Adaptive PK Sampling for Parameter Identification

Why Sampling Time Is a Design Choice

A pharmacokinetic sample is informative not simply because it measures concentration, but because its timing makes the concentration respond differently to the parameters of interest. Early samples can be sensitive to absorption; late samples can be sensitive to elimination. If all samples are taken in a similar part of the concentration-time curve, different parameter combinations may explain them almost equally well. The resulting estimates can be unstable even when every concentration is measured precisely.

In an adaptive trial, interim concentration data can update the parameter estimates used to plan later samples. The design question is then: which permitted future sampling times are expected to add the most information about the parameters that matter? This tutorial develops a local, model-based answer. It concerns identification of pharmacokinetic parameters, not decisions based on clinical outcomes.

A Model and Its Sensitivities

Begin with a single oral dose \(D\) under a one-compartment model with first-order absorption and elimination. Assume bioavailability \(F\), apparent volume \(V\), absorption rate constant \(k_a\), and elimination rate constant \(k_e\), with \(k_a>k_e>0\). The predicted concentration at time \(t>0\) is

For the calculations below, assume the dose is known, the model is correctly specified, the parameters are constant over the modeled dosing interval, and concentration errors are independent with a common standard deviation on the log-concentration scale. Treat \(F\) and \(V\) as known when estimating \(k_a\) and \(k_e\). If they are unknown, their effects must be included in the information calculation; in particular, concentration data alone generally do not separate \(F\) from \(V\) without further assumptions or information.

The sensitivity of the predicted log concentration to a parameter describes the local change in log concentration for a small change in that parameter. For this model, differentiating the log of \(C(t)\) gives

Each derivative has units of time because the rate constants have units of inverse time. A large absolute sensitivity means a small rate-constant change can produce a comparatively large log-concentration change at that time. Sensitivity alone is not enough, however: if two parameters affect the concentration in nearly the same way across all chosen times, they remain difficult to distinguish.

From Sensitivities to a Design Criterion

Let \(g(t)\) denote the column vector of log-concentration sensitivities for the parameters being estimated. If the log measurement error at each sample has variance \(\sigma^2\), independent across samples, the information matrix for a design with times \(t_1,\ldots,t_m\) is

This expression follows from the normal log-concentration model: the derivative of the mean log concentration is \(g(t_j)\), and a normal observation with variance \(\sigma^2\) contributes its mean-derivative outer product divided by \(\sigma^2\). Independent observations contribute additively. If \(I\) is nonsingular, its inverse approximates the covariance matrix of the parameter estimates locally. If it is singular, at least one parameter combination is not locally identifiable from those samples.

A common design criterion is D-optimality: choose the permitted sampling plan that maximizes \(\det(I)\). For a fixed number of parameters, a larger determinant corresponds to a smaller approximate joint confidence-ellipsoid volume. It rewards information across the parameter space, not just sensitivity to one parameter. When one parameter or parameter combination is the priority, a targeted criterion can instead minimize its approximate variance, obtained from the relevant element or contrast of \(I^{-1}\).

Choosing an Adaptive Sampling Plan

At an interim analysis, use the updated parameter estimates to calculate sensitivities at candidate future times. Add each candidate sample's expected information to the information already accumulated, then compare plans using the prespecified criterion. In an operational trial, candidate times must also satisfy dose-timing, visit, assay, blood-volume, and participant-burden constraints. Adaptation should follow a defined rule rather than informal inspection of individual concentration values.

The following calculation illustrates how existing information can change the value of a candidate sample. Suppose the current information for \((k_a,k_e)\), in the same rate-constant parameterization, is

Use the parameter values and \(\sigma=0.20\) from the preceding example. For a candidate sample with sensitivity \(g\), the updated information is \(I_0+gg'/0.20^2\). The matrix determinant identity gives the determinant ratio

This identity follows by factoring out \(I_0\) inside the determinant and using \(\det(A+uv')=\det(A)(1+v'A^{-1}u)\) for nonsingular \(A\). It provides a simple score for each candidate time, conditional on the current local information matrix.

Absorption, Elimination, and Distribution

For an oral one-compartment model, very early samples can be useful for absorption, samples around the rising limb and peak can help distinguish absorption from elimination, and later samples can be strongly sensitive to elimination. But a single-compartment model has no separate distribution phase. It cannot identify a distinct distribution rate or intercompartmental clearance merely by taking more samples.

If distribution is a design target, the model must represent it. For example, an intravenous two-compartment concentration curve can be written as \(C(t)=A e^{-\alpha t}+B e^{-\beta t}\), with \(\alpha>\beta>0\). The faster component is associated with the early distribution phase and the slower component with the terminal phase. Samples only in the terminal phase may characterize the slower decline but provide little information to distinguish the fast component. Sampling must cover the early phase as well as later times, and the model’s sensitivities to the underlying volumes and clearances must be evaluated. A schedule designed under the one-compartment model cannot be assumed adequate for this purpose.

Sampling intensity can also be adapted by allocating more observations to a poorly identified phase, if trial logistics allow. Repeated samples at nearly identical times may improve precision under independent error, but they often add less information about parameter separation than samples at distinct phases. Additional observations also have costs, and the design criterion should reflect the actual objective and constraints rather than maximize information without regard to feasibility.

Diagnostics, Alternatives, and Limitations

Information-based design is a local approximation: it evaluates sensitivity near a particular parameter estimate. Check whether the proposed plan is robust to plausible parameter values, not just the interim point estimate. Useful warning signs include a nearly singular information matrix, high estimated parameter correlations, large changes in the preferred schedule under modest parameter perturbations, and systematic residual patterns that suggest the concentration model is inadequate. Assay limits matter too: a nominally informative late sample may be below quantification, in which case a normal-error information calculation for an observed concentration is inappropriate.

D-optimality is one option, not a default answer to every design question. A targeted variance criterion is more aligned with a single parameter or parameter contrast. A Bayesian expected-information criterion can average over uncertainty in the parameters instead of evaluating only one estimate, but requires a specified distribution over plausible parameter values. A fixed, conventional schedule may be simpler to run and less sensitive to interim model instability. Whichever approach is used, prespecify the candidate times, objective, constraints, and rule for updating the design.

The information calculation also assumes the model and error structure used to construct it are suitable. If concentrations are censored, error variance changes with concentration, or observations are correlated, the likelihood and information contributions must reflect those features. Adaptive sampling can improve precision under a sound model; it cannot make a misspecified model correct or recover a parameter that the design and data do not identify.