Why Check Sensitivity to a Dose History?
A fitted pharmacokinetic model can give a precise prediction for the data supplied to it, yet the recorded dose history may be uncertain. A dose might have been taken later than recorded, partly administered, or missed. If the predicted concentration or a resulting clinical classification changes materially across plausible versions of that history, the conclusion depends on information that is not firmly known. Sensitivity analysis makes that dependence visible.
The central comparison is between predictions from the same fitted model under specified alternative dose histories. This isolates how much the history alone changes predictions, provided the model parameters and other inputs are held fixed. It does not establish which history actually occurred, nor does it by itself assign probabilities to the alternatives.
Model and Comparison Setup
For a transparent calculation, consider a one-compartment model with instantaneous intravenous doses and first-order elimination. Let \(D_j\) be the amount of dose event \(j\), \(t_j\) its administration time, \(V\) the apparent volume of distribution, and \(k\) the elimination rate constant. For a prediction time \(t\), the concentration is
where \(H\) denotes the complete dose history. The formula assumes a known dose amount and time for each included event, linear one-compartment disposition, instantaneous distribution of each dose, first-order elimination, and no additional input between dose events. Concentrations are in amount per volume when \(D_j\) and \(V\) use compatible units; \(k\) must be in inverse time units consistent with \(t\) and \(t_j\).
In an NLME analysis, \(V\) and \(k\) may be individual parameters derived from population parameters and individual random effects. For a dose-history comparison, use the same fitted parameter values in each scenario unless the purpose is explicitly to assess the effect of refitting. Keep observation times, covariate values, and other model assumptions fixed as well. Then changes in predictions can be attributed to the specified change in history.
A useful scenario set should be grounded in what is plausible from the available records: for example, an administration-time window supported by a diary, an amount range supported by tablet counts, or a missed dose supported by an adherence record. It should not be an arbitrary collection of extreme possibilities. If the alternatives have no defensible probabilities, report their predictions as scenarios or a range, not as a probability distribution.
Worked Example: Uncertain Administration Time
Suppose a hypothetical patient received a 100 mg intravenous dose, and the model uses \(V=20\) L and \(k=0.10\ \mathrm{h}^{-1}\). A concentration is to be predicted at 10 h. The recorded administration time is 0 h, but a plausible alternative is 2 h. Assume these are the only dose events and all model parameters are held fixed.
Under the recorded history, elapsed time since dosing is 10 h:
Under the alternative history, elapsed time is 8 h:
Using unrounded calculations, the second prediction exceeds the first by approximately \(0.407247615\) mg/L, or \(0.407248\) mg/L to six decimal places and \(0.407\) mg/L to three decimal places. The difference arises because the later dose has had less time to be eliminated. These are conditional predictions under two histories, not evidence that either administration time is more likely.
Summarizing the Sensitivity
For a finite set of histories \(H_1,\ldots,H_m\), calculate the prediction \(C_i=C(t;H_i,\theta)\) under each history using the same parameter vector \(\theta\). The scenario envelope is the smallest-to-largest prediction, and its width describes the absolute spread:
The envelope is not a confidence interval or a prediction interval: it does not account for sampling uncertainty, parameter uncertainty, or residual variability. It reports only the range generated by the chosen histories at the fixed model inputs. If values are positive and a relative comparison is useful, one may also report the range as a percentage of a clearly stated reference prediction, such as the nominal-history prediction.
Worked Example: A Plausible Time Window
Retain the 100 mg dose, \(V=20\) L, \(k=0.10\ \mathrm{h}^{-1}\), and prediction time 10 h. Suppose the administration time is uncertain but judged plausible to lie between 3 h and 5 h. Calculate the endpoint predictions and compare them with the width of the resulting envelope.
For administration at 3 h, elapsed time is 7 h, so
For administration at 5 h, elapsed time is 5 h, so
Because this concentration increases as the administration time moves later, the scenario envelope is approximately 2.482927 to 3.032653 mg/L. Its width from the unrounded predictions is
or 0.549727 mg/L to six decimal places. Subtracting the displayed rounded endpoints gives 0.549726 mg/L because each endpoint has already been rounded; the reported width above is calculated from the underlying values. The range describes the specified 3-to-5-hour uncertainty only. It does not imply that every intermediate time is equally likely.
Worked Example: Uncertain Dose Amount and a Threshold
Suppose a hypothetical 100 mg dose at time 0 is uncertain because the patient may have received between 80 mg and 120 mg. Use \(V=20\) L, \(k=0.10\ \mathrm{h}^{-1}\), and predict concentration at 8 h. For illustration, consider whether the prediction is above a decision threshold of 2.0 mg/L. The threshold is stipulated for this example; it is not a treatment recommendation.
For 80 mg, the predicted concentration is
For the nominal 100 mg amount and the upper alternative of 120 mg:
In this linear model, concentration is directly proportional to dose amount, so the three predictions form an envelope from approximately 1.797316 to 2.695974 mg/L. The 80 mg scenario is below 2.0 mg/L, while the 100 mg and 120 mg scenarios are above it. Therefore, the threshold classification is not robust to the assessed amount uncertainty, even though the prediction range is clear.
Adherence Scenarios and Robust Conclusions
Adherence uncertainty can be represented by adding, removing, or modifying dose events in the history. It is important to distinguish a known zero dose from an unknown record: a scenario that omits a dose represents a hypothesized missed administration, not proof that no dose was taken. The scenarios should reflect the evidence and the specific uncertainty being evaluated.
Worked Example: Possible Missed Dose Before a Trough
Consider hypothetical 100 mg intravenous doses scheduled at 0, 12, and 24 h, with \(V=20\) L and \(k=0.10\ \mathrm{h}^{-1}\). Predict concentration at 36 h, just before any subsequent dose. Compare the recorded-history scenario, in which all three doses were given, with a scenario in which the 24 h dose was missed.
With all doses given, the elapsed times are 36, 24, and 12 h. Thus
If the 24 h dose was missed, only the doses at 0 and 12 h contribute:
At a hypothetical minimum-concentration threshold of 1.0 mg/L, the all-doses scenario is above the threshold and the missed-dose scenario is below it. The classification is therefore sensitive to whether the 24 h dose was administered. The calculation does not determine adherence; it shows why resolving the dose record could matter for this prediction.
Justification: Why the Dose Contributions Add
The sum in the concentration formula follows from the assumed model dynamics. Between dose events, concentration satisfies \(dC/dt=-kC\), whose solution from a starting concentration \(C(t_0)\) is \(C(t)=C(t_0)e^{-k(t-t_0)}\). An instantaneous dose \(D_j\) increases concentration by \(D_j/V\). Therefore, at a later time \(t\), that dose contributes \((D_j/V)e^{-k(t-t_j)}\). Under linear one-compartment dynamics, each dose contribution evolves independently and the total concentration is their sum.
This derivation also identifies the limits of the examples. If absorption is delayed, disposition is nonlinear, or the compartmental structure differs, the simple exponential contributions are not adequate. The relevant fitted model must be evaluated with each alternative event history; the scenario-comparison principle remains the same, but the prediction calculation must match that model.
Diagnostics, Alternatives, and Limitations
A sensitivity analysis is only as informative as its scenario construction. Check that each alternative is supported by a plausible account of adherence, timing, or amount; that event ordering and time units are correct; and that the same prediction time and non-dose inputs are used in every comparison. For each scenario, inspect the resulting concentration-time predictions for discontinuities or implausible behavior. A surprising result may signal an encoding error, an inconsistent unit, an incorrectly placed event, or a model assumption that is inappropriate for the situation.
The main approaches answer related but different questions:
| Approach | What it changes | Interpretation |
|---|---|---|
| One-at-a-time scenarios | One uncertain feature, such as time or amount, at a time | Shows the separate effect of each specified change, but can miss interactions. |
| Joint scenario set | Several uncertain features together | Shows predictions under coherent combinations; conclusions depend on which combinations are included. |
| Probability-weighted analysis | Histories are assigned defensible probabilities | Can summarize a distribution of predictions, but requires a justified probability model for the histories. |
| Refitting under alternatives | The model is re-estimated for each assumed history | Assesses how the history affects fitted parameters and predictions together, rather than isolating prediction sensitivity at fixed parameters. |
A probability-weighted average is not justified merely because several scenarios are listed. It requires probabilities that represent uncertainty about the actual history. Likewise, the midpoint of a scenario envelope is not automatically the best prediction. If parameter uncertainty is also important, it should be assessed separately or jointly and reported as such; the fixed-parameter ranges above do not include it.
A common mistake is to say, “The model prediction is 2.25 mg/L, so the conclusion is certain,” when that value was computed from only one assumed administration time. The correction is to state which history produced the prediction, calculate the alternatives supported by the records, and check whether the substantive conclusion changes across them. Conversely, an envelope cannot demonstrate robustness to histories that were never considered. Report the assumptions and the scope of the scenario set explicitly.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- For the first worked example, why does the 2 h administration scenario predict a higher concentration at 10 h than the 0 h scenario?
- In the dose-amount example, does the threshold classification remain robust across 80–120 mg? Explain using the computed predictions.
- What does the 0.549727 mg/L width in the time-window example represent, and what does it not represent?
- Why is a missed-dose scenario not evidence that the dose was actually missed?
- If parameters are re-estimated separately for each dose history, what additional source of change enters the comparison?
Solutions. (1) The later administration leaves less elapsed time for elimination: 8 h rather than 10 h, giving approximately 2.246645 rather than 1.839397 mg/L. (2) No. The 80 mg prediction is approximately 1.797316 mg/L, below 2.0 mg/L, whereas the 100 mg and 120 mg predictions are approximately 2.246645 and 2.695974 mg/L, above the threshold. (3) It is the difference between the largest and smallest predictions for the specified 3-to-5-hour administration-time alternatives at fixed parameters. It is not a confidence interval or a probability distribution. (4) It is a modeled alternative used to examine consequences; only the dose records or other evidence can address whether the dose occurred. (5) Differences may reflect both the changed dose history and changes in the estimated model parameters, so the comparison no longer isolates history effects on predictions at fixed fitted parameters.