Applying 3+3 Rules to Combination Dose Escalation

Why Combination Escalation Needs a Path

With one drug, dose levels can usually be listed from lower to higher. With two drugs, a candidate dose is a combination: one specified dose of drug A paired with one specified dose of drug B. Increasing A while holding B fixed and increasing B while holding A fixed are different moves. A collection of candidate combinations is therefore not automatically one ordered ladder.

A 3+3 rule can still be applied, but the protocol must specify which combination is tested next and which dose component changes. The resulting recommendation depends on that prespecified path. The rule does not, by itself, compare every combination in the grid or estimate the effects of drug interaction.

Before applying the rule, the protocol must define the dose grid, the escalation path, the DLT (dose-limiting toxicity) definition and assessment window, and how outcomes are classified. Each participant in a cohort receives the same combination and schedule. The standard rule assumes that the cohort’s DLT outcomes are available and evaluable under those definitions before the next decision is made. It also presumes that the combination and schedule are delivered consistently enough for a cohort-level decision to be meaningful.

For interpreting observed proportions, a simple working description is that each participant at a particular combination has the same probability \(p_{ij}\) of a DLT in the specified window, and that participants’ DLT outcomes are independent. Then \(X_{ij}\), the number of DLTs in a cohort of size \(n\) at \(C_{ij}\), has a binomial distribution with parameters \(n\) and \(p_{ij}\). This description is useful for understanding uncertainty; the 3+3 decision itself is a count-based algorithm, not a fitted dose-toxicity model.

The 3+3 Decision at One Combination

At each combination on the path, begin with three evaluable participants. The usual rule is:

• If 0 of 3 have a DLT, move to the next combination permitted by the path.
• If 1 of 3 has a DLT, treat three more participants at the same combination. If no more than 1 of all 6 has a DLT, move to the next permitted combination. If at least 2 of 6 have a DLT, stop escalation; the preceding path combination is the rule-based recommendation, if one exists.
• If at least 2 of the first 3 have a DLT, stop escalation without enrolling three more at that combination. The preceding path combination is the rule-based recommendation, if one exists.

Thus, “no more than 1 of 6” means one DLT in total across both cohorts, not one DLT in each cohort. A protocol may have additional constraints on which combinations can be entered or what to do if no preceding combination exists. Those constraints must be specified rather than inferred from the 3+3 counts.

Why the Recommendation Is Path-Dependent

The escalation path is part of the design, not a cosmetic ordering of the data. In a two-drug grid, the next step after \(C_{11}\) could be an increase in A, an increase in B, or a protocol-defined move that changes both. Those choices expose participants to different combinations and yield different observations. The 3+3 counts do not determine which move is appropriate; that is a design decision made in advance.

What Sparse Combination Data Can and Cannot Show

At a given combination, the observed DLT proportion is \(x/n\), where \(x\) is the number of DLTs and \(n\) is the number of evaluable participants at that combination. It is a description of those observations, not a guarantee about the underlying DLT probability \(p_{ij}\). Small cohorts leave substantial uncertainty, and results at one combination do not fill in missing cells elsewhere in the grid.

A simple likelihood calculation illustrates this point. Under the binomial working description, the probability of observing no DLTs in three participants is \((1-p_{ij})^3\). If the true DLT probability were 0.10, this probability would be \(0.90^3=0.729\). If it were 0.30, it would be \(0.70^3=0.343\). Thus, 0 DLTs among 3 participants is possible under either probability. It supports following the specified rule; it does not prove that the combination is safe or that its DLT probability is zero.

Checks, Alternatives, and Limitations

Before acting on a combination 3+3 result, verify that the observations match the design conditions: the intended dose and schedule were administered, the DLT window was defined consistently, and the cohort counts use the protocol’s evaluability and outcome-classification rules. If outcomes are still pending, classifications are ambiguous, or the actual dose differs from the assigned dose, the count should not be treated as if it were a complete, comparable cohort. The protocol must state how such circumstances affect decisions.

Also check the path itself. If a proposed next combination changes both drugs when the protocol permits only one component to change, or if it is not among the permitted next steps, the ordinary rule does not authorize that move. Nonmonotone observed patterns across combinations may occur, particularly when combinations differ in more than one component. They are a reason to describe the observed cells accurately, not a reason to invent an ordering after seeing the outcomes.

A one-drug-at-a-time escalation path is a straightforward way to apply 3+3, but it examines only the combinations on that path. A design that varies one drug while holding the other at a fixed dose answers a narrower question about that chosen background dose. Other dose-finding approaches can use a broader model of dose and combination effects, but require different assumptions and procedures. The simple 3+3 rule is transparent and operationally easy to specify; it is limited by small cohorts, path dependence, and weak information about untested combinations. It should not be presented as a map of the full dose-toxicity surface.