From Exposure–Outcome Evidence to a Candidate Interval
A target concentration is not established merely by finding a concentration associated with a favorable outcome. The evidence must support a clinically interpretable balance: exposure should be high enough to make benefit sufficiently likely and low enough to keep toxicity sufficiently unlikely. This tutorial develops that logic from exposure–response and exposure–toxicity relationships. It builds on the introductory material on defining a drug-specific target concentration by showing how the two relationships jointly constrain a candidate interval.
Let \(C\) denote the prespecified drug exposure measure used for this decision, expressed in a stated unit such as mg/L. The specific exposure measure must be appropriate to the drug and clinical question; choosing among measures is a separate task. Define \(p_R(c)\) as the probability of a specified beneficial response at exposure \(c\), and \(p_T(c)\) as the probability of a specified toxicity by a specified time or under a specified assessment. These probabilities are conditional on the population and conditions represented by the evidence. They are not guarantees about an individual patient.
This definition makes the decision explicit. A model describes how outcome probabilities vary with exposure; clinical criteria state what probabilities are acceptable. Neither curve alone defines the target. The candidate set can be an interval, a single point, or empty, depending on the curves, criteria, and admissible domain.
Deriving Cutoffs Under Increasing Relationships
A useful starting model assumes that both the probability of response and the probability of toxicity increase with exposure. For illustration, suppose each probability follows a logistic curve:
where \(\operatorname{logit}(p)=\ln\{p/(1-p)\}\), and \(b_R>0\) and \(b_T>0\). The positive slopes encode the increasing-probability assumption. If the criteria satisfy \(0<q_R<1\) and \(0<q_T<1\), the corresponding finite cutoffs are
Because the response curve is increasing, \(p_R(c)\ge q_R\) holds exactly when \(c\ge C_R\). Because the toxicity curve is increasing, \(p_T(c)\le q_T\) holds exactly when \(c\le C_T\). Thus the candidate set is the intersection of the exposure domain with both constraints:
If \(C_R>C_T\), there is no exposure satisfying both criteria under these assumptions. If \(C_R\le C_T\), the interval still must be intersected with \(D\); it is empty if that intersection is empty. In particular, with nonnegative concentrations the result is not allowed to include negative values.
The finite-cutoff formulas require the criteria to be strictly between zero and one. Under a logistic model with finite coefficients, the probability is strictly between zero and one at every finite \(c\). Therefore, a criterion such as \(q_R=0\) makes the response requirement automatically satisfied, while \(q_R=1\) cannot be met at any finite exposure. For toxicity, \(q_T=1\) makes the toxicity constraint automatic, while \(q_T=0\) cannot be met at finite exposure. When endpoint criteria are allowed, apply the probability inequalities directly rather than substituting into a logit formula.
The cutoff result follows directly from monotonicity. Since the logistic function is strictly increasing, applying its inverse preserves the direction of an inequality. For example:
where the final inequality keeps its direction because \(b_R>0\). For toxicity, rearranging \(p_T(c)\le q_T\) gives \(c\le C_T\). If a slope is negative, or the relationship is nonmonotone, these same interval inequalities do not follow; the feasible set must instead be obtained from the probability inequalities over the domain.
Worked Example: Calculating a Narrow Candidate Interval
Suppose hypothetical evidence is summarized by \(\operatorname{logit}\{p_R(c)\}=-2+0.8c\) and \(\operatorname{logit}\{p_T(c)\}=-5+0.9c\), with \(c\) in mg/L. The clinical criteria are \(q_R=0.70\) and \(q_T=0.15\), and the admissible domain includes the resulting nonnegative concentrations. Both slopes are positive, and both criteria lie strictly between zero and one, so the finite-cutoff derivation applies.
For response, \(\operatorname{logit}(0.70)=\ln(0.70/0.30)=0.8473\). Therefore,
For toxicity, \(\operatorname{logit}(0.15)=\ln(0.15/0.85)=-1.735\). Thus,
Since \(3.559<3.628\), the point-estimate candidate interval is \([3.559,3.628]\) mg/L. It is narrow: its width is \(3.628-3.559=0.069\) mg/L. At \(c=3.59\) mg/L, the response linear predictor is \(-2+0.8(3.59)=0.872\), giving \(p_R(3.59)=1/(1+e^{-0.872})\approx0.705\). The toxicity predictor is \(-5+0.9(3.59)=-1.769\), giving \(p_T(3.59)\approx0.146\). These estimates satisfy both criteria, but their proximity to the boundaries means modest model or estimation error could matter.
Uncertainty Can Change the Decision
The calculated cutoffs are usually estimates, not known constants. Uncertainty can arise from limited data, uncertainty in model parameters, model misspecification, and differences between the evidence population and the patients for whom the target will be used. A point-estimate interval answers what the fitted curves imply at their estimated parameter values. It does not show how reliably the criteria will be met.
One way to assess this uncertainty is to construct confidence bounds for the probability curves. Let \(L_R(c)\) be a lower confidence bound for response probability and \(U_T(c)\) an upper confidence bound for toxicity probability. A conservative, uncertainty-aware set can be defined as
This rule requires evidence to support the response criterion even at the lower bound and the toxicity criterion even at the upper bound. The interpretation depends on how the bounds were constructed. Pointwise confidence bounds have their stated coverage at each fixed exposure, not necessarily simultaneously over the entire interval. A claim that the full interval meets both criteria with a specified confidence level requires suitable joint or simultaneous uncertainty assessment. Even then, the result concerns uncertainty in the estimated curves under the model; it does not remove individual outcome variability.
An alternative is to calculate uncertainty intervals for the cutoffs. For increasing curves, a high response cutoff is unfavorable, while a low toxicity cutoff is unfavorable. A conservative interval therefore uses an upper uncertainty bound for \(C_R\) and a lower uncertainty bound for \(C_T\), provided these are constructed with appropriate joint coverage. If the upper response cutoff exceeds the lower toxicity cutoff, there is no interval certified by that conservative rule. That finding is not proof that no patient can benefit safely; it indicates that the evidence does not establish a robust interval at the chosen confidence standard.
Worked Example: Point-Estimate Overlap but No Conservative Interval
In a hypothetical analysis, the estimated response cutoff is \(3.4\) mg/L, with a 95% uncertainty interval from \(3.0\) to \(3.9\) mg/L. The estimated toxicity cutoff is \(4.2\) mg/L, with a 95% uncertainty interval from \(3.7\) to \(4.8\) mg/L. For simplicity, suppose these cutoff bounds have been constructed to support a joint conservative comparison; separate marginal 95% intervals would not by themselves guarantee 95% joint coverage.
Using only the point estimates gives \([3.4,4.2]\) mg/L, which has width \(4.2-3.4=0.8\) mg/L. For the conservative comparison, use the upper response cutoff, \(3.9\) mg/L, and the lower toxicity cutoff, \(3.7\) mg/L. Since \(3.9>3.7\), the conservative interval is empty. There is no concentration that is at least \(3.9\) mg/L to meet the conservative response requirement and at most \(3.7\) mg/L to meet the conservative toxicity requirement.
The correct conclusion is that the point estimates show overlap, but the specified uncertainty-based criterion does not establish a robust target interval. It would be incorrect to report \([3.4,4.2]\) mg/L as though both probability criteria were assured throughout that interval. The next decision may involve additional evidence, reconsideration of the clinical thresholds, or an explicit benefit–risk decision; it should not be hidden by rounding the cutoffs.
Outcome Overlap, Diagnostics, and Alternatives
Response and toxicity are probabilistic outcomes, so they can occur at the same exposure. A concentration meeting both probability thresholds does not mean that response and toxicity are mutually exclusive, nor does it guarantee a favorable outcome for every person. It means only that the modeled response probability meets its minimum criterion and the modeled toxicity probability meets its maximum criterion in the population and setting represented by the model.
Check whether the estimated relationships are credible before treating their intersection as a target. Important diagnostics include whether the data span the proposed interval, whether observed outcomes are consistent with the fitted curves, whether uncertainty widens substantially near either boundary, and whether the assumed monotonicity is supported. Sparse observations near a cutoff can make that boundary especially uncertain. A curve that appears flat, bends, or reverses direction may not support the simple increasing-cutoff calculation. Differences in follow-up, outcome definition, concomitant treatment, or patient characteristics can also make a pooled exposure–outcome relationship misleading for a particular decision.
The table summarizes three common ways to express the result. They answer different questions and should not be presented as interchangeable.
| Approach | What it reports | Main limitation |
|---|---|---|
| Point-estimate intersection | Exposures satisfying both criteria on the fitted curves | Does not show estimation uncertainty |
| Uncertainty-aware intersection | Exposures satisfying criteria using chosen confidence bounds | Depends on the bounds and their joint coverage; may be empty |
| Explicit benefit–risk decision | A choice that weighs the consequences of response and toxicity under stated preferences | Requires defensible value judgments and may not yield a universal interval |
When the response and toxicity constraints do not overlap, adjusting the criteria is not a statistical repair; it changes the clinical decision. A different approach may be needed when no concentration meets both requirements, but any such choice should state which outcome is being prioritized and why. Likewise, a narrow overlap is not automatically unusable, but it may be too fragile to support a practical target if uncertainty, measurement variation, or model applicability is not addressed.
Worked Example: Intersecting Cutoffs With the Admissible Domain
Consider another hypothetical logistic response model, \(\operatorname{logit}\{p_R(c)\}=-1+c\), with a response criterion \(q_R=0.12\). Let the toxicity model be \(\operatorname{logit}\{p_T(c)\}=-5+c\), with a toxicity criterion \(q_T=0.50\). Assume both slopes are positive and the admissible domain is \(D=[0,8]\) mg/L.
For response, \(\operatorname{logit}(0.12)=\ln(0.12/0.88)=-1.992\), so
For toxicity, \(\operatorname{logit}(0.50)=0\), giving
The curve constraints alone give \([-0.992,5.000]\) mg/L, but negative concentrations are not admissible. Intersecting with the stated domain gives \([ -0.992,5.000]\cap[0,8]=[0,5.000]\) mg/L. Thus the candidate target set is \([0,5.000]\) mg/L, not an interval that includes negative exposure. If the domain had instead been \([1,8]\) mg/L, the same cutoff calculation would yield \([1,5.000]\) mg/L.
Common Mistakes and Practical Limits
- Treating fitted cutoffs as certainty. The estimated intersection is conditional on the model and its estimated parameters. Report and assess uncertainty rather than presenting cutoffs as exact biological boundaries.
- Forgetting the allowable domain. The result is \(D\cap[C_R,C_T]\), not automatically \([C_R,C_T]\). A negative cutoff does not make negative concentration clinically admissible.
- Using a finite logit cutoff at an endpoint criterion. The logit is finite only for probabilities strictly between zero and one. For endpoint criteria, apply the probability inequalities directly.
- Assuming overlap guarantees individual success. The criteria concern probabilities for the modeled population, not certainty for a particular patient.
- Assuming the curves must be monotone. The cutoff interval follows from the stated increasing relationships. If a curve is nonmonotone, solve the constraints over the domain rather than forcing the result into one interval.
These methods also have limits. The probability thresholds embody clinical judgments, and different consequences of response and toxicity may make a simple pair of thresholds inadequate. The model may not capture important patient-level differences or confounding, especially when exposure is influenced by disease severity or treatment changes. A target supported by exposure–outcome evidence is therefore a structured decision aid, not a substitute for checking whether the evidence applies to the patient and clinical setting.
Check Your Understanding
Answer each question, then compare with the solutions below.
- For increasing response and toxicity curves, the estimated cutoffs are \(C_R=2.8\) mg/L and \(C_T=3.6\) mg/L. If \(D=[0,3.2]\) mg/L, what is the candidate target set?
- Why can a point-estimate interval exist while a conservative uncertainty-based interval is empty?
- For a finite-coefficient logistic model, what happens to the response constraint if \(q_R=0\), and why does the finite logit cutoff formula not apply?
- At one exposure, the model estimates \(p_R=0.80\) and \(p_T=0.10\). Explain what this does and does not imply about an individual patient.
Solutions. (1) First intersect \([2.8,3.6]\) with the domain \([0,3.2]\). The result is \([2.8,3.2]\) mg/L. (2) The point estimates may satisfy \(C_R<C_T\), but uncertainty can make the upper plausible response cutoff larger than the lower plausible toxicity cutoff. The conservative constraints then have no common exposure, indicating insufficient support at that uncertainty standard rather than proving that no safe, effective exposure exists. (3) Since probabilities are nonnegative, \(p_R(c)\ge0\) is automatically satisfied throughout the domain. The logit of zero is not finite, so the finite inversion formula is not defined. (4) These are modeled probabilities for the population and conditions represented by the evidence. They do not guarantee response or absence of toxicity for an individual, and the two outcomes are not necessarily mutually exclusive.