Why an Interim Concentration Is Not an Exact Exposure
An interim drug concentration is an observation produced by both a patient’s pharmacokinetics and a measurement process. Even a technically precise assay does not make a concentration measured at one or two times an exact summary of the patient’s full exposure. Assay error adds uncertainty to each observed value; sparse sampling adds uncertainty about the concentration-time profile between and beyond those observations. If an adaptation rule treats a fitted concentration or exposure estimate as known, it can make a decision that is more certain than the data justify.
The earlier material on Bayesian therapeutic drug monitoring addresses using interim concentration data to inform within-trial decisions. Here the emphasis is narrower: how assay error and sparse sampling enter those data, and why uncertainty must be propagated into a decision. Concentration units and exposure units should always be stated separately: a concentration might be measured in mg/L, while an area under the concentration-time curve (AUC) might be measured in mg·h/L.
Modeling Assay Error
Drug concentrations are positive, and assay error is often usefully represented as multiplicative rather than additive. Let \(C_{ij}\) be the underlying concentration for participant \(i\) at sampling time \(t_{ij}\), and let \(Y_{ij}\) be the assay result. A basic lognormal measurement model is
Here \(\sigma_a\) is the assay’s log-scale standard deviation. Conditional on the true concentration, the log assay result has mean \(\log C_{ij}\) and variance \(\sigma_a^2\). The multiplicative error is centered at one on the original scale in the sense that the median of \(Y_{ij}/C_{ij}\) is one. Its coefficient of variation is \(\sqrt{\exp(\sigma_a^2)-1}\). This model assumes that the log-scale errors are appropriately described by a normal distribution, are independent across measurements unless modeled otherwise, and have a known or estimated error variance applicable to the assay conditions.
Independence and constant precision are substantive assumptions, not automatic properties of a laboratory result. Replicates may share batch effects, and assay precision can vary with concentration. Calibration data, quality-control samples, and replicate measurements can help assess these assumptions. If a result is below the limit of quantification, replacing it with zero or with the limit itself is generally not equivalent to using the information in a censored observation; the observation process must be handled consistently with the analysis model.
Worked Example: Assay Uncertainty Around One Concentration
Suppose, hypothetically, a specimen is reported as 10.0 mg/L. The assay has a 10% coefficient of variation, and assume the lognormal error model above. For this illustration, also assume a flat prior for the true log concentration, so that the observed log result is the center of the conditional distribution for the true log concentration.
First convert the coefficient of variation to a log-scale standard deviation:
Under the stated flat-prior assumption, an approximate 95% interval for the true concentration is obtained by exponentiating the log-scale interval:
Thus the reported value is 10.0 mg/L, but the model does not support treating the underlying specimen concentration as known to that precision. This interval concerns the specimen concentration under the stated model; it does not include uncertainty about the participant’s concentration at other times, the PK parameters, or the assay calibration itself.
From Sparse Samples to Exposure Uncertainty
A PK model links concentrations across time. Write the model-predicted concentration as \(C(t;\theta)\), where \(\theta\) denotes the relevant PK parameters, such as an initial concentration and an elimination rate. Given sparse noisy observations, the fitted values of \(\theta\) are uncertain. A derived quantity \(g(\theta)\), such as AUC, is therefore uncertain as well. A point estimate \(g(\hat\theta)\) alone does not describe that uncertainty.
One useful local approximation shows how parameter uncertainty carries through a derived exposure. If \(\hat\theta\) estimates \(\theta\), and the estimated parameter covariance matrix is \(V\), then a first-order Taylor expansion gives
This approximation requires that the function be sufficiently smooth near \(\hat\theta\) and that local uncertainty be described reasonably by \(V\). It can be poor when the parameter distribution is highly skewed, the model is weakly identified, or the derived quantity is strongly nonlinear over the plausible parameter range. In those settings, integrating over the parameter distribution or using posterior simulation is generally preferable. In either case, assay error belongs in the likelihood used to estimate the PK parameters; it should not be silently discarded after fitting.
Worked Example: Two Sparse Samples and an AUC Interval
Consider hypothetical concentrations of 8.19 mg/L at 1 hour and 3.68 mg/L at 5 hours after a dose. Assume a one-compartment decline model with no further input during this interval, \(C(t)=C_0\exp(-kt)\), and independent log-scale assay errors with known standard deviation 0.10. The model also assumes the samples are timed correctly and that the exponential decline is appropriate over the period. The two observed concentrations provide enough information to estimate the two parameters in this simplified model, but their assay error leaves both estimates uncertain.
Let \(z_1=\log(8.19)\approx2.1029\) and \(z_2=\log(3.68)\approx1.3029\). The fitted elimination rate and log initial concentration are
The estimated initial concentration is \(\hat C_0=\exp(2.3029)\approx10.00\) mg/L. For this model, AUC from time zero to infinity is \(C_0/k\), so the fitted exposure is
To quantify approximate uncertainty, the log-error model gives \(\operatorname{Var}(z_1)=\operatorname{Var}(z_2)=0.01\). From the coefficient formulas above, \(\operatorname{Var}(\hat a)=0.01625\), \(\operatorname{Var}(\hat k)=0.00125\), and \(\operatorname{Cov}(\hat a,\hat k)=0.00375\). Since \(\log(\mathrm{AUC})=a-\log k\), the delta-method variance is
The approximate standard error on the log-AUC scale is \(\sqrt{0.01000}=0.1000\). Exponentiating the log-scale interval gives
The fitted AUC is about 50.02 mg·h/L, but the interval is wide relative to that point estimate. This calculation is conditional on the simplified model and known assay precision; uncertainty in the model structure, assay variance, or sampling times would require additional modeling. With only two samples, a more complex PK model would not be identified reliably by these data alone.
Why Plug-In Decisions Can Be Miscalibrated
A plug-in decision substitutes a point estimate for an uncertain quantity and then applies a rule as if that estimate were exact. For example, a rule might define an exposure range as acceptable and require at least a prespecified probability of being in that range before adapting a regimen. The relevant probability is calculated over uncertainty in exposure conditional on the interim data, not by checking whether a single fitted value lies inside the range.
The distinction matters because threshold rules are nonlinear. A point estimate just inside a target can coexist with substantial probability outside it. Conversely, a point estimate just outside the target can coexist with substantial probability inside it. A posterior target-attainment rule uses the distribution of exposure, including uncertainty carried forward from the assay and sparse sampling. The probability threshold is a design choice, not a universal constant, and should be set in advance as part of the adaptation rule.
Worked Example: A Target-Range Probability Versus a Plug-In Classification
Suppose, hypothetically, the interim analysis yields a posterior distribution for AUC such that \(\log(\mathrm{AUC})\) is approximately normal with mean \(\log(41.94)\) and standard deviation 0.07868. The target range is 40 to 50 mg·h/L, and the prespecified adaptation rule requires at least 0.80 posterior probability that AUC is in range.
The posterior probability is calculated on the log scale:
The posterior center, 41.94 mg·h/L, lies inside the target range. A plug-in classification would therefore label the exposure as in range. But the posterior probability of being in range is about 0.7137, or 71.4%, below the prespecified 0.80 criterion. Under this hypothetical rule, the probability requirement is not met. That does not establish that AUC is outside the target; it establishes that the interim data do not support the required degree of confidence in target attainment.
Assumptions, Diagnostics, and Alternatives
An uncertainty-aware analysis is only as reliable as its observation and PK models. Check the assay’s scale, precision, calibration, and possible batch dependence. Examine whether residuals on the log scale are compatible with the assumed error distribution and whether their spread changes with concentration. Check that sampling times and dose histories are accurate. For the PK component, assess whether observed concentration-time patterns are compatible with the model and whether the available sampling times identify the parameters needed for the decision. The earlier material on predictive diagnostics is relevant to model checking, but no diagnostic can make uninformative sparse data informative.
Useful warning signs include large changes in the estimated exposure when one sample is removed, implausibly narrow intervals despite few observations, strong posterior dependence between PK parameters, and systematic concentration residuals. A concentration below quantification, a batch shift, or a sample collected at an uncertain time can also invalidate a simple independent-error analysis. If uncertainty is omitted, the resulting adaptation may be overconfident; the direction and size of the error depend on the model, data, and decision rule.
| Approach | What it uses | Main limitation or strength |
|---|---|---|
| Plug-in exposure estimate | A single fitted concentration or exposure | Simple, but ignores uncertainty when classifying a threshold or target range. |
| Replicate assay measurements | Repeated measurements of a specimen | Can estimate or reduce assay noise, but does not resolve sparse time-course information. |
| Additional informative sampling | More concentrations at times that distinguish PK parameters | Can reduce profile uncertainty, at the cost of additional sampling and operational burden. |
| Uncertainty-integrated decision | A distribution for exposure conditional on interim data | Propagates modeled uncertainty into the rule, but depends on credible assay and PK models. |
These approaches address different problems and can be combined. Replicate assays improve information about analytical error; carefully selected sampling times improve information about the concentration-time profile; and uncertainty integration prevents a point estimate from being mistaken for certainty. None eliminates structural model uncertainty. Where plausible models yield different adaptation decisions, that sensitivity should be investigated and reported rather than hidden inside a single narrow interval.
Common Pitfall
- Wrong reasoning: “The estimated AUC is inside the target, so the participant has attained the target with adequate certainty.”
- Correction: A point estimate’s location does not give the probability that the true or posterior exposure lies in the target range. Calculate the target-range probability under the stated model and compare it with the prespecified decision threshold.
- Wrong reasoning: “The assay is precise, so the sparse PK estimate is precise.”
- Correction: Assay precision concerns measurement of sampled specimens. It does not guarantee that a limited number of samples identify the individual PK profile or its derived exposure.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- In a lognormal assay model, what does \(\sigma_a\) describe, and why is it not a measure of uncertainty about AUC by itself?
- For the two-sample example, what is the estimated elimination rate, and what assumptions make that estimate interpretable?
- If an interim AUC estimate lies inside a target range but its target-attainment probability is below the prespecified threshold, what conclusion follows under that rule?
- Name one diagnostic warning sign of sparse-sampling or model problems and explain what it may indicate.
Solutions. (1) \(\sigma_a\) is the standard deviation of log assay error conditional on the underlying specimen concentration. It does not describe how well the sampling schedule identifies the whole concentration-time curve, so it is not AUC uncertainty by itself. (2) The estimate is \((\log 8.19-\log 3.68)/4\approx0.2000\ {\rm h}^{-1}\). It is interpretable under the assumed exponential decline, correct sampling times, and stated independent log-error model. (3) The required probability criterion is not met; this does not prove that AUC is outside the target. (4) For example, a large change in estimated exposure when one sample is removed may indicate that the sparse data weakly identify the exposure or that one measurement has undue influence.