Why Insert an Intermediate Dose?
An escalation ladder may contain a large gap between consecutive planned doses. If the current dose has met the protocol’s escalation criterion, moving directly to the next planned dose could make a large jump. A protocol can address this by specifying an intermediate dose and the conditions under which it enters the escalation sequence. The goal is not to choose a dose that looks attractive after seeing outcomes; it is to make a reproducible decision that follows a rule set before the decision is made.
Insertion changes the set and order of doses that participants may receive. It therefore needs more than a calculation of a midpoint. The protocol must identify the trigger, the candidate dose, the eligibility conditions, and what happens next. The rule must also preserve the trial’s existing dose-limiting toxicity (DLT) definition and observation window, rather than changing how outcomes are counted to justify a new dose.
Write the Rule Before Using It
Let \(d_L\) be the current lower dose and \(d_U\) the next planned higher dose, with \(d_L < d_U\). An insertion rule should distinguish two questions. First, does the protocol call for an intermediate level in this gap? Second, is escalation from the current level permitted by the toxicity decision rule? A large gap may satisfy the first condition while toxicity findings fail the second. The dose gap alone never authorizes escalation.
A fully specified rule should state, at minimum:
- The triggering gap condition. For example, the protocol may require insertion when \(d_U/d_L\) exceeds a stated maximum ratio. A threshold is a design choice, not a universal clinical standard.
- The dose calculation and rounding rule. State whether interpolation is arithmetic or geometric, and how a calculated value is mapped to a dose that can actually be prepared and administered.
- The safety eligibility condition. The current cohort must satisfy the protocol’s rule for escalation, with required DLT observation complete. If the rule calls for expansion or stopping instead, insertion does not override that decision.
- The sequence after insertion. State whether the new dose becomes the next level, and whether the original upper level remains unavailable until the inserted level has been evaluated under the ordinary decision rule.
- Operational and governance conditions. Confirm that the dose is deliverable, that the protocol and pharmacy instructions specify it, and that any required review or approval occurs before participants are assigned to it.
The threshold, formula, rounding, and escalation path must be selected for the study before the relevant outcomes are reviewed. A protocol amendment made after data are observed can sometimes be necessary, but it is not equivalent to a preplanned insertion rule. It should be documented and reviewed as an amendment, with the reason and the information available at the time recorded.
Choosing a Dose Between Two Levels
The appropriate interpolation depends on how doses are spaced and how the protocol defines a meaningful increment. If the protocol limits absolute increases, an arithmetic midpoint may be suitable. If it limits relative or multiplicative increases, a geometric midpoint gives equal ratios to the two endpoints.
The formula follows directly from the equal-ratio requirement:
For an arithmetic midpoint, equal absolute spacing gives \(d_M-d_L=d_U-d_M\), so \(d_M=(d_L+d_U)/2\). Neither formula is automatically preferable in every study. The protocol should use a scale that is justified for the dose units and administration context, and it must specify what to do if the computed value is not practically deliverable. Rounding should not silently produce a dose outside the permitted interval or violate the protocol’s stated increment constraint.
Worked Example: A Large Multiplicative Gap
Suppose a hypothetical protocol has planned doses of 40 mg and 90 mg. It specifies that a geometric midpoint is inserted when the ratio of the upper planned dose to the current dose exceeds 1.8. The current cohort’s DLT observation is complete and meets the protocol’s escalation criterion. Determine whether insertion is triggered and calculate the candidate dose.
The gap ratio is \(90/40=2.25\), which exceeds 1.8. The rule therefore triggers insertion, subject to the stated escalation eligibility condition, which is met here. The geometric midpoint is:
The resulting ratios are \(60/40=1.5\) and \(90/60=1.5\), so the intermediate dose divides the original multiplicative gap equally. Under the protocol, 60 mg becomes the next level to evaluate. The 90 mg dose is not automatically authorized; the 60 mg outcome must first be handled by the ordinary escalation rule.
This calculation establishes only the candidate dose and the sequence. It does not establish that 60 mg is safe, tolerated, or optimal. Those questions require observed participant outcomes under the protocol’s DLT definition and observation window.
Insertion Must Follow the Safety Decision
Dose selection and dose escalation are related but distinct. An insertion algorithm can refine the ladder, but it cannot replace the rule that determines whether escalation is allowed. For example, if the current cohort’s DLT count requires additional participants at that dose, the next action is the required evaluation at the current level—not enrollment at a newly calculated intermediate level. Likewise, a stopping or de-escalation decision is not converted into permission to insert by noting that a large gap exists.
The following order makes the distinction explicit:
Apply the protocol’s DLT definition, observation window, and escalation rule. Determine whether escalation is permitted, further evaluation is required, or escalation must stop or move downward.
Only if escalation is permitted, compare the current and next planned doses using the prespecified gap condition.
Use the specified interpolation and rounding rule. Confirm that the dose is within the intended interval and is operationally deliverable.
Record the inserted level and evaluate it under the same protocol decision rules. Do not bypass it to reach the original upper dose unless a separate prespecified rule allows that action.
Worked Example: Insertion After an Escalation-Eligible Cohort
In a hypothetical study, the planned ladder includes 25 mg followed by 100 mg. The protocol inserts a geometric midpoint whenever the ratio of adjacent planned levels exceeds 2.0. At 25 mg, three participants complete the DLT observation window, with zero DLTs. Assume the protocol’s escalation rule permits escalation after zero DLTs among three participants. Find the inserted dose and describe the next decision.
First, the gap ratio is \(100/25=4\), which exceeds the threshold of 2.0. The cohort is escalation-eligible under the stated rule, so the insertion condition may be applied. The geometric midpoint is:
The ratios are \(50/25=2\) and \(100/50=2\). The next cohort is assigned to 50 mg, not directly to 100 mg. Suppose, hypothetically, that one of the first three participants at 50 mg has a DLT. If the governing escalation rule then requires expansion at 50 mg, that expansion takes place before any move to 100 mg. The fact that 100 mg was the original next planned dose does not bypass the decision at the inserted level.
The zero DLTs at 25 mg explain why escalation was permitted in this example; they do not imply that the true DLT probability at 25 mg is zero. Nor does the midpoint calculation predict the DLT probability at 50 mg.
When No Insertion Is Permitted
A well-defined rule also specifies when not to insert. The following hypothetical case illustrates why the toxicity decision comes first. A protocol has a current dose of 40 mg and a planned next dose of 80 mg. It would insert a geometric midpoint if the dose ratio exceeds 1.5. At 40 mg, three participants complete observation and one has a DLT. Assume the protocol requires additional evaluation at 40 mg after this result rather than immediate escalation.
Worked Example: A Large Gap Does Not Override the Decision Rule
The planned-dose ratio is \(80/40=2\), which exceeds the insertion threshold of 1.5. If the gap were considered alone, the candidate geometric midpoint would be:
However, the assumed rule requires additional evaluation at 40 mg after one DLT among three. Escalation is therefore not currently permitted. The insertion condition cannot be acted on, and no participant is assigned to approximately 56.57 mg at this stage. The required evaluation at 40 mg comes first. If its result later permits escalation, the protocol’s prespecified insertion rule can then be applied if it still calls for insertion.
This is a useful distinction in protocol review: a calculated number can be mathematically valid while its use is procedurally invalid. A candidate dose is not an instruction to treat.
Auditability, Diagnostics, and Alternatives
An insertion rule is working as intended when an independent reviewer can reconstruct the decision from the protocol and the recorded data. For each proposed insertion, the record should show the current and next planned doses, completed DLT evaluations, the escalation decision, the gap calculation, the interpolation and rounding, and the authorization to use the resulting dose. A missing observation window or unresolved cohort means the escalation decision may not yet be final.
Warning signs include choosing among several possible intermediate doses after reviewing the observed DLT pattern; changing the ratio threshold for a particular gap; rounding differently depending on whether the result seems concerning; or advancing past an inserted level without applying the ordinary decision rule. These practices make the path outcome-dependent in ways the protocol did not specify. A nonmonotone observed DLT pattern can also prompt concern, but it does not justify informally changing the insertion formula or dose sequence. The response should follow the protocol and any applicable review process.
Insertion is one option among several, and each has limitations:
| Approach | What it does | Main limitation |
|---|---|---|
| Keep the original ladder | Moves to the next planned dose when the escalation rule permits. | A large planned increment remains; the study does not evaluate a dose in the gap. |
| Insert a prespecified intermediate dose | Reduces a large gap according to a fixed trigger and calculation. | Adds a dose decision and may lengthen evaluation; it does not remove uncertainty about toxicity. |
| De-escalate or stop under the existing rule | Responds to the current safety decision rather than the size of the upward gap. | Does not itself identify a suitable dose within an untested interval. |
Interpolation is a design convenience, not a dose-response model. It does not establish that toxicity changes smoothly between tested levels, and it does not guarantee that the inserted dose will be tolerable. A calculated value may also be unsuitable for a formulation, available capsule strength, or administration procedure. If the value cannot be delivered as specified, the protocol needs a prespecified resolution or formal amendment—not an improvised substitute.
Check Your Understanding
Answer each question using the rule order and calculations above. Short solutions follow.
- A protocol uses a geometric midpoint if the upper-to-current dose ratio exceeds 2.0. The doses are 30 mg and 90 mg, and escalation is permitted. Is insertion triggered, and what is the candidate dose?
- Why does the geometric midpoint balance relative increments rather than absolute increments?
- A gap meets the insertion threshold, but the current cohort’s outcome requires expansion at the current dose. May the next cohort receive the candidate intermediate dose?
- A calculated intermediate dose is 47 mg, but the protocol’s specified formulation can deliver only 45 mg or 50 mg. What should the team do?
- State two ways that selecting an intermediate dose after reviewing outcomes could undermine the rule-based design.
Solutions. (1) The ratio is \(90/30=3\), greater than 2.0, so insertion is triggered. The candidate is \(\sqrt{30\times90}\text{ mg}=\sqrt{2700}\text{ mg}\approx51.96\text{ mg}\), subject to the protocol’s rounding and deliverability rule. (2) It satisfies \(d_M/d_L=d_U/d_M\), so the ratios on either side match; equal absolute differences are not generally implied. (3) No. The required expansion at the current dose takes priority, so escalation and insertion are not yet authorized. (4) Follow the prespecified rounding or dose-mapping rule. If none applies or neither option meets the rule, do not choose informally; use the stated protocol resolution or seek a formal amendment before assigning participants. (5) Examples include selecting a different midpoint formula after seeing DLTs, altering the threshold for a particular gap, or rounding differently to favor a desired next dose. Each makes the path depend on observed outcomes beyond the prespecified rule.