Why Target Definitions Deserve a Sensitivity Analysis
A probability of target attainment (PTA) is conditional on how the target is defined. A target based on area under the concentration-time curve (AUC) can give a different PTA from one based on trough concentration (Cmin), even when both are calculated from the same simulated subjects. Likewise, changing an exposure threshold or the limits of a therapeutic window can alter both the PTA curve and the dose chosen from it.
Sensitivity analysis makes this dependence visible. The question is not whether one definition is correct in the abstract: the exposure metric, threshold, and any upper limit should be supported by the pharmacologic and clinical rationale for the application. The analysis asks how conclusions change across plausible alternatives, while holding the PK model, simulated cohort, and dose regimens fixed. This isolates the effect of the target definition; it does not assess uncertainty in the model parameters.
Calculation and Interpretation
Let \(N\) denote the number of subjects in the fixed simulated cohort, \(d\) a candidate dose, \(m\) an exposure metric, and \(s\) a target scenario. Write \(E_i(d;m)\) for the exposure metric calculated for subject \(i\) at dose \(d\). Define \(I_{i,s}(d)\) to equal 1 if that subject's exposure meets scenario \(s\), and 0 otherwise. For a lower-bound threshold \(L_s\), for example, \(I_{i,s}(d)=1\) when \(E_i(d;m)\geq L_s\). For a therapeutic window with limits \(L_s\) and \(U_s\), it equals 1 when \(L_s\leq E_i(d;m)\leq U_s\).
This is the proportion of simulated subjects meeting the target under scenario \(s\) at dose \(d\). It estimates the probability of attainment represented by the simulation design and model. When comparing definitions, compute this proportion on the same subjects at each dose. A useful direct summary is the difference in PTA between two scenarios, expressed in percentage points, at a given dose. A separate summary is the change in the dose selected by a prespecified PTA criterion.
For instance, if the dose criterion is a PTA of at least \(c\), the selected dose is the smallest candidate dose whose PTA is at least \(c\). If no candidate dose reaches \(c\), report that no dose in the evaluated set meets the criterion. Do not silently extrapolate beyond the simulated doses.
Worked Examples
Worked Example: Changing a Lower-Bound Threshold
Suppose a hypothetical simulation uses the same model and cohort of 10 subjects at three candidate doses. The exposure metric is AUC in mg·h/L. The simulated AUC values are:
| Dose (mg) | Simulated AUC values (mg·h/L) |
|---|---|
| 100 | 70, 75, 78, 82, 84, 86, 89, 91, 95, 102 |
| 125 | 80, 84, 87, 91, 94, 97, 101, 105, 110, 118 |
| 150 | 90, 96, 100, 104, 108, 112, 116, 120, 126, 135 |
Compare lower-bound targets of AUC at least 80 mg·h/L and at least 90 mg·h/L. With the 80 mg·h/L threshold at 100 mg, seven of the 10 subjects meet the target, so \(\widehat{\mathrm{PTA}}=7/10=0.70\). At 125 mg, all 10 meet it, giving \(10/10=1.00\). With the 90 mg·h/L threshold, three subjects meet the target at 100 mg, giving \(3/10=0.30\), and seven meet it at 125 mg, giving \(7/10=0.70\). At 150 mg, all 10 meet either threshold.
If the prespecified criterion is PTA of at least 0.80, the 80 mg·h/L definition selects 125 mg: 100 mg gives 0.70 and 125 mg gives 1.00. The 90 mg·h/L definition selects 150 mg: PTA is 0.30, 0.70, and 1.00 at 100, 125, and 150 mg, respectively. Thus this plausible change in threshold shifts the selected dose from 125 to 150 mg in this hypothetical analysis.
Worked Example: Changing the Exposure Metric
Consider a hypothetical fixed cohort of 10 subjects simulated at 100 mg and 125 mg. For illustration, compare AUC in mg·h/L against a threshold of 100 mg·h/L with Cmin in mg/L against a threshold of 1.00 mg/L. The paired results are:
| Subject | AUC at 100 mg | Cmin at 100 mg | AUC at 125 mg | Cmin at 125 mg |
|---|---|---|---|---|
| 1 | 120 | 1.40 | 150 | 1.80 |
| 2 | 115 | 1.30 | 144 | 1.70 |
| 3 | 110 | 1.20 | 138 | 1.50 |
| 4 | 108 | 1.10 | 135 | 1.40 |
| 5 | 105 | 1.05 | 131 | 1.30 |
| 6 | 103 | 1.02 | 129 | 1.20 |
| 7 | 101 | 0.98 | 126 | 1.05 |
| 8 | 100 | 0.95 | 124 | 1.00 |
| 9 | 95 | 0.90 | 119 | 0.98 |
| 10 | 88 | 0.80 | 98 | 0.90 |
At 100 mg, eight subjects meet the AUC threshold: \(8/10=0.80\). Six meet the Cmin threshold: \(6/10=0.60\). At 125 mg, nine meet the AUC threshold: \(9/10=0.90\), while eight meet the Cmin threshold: \(8/10=0.80\). If the dose criterion is at least 0.80, AUC selects 100 mg, whereas Cmin selects 125 mg. The calculations do not show that one metric is inherently preferable; they show that the metric is consequential and should be justified by the intended pharmacologic target.
Worked Example: Changing Therapeutic-Window Limits
Suppose the target is AUC within a therapeutic window, and the same 10 hypothetical subjects have these exposures at each dose:
| Subject | AUC at 75 mg (mg·h/L) | AUC at 90 mg (mg·h/L) |
|---|---|---|
| 1 | 70 | 75 |
| 2 | 75 | 80 |
| 3 | 80 | 90 |
| 4 | 82 | 92 |
| 5 | 84 | 94 |
| 6 | 86 | 96 |
| 7 | 88 | 98 |
| 8 | 89 | 100 |
| 9 | 95 | 105 |
| 10 | 100 | 110 |
For the broad window from 80 to 120 mg·h/L, inclusive, eight subjects are within range at 75 mg. Thus PTA is \(8/10=0.80\), which meets an 80% criterion. At 90 mg, nine subjects are within range (all except the subject with AUC 75 mg·h/L), so PTA is \(9/10=0.90\). The lowest dose meeting the criterion is 75 mg.
For the narrower window from 90 to 110 mg·h/L, inclusive, two subjects are within range at 75 mg (AUC 95 and 100), giving \(2/10=0.20\). At 90 mg, eight subjects are within range (AUC 90, 92, 94, 96, 98, 100, 105, and 110), giving \(8/10=0.80\). Under this definition, 90 mg is the lowest dose meeting the same criterion. The upper limit matters: for a window target, attaining more exposure is not automatically better.
Why the Calculation Works, and What to Check
For each simulated subject, the indicator \(I_{i,s}(d)\) records whether the subject meets the specified rule. Summing these indicators counts the subjects who meet it; dividing by \(N\) converts that count to a proportion. Equivalently, if a subject is selected uniformly from the fixed simulated cohort, the average of the indicators is the fraction of that cohort meeting the target. This gives the PTA estimate for that cohort and scenario. Changing only the scenario changes the classification rule, not the simulated exposures.
A disciplined comparison changes one definition at a time when the purpose is to identify which choice drives a result. For example, compare two AUC thresholds while holding the metric and all other rules fixed, then compare the metric itself under separately justified target definitions. A factorial set of scenarios can also evaluate combinations, but report the exact metric and limits for every scenario; otherwise the contrast is difficult to interpret.
Before interpreting results, verify that the same subjects, dose histories, and simulation outputs were used across scenarios. Check that metrics have appropriate units, thresholds use those units, and boundary conventions are explicit. In the examples above, window limits are inclusive. A strict inequality would change the classification of a subject exactly at a boundary. Also check that the dose-selection criterion and the candidate-dose set are unchanged. Report a PTA difference such as 0.80 versus 0.60 as a difference of 20 percentage points, not as an unqualified “20% increase.”
Some diagnostics depend on the target. For a lower-bound target, PTA would generally be expected not to decrease as the threshold is lowered, when exposures and the cohort are unchanged. For nested windows, a narrower window cannot include more subjects than a wider window at the same dose. Violations often indicate a coding, units, or boundary-rule inconsistency. By contrast, PTA for a therapeutic window can rise and then fall with increasing dose as subjects first enter and later exceed the window. A nonmonotone curve is not, by itself, evidence of an error.
Comparison, Limitations, and Common Pitfall
The central comparison is between target definitions applied to a common set of simulated exposures. This paired design makes differences in PTA attributable to the classification rules, rather than to a different cohort being simulated for each target. A one-scenario analysis is simpler and may be appropriate when a target is firmly established, but it cannot show how sensitive the dose recommendation is to plausible alternatives. Re-simulating a new cohort for each scenario can introduce avoidable differences from cohort composition; if simulations must be rerun, use a design that keeps the underlying cohort comparable.
The results remain conditional on the selected PK model, its inputs, the simulated cohort, the evaluated regimens, and the target scenarios. A sensitivity analysis does not establish that an alternative threshold is clinically justified, and it does not quantify uncertainty in PK parameters. The latter is a distinct question. Nor does a stable dose recommendation prove that every modeling or clinical assumption is correct: different scenarios may happen to select the same candidate dose despite meaningful changes in PTA.
- Common mistake: changing the cohort along with the threshold. A student simulates new subjects for each target and attributes every PTA difference to the target definition. Correction: apply each plausible definition to the same simulated subjects and dose outputs, so the comparison isolates the classification rule.
- Common mistake: assuming PTA must increase with dose. This can be reasonable to examine for a lower-bound target but is not a general rule for therapeutic windows, which include an upper limit. Correction: interpret the target rule before judging the shape of the PTA curve.
- Common mistake: reporting a selected dose without its criterion. A dose recommendation depends on both the target scenario and the prespecified PTA cutoff. Correction: report the threshold or window, the PTA criterion, the evaluated doses, and whether any dose met it.
Check Your Understanding
Answer each question, then compare with the solutions below.
- At a dose, 18 of 30 simulated subjects meet a specified target. Calculate PTA.
- At a fixed dose, 12 subjects meet a lower-bound threshold of 50 units and 8 meet a stricter threshold of 60 units. Is that ordering consistent with nested lower-bound targets?
- For an inclusive window from 10 to 20 mg·h/L, do exposures of 10, 15, and 20 mg·h/L meet the target?
- Why should the same simulated cohort be used when comparing target definitions?
- A therapeutic-window PTA falls as dose increases. Does that alone demonstrate a simulation error?
Solutions. (1) \(\widehat{\mathrm{PTA}}=18/30=0.60\), or 60%. (2) Yes. Every exposure at least 60 units is also at least 50 units, so the stricter threshold cannot have more attainments; 8 is no greater than 12. (3) All three meet the inclusive rule because each value satisfies \(10\leq E\leq20\). (4) Holding the cohort fixed avoids confounding the target-definition comparison with changes in simulated subjects; observed PTA differences then reflect the changed classification rule. (5) No. If increasing dose moves subjects above the window's upper limit, fewer may remain within it. Check the exposure values and target implementation, but a decline can be consistent with the definition.