BOIN12 Decisions With Joint Benefit And Toxicity

Why Combine Benefit and Toxicity?

In dose finding, a dose can produce more efficacy and more toxicity at the same time. Ranking doses by response rate alone can therefore favor a dose with an unacceptable safety profile; ranking them by toxicity alone can discard a useful dose. BOIN12 addresses this tradeoff by combining joint efficacy-toxicity outcomes with a prespecified desirability scale, while retaining a safety screen.

A BOIN12 decision depends on the clinical priorities encoded in its utility scores and on its protocol-defined safety and allocation rules. There is no universal utility table or cutoff that is appropriate for every trial. The calculations below use a transparent illustrative specification to show how the framework works, not to prescribe settings for a real trial.

Model and Decision Ingredients

For a basic calculation, assume each participant has a fully observed binary efficacy outcome and binary toxicity outcome, classified using the same definitions and follow-up windows at every dose. At a given dose, assume participants' joint outcomes are independent draws from a common four-category probability vector. The category probabilities may reflect dependence between efficacy and toxicity; the model does not require those two outcomes to be independent.

One convenient model assigns a Dirichlet prior to the four category probabilities. With a symmetric Dirichlet prior having parameter \(0.5\) for each category, the posterior parameters are the observed category counts plus \(0.5\). This prior gives a uniform Beta(1,1) marginal prior for toxicity probability: the two toxicity categories together have prior parameter 1, as do the two non-toxicity categories. If \(x_j\) participants among \(n_j\) treated at dose \(j\) experience toxicity, the posterior marginal distribution is therefore:

Here \(p_{T,j}\) is the toxicity probability at dose \(j\). A protocol may declare a dose inadmissible when the posterior probability that its toxicity probability exceeds a specified limit is too high. For illustration, use a toxicity limit of \(0.30\) and exclude a dose if \(P(p_{T,j}>0.30\mid\text{data})>0.95\). These two cutoffs are design choices, not universal constants.

For the utility calculation, use scores of 100 for efficacy without toxicity, 40 for neither outcome, 60 for both outcomes, and 0 for toxicity without efficacy. Larger scores represent more desirable outcomes. Other clinically justified scores could lead to different decisions, so the utility table must be chosen before using trial results to guide allocations.

From Joint Outcomes to Posterior Desirability

Let \(N_{jk}\) be the observed count in category \(k\) at dose \(j\), and let \(u_k\) be that category's utility. Under the stated Dirichlet prior, the posterior mean probability for category \(k\) is its observed count plus \(0.5\), divided by the total count plus 2. Because expectation is linear, the posterior expected desirability is the utility-weighted sum of these posterior mean probabilities.

This is a short justification for the calculation: if \(U_j=\sum_k u_k\pi_{jk}\), where \(\pi_{jk}\) is the probability of category \(k\), then \(E(U_j\mid\text{data})=\sum_k u_kE(\pi_{jk}\mid\text{data})\). The Dirichlet posterior mean is \(E(\pi_{jk}\mid\text{data})=(N_{jk}+0.5)/(n_j+2)\), yielding the displayed expression. No independence assumption between efficacy and toxicity is used in this utility calculation.

The safety screen and desirability score answer different questions. The screen rules out doses whose toxicity is too concerning under the chosen posterior criterion. Desirability then compares the remaining doses using both outcomes. A protocol must also specify how allowable doses constrain allocation, such as restricting the next cohort to the current dose or an adjacent dose. The following examples state a simple local rule explicitly: allocate to the dose with the largest posterior expected desirability among the admissible current dose and its immediately adjacent admissible dose or doses. A complete trial protocol may add boundaries, stopping rules, or other constraints.

Updating a Decision After New Outcomes

Dose decisions are recalculated as outcomes accumulate. The following example shows why a posterior safety calculation and a utility calculation should both be updated rather than relying on a previous allocation or a raw observed rate alone.

Utility Choices, Comparisons, and Diagnostics

A desirability score makes clinical tradeoffs explicit, but does not make them objective. In the example, efficacy with toxicity receives 60 points, more than neither outcome's 40 points. If the clinical team considers toxicity especially harmful, it might assign a lower score to efficacy with toxicity. For example, changing that score to 30 changes the three initial posterior expected desirabilities to \(51.92\), \(56.82\), and \(40.94\), respectively. Dose 2 remains the preferred admissible dose in this dataset, but the scores and possibly the ranking would change in other data.

The principal alternatives answer different questions. A toxicity-only interval rule focuses on controlling toxicity and does not directly reward efficacy. Selection by observed efficacy rate ignores uncertainty and may favor a dose with unacceptable toxicity. A constrained optimal-biological-dose rule can impose explicit efficacy and toxicity conditions before optimizing another target. BOIN12's utility-based comparison instead summarizes the joint outcomes according to prespecified preferences, alongside its safety restrictions. No one approach removes the need to justify the clinical objective.

Before applying the calculation, verify that outcome categories are mutually exclusive and exhaustive, follow-up is sufficient for classifying outcomes, and the same definitions are used across doses. Check that the four counts sum to \(n_j\), that toxicity counts agree with the categories containing toxicity, and that the utility ordering reflects the protocol's clinical priorities. A high posterior expected utility at a dose that fails the safety screen is not a reason to ignore the screen.

The model also has limits. A simple dose-specific Dirichlet model does not borrow information across doses or impose a monotone dose-response relationship. A decision based on posterior means does not show how uncertain the utility ranking is; two doses can have similar means but very different posterior uncertainty. Delayed or unevaluable outcomes require additional handling rather than being silently counted as no toxicity or no efficacy. Finally, allocations made using accumulated outcomes can affect later evidence, so the entire decision algorithm should be evaluated before a trial begins.