Why “the target dose” is not one definition
In an early-phase trial, selecting a dose requires more than asking which dose can be given without unacceptable toxicity. A dose may be tolerable but ineffective; efficacy may continue to improve after a biological response has plateaued; and the dose that looks best on one benefit-harm criterion may not be best under another. The maximum tolerated dose, minimum effective dose, optimal biological dose, and benefit-risk target name different decision goals. They should not be used interchangeably.
The dose-level estimands introduced in the earlier material on estimands for dose-level benefit and harm are quantities defined for a specified population, outcome, and assessment period. For the profiles here, let \(p_T(d)\) be the probability of the protocol-defined toxicity outcome by its specified assessment time at dose \(d\), and let \(p_E(d)\) be the probability of the specified efficacy outcome by its assessment time. These probabilities are not universal properties of a dose: they depend on the population, endpoint definitions, and time horizon.
Four target-dose concepts
The definitions below describe common operational choices. Clinical protocols and designs may use different criteria, especially for the MTD and OBD. A target should therefore be reported with its rule, not just its acronym.
- Maximum tolerated dose (MTD). A common operational MTD is the highest candidate dose whose estimated toxicity probability does not exceed a prespecified tolerability threshold \(\tau_T\). In a design using a posterior safety rule, for example, a dose may be considered acceptable only when the posterior probability that its toxicity probability exceeds \(\tau_T\) is below a specified limit. These are distinct rules: the former compares an estimate with a threshold; the latter also incorporates uncertainty. Some designs instead seek a dose with toxicity probability closest to a target rate. The design’s stated rule determines which dose is meant by MTD.
- Minimum effective dose (MED). A common rule is the lowest candidate dose whose estimated efficacy probability reaches a prespecified minimum \(\tau_E\). This definition requires an efficacy endpoint and a clinically justified threshold. It does not say that the MED is the dose with greatest efficacy, nor does efficacy alone establish that the dose is acceptably safe.
- Optimal biological dose (OBD). The OBD is selected using a prespecified measure of biological activity, such as a biomarker response or target engagement, rather than simply maximizing dose. One operational rule is the lowest dose that achieves a defined near-maximum biological effect or enters a prespecified plateau. Other rules can select the dose maximizing a biological endpoint. The rule must name the biological measure and define what counts as optimal.
- Benefit-risk target. This is a dose selected by a prespecified criterion that considers benefit and harm together, possibly subject to safety constraints. The criterion could be based on a joint outcome or a clinically specified comparison of outcomes. There is no single universal benefit-risk target: different defensible criteria can select different doses from the same profile.
These definitions answer different questions. The MTD is a tolerability boundary, the MED is an efficacy threshold, and the OBD concerns biological activity. A benefit-risk target is a decision based on an explicit combined criterion. None is guaranteed to be the dose with the largest administered amount, and none should be called “optimal” without specifying what is being optimized.
Worked Example: MTD and MED from one profile
Worked Example: MTD and MED from one profile
Hypothetical data. Consider four candidate doses, in milligrams per day. Suppose the estimated toxicity probabilities by the protocol-defined toxicity assessment time are 0.05, 0.12, 0.22, and 0.35, respectively. The estimated efficacy probabilities by the protocol-defined efficacy assessment time are 0.10, 0.35, 0.55, and 0.60. For illustration, define the MTD as the highest dose with \(p_T(d)\leq 0.25\), and the MED as the lowest dose with \(p_E(d)\geq 0.50\).
| Dose (mg/day) | Estimated toxicity probability | Estimated efficacy probability |
|---|---|---|
| 10 | 0.05 | 0.10 |
| 20 | 0.12 | 0.35 |
| 30 | 0.22 | 0.55 |
| 40 | 0.35 | 0.60 |
MTD calculation. The acceptable doses under the stated rule are 10, 20, and 30 mg/day, because their estimated toxicity probabilities are no greater than 0.25. The 40 mg/day dose is excluded because \(0.35>0.25\). The highest acceptable candidate is therefore 30 mg/day.
MED calculation. The doses meeting the stated efficacy threshold are 30 and 40 mg/day, because \(0.55\geq0.50\) and \(0.60\geq0.50\). The lowest dose meeting it is therefore 30 mg/day.
Under these particular rules, the MTD and MED coincide at 30 mg/day. This coincidence is a property of this hypothetical profile and these thresholds, not a general result. The probabilities are assumed to be valid estimates for the stated population and assessment times; a real decision must also account for estimation uncertainty and the trial’s prespecified safety procedure.
Worked Example: A biological plateau need not be a clinical target
Worked Example: A biological plateau need not be a clinical target
Hypothetical data. At doses of 5, 10, 20, and 30 mg/day, suppose a measured biomarker response is 18%, 52%, 78%, and 80% from baseline. For this example, define the biological plateau as reaching at least 95% of the largest observed response, and define the OBD as the lowest candidate dose meeting that rule.
The largest observed response is 80%. The threshold for a plateau under the example rule is
The 5 mg/day dose has an 18% response and the 10 mg/day dose has a 52% response; neither reaches 76%. The 20 mg/day dose has a 78% response, which does reach the threshold. Thus the OBD under this operational definition is 20 mg/day. The 30 mg/day dose produces only a further 2 percentage-point increase over 20 mg/day.
This calculation identifies a plateau in the measured biomarker under the chosen rule. It does not establish that 20 mg/day maximizes clinical benefit, meets an efficacy threshold, or has acceptable toxicity. Those conclusions require their own endpoints and decision rules. If the biomarker is not a valid indicator of clinical benefit, a biomarker plateau alone is not a sufficient clinical dose-selection argument.
Benefit-risk targets require an explicit criterion
A benefit-risk target does not follow from the toxicity and efficacy probabilities alone unless the decision rule connecting them has been stated. For example, a protocol could first exclude doses that exceed a safety threshold and then select among the remaining candidates using a prespecified joint outcome. Another protocol could use a different clinically justified comparison. Those choices answer different questions and can lead to different selected doses.
The following calculation illustrates one possible rule, not a universal definition. Suppose the relevant joint benefit is the probability of efficacy without the protocol-defined toxicity, \(q(d)=P(E=1,T=0\mid d)\), and the rule is to select the dose with the largest \(q(d)\) among doses satisfying \(p_T(d)\leq0.25\). The joint probabilities below are stipulated hypothetical inputs; they are not calculated from marginal probabilities, which would require additional assumptions or data about the relationship between efficacy and toxicity.
Worked Example: A benefit-risk criterion selects a different dose
Hypothetical data. At doses of 10, 20, and 30 mg/day, suppose the estimated toxicity probabilities are 0.08, 0.18, and 0.24, and the stipulated estimated probabilities of efficacy without toxicity are 0.22, 0.38, and 0.34. Use the safety constraint \(p_T(d)\leq0.25\), then choose the dose with the largest estimated probability of efficacy without toxicity.
All three doses satisfy the safety constraint: \(0.08\leq0.25\), \(0.18\leq0.25\), and \(0.24\leq0.25\). The criterion compares the joint-benefit probabilities 0.22, 0.38, and 0.34. Since \(0.38\) is largest, this particular benefit-risk rule selects 20 mg/day.
The result depends on the selected joint endpoint and safety rule. A different patient-relevant outcome, constraint, or decision criterion could select another dose. In particular, the example does not claim that the joint probabilities can be inferred by multiplying the marginal efficacy and toxicity probabilities.
A short justification for separating the rules
The target concepts can be expressed as different operations on the same finite set of candidate doses. Let \(\mathcal{D}\) denote that set, and suppose the profile quantities have been defined for each dose. Under the operational threshold definitions above, the MTD and MED are
The first selects an extreme dose from the set satisfying a toxicity constraint; the second selects an extreme dose from the set satisfying an efficacy constraint. They coincide only if the largest dose in the first set is also the smallest dose in the second set. That equality is not implied by either definition. A benefit-risk rule instead chooses a dose by comparing a specified criterion across an eligible set, and an OBD uses a biological endpoint and its own rule. Since the sets and comparison criteria differ, one cannot generally substitute one target concept for another.
The expressions also expose an important edge case: if no dose satisfies the relevant condition, the set is empty and the rule does not select a dose. It is not appropriate to silently label the lowest or highest tested dose as the target. The protocol needs a separate action for that situation, such as concluding that no tested dose meets the criterion or considering further evidence or dose levels.
Assumptions, diagnostics, and limitations
These rules are useful only if the quantities being compared are relevant and adequately estimated. At minimum, a dose-selection interpretation assumes that the candidate doses, population, endpoint definitions, and assessment windows are specified; that the estimated profile refers to those same definitions; and that the threshold or criterion has clinical justification. In a trial, estimates are uncertain, so a point estimate just below a safety threshold does not by itself prove that the dose is safe. The design must state how uncertainty affects eligibility and selection.
Many profiles are expected to increase with dose, particularly for toxicity, but monotonicity is not guaranteed. Sampling variability, changing exposure, or other features of the treatment can produce nonmonotone estimates. Do not force a monotone interpretation just because it seems clinically plausible. Inspect the dose-level estimates and uncertainty, and investigate whether a model’s assumed shape is compatible with the observed data. If estimated efficacy peaks and then falls, the highest dose is not necessarily the most effective; if estimated toxicity falls at a higher dose, that pattern may reflect uncertainty rather than a genuine reversal.
Another diagnostic is whether a proposed target is driven by a single noisy estimate or an arbitrary threshold. Examine how the selected dose changes under reasonable alternatives for the threshold, endpoint definition, or assessment time. A target that changes substantially is sensitive to those choices and should be reported as such. This sensitivity check does not identify a universally correct threshold; it makes the dependence of the decision visible.
The principal alternative approaches differ in their objective. A toxicity-only MTD rule is straightforward when the clinical goal is to identify a tolerable dose, but it does not ensure adequate efficacy. An MED rule identifies where a minimum efficacy criterion is first met, but does not by itself control toxicity. An OBD can avoid escalating beyond a biological plateau, but its relevance depends on the biomarker and its relationship to clinical outcomes. A benefit-risk rule can address outcomes together, but requires a defensible and prespecified criterion. A trial may report several of these quantities to show how conclusions depend on the goal; it should not imply that they are interchangeable.
| Concept | Primary question | What must be specified | What it does not establish alone |
|---|---|---|---|
| MTD | Which candidate is highest under the stated tolerability rule? | Toxicity outcome, assessment time, threshold, uncertainty rule | Adequate efficacy or best overall benefit-risk |
| MED | Which candidate is lowest while meeting the efficacy threshold? | Efficacy outcome, assessment time, threshold | Acceptable toxicity or maximum benefit |
| OBD | Which candidate meets the stated biological activity goal? | Biological measure and optimality or plateau rule | Clinical benefit or acceptable toxicity |
| Benefit-risk target | Which eligible candidate best meets the combined criterion? | Outcomes, constraints, and comparison criterion | A choice independent of clinical priorities |
Common mistake: treating the MTD as the universally best dose
Incorrect reasoning: “The MTD is the highest dose that meets the safety rule, so it must be the best dose to take forward.” This confuses a toxicity boundary with an overall treatment objective. A higher tolerable dose may add little efficacy, may not meet a relevant efficacy threshold, or may compare unfavorably under a combined benefit-risk criterion.
Correction: Name the objective before naming the target. If the question is tolerability, apply the stated MTD rule. If it is minimum efficacy, apply the MED rule. If biological activity is the goal, specify the OBD measure and rule. If benefits and harms are to be considered together, state the joint criterion and any safety constraints. Report the selected dose as a result of that rule, not as universally optimal.
Check Your Understanding
Answer each question using the definitions and rules stated here. Brief solutions follow the questions.
- Candidate doses are 5, 10, and 20 mg/day, with toxicity probabilities 0.06, 0.16, and 0.29. If the MTD rule is the highest dose with toxicity probability no greater than 0.20, which dose is the MTD?
- Efficacy probabilities at the same doses are 0.18, 0.42, and 0.58. If the MED rule is the lowest dose with efficacy probability at least 0.50, which dose is the MED?
- A biomarker response is 40%, 70%, and 72% across increasing doses. Under a rule that defines the plateau as at least 95% of the largest observed response and chooses the lowest dose meeting it, which dose is the OBD?
- Why can’t the probability of efficacy without toxicity generally be obtained by multiplying the marginal efficacy probability by one minus the marginal toxicity probability?
- Can a dose with the largest toxicity probability be called the benefit-risk target solely for that reason? Explain.
Solutions. (1) The 5 and 10 mg/day doses pass the rule, but 20 mg/day does not because \(0.29>0.20\); the MTD is 10 mg/day. (2) Only 20 mg/day reaches 0.50, so it is the MED. (3) The largest response is 72%, and \(0.95\times72\%=68.4\%\). The 70% response at the second dose exceeds 68.4%, so the OBD is the second dose. (4) Multiplication requires an independence assumption for efficacy and toxicity at that dose. Without independence, the joint probability depends on their association and cannot be determined from the marginal probabilities alone. (5) No. A high toxicity probability alone gives no evidence that the dose best meets a combined benefit-risk criterion; that criterion and its outcomes must be specified and applied.