Estimating The Therapeutic Window

From Dose-Response Curves to an Acceptable Interval

A dose-finding analysis may identify doses with a sufficient chance of efficacy and an acceptably low chance of toxicity. Those are two different criteria, and neither alone defines an acceptable dose. The therapeutic window is the set of doses meeting both criteria at once. It may contain several doses, one dose, or none within the range studied.

The window depends on the clinical thresholds specified for the trial and on the estimated dose-response curves. It is therefore not a biological constant that can be read from a curve without making choices. Before calculating it, state the efficacy probability considered adequate and the toxicity probability considered acceptable. These thresholds express clinical priorities; the fitted curves describe how the probabilities vary with dose.

Definition: Let $p_E(d)$ be the probability of the prespecified efficacy outcome at dose $d$, and let $p_T(d)$ be the probability of the prespecified toxicity outcome. Given an efficacy threshold $q_E$ and an upper toxicity threshold $q_T$, the therapeutic window is $$ \mathcal{W}=\{d\in\mathcal{D}:p_E(d)\ge q_E\ \text{and}\ p_T(d)\le q_T\}, $$ where $\mathcal{D}$ is the dose range under consideration. An estimated window substitutes fitted or estimated probabilities for the unknown probabilities.

This definition uses marginal probabilities: efficacy and toxicity are each assessed separately in the target population and follow-up period. A dose meeting both marginal criteria does not, by itself, specify the probability that an individual patient experiences efficacy without toxicity. That is a different joint-outcome quantity.

Calculating the Boundary Doses

For a simple continuous-dose analysis, suppose both outcomes are binary and their marginal probabilities follow logistic curves:

\operatorname{logit}\{p_E(d)\}=\beta_{0E}+\beta_{1E}d,\qquad \operatorname{logit}\{p_T(d)\}=\beta_{0T}+\beta_{1T}d,

where $\operatorname{logit}(p)=\log\{p/(1-p)\}$. Assume the dose is measured in a stated unit, the curves are correctly specified over the dose range of interest, and both slopes are positive. Thus, increasing dose increases each modeled probability. The patients and outcome definitions used to estimate the curves should correspond to the population and observation period to which the thresholds apply. The marginal dose-response models need not assume that efficacy and toxicity are independent of one another.

The efficacy boundary is the dose at which the efficacy probability reaches $q_E$. The toxicity boundary is the dose at which the toxicity probability reaches $q_T$. Since the logit function is strictly increasing, applying it to each criterion and rearranging gives:

d_E=\frac{\operatorname{logit}(q_E)-\beta_{0E}}{\beta_{1E}}, \qquad d_T=\frac{\operatorname{logit}(q_T)-\beta_{0T}}{\beta_{1T}}.

If the slopes are positive, efficacy meets its criterion for $d\ge d_E$, and toxicity meets its criterion for $d\le d_T$. Therefore the model-based window is $[d_E,d_T]$ when $d_E\le d_T$, after intersecting that interval with $\mathcal{D}$. If $d_E>d_T$, there is no dose satisfying both criteria under these fitted curves.

Key result: With increasing efficacy and toxicity curves, the lower boundary is set by efficacy and the upper boundary by toxicity. The calculated interval is acceptable only where it overlaps the dose range under consideration. If either boundary falls outside that range, report the truncation and do not imply that the unstudied doses were tested.

This inversion is the essential calculation. It does not require the two outcomes to be independent; it does require each marginal curve to estimate the relevant probability adequately. If a slope is zero, the fitted probability is constant in dose and a unique boundary does not exist: the criterion is met at every dose if the constant probability satisfies it (for efficacy, $p_E\ge q_E$; for toxicity, $p_T\le q_T$), and at no dose otherwise. If a slope is negative, the inequality direction changes, so the increasing-curve formulas for which side of the boundary is acceptable cannot be used without adjustment.

Worked Example: Estimating Both Boundaries

Suppose a hypothetical trial models efficacy and toxicity probabilities for doses in mg/kg. The fitted efficacy curve is $\operatorname{logit}\{p_E(d)\}=-2.8+0.8d$, and the fitted toxicity curve is $\operatorname{logit}\{p_T(d)\}=-5.2+0.9d$. The prespecified criteria are $q_E=0.40$ and $q_T=0.25$. Assume the dose range under consideration includes both calculated boundaries.

First calculate the logits of the thresholds: $\operatorname{logit}(0.40)=\log(0.40/0.60)\approx-0.4055$, and $\operatorname{logit}(0.25)=\log(0.25/0.75)\approx-1.0986$.

The efficacy boundary is $d_E=\{-0.4055-(-2.8)\}/0.8=2.3945/0.8\approx2.993$ mg/kg. At doses at or above this value, the fitted efficacy probability meets or exceeds 0.40.

The toxicity boundary is $d_T=\{-1.0986-(-5.2)\}/0.9=4.1014/0.9\approx4.557$ mg/kg. At doses at or below this value, the fitted toxicity probability is at most 0.25. Because $2.993<4.557$, the estimated window is approximately $[2.993,4.557]$ mg/kg, with width $4.557-2.993=1.564$ mg/kg.

As a check, at the efficacy boundary the fitted toxicity logit is $-5.2+0.9(2.993)\approx-2.506$, giving a toxicity probability of about 0.0755, below 0.25. At the toxicity boundary, the fitted efficacy logit is $-2.8+0.8(4.557)\approx0.8456$, giving an efficacy probability of about 0.700, above 0.40. The two criteria are therefore consistent across this interval under the increasing-curve assumptions.

When the Estimated Window Is Empty

An empty estimated window is an informative result, not a calculation to repair by choosing whichever boundary seems more attractive. It means that, under the fitted curves and the stated criteria, the dose needed to meet the efficacy requirement is higher than the dose allowed by the toxicity requirement. It can reflect a genuine efficacy-toxicity conflict, imprecise estimates, unsuitable model shapes, or thresholds that cannot both be met in the available dose range.

Worked Example: No Dose Meets Both Criteria

In a second hypothetical trial, let $\operatorname{logit}\{p_E(d)\}=-3+0.7d$ and $\operatorname{logit}\{p_T(d)\}=-3+d$, with dose in mg/kg. Let the minimum efficacy probability be $q_E=0.50$, and the maximum toxicity probability be $q_T=0.20$. Both slopes are positive, as assumed for the boundary calculation.

Because $\operatorname{logit}(0.50)=0$, the efficacy boundary is $d_E=(0-(-3))/0.7=3/0.7\approx4.286$ mg/kg. For toxicity, $\operatorname{logit}(0.20)=\log(0.20/0.80)\approx-1.3863$, so $d_T=\{-1.3863-(-3)\}/1\approx1.614$ mg/kg.

Efficacy requires $d\ge4.286$ mg/kg, while toxicity requires $d\le1.614$ mg/kg. These conditions cannot both hold, so the fitted window is empty. For example, at 2 mg/kg the efficacy logit is $-3+0.7(2)=-1.6$, giving an efficacy probability of about 0.168, below 0.50; the toxicity logit is $-3+2=-1$, giving a toxicity probability of about 0.269, above 0.20. That dose fails both criteria in this hypothetical model.

A continuous interval is appropriate only when the dose is treated as continuous and the fitted curves support interpolation over the interval. If the trial investigates only a set of prespecified dose levels, report which of those levels meet both criteria. Do not treat an interpolated boundary as an observed dose, and do not assume that an untested dose is safe or effective merely because it lies between tested levels.

Uncertainty in the Boundaries

The plug-in boundaries are estimates because the model coefficients are estimated. Reporting only a point interval can make the result appear more precise than the data support. One way to describe uncertainty in a boundary is the delta method, which approximates the sampling variance of a smooth function of estimated model coefficients.

For either logistic curve, write a boundary as $d=(\operatorname{logit}(q)-\beta_0)/\beta_1$, where the threshold $q$ is treated as fixed. The gradient with respect to $(\beta_0,\beta_1)$ is $(-1/\beta_1,-d/\beta_1)$. If $\widehat{\Sigma}$ is the estimated covariance matrix of the coefficient estimates, the approximate variance of the estimated boundary is the gradient evaluated at the estimates, multiplied on the left and right by $\widehat{\Sigma}$ and its transpose:

\widehat{\operatorname{Var}}(\hat d) \approx \hat{\boldsymbol g}^{\,T}\widehat{\Sigma}\hat{\boldsymbol g}, \qquad \hat{\boldsymbol g}= \begin{pmatrix} -1/\hat\beta_1\\ -\hat d/\hat\beta_1 \end{pmatrix}.

This approximation uses the joint covariance of the intercept and slope estimates; omitting their covariance can give a materially different standard error. A large-sample approximate 95% confidence interval is $\hat d\pm1.96\,SE(\hat d)$. It describes repeated-sampling uncertainty in the boundary under the model and its assumptions; it is not a 95% probability statement about the fixed boundary.

Worked Example: Approximate Uncertainty for an Efficacy Boundary

Suppose a hypothetical efficacy model has estimates $\hat\beta_{0E}=-2.0$ and $\hat\beta_{1E}=0.7$, and the efficacy criterion is $q_E=0.50$. The estimated boundary is $\hat d_E=\{0-(-2.0)\}/0.7\approx2.857$ mg/kg. Assume the estimated standard errors of the intercept and slope are 0.30 and 0.12, respectively, and their estimated covariance is $-0.0072$.

The gradient is $\hat{\boldsymbol g}=(-1/0.7,-2.857/0.7)\approx(-1.429,-4.082)$. Thus, $\widehat{\operatorname{Var}}(\hat d_E)\approx(-1.429)^2(0.30^2)+(-4.082)^2(0.12^2)+2(-1.429)(-4.082)(-0.0072)\approx0.3397$. The approximate standard error is $\sqrt{0.3397}\approx0.583$ mg/kg.

The approximate 95% confidence interval is $2.857\pm1.96(0.583)$, or about $[1.715,3.999]$ mg/kg. This interval concerns the efficacy boundary alone. It does not establish that a therapeutic window exists: that conclusion also depends on uncertainty in the toxicity boundary and on their joint uncertainty. A separate confidence interval for each boundary should not be interpreted as a confidence interval for the entire window.

Diagnostics, Alternatives, and Limitations

Boundary estimates are only as credible as the dose-response models and the trial data that support them. Inspect fitted probabilities against observed response proportions at each dose, along with residual or other model-adequacy checks appropriate to the fitted model. Look for systematic departures from the curve, sparse observations near a threshold, or a boundary that is driven mainly by extrapolation beyond studied doses. Check that the endpoint definitions, assessment times, and analysis population match the clinical interpretation of both thresholds. If those differ between outcomes, the resulting window may not describe one coherent clinical decision.

Monotonicity also matters. The logistic models above force probabilities to change monotonically with dose. If the observed pattern contradicts that assumption, a single lower and upper boundary may misrepresent the acceptable set. A plateau in efficacy may still yield a contiguous window when toxicity rises, but a nonmonotone curve can yield separated acceptable dose regions. In that case, define the acceptable set from the fitted curves over the supported dose range and report its components rather than forcing it into one interval. Sparse data can make either the shape or the boundary unstable; uncertainty intervals that are very wide, or boundaries that shift substantially under plausible model choices, are warnings against a confident interpretation.

A therapeutic window is not the same as a single recommended dose. The window answers, “Which doses meet these two stated criteria?” It does not rank all acceptable doses or account for other considerations such as administration burden, uncertainty, or a preference for a particular balance of outcomes. A target dose, utility-based choice, or other decision rule may select one dose using additional priorities. Conversely, reporting only one selected dose hides whether nearby doses also satisfy the thresholds. A useful report gives the criteria, boundary estimates, dose range and model used, and uncertainty, then distinguishes the acceptable set from any subsequent dose recommendation.

The main limitation is that thresholds and fitted probabilities are not certainty. A point estimate just meeting a threshold does not guarantee the true probability meets it. The next step in a decision analysis may therefore account explicitly for uncertainty, but the point-estimated window itself should not be described as having a particular probability of being acceptable. Finally, if the objective is a joint event such as efficacy without severe toxicity, marginal thresholds alone are insufficient; use a joint-outcome estimand for that question.

Threshold mismatch: thresholds chosen after seeing the fitted curves can make the window look more favorable than a prespecified analysis warrants.
Sign error: $\operatorname{logit}(q)$ can be negative. Keep the subtraction in $\operatorname{logit}(q)-\hat\beta_0$ explicit before dividing by the slope.
False precision: decimal boundary values do not imply that doses can be practically administered or clinically distinguished at that precision.
Confusing interval with recommendation: an acceptable range does not identify a uniquely best dose.

Key takeaway: Estimate the efficacy and toxicity crossing doses from their fitted marginal curves, check that they overlap in the supported dose range, and report uncertainty. The resulting window is a set of doses meeting prespecified criteria, not a single dose recommendation.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

A fitted efficacy curve is $\operatorname{logit}\{p_E(d)\}=-2+0.5d$, and the required efficacy probability is 0.50. What is the efficacy boundary?
If the efficacy boundary is 3 mg/kg and the toxicity boundary is 2 mg/kg, is the estimated therapeutic window nonempty under increasing curves? State why.
Why does a dose meeting separate marginal efficacy and toxicity criteria not necessarily meet a criterion about the probability of efficacy without toxicity?
What does a 95% delta-method confidence interval for one boundary describe, and what does it not establish about the full window?

Solutions. (1) Since $\operatorname{logit}(0.50)=0$, $d_E=\{0-(-2)\}/0.5=4$ mg/kg. (2) No. Efficacy requires $d\ge3$ mg/kg and toxicity requires $d\le2$ mg/kg, so no dose satisfies both. (3) Marginal probabilities do not determine the joint probability of efficacy and toxicity without information about their dependence. (4) It describes approximate repeated-sampling uncertainty in that boundary under the fitted model; alone it does not establish whether the efficacy and toxicity boundaries overlap or quantify uncertainty in the full window.