Uncertain Clinical Targets and Moving Thresholds

Why a Clinical Target Can Move

A dose calculation often begins with a clinical threshold: the response level that is considered adequate. That threshold may itself be uncertain. Evidence may support a range of clinically meaningful cutoffs, or a decision-maker may have uncertainty about which response level should count as success. This is distinct from uncertainty about the dose-response model. Even with the model parameters fixed, a higher target generally requires a higher dose; even with the target fixed, uncertainty in the model changes the dose needed to reach it.

These uncertainties should be represented separately before they are combined. Let \(T\) denote the clinical response threshold and let \(\theta\) denote the model parameter vector. For a dose \(d\), write the model-predicted response as \(m(d;\theta)\). The required dose is a function of both quantities, \(D^*(T,\theta)\). Its uncertainty therefore reflects the uncertainty distribution assigned to \(T\), the uncertainty distribution assigned to \(\theta\), and any dependence between them.

The probability statements in this tutorial are conditional on the adopted uncertainty distributions for \(T\) and \(\theta\). They describe uncertainty about the dose calculation, not a guarantee about an individual patient's outcome. The response function is treated as the modeled quantity being compared with the threshold. Adding residual outcome variation or patient-to-patient heterogeneity would require those sources to be represented explicitly as well.

Inverse Dose and the Moving Threshold

For a concrete derivation, consider an increasing Emax response model with baseline response \(E_0\), positive maximum increment \(E_{\max}\), and positive half-maximal dose \(ED_{50}\):

Here \(\theta=(E_0,E_{\max},ED_{50})\). The response is continuous and nondecreasing, and its limiting value as dose increases without bound is \(E_0+E_{\max}\). If \(T\leq E_0\), dose zero already meets the threshold. If \(E_0<T<E_0+E_{\max}\), solve \(m(d;\theta)=T\) to find the smallest dose that reaches it. If \(T\geq E_0+E_{\max}\), no finite dose reaches the threshold: equality with the asymptote is approached only in the limit.

The final expression applies only when \(E_0<T<E_0+E_{\max}\), so its numerator and denominator are both positive. The complete rule, including its boundary cases, is:

Thus, for this model, a finite nonnegative dose can meet the threshold exactly when \(T<E_0+E_{\max}\). A threshold below baseline is not infeasible: zero dose already meets it. As \(T\) approaches the asymptote from below, the required dose grows rapidly.

Propagating Both Sources of Uncertainty

Suppose uncertainty is represented by a joint distribution for \((T,\theta)\). For each draw from that distribution, calculate \(D^*(T,\theta)\) using the full piecewise rule. The resulting collection of doses represents uncertainty in the required dose under the specified model and uncertainty assumptions. Using paired draws matters: if target and parameter uncertainty are dependent, independently reshuffling one set of draws would discard that dependence and could change the result.

For a continuous nondecreasing response function, the threshold is attained at the infimum dose whenever a finite threshold-attaining dose exists. Consequently, at a fixed candidate dose \(d\),

The continuity condition is important. Monotonicity alone is not enough: a discontinuous increasing response could have threshold-attaining doses arbitrarily close above an infimum without attaining the threshold at that infimum. The Emax model above is continuous, so the equivalence holds for it. Averaging the indicator of this event over joint uncertainty draws estimates the probability that the required dose is no greater than \(d\). This is a probability over the specified uncertainty distribution, not a population success rate unless the distribution and model explicitly represent population variation.

Target Uncertainty by Itself

When \(\theta\) is fixed, uncertainty in \(T\) alone can still create a nontrivial distribution of required doses. Because \(m(d;\theta)\) is continuous and increasing, the event \(D^*\leq d\) is equivalent to \(T\leq m(d;\theta)\). Therefore the dose distribution can be obtained by transforming the target distribution, while respecting the zero-dose region and any infeasible targets.

Interpretation, Diagnostics, and Alternatives

A useful result is not just a single dose. Depending on the decision, report the distribution of required dose, the probability of meeting a specified dose limit, and how often the target is infeasible under the model draws. If a quantile of required dose is reported, describe it as a quantile under the adopted joint uncertainty distribution. It is not a confidence statement about a fixed unknown dose unless the underlying inferential construction justifies that interpretation.

Check calculations and assumptions with targeted diagnostics:

• Confirm that target and response use the same endpoint scale and units. A numeric threshold cannot be compared directly with a response expressed on a different scale.
• Count draws with \(T\leq E_0\), \(E_0<T<E_0+E_{\max}\), and \(T\geq E_0+E_{\max}\). Many infeasible draws may mean the model cannot support the requested target.
• Inspect doses near the asymptote. A small change in target or parameters can cause a very large dose change; this is a mathematical feature of the inverse, not necessarily a coding error.
• Verify selected draws directly by substituting their calculated doses into \(m(d;\theta)\). For a feasible interior target, the predicted response should equal the target up to numerical precision.
• Check that joint draws preserve parameter correlation and any target-parameter dependence. Also increase the number of simulation draws and compare summaries to assess Monte Carlo stability.

A plug-in calculation uses one target and one parameter estimate. It is simple, but hides the spread, zero-dose mass, and infeasible cases; moreover, the dose evaluated at average inputs need not equal the average required dose because the inverse is nonlinear. Propagating target uncertainty alone or parameter uncertainty alone can be informative as a sensitivity comparison, but neither represents joint uncertainty when both sources matter. Joint propagation retains both. A conservative single target can be used when a decision requires a fixed planning threshold, but it should be identified as a chosen rule rather than mistaken for the full uncertainty distribution.

Common mistake: Applying the inverse formula whenever a target is “near” the response range, without checking its relation to baseline and the asymptote. If \(T<E_0\), the inverse expression is not the required dose; zero dose already meets the threshold. If \(T=E_0+E_{\max}\), the denominator is zero and no finite dose attains that threshold. The correction is to check feasibility and the baseline case first, then use the inverse only for \(E_0<T<E_0+E_{\max}\).

The uncertainty distribution itself is an assumption that deserves scrutiny. A threshold distribution may encode uncertainty from evidence, clinical judgment, or a decision rule; these are not interchangeable meanings. Likewise, parameter draws must represent the intended model uncertainty. Results can change substantially if plausible threshold values cluster near the model asymptote. Report the assumptions and units that determine those distributions, rather than presenting a dose summary as independent of them.