Why Use a Limited-Sampling Strategy?
AUC is often the exposure measure of interest, but estimating it from a complete concentration-time profile can require many samples. In therapeutic drug monitoring, a prespecified set of one or a few samples may offer a practical alternative: use concentrations drawn at informative times to predict AUC over a dosing interval. This can reduce sampling burden, but the prediction is only useful when its design and validation match the intended patients, dosing conditions, and monitoring decision.
A limited-sampling strategy is not simply “measure a concentration and infer exposure.” It is a defined prediction rule: specify the sampling times, the population and conditions for which the rule is intended, and the calculation that maps the sampled concentrations to AUC. The AUC target and the timing conventions must be explicit. For steady-state dosing, for example, the target may be AUC over one dosing interval, denoted \(AUC_{\tau}\), where \(\tau\) is the interval between doses.
The preceding material on recording sample time, steady-state sampling, and interpreting peak-and-trough samples matters here because an AUC prediction is conditional on the circumstances under which its samples were collected. A concentration recorded without a reliable time relative to dose cannot be treated as though it came from the prescribed sampling time.
Prediction Rules and Their Assumptions
One common strategy uses a regression equation fitted in patients who have both the limited samples and a reference AUC. For two sampling times \(t_1\) and \(t_2\), a linear rule might be
Here \(C(t_j)\) is the measured concentration at the prespecified time \(t_j\); \(\widehat{AUC}_{\tau}\) is the predicted AUC; \(\beta_0\) is an intercept; and \(\beta_1,\beta_2\) are fitted coefficients. If concentration is expressed in mg/L and AUC in mg·h/L, the concentration coefficients have units of hours, while the intercept has units of mg·h/L. The equation and coefficients are specific to the population and conditions used to develop and validate the strategy.
Before applying such a rule, check that the development data and intended use are sufficiently aligned. Relevant conditions include dose and dosing interval, formulation and route, steady-state status, sample timing, assay, and patient characteristics that influence concentration-time profiles. The strategy also assumes that the chosen samples contain enough information about AUC in the intended setting. Two patients may have similar concentrations at one time but different AUCs if their profiles differ elsewhere; a single sample cannot resolve that difference unless the relationship is reliably predictable in the target population.
A fitted rule should not be treated as a physical identity. Regression may summarize an empirical relationship rather than describe a complete pharmacokinetic model. If the rule is linear, it assumes that the chosen linear combination predicts AUC adequately over the range of concentrations and patient characteristics for which it will be used. Predictions outside that range are extrapolations, not validated estimates.
How the Prediction Rule Is Calibrated
Calibration requires a development set with both the limited concentrations and a reference AUC for each individual. The reference AUC is obtained using a suitably justified method based on richer concentration information or an established model. Its uncertainty matters: a prediction rule cannot be more trustworthy than the reference values used to assess it.
For a linear rule, the coefficients can be chosen to minimize the sum of squared differences between reference AUCs and predicted AUCs in the development data. Let \(y_i\) be the reference AUC for individual \(i\), and let the row \(x_i\) contain an intercept and that individual’s sampled concentrations. Then the fitted coefficients minimize \(\sum_i (y_i-x_i\beta)^2\). If the columns of the design matrix \(X\) are linearly independent, the unique least-squares solution is \(\hat{\beta}=(X^\mathsf{T}X)^{-1}X^\mathsf{T}y\).
The justification follows by differentiating the objective with respect to \(\beta\). Setting its gradient to zero gives \(X^\mathsf{T}X\hat{\beta}=X^\mathsf{T}y\); when \(X^\mathsf{T}X\) is invertible, multiplying by its inverse gives the stated solution. This derivation explains how coefficients are estimated, not why a particular set of sampling times is informative. That must be assessed from concentration-time behavior and from prediction performance in data not used to fit the coefficients.
Worked Example: Applying a Two-Sample Rule
Suppose, hypothetically, that a strategy has been calibrated for a defined group of patients at steady state. Its rule is \(\widehat{AUC}_{\tau}=4.0+3.0C(2)+2.0C(8)\), with AUC in mg·h/L, concentrations in mg/L, and sample times in hours after the dose. For one patient, the concentrations are 5.0 mg/L at 2 hours and 2.0 mg/L at 8 hours. Assume the dose history, steady-state condition, and sampling times meet the strategy’s specifications.
Substitute the measured concentrations:
The predicted AUC is 23.0 mg·h/L. This is a model-based prediction, not a directly observed AUC and not a guarantee that the patient’s true AUC equals 23.0 mg·h/L. Its clinical meaning depends on the strategy’s prediction error and whether this patient is represented by the validation evidence.
Choosing Times and Validating Performance
Sampling times should be selected before routine use, based on the expected concentration-time profiles and the information needed to predict the target AUC. A sample early after dosing may reflect absorption and peak behavior; a later sample may help characterize the decline or the concentration near the next dose. The most useful combination depends on the drug, route, dosing interval, and patient population. Times should not be chosen only because they are convenient if that choice leaves important profile differences unrepresented.
Evaluate performance in a validation set that was not used to fit the rule. Ideally, validation is independent and includes patients and conditions resembling the intended monitoring setting. For each individual, calculate the prediction error as predicted AUC minus reference AUC. Summarize both systematic error (bias) and the spread or size of errors. Examine whether errors depend on AUC level, concentration pattern, dose, or patient subgroup. A small average bias alone is not sufficient: overprediction in some patients can cancel underprediction in others.
Worked Example: Summarizing Validation Errors
In a hypothetical validation set of five patients, reference AUC values are 20, 24, 30, 35, and 40 mg·h/L. The limited-sampling rule predicts 22, 21, 33, 32, and 44 mg·h/L. Define each error as prediction minus reference.
The errors are \(2,-3,3,-3,\) and \(4\) mg·h/L. The mean prediction error, or bias, is
The mean absolute error is
The root mean squared error is
The mean error is near zero, but individual errors reach 4 mg·h/L in magnitude. In this small example, all five absolute errors are at most 20% of their corresponding reference AUCs; that observation alone does not establish adequate performance. Five validation patients provide weak evidence about performance across a broader clinical population. Adequacy must be judged against a prespecified, clinically justified error tolerance and with enough representative validation data.
Validation should consider the decision the estimate will support. If a small error could change a dose recommendation near a target boundary, average performance may obscure clinically important misclassification. Check the distribution of errors, the frequency of errors beyond a clinically acceptable limit, and whether error differs across relevant subgroups or across the exposure range. The acceptable limits should be set for the intended use rather than selected after seeing the results.
Using a Prediction for a Monitoring Decision
A limited-sampling estimate can inform a decision only if the target and the prediction uncertainty are considered together. For example, a point estimate near an individual AUC target may not justify a confident dose change if the strategy has substantial prediction error. In addition, translating exposure into a dose adjustment requires assumptions beyond the sampling rule, including that dose and AUC are sufficiently proportional over the proposed change and that the patient’s relevant conditions remain stable.
Worked Example: AUC Estimate Near a Target
Suppose, hypothetically, a patient is receiving 600 mg per dosing interval, and the limited-sampling rule predicts an AUC of 36 mg·h/L. The individualized AUC target is 30 mg·h/L. If pharmacokinetics are dose proportional over the contemplated range and other relevant conditions remain unchanged, a proportional dose calculation gives
The arithmetic yields a hypothetical proportional dose of 500 mg per interval. It does not establish that this change is clinically appropriate. The AUC prediction may be uncertain, the dose proportionality assumption may not hold, and other clinical information may affect the decision. If the estimated AUC were close to the target, the prediction uncertainty and consequences of changing the dose would be particularly important.
Alternatives, Diagnostics, and Limitations
A limited-sampling regression rule is one approach among several. A rich-profile approach uses more samples to characterize exposure directly, at the cost of greater sampling burden. A model-based approach can use a population pharmacokinetic model and selected concentrations to estimate an individual’s exposure; it may use prior population information but depends on the model being appropriate for the patient and data. These approaches are not interchangeable by default: each has its own data requirements, assumptions, and validation needs.
Warning signs that a limited-sampling strategy may be failing include sample times that differ materially from the specified times, missing or uncertain dose histories, use before the required steady state, concentrations outside the development range, or prediction errors that vary systematically by dose or patient subgroup. A rule may also fail after a change in formulation, assay, route, or clinical population if that change alters the relationship between sampled concentrations and AUC.
A common mistake is to assume that a high correlation between predicted and reference AUC proves that estimates are sufficiently accurate. Correlation measures how well values move together, not whether predictions are close to the reference. A rule can have high correlation while consistently overpredicting, underpredicting, or making errors too large for the clinical decision. Examine bias and the size and pattern of errors, and assess whether the evidence is precise enough for the intended use.
Another mistake is to substitute a convenient sample time into a rule calibrated at a different time. For example, a concentration at 9 hours is not automatically an acceptable substitute for \(C(8)\). The coefficient was calibrated for the specified time, and the concentration may be changing substantially over that hour. Use the rule only when timing meets its prespecified requirements, or use a strategy validated for the actual timing.
Check Your Understanding
Try each question, then compare your reasoning with the solutions.
- A rule predicts AUC over a dosing interval from samples collected at 1 and 6 hours. Why is a concentration drawn at 7 hours not automatically interchangeable with the 6-hour sample?
- A validation set has a mean prediction error of zero. Does that show that every prediction is accurate? Explain.
- A hypothetical rule is \(\widehat{AUC}_{\tau}=2+4C(3)\), where AUC is in mg·h/L and concentration is in mg/L. What is the predicted AUC when \(C(3)=5\) mg/L?
- Name two types of evidence or diagnostics, beyond correlation, that are useful when evaluating a limited-sampling strategy.
Solutions. (1) The rule’s coefficient was calibrated for a 6-hour concentration. Concentrations can change between 6 and 7 hours, so substituting the later measurement may introduce an unvalidated error. (2) No. Positive and negative errors can cancel in the mean; individual errors may still be large. Inspect error magnitude and pattern, including errors relevant to the clinical decision. (3) \(\widehat{AUC}_{\tau}=2+(4\ \text{h})(5\ \text{mg/L})=22\ \text{mg·h/L}\). (4) Examples include mean prediction error, MAE or RMSE, errors relative to a prespecified clinical tolerance, and error patterns across exposure levels or patient subgroups.