What the 3+3 Rule Calls the MTD
The 3+3 design turns dose-finding decisions into a prespecified sequence of cohort decisions. Its maximum tolerated dose (MTD) convention is not simply “the dose with the most toxicity that was tried.” The selection depends on where the trial stops and on the observed dose-limiting toxicity (DLT) counts at that dose and the doses below it.
This tutorial uses the conventional rule-based design: participants are treated in cohorts of three at ordered dose levels, and escalation proceeds only after the DLT count at the current level meets the rule’s criterion. A count of one DLT among the first three leads to expansion at that same dose; a count of at least two among three, or at least two among six after expansion, stops escalation. When this first excessive-toxicity result occurs above a dose that has passed its applicable rule, the MTD convention selects that preceding, highest cleared dose.
Here, “excessive toxicity” is an operational decision threshold, not a claim that a dose’s true toxicity probability has been measured precisely. At a dose with 0 DLTs among 3, the rule permits escalation. With 1 DLT among 3, it pauses escalation and treats 3 more participants at that dose. After six have been assessed, 0 or 1 DLT permits escalation; 2 or more ends escalation. The observation window and the definition of a DLT must be specified before applying these counts, so that participants’ outcomes are judged consistently.
Thresholds and Selection Logic
Let \(x\) denote the number of participants with a DLT at a dose, and \(n\) the number assessed there. The rule’s decisions can be summarized as follows. Counts are cumulative at that dose: after expansion, the count includes all six participants, not just the additional three.
| Participants assessed at a dose | Observed DLT count | 3+3 action |
|---|---|---|
| 3 | 0 | Clear the dose; escalate to the next planned level. |
| 3 | 1 | Expand to 6 participants at this dose; do not yet escalate. |
| 3 | 2 or 3 | Stop escalation; select the highest preceding dose that passed its rule, if one exists. |
| 6 | 0 or 1 | Clear the dose; escalate to the next planned level. |
| 6 | 2 or more | Stop escalation; select the highest preceding dose that passed its rule, if one exists. |
The numerical cutoffs follow directly from the rule’s permitted counts. Among six participants, the largest count that clears a dose is one, giving an observed fraction of \(1/6 = 0.1667\), or about 16.7%. Two DLTs among six give \(2/6 = 0.3333\), or about 33.3%, and trigger stopping. Among the first three, one DLT is not yet a stopping result: it triggers expansion, because the rule requires more observations at that dose before deciding whether to escalate.
The MTD selection follows from the stopping location. Suppose dose levels are ordered from lower to higher. Once level \(j\) meets a stopping criterion, the candidate MTD is level \(j-1\), provided that level \(j-1\) had previously cleared its applicable criterion. The selection is not the stopping dose itself: that dose is the first one with excessive observed toxicity under the algorithm.
Why the Selection Follows from the Rule
The 3+3 procedure has two linked operations: it classifies each dose as cleared or excessively toxic based on its observed count, then uses the ordering of the dose ladder to make a selection when escalation stops. The justification is operational rather than a probability model: each decision is the direct consequence of the prespecified count thresholds.
With 0 of 3 DLTs, move up. With 1 of 3, enroll 3 more at the same level. With at least 2 of 3, stop immediately.
For 6 participants in total at that dose, 0 or 1 DLT clears the dose for escalation; 2 or more triggers stopping.
If stopping occurs at a dose above the starting level, select the immediately preceding level that passed its applicable criterion. If stopping occurs at the starting level, no MTD is identified.
The rule does not ask whether the observed DLT fraction is below a separately estimated target, nor does it fit a dose-toxicity curve. It uses a small set of integer counts. The simple arithmetic helps make the decision reproducible, but it also limits what the selected dose can establish about the underlying toxicity probability.
Worked Example: One DLT at the Intermediate Dose
Suppose a hypothetical trial uses dose levels of 10, 20, 40, and 80 mg/m². At 10 mg/m², 0 of 3 participants have a DLT. At 20 mg/m², 1 of the first 3 has a DLT, and 1 of the 3 additional participants has a DLT. At 40 mg/m², 0 of 3 have a DLT. At 80 mg/m², 2 of the first 3 have a DLT. Assume each participant completed the prespecified DLT observation window and that the counts are correctly recorded.
At 10 mg/m², \(x=0\) among \(n=3\), so the dose clears for escalation. At 20 mg/m², the first cohort has \(x=1\) among \(n=3\), so the rule requires expansion. The cumulative count after expansion is \(1+1=2\) DLTs among 6 participants:
Because \(2\) is at least \(2\), 20 mg/m² does not clear for escalation. Under the rule, escalation would stop there; the later results at 40 and 80 mg/m² therefore cannot belong to a trial that followed this conventional algorithm. The MTD would be the preceding level, 10 mg/m², if the observed sequence had stopped at 20 mg/m². This check reveals that the described sequence is internally inconsistent with the 3+3 escalation rule.
To make the sequence consistent, suppose instead the expanded cohort at 20 mg/m² had 0 additional DLTs. Then the cumulative result would be \(1+0=1\) DLT among 6, and the trial could escalate. At 40 mg/m², 0 of 3 clears the dose. At 80 mg/m², 2 of 3 triggers stopping, so 40 mg/m² is selected as the MTD under the convention.
Worked Example: Stopping After Expansion at the Highest Dose Reached
In a hypothetical study, the planned ladder is 5, 10, 15, and 20 mg/m². The observed counts are 0 of 3 at 5 mg/m², 0 of 3 at 10 mg/m², and 1 of the first 3 at 15 mg/m². After expansion at 15 mg/m², the cumulative count is 2 DLTs among 6. All outcomes are assessed in the same prespecified window.
At 5 and 10 mg/m², each cohort has 0 of 3 DLTs, so both levels clear for escalation. At 15 mg/m², the first result, \(1/3\), calls for expansion rather than immediate stopping. After the additional 3 are assessed, the total is \(2/6\):
Since the cumulative count is at least 2, escalation stops at 15 mg/m². The highest preceding level that cleared the rule is 10 mg/m², so the conventional MTD selection is 10 mg/m². The 15 mg/m² dose is not selected: it is the dose at which the stopping threshold was reached. The 20 mg/m² dose is not considered because the trial stops before treating a cohort there.
Worked Example: Excessive Toxicity at the Starting Dose
Consider a hypothetical trial whose lowest planned dose is 2 mg/kg. Two of the first three participants at that dose have a DLT during the observation window. The cohort is complete and the DLT assessments are evaluable.
Here, \(x=2\) and \(n=3\), so \(x\geq 2\). The rule stops escalation immediately; it does not require three more participants at this dose. Because 2 mg/kg is the starting dose, there is no preceding level that passed the rule:
The correct conclusion is that this 3+3 run identifies no MTD. It would be incorrect to report 2 mg/kg as the MTD, since that is the level that triggered stopping. A protocol could specify another course of action, such as testing a lower dose, but that action would require a prespecified rule beyond the conventional selection described here.
Conditions, Interpretation, and Alternatives
A 3+3 MTD selection is meaningful only if the trial’s operational conditions support the counts and the dose ordering. The rule assumes that dose levels are specified in advance and ordered, that participants in a cohort receive the same dose, and that escalation decisions are made using the DLT outcomes required by the protocol. In particular, a participant whose DLT window is incomplete cannot simply be counted as having no DLT. The protocol must state how to handle unresolved outcomes before the rule is applied.
The rule can always be mechanically applied to the observed counts, but the clinical interpretation also relies on a sensible dose ladder and consistent DLT assessment. A generally increasing toxicity risk with dose is often expected in dose finding, but the rule’s arithmetic does not verify that pattern. An unexpected count at a lower dose is a reason to check data, DLT adjudication, and protocol conduct rather than to treat the highest observed dose automatically as the MTD.
It is also important to distinguish “MTD selected by the rule” from “true toxicity rate known to be acceptable.” For example, 0 DLTs among 3 does not prove that the population DLT probability at that dose is small. It is only the observed count that allows escalation under this algorithm. The selected MTD is therefore a design-based decision from the trial’s data, not a guarantee that future patients will have a particular risk.
| Approach | How the dose decision is made | Relation to the 3+3 convention |
|---|---|---|
| Conventional 3+3 | Uses prespecified cohort sizes and integer DLT thresholds. | Selection is transparent and rule-based, but uses only a small amount of information at each level. |
| Model-based dose finding | Uses a specified statistical model to relate dose to toxicity and guide dose selection. | Can use information across dose levels, but requires model specification and model-based decisions. |
| Target-toxicity selection | Frames the goal in terms of a prespecified toxicity probability or acceptable range. | Clarifies the clinical target, but the conventional 3+3 thresholds do not directly estimate or guarantee that target. |
The 3+3 rule is attractive when a simple, auditable escalation algorithm is needed. Its simplicity should not be confused with precision: the selected dose may depend strongly on a few observed DLT outcomes, and the procedure does not use all possible information about dose and toxicity. The introductory material on the conventional 3+3 escalation algorithm describes the cohort decisions; the key point here is how the first stopping level determines the MTD convention.
Common Pitfalls
- Calling the stopping dose the MTD. The stopping dose is the first level that reaches the excessive-toxicity threshold. The convention selects the highest preceding level that cleared its rule, if such a level exists.
- Treating 1 of 3 as an automatic stop. One DLT among the first three calls for expansion to six at that dose. The decision is made from the cumulative count after expansion.
- Ignoring the cohort denominator. The rule distinguishes 2 of 3 from 1 of 3 and 2 of 6 from 1 of 6. State both the numerator and denominator before choosing an action.
- Reporting an MTD when the first dose is too toxic. If the starting level triggers stopping, there is no lower cleared level to select; the rule identifies no MTD.
- Interpreting the result as a precise toxicity-rate estimate. The rule classifies observed counts. It does not establish that the selected dose has a particular true DLT probability.
Check Your Understanding
Answer each question using cumulative DLT counts at the dose level.
- At a dose, 1 of the first 3 participants has a DLT. What does the rule require next?
- A dose has 1 DLT among 6 after expansion, and the next dose has 2 DLTs among its first 3 participants. Which dose is selected?
- The starting dose produces 2 DLTs among 3 participants. What is the MTD under the conventional rule?
- Why does 0 DLTs among 3 not prove that the true DLT probability at that dose is zero?
Solutions. (1) Expand to 6 participants at the same dose; do not yet escalate. (2) The first dose clears because \(1/6\) is no more than 1 DLT among 6. The next dose triggers stopping at \(2/3\), so the preceding cleared dose is selected. (3) No MTD is identified because the stopping dose is the lowest planned dose and there is no lower cleared level. (4) Zero observed events in three participants is a sample result; other participants could experience DLTs, so the count does not determine the true population probability.