding Design (Two Agents)

Trial Inputs

The implementation follows the published Bayesian copula design. The calculator is intended for the model-based phase after the published start-up rule has supplied initial toxicity information in both dose directions.

Target toxicity φ

Escalation cutoff c_e

De-escalation cutoff c_d

Agent A prior toxicity skeleton p_j

Agent B prior toxicity skeleton q_k

Enter increasing probabilities separated by commas. The published design treats the highest single-agent dose as the previously established single-agent MTD and uses these probabilities as the prior skeleton.

Current Agent A dose

Current Agent B dose

Observed combination data

For each combination enter n = patients treated and x = DLTs. Unentered combinations are treated as n = 0.

Recommendation

Posterior inference under the two-agent Clayton copula regression model, using numerical integration over the three model parameters.

Enter the trial inputs and click Get Dose Recommendation.

Methodology

The calculator implements the Bayesian two-agent combination dose-finding design of Yin and Yuan (2009). Each agent has a prespecified single-agent toxicity skeleton, p_j for Agent A and q_k for Agent B. The uncertainty in those skeletons is represented by positive power parameters α and β, so the marginal toxicity probabilities are p_j^α and q_k^β.

π_jk = 1 − { (1 − p_j^α)^−γ + (1 − q_k^β)^−γ − 1 }^−1/γ

The joint toxicity probability π_jk is linked through a Clayton-type copula. γ is the positive interaction parameter. The observed data contribute a binomial likelihood across all dose combinations:

L(α,β,γ | data) ∝ ∏_j,k π_jk^x_jk(1 − π_jk)^n_jk−x_jk

The published implementation uses independent Gamma(2,2) priors for α and β and a vague Gamma(0.1,0.1) prior for γ. The original paper obtains the posterior with Gibbs sampling; this browser implementation evaluates the same likelihood and priors with deterministic numerical quadrature instead, so that the result is reproducible without an external server or random-number generator.

Dose decision rule

If Pr(π_jk < φ | data) > c_e, escalate one dose level to an adjacent combination whose posterior toxicity estimate is higher than the current combination and closest to φ.
If Pr(π_jk > φ | data) > c_d, de-escalate one dose level to an adjacent combination whose posterior toxicity estimate is lower than the current combination and closest to φ.
Otherwise, stay at the current combination.
Diagonal moves that change both agents simultaneously are not allowed. At the lowest combination, a de-escalation signal terminates the trial; at the highest combination, an escalation signal leaves the dose unchanged.

The published design also uses a conservative start-up rule: patients are first escalated vertically through combinations (A₁,B₁), (A₁,B₂), … until a toxicity is observed, then horizontally through (A₂,B₁), (A₃,B₁), … until a toxicity is observed. This calculator begins with the model-based decision stage rather than simulating that start-up sequence.

Published worked configuration and validation

Fixed validation target: the published simulation configuration uses φ = 0.40, c_e = 0.80, c_d = 0.45, Agent A skeleton (0.08, 0.16, 0.24, 0.32, 0.40), and Agent B skeleton (0.075, 0.15, 0.225, 0.30). The browser implementation was checked with a constructed three-patient cohort at A₂B₂ with 0 DLTs and no other observed data. Using the same priors and quadrature grid, the posterior mean toxicity at A₂B₂ is 0.2465, Pr(π₂₂ < 0.40 | data) = 0.8440, and the decision is Escalate to A₃B₂. The adjacent posterior means are 0.2955 for A₃B₂ and 0.2920 for A₂B₃, so A₃B₂ is the closer eligible escalation.

References

Yin, G. & Yuan, Y. (2009). Bayesian dose finding in oncology for drug combinations by copula regression. Journal of the Royal Statistical Society: Series C (Applied Statistics), 58(2), 211–224. The paper specifies the copula model, binomial likelihood, Gamma priors, escalation/de-escalation cutoffs, one-level non-diagonal dose transitions, and start-up rule.

Yin, G. & Yuan, Y. (2009). A latent contingency table approach to dose finding for combinations of two agents. Biometrics, 65(3), 866–875.

For broader combination-trial context, the NCI Investigational Drug Steering Committee notes that model-based approaches can be useful when escalating both agents and emphasizes the need to account for potential pharmacologic interactions and overlapping toxicity.

Combination Dose-Finding Design (Two Agents)

Trial Inputs

Recommendation

Methodology

Dose decision rule

Published worked configuration and validation

References

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