Exposure-Response Therapeutic Windows

What an Exposure-Response Window Represents

A therapeutic window in an exposure-response analysis is a range of exposure values associated with prespecified efficacy and safety criteria. It is not automatically a range of doses, nor does it establish that changing a patient's dose to place exposure inside that range will cause the desired outcome. The result depends on the population, outcome definitions, follow-up period, exposure measure, and model used.

The key task is to make the criteria explicit. For example, a team might require the probability of an efficacy response by week 12 to be at least 0.70 and the probability of a specified adverse event by week 12 to be no greater than 0.20. These are illustrative thresholds, not universal standards. They should be clinically justified before examining which exposure values satisfy them.

Definition: Let $x$ denote a specified exposure measure, such as steady-state area under the concentration-time curve, in a defined population and over a defined outcome horizon. Let $p_E(x)$ be the model-estimated probability of the target efficacy outcome and $p_T(x)$ the model-estimated probability of the specified toxicity outcome. Given efficacy target $c_E$ and maximum acceptable toxicity probability $c_T$, the criterion-based exposure window is the set $$ \mathcal{W}=\{x:p_E(x)\ge c_E \text{ and }p_T(x)\le c_T\}. $$ The window may be empty, a single point, one interval, or multiple disjoint ranges.

This is a statement about model-estimated outcome probabilities at exposure values. If efficacy rises monotonically with exposure and toxicity also rises monotonically, the criteria usually yield a lower efficacy boundary and an upper safety boundary. Their intersection is an interval only when the lower boundary does not exceed the upper boundary. With nonmonotone or more complex response shapes, the set itself must be examined rather than assumed to be one interval.

Deriving Boundaries Under a Simple Model

Suppose the conditional probabilities are described by logistic models:

\operatorname{logit}\{p_E(x)\}=\beta_{E0}+\beta_{E1}x,\qquad \operatorname{logit}\{p_T(x)\}=\beta_{T0}+\beta_{T1}x, \qquad \operatorname{logit}(p)=\log\left(\frac{p}{1-p}\right).

When $\beta_{E1}>0$, efficacy probability increases with exposure. Solving $p_E(x)=c_E$ gives the minimum exposure meeting the efficacy target:

L_E=\frac{\operatorname{logit}(c_E)-\beta_{E0}}{\beta_{E1}}.

When $\beta_{T1}>0$, toxicity probability increases with exposure. Solving $p_T(x)=c_T$ gives the maximum exposure meeting the safety criterion:

U_T=\frac{\operatorname{logit}(c_T)-\beta_{T0}}{\beta_{T1}}.

These formulas require positive slopes and a meaningful model over the exposure region being interpreted. If a slope is zero, the corresponding probability may either meet its criterion at all modeled exposures or meet it at none. If a slope has the opposite sign, the direction of the boundary changes; it should not be forced into a “lower efficacy, upper toxicity” interpretation.

Worked Example: Calculating a Criterion-Based Window

Consider hypothetical exposure-response fits over an observed exposure range of 50 to 100 exposure units. Let $x$ be steady-state exposure in mg·h/L. Suppose the fitted week-12 efficacy and toxicity models are

\operatorname{logit}\{p_E(x)\}=-2.00+0.0400x,\qquad \operatorname{logit}\{p_T(x)\}=-4.00+0.0300x.

Use $c_E=0.70$ and $c_T=0.20$. Both slopes are positive, so the first model gives a lower boundary and the second an upper boundary. The efficacy boundary is

L_E=\frac{\log(0.70/0.30)-(-2.00)}{0.0400} =\frac{0.8473+2.00}{0.0400} =71.18\ \text{mg·h/L}.

The toxicity boundary is

U_T=\frac{\log(0.20/0.80)-(-4.00)}{0.0300} =\frac{-1.3863+4.00}{0.0300} =87.12\ \text{mg·h/L}.

Since $71.18\le87.12$, the estimated criterion-based window is approximately 71.2 to 87.1 mg·h/L, with width $87.12-71.18=15.94$ mg·h/L. At $x=80$ mg·h/L, the efficacy logit is $1.20$, giving $p_E(80)=1/(1+\exp(-1.20))=0.7685$. The toxicity logit is $-1.60$, giving $p_T(80)=1/(1+\exp(1.60))=0.1680$. Both point estimates meet their respective criteria. This calculation establishes neither that the criteria are clinically appropriate nor that the interval is estimated precisely.

Overlapping Criteria and Joint Response Probability

The intersection of the two marginal criteria is not the same quantity as the probability that one person experiences both the target efficacy outcome and acceptable safety. Define $q(x)=P(E=1,T=0\mid x)$, where $E=1$ denotes the target efficacy outcome and $T=1$ denotes the specified toxicity outcome. A joint model can estimate $q(x)$ directly; alternatively, it can combine a marginal efficacy probability with a conditional safety probability, provided that the conditional probability is part of a coherent joint model.

Marginal probabilities alone do not determine $q(x)$. In particular, multiplying $p_E(x)$ by $1-p_T(x)$ assumes conditional independence of efficacy and toxicity at that exposure. That assumption is not automatic: shared patient characteristics or biological mechanisms can make the outcomes dependent. A joint model, as in the earlier material on joint models of efficacy and safety by exposure, represents that dependence.

Interpretation: The marginal window $\mathcal{W}$ identifies exposures at which each separate probability threshold is met. If a joint-response target is also required, such as $q(x)\ge c_J$, define and report the additional set $\{x:q(x)\ge c_J\}$ and its intersection with $\mathcal{W}$. State clearly whether the reported “window” is based on marginal criteria, a joint criterion, or both.

Worked Example: Marginal Overlap Is Not Joint Success

At the hypothetical exposure $x=80$ mg·h/L in the preceding example, the fitted marginal estimates are $p_E=0.7685$ and $p_T=0.1680$. Suppose a fitted joint model estimates the probability of both efficacy and no specified toxicity as $q(80)=0.660$. The joint estimate is feasible because it satisfies the Fréchet bounds: $\max(0,p_E+(1-p_T)-1)=0.6005\le q(80)=0.660\le\min(p_E,1-p_T)=0.7685$.

If independence were imposed, the product would be

p_E(80)\{1-p_T(80)\} =0.7685(1-0.1680) =0.7685(0.8320) =0.6394.

That value differs from the joint-model estimate of 0.660. The difference illustrates why a product should not be presented as the joint probability unless independence is an explicit, defensible model assumption. With a prespecified joint target of $c_J=0.60$, the joint-model point estimate at 80 mg·h/L meets that target; it does not by itself show that the target is met throughout the entire marginal window.

Quantifying Uncertainty in the Window

Boundary estimates depend on fitted model parameters, which are uncertain. A practical approach is to refit the model in each of many bootstrap resamples of the analysis units, using the original sampling structure (for example, resampling patients rather than individual records when patients contribute repeated observations). Each refit produces $L_E^{(b)}$, $U_T^{(b)}$, and, where applicable, a joint probability curve $q^{(b)}(x)$. Using paired refits preserves the estimated dependence between efficacy and safety parameters.

Resample and refit.
Resample the appropriate independent units and fit the same prespecified model or joint model to each resample.

Recalculate criteria.
For each refit, solve for the boundaries or evaluate the criteria on a defined exposure grid. Record whether the resulting set is empty, a singleton, or has positive width.

Summarize uncertainty.
Report boundary distributions or percentile intervals, the proportion of refits with a nonempty window, and a clearly labelled width summary. Summarize joint-response probability uncertainty at exposures relevant to the decision.

For a refit with boundaries, a nonempty window occurs when $U_T^{(b)}\ge L_E^{(b)}$; equality is a singleton and has zero width. A positive-width window instead requires $U_T^{(b)}>L_E^{(b)}$. These are different summaries and should be labelled accordingly. If some refits produce no finite boundary, or if the fitted curve is nonmonotone, calculate the criterion set directly rather than treating a boundary formula as universally available.

Worked Example: Bootstrap Windows Including a Singleton

For clarity, consider 10 hypothetical paired bootstrap refits, with boundaries in mg·h/L. The $L_E^{(b)}$ values are 68, 72, 75, 79, 66, 70, 74, 81, 77, and 69. The corresponding $U_T^{(b)}$ values are 84, 80, 75, 76, 90, 88, 82, 86, 73, and 85.

Eight refits have $U_T^{(b)}\ge L_E^{(b)}$: refits 1, 2, 3, 5, 6, 7, 8, and 10. Thus, the observed proportion with a nonempty window is $8/10=0.80$. Refit 3 has equal boundaries, 75 mg·h/L, so it is a nonempty singleton, not an empty window. Only seven refits have positive width. Their widths, together with the singleton width, among all eight nonempty refits are 16, 8, 0, 24, 18, 8, 5, and 16 mg·h/L. The mean width among nonempty refits is

\frac{16+8+0+24+18+8+5+16}{8} =\frac{95}{8} =11.88\ \text{mg·h/L}.

If the intended summary is mean width among positive-width windows only, it is $95/7=13.57$ mg·h/L. The denominator and definition must accompany either summary. With only 10 refits, 0.80 is an illustration of the calculation, not a stable uncertainty estimate. In a real analysis, use enough refits to assess the relevant sampling variability and report boundary intervals as well.

A bootstrap proportion is a feature of the resampling distribution under the fitted analysis; it is not the probability that the true therapeutic window exists. Likewise, a confidence interval for a boundary does not mean there is a 95% probability that the fixed boundary lies in that particular interval. A percentile interval is an approximate frequentist interval whose coverage depends on the model, data, and resampling scheme.

Worked Example: Uncertainty in a Joint-Response Probability

Suppose the joint-model estimate at 80 mg·h/L is $\hat q(80)=0.660$, and the 2.5th and 97.5th percentiles of 1,000 paired bootstrap estimates are 0.580 and 0.730. The resulting 95% bootstrap percentile interval is 0.580 to 0.730. In this hypothetical analysis, 760 of the 1,000 refits estimate $q(80)\ge0.60$, so the bootstrap fraction meeting the joint target is $760/1000=0.760$.

The point estimate is above the target of 0.60, but the interval includes values below it. The bootstrap fraction describes how often refitted estimates meet the target under this resampling procedure; it is not a posterior probability that the true joint-response probability exceeds 0.60. The interval and fraction answer different uncertainty questions and should be reported as such.

Diagnostics, Alternatives, and Limits

A window can look precise while depending on unsupported assumptions. Check the exposure range represented by the data, the adequacy of the fitted curves, and whether the outcome horizon and exposure summary are clinically meaningful. The earlier material on diagnostics for exposure-response model fit is relevant: poor fit, sparse observations near a boundary, influential observations, or a misspecified response shape can make a calculated threshold unreliable. Extrapolating a boundary beyond the observed exposure range is especially fragile and should be identified rather than hidden by a numerical estimate.

Other warning signs include boundary estimates that vary substantially across bootstrap refits, a high frequency of empty windows, implausible predicted probabilities, and sensitivity to reasonable model specifications. If monotonicity is not supported, evaluate the criterion set over the exposure range and report any disconnected regions. A single lower and upper bound can misrepresent such a set. For joint probabilities, assess whether the fitted joint model is coherent and whether dependence is informed by the observed data.

An alternative to reporting threshold boundaries is to show the fitted efficacy and toxicity curves with uncertainty bands over the supported exposure range. This makes trade-offs visible without implying a sharp change in clinical value at a single cutoff. A joint-response curve can add information about co-occurrence, but it does not replace the separate marginal curves when separate efficacy and safety criteria are important. The appropriate display depends on the estimand and the intended interpretation.

Most importantly, an exposure window is descriptive or model-based evidence about outcomes associated with exposure under the analysis assumptions. Exposure may be related to patient characteristics, adherence, disease severity, or dose modifications. Therefore, an exposure-response association is not automatically a causal effect of assigning an exposure. Nor does an exposure range directly translate to a dose range: dose-to-exposure variability, pharmacokinetics, treatment feasibility, and the broader clinical evidence must be considered separately. The window is one piece of evidence, not a dosing recommendation.

Do not select thresholds after seeing the most favorable interval. Criteria should be specified and justified; exploring many alternatives raises additional interpretive concerns.
Do not call equality empty. If $U_T=L_E$, the set contains one exposure value and has zero width. Report nonempty and positive-width proportions distinctly.
Do not multiply marginal probabilities without an independence assumption. Use a joint model or state and defend the assumption.
Do not interpret a model-based interval as a proven safe or effective dose range. It describes fitted probabilities for a defined population, exposure measure, and horizon.

Key takeaway: Define the efficacy and safety criteria first, derive or calculate their intersection on the exposure scale, and quantify uncertainty with a method that respects the model and sampling structure. Report marginal overlap, joint-response probabilities, and their limitations as distinct results.

Check Your Understanding

Answer each question, then compare with the brief solutions.

If the estimated efficacy boundary is 75 mg·h/L and the toxicity boundary is 75 mg·h/L, is the window empty? Does it have positive width?
For $p_E(x)=\operatorname{logit}^{-1}(-1.50+0.0300x)$, find the exposure at which $p_E(x)=0.80$.
Why is $p_E(x)\{1-p_T(x)\}$ not generally the joint probability of efficacy and no toxicity?
In 500 paired bootstrap refits, 435 have $U_T^{(b)}\ge L_E^{(b)}$, while 420 have $U_T^{(b)}>L_E^{(b)}$. Report both proportions and explain their difference.
A model-based window extends beyond the highest exposure well represented in the data. What is the main interpretive concern?

Solutions. (1) It is nonempty because the boundaries are equal, but it is a singleton with zero width. (2) $\operatorname{logit}(0.80)=\log(4)=1.3863$, so $x=(1.3863+1.50)/0.0300=96.21$ exposure units. (3) The product assumes efficacy and toxicity are conditionally independent at that exposure; dependence requires a joint model or another justified specification. (4) The nonempty-window proportion is $435/500=0.870$; the positive-width proportion is $420/500=0.840$. The difference consists of 15 refits with equal boundaries. (5) The boundary may rely on extrapolation and weakly supported model shape, so it is uncertain and may not represent the exposure-response pattern in that region.