From a PTA Curve to a Dose Decision
A probability-of-target-attainment (PTA) curve summarizes, across candidate doses, the probability that a simulated individual meets a prespecified exposure target. The curve helps organize dose comparisons, but it does not by itself specify which dose to choose. A decision rule must state what level of PTA is required and how simulated uncertainty will be handled. For example, one rule may select the lowest candidate dose whose estimated PTA is at least 0.90.
The selected dose depends on the set of doses considered. If doses are evaluated only every 25 mg, the result can differ from one obtained by checking every 5 mg. It can also depend on whether the rule compares the threshold with the estimated PTA alone or requires evidence that PTA exceeds the threshold after allowing for Monte Carlo error. These are distinct choices and should be prespecified rather than chosen after inspecting the curve.
This is a model-based selection, conditional on the target definition and the assumptions used to simulate virtual individuals. It is not, by itself, a clinical recommendation: other constraints, such as safety, feasibility, or approved dose options, may also matter. Here the focus is narrower: applying a PTA criterion consistently to the simulated candidates.
Specify the Rule Before Comparing Doses
A complete selection rule should state the target event, the PTA threshold \(q\), the eligible dose set, and the statistic used to decide whether each dose qualifies. If the target is that an exposure metric reaches a threshold, define that event exactly as in the earlier material on exposure targets. For each candidate dose, the simulated target-attainment indicators determine \(X(D)\), and the estimated PTA is \(X(D)/N\).
The simplest rule compares \(\hat{\pi}(D)\) directly with \(q\). It is transparent and directly answers whether the observed Monte Carlo estimate meets the threshold. However, \(\hat{\pi}(D)\) varies across simulation runs even when the underlying model and dose are unchanged. A slightly different random sample can move a dose from just below to just above \(q\). An alternative is to require a one-sided Monte Carlo lower confidence bound to meet \(q\). That rule is more conservative about simulation error, but it is a different decision criterion, not a correction to the point-estimate rule.
For either rule, “smallest qualifying dose” means the smallest dose in the eligible set, not necessarily the smallest dose on a continuous scale. If no candidate qualifies, the result is that the evaluated grid contains no dose meeting the chosen criterion. It is not appropriate to report the largest tested dose as qualifying merely because it was closest to the threshold.
Worked Example: Applying a Point-Estimate Criterion
Suppose a hypothetical simulation evaluates five eligible doses. At each dose, \(N=1{,}000\) virtual individuals are simulated, and the target is attained by the following numbers. The prespecified threshold is \(q=0.90\).
| Dose (mg) | Attain target, \(X(D)\) | Estimated PTA, \(X(D)/1{,}000\) |
|---|---|---|
| 50 | 720 | 0.720 |
| 75 | 810 | 0.810 |
| 100 | 875 | 0.875 |
| 125 | 925 | 0.925 |
| 150 | 955 | 0.955 |
At 100 mg, \(\hat{\pi}(100)=875/1{,}000=0.875\), which is below 0.90. At 125 mg, \(\hat{\pi}(125)=925/1{,}000=0.925\), which meets the criterion. The higher dose also meets it, but the rule calls for the lowest qualifying dose. Therefore, among the tested candidates, the selected dose is 125 mg.
This conclusion uses the point-estimate rule only. It does not show that the true PTA is at least 0.90 with certainty, nor does it account for uncertainty in the PK model or its parameters.
What Dose-Grid Resolution Can and Cannot Guarantee
A finite grid limits how closely the selected candidate can approach a continuous-scale crossing. To state a useful bound, assumptions matter. Suppose the true PTA curve \(\pi(D)\) is nondecreasing over a dose interval, the relevant target threshold is \(q\), and there is at least one dose in the interval with \(\pi(D)\ge q\). Define the first crossing as the infimum of all qualifying doses:
The infimum is used because a qualifying set need not have a smallest member on a continuous dose scale. For example, a nondecreasing curve could jump from below \(q\) to at least \(q\) immediately above a dose \(a\). The set of qualifying doses would then have infimum \(a\), without including \(a\) itself.
Now suppose adjacent grid doses \(D_{j-1}\) and \(D_j\) bracket this infimum, with spacing no greater than \(\Delta\), and the grid includes a qualifying dose at or above the crossing. The first grid point at or above the crossing is no more than one grid spacing above it:
This is a bound on dose-grid resolution for the true curve, not a guarantee about a selection made from noisy estimates. It also does not say that the selected dose’s PTA is close to \(q\): a steep curve can rise substantially within one grid interval. If the true PTA curve is continuous and crosses \(q\) between adjacent doses, the crossing lies strictly above the lower endpoint and at or below the upper endpoint. The non-strict infimum bound above remains valid without relying on continuity.
In practice, PTA is estimated from simulation, so an observed bracket such as \(\hat{\pi}(D_{j-1})<q\le\hat{\pi}(D_j)\) does not establish a bracket for the true curve. Monte Carlo variation may put either estimate on the other side of \(q\). Refining the dose grid can reduce discretization, but increasing the number of simulated individuals or using an uncertainty-aware rule addresses a different issue: Monte Carlo error.
Worked Example: Refining a Coarse Dose Grid
Consider a hypothetical monotone PTA curve and a point-estimate threshold of \(q=0.90\). On an initial grid with 20 mg spacing, simulation gives \(\hat{\pi}(80\text{ mg})=0.84\) and \(\hat{\pi}(100\text{ mg})=0.91\). The point-estimate rule therefore selects 100 mg from this grid.
If 2.5 mg increments are feasible and a finer simulation gives \(\hat{\pi}(95\text{ mg})=0.895\) and \(\hat{\pi}(97.5\text{ mg})=0.905\), then 95 mg fails and 97.5 mg qualifies. The selected candidate on the refined grid is 97.5 mg. The difference from the coarse-grid selection is \(100-97.5=2.5\) mg. This illustrates how a coarse grid can miss a lower qualifying candidate; it does not establish that the true crossing is exactly 97.5 mg.
For the true curve, if the first crossing lies between 80 and 100 mg and the curve is nondecreasing, the coarse grid’s upper endpoint is at most 20 mg above the crossing. That bound concerns dose spacing only. The estimates 0.84, 0.91, 0.895, and 0.905 are Monte Carlo results, so they do not provide a deterministic bound on the distance between the selected candidate and the true crossing.
A finer grid is useful only if the additional doses are eligible and meaningful to evaluate. If the available formulation permits only 25 mg increments, a candidate at 97.5 mg is not an actionable option; the eligible set should reflect that constraint from the start.
How the Decision Criterion Changes the Selection
A point-estimate rule and a lower-confidence-bound rule answer different questions. The point-estimate rule asks whether the simulated proportion attained the target at the chosen dose. A Monte Carlo lower-bound rule asks whether a one-sided lower confidence bound for the underlying PTA, based on the simulated Bernoulli outcomes, is at least \(q\). The latter aims to reduce the chance of selecting a dose whose true PTA is below \(q\) because of Monte Carlo sampling variation.
For a count \(X(D)\) from \(N\) independent simulated individuals, the estimated PTA is a sample proportion. A confidence bound for this proportion is an approximation to uncertainty from finite simulation, conditional on the simulation model and the chosen dose. It does not include uncertainty in model parameters unless that uncertainty has been incorporated into the simulation design. A Wilson score bound is often preferable to a simple normal bound, especially when proportions are near 0 or 1 or the sample size is small.
Worked Example: Point Estimate Versus a Monte Carlo Lower Bound
In a hypothetical simulation, \(N=400\) independent virtual individuals are evaluated at each dose, and \(q=0.90\). The target is attained by 350 individuals at 90 mg, 365 at 100 mg, and 374 at 110 mg. Thus, the estimated PTAs are \(350/400=0.875\), \(365/400=0.9125\), and \(374/400=0.935\), respectively.
Under the point-estimate rule, 90 mg fails and 100 mg qualifies, so the selected dose is 100 mg. Now consider an alternative rule requiring a one-sided 95% Wilson lower confidence bound to be at least 0.90. For \(\hat{p}=x/n\) and \(z=1.645\), the Wilson lower bound is:
At 100 mg, substituting \(\hat{p}=0.9125\) and \(n=400\) gives:
Because 0.8864 is below 0.90, 100 mg does not meet the lower-bound rule. At 110 mg, substituting \(\hat{p}=0.935\) gives:
This lower bound exceeds 0.90. Since the 90 mg estimate is already below the threshold, the lower-bound rule selects 110 mg from these three candidates. Thus, the same simulation results select 100 mg under the point-estimate rule and 110 mg under the specified uncertainty-aware rule. The difference reflects the decision criteria, not a change in the simulated data.
Diagnostics, Alternatives, and Limitations
Before reporting a selected dose, check that the rule has been applied to every eligible candidate and that the dose ordering is sensible. A PTA curve may be expected to rise with dose for some targets, but monotonicity should not be assumed automatically. Nonlinear PK, competing exposure constraints, or simulation variability can produce nonmonotone estimates. When estimates rise and fall irregularly, inspect the simulated outcomes and Monte Carlo precision at each dose; do not silently smooth the curve and treat the smoothed values as observed results.
The main alternatives have different purposes:
- Point-estimate threshold: straightforward and easy to communicate, but sensitive to Monte Carlo variation near \(q\).
- Lower-confidence-bound threshold: guards against qualifying a dose based on an optimistic Monte Carlo estimate, but can select a higher dose and depends on the chosen confidence-bound method and level.
- Finer dose grid: reduces dose discretization when lower doses may qualify, but does not by itself reduce Monte Carlo uncertainty.
- More simulation replicates: generally improves Monte Carlo precision, but does not resolve a coarse dose grid or uncertainty in the model itself.
If simulation estimates are close to the threshold, report the number of replicates, the estimated PTA, and the criterion used. A dose just above \(q\) under the point-estimate rule may be a fragile selection. If no candidate meets the rule, report that outcome and consider whether further simulation or additional candidate doses are warranted; do not extrapolate beyond the evaluated range as if the criterion had been verified there.
A prespecified criterion prevents the selection from being tuned to whichever dose looks most favorable after viewing the results. It also makes comparisons reproducible: another analyst can apply the same target, threshold, eligible grid, and uncertainty rule to the same simulation outputs. The selected dose remains conditional on the specified model and simulation assumptions, so the precision of a PTA estimate should not be confused with certainty that the model represents clinical outcomes accurately.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A point-estimate rule uses \(q=0.85\). On a grid of 60, 80, and 100 mg, estimated PTAs are 0.82, 0.85, and 0.93. Which dose is selected?
- What does a dose-grid bound of 10 mg say, and what does it not say?
- Why can a finer dose grid fail to resolve an estimate that lies close to the PTA threshold?
- Under a lower-confidence-bound rule, why might the selected dose exceed the one selected by a point-estimate rule?
Solutions. (1) The smallest candidate meeting \(\hat{\pi}(D)\ge0.85\) is 80 mg. (2) Under the stated monotonicity and crossing assumptions, the first grid point at or above the true crossing is no more than 10 mg above the crossing. This does not bound Monte Carlo error or model uncertainty. (3) More dose points reduce discretization but do not necessarily make the PTA estimate at any point more precise; that depends on the number and design of simulations. (4) A lower confidence bound is below the point estimate, so a dose whose estimate barely exceeds \(q\) may fail the lower-bound criterion, requiring a higher candidate dose before the bound reaches \(q\).