Common Errors in Individual PK Sampling Design

Why Plausible Designs Can Mislead

A sampling schedule can look attractive because it includes a sharp concentration change, spans a wide time range, or produces a large numerical design score. None of these features alone guarantees useful parameter estimates. The central check is whether the observations provide appropriately weighted, distinct information about the parameters that matter.

For an individual PK model, design calculations are usually local: sensitivities are evaluated at specified nominal parameter values, and a local information matrix summarizes the information expected from the proposed sampling times. Errors in the sensitivities, residual-error model, parameter scaling, or interpretation of the matrix can make a formally optimized schedule practically poor. This tutorial focuses on recognizing those errors rather than replacing the design criteria introduced in the earlier material on D-, A-, and c-optimal sampling.

When the residual variance is constant, every observation has the same variance weight. When it differs by time or predicted concentration, the weights differ. If residual variances are estimated, depend on the mean, or errors are correlated, the likelihood and its information must reflect those features; the simple sum above is not automatically the full Fisher information. Likewise, local sensitivities depend on the nominal parameter values, so a schedule should be checked over plausible values rather than treated as universally optimal.

Error 1: Optimizing Unweighted Sensitivities

A sensitivity measures how much the predicted concentration changes for a small change in a parameter. A large absolute derivative can appear valuable, but it is only informative relative to the measurement noise. Under the model above, an observation's contribution is \(g_i^\mathsf{T}g_i/\sigma_i^2\): its sensitivity is effectively divided by the residual standard deviation before it contributes information.

The diagnostic is direct: compare the residual-error model used in the design calculation with the assay and model assumptions for the proposed times. If variability increases at low or high concentrations, a calculation that assigns equal weight to all sensitivities may favor a time whose apparent signal is overwhelmed by noise. The earlier material on residual error models and weighting for assay precision covers how to construct these variance weights; the practical error here is to omit them or apply the wrong ones.

Error 2: Mixing Parameter Scales

Sensitivities have units that depend on how parameters are expressed. A derivative with respect to clearance measured in litres per hour is numerically different from a derivative with respect to clearance measured in millilitres per hour. Raw sensitivity magnitudes, matrix traces, and condition numbers can therefore change simply because a parameter's units or scale changed.

For a fixed, nonsingular reparameterization \(\theta=\theta(\phi)\), let \(G=\partial\theta/\partial\phi\). The information matrix in the new coordinates is \(I_\phi=G^\mathsf{T}I_\theta G\). Its determinant is \(\det(I_\phi)=\det(G)^2\det(I_\theta)\). Since the same transformation is applied to every candidate schedule, it multiplies every determinant by the same positive factor. Thus the ranking by D-optimality is preserved under such a fixed transformation, while unscaled summaries such as the trace need not be. Parameter-specific uncertainty must still be interpreted in the chosen parameterization and units.

Error 3: Treating a Large Determinant as Precision for Every Parameter

For a locally regular model with information matrix \(I\), the inverse \(I^{-1}\) approximates the covariance matrix of an efficient estimator. D-optimality uses \(\det(I)\), which summarizes the inverse volume of the joint uncertainty ellipsoid. It is a joint measure, not a promise that each diagonal variance is small. One direction can be highly informative while another is poorly informed.

Check the individual diagonal entries of \(I^{-1}\), relevant parameter contrasts, and parameter correlations, not only the determinant. If the design goal is precise estimation of one parameter or a specified function of parameters, the earlier material on c-optimal sampling provides a more targeted criterion. A-, D-, and c-optimal criteria answer different questions; no single scalar score substitutes for examining the uncertainty that matters.

Error 4: Accepting Redundant Sampling Times

A sample is redundant in the design sense when its sensitivity contributes little new information about parameter directions that are otherwise weakly identified. Two times can have different predicted concentrations yet have nearly parallel sensitivity vectors. Conversely, repeating a time is not automatically useless: with independent errors, a replicate can increase precision along the sensitivity direction it shares. The issue is whether the design gains needed directions of information, not whether the clock time is unique.

In practice, inspect sensitivity directions, the rank or near-singularity of the information matrix, parameter correlations, and changes in uncertainty when each time is removed. A very small eigenvalue or a large parameter correlation can signal that several samples are contributing nearly the same direction. Also check operational constraints such as minimum spacing and clinic windows; a mathematically distinct time may not be feasible or may not be reliably measured.

A Short Justification and a Practical Review

The weighting and redundancy checks follow from the likelihood under the stated independent normal-error model. For one observation, the log-likelihood, apart from constants, is \(-[Y_i-f(t_i,\theta)]^2/(2\sigma_i^2)\). Differentiating with respect to \(\theta\) gives the score \(g_i^\mathsf{T}[Y_i-f(t_i,\theta)]/\sigma_i^2\). Taking the expected outer product of this score gives \(g_i^\mathsf{T}g_i/\sigma_i^2\), because the residual has mean zero and variance \(\sigma_i^2\). Independent observations have additive information, yielding the sum used above. If \(g_i\) is parallel to existing sensitivity rows, its information adds along those same directions; it cannot by itself raise the rank.

These checks are diagnostics, not a proof that a design will perform well in a real study. A local information calculation can fail when the nominal values are far from an individual's values, the model is misspecified, the likelihood is strongly nonlinear over the relevant parameter range, or estimates are constrained by boundaries. Simulation-based evaluation and robust or Bayesian design approaches can address some of these concerns, but they answer broader questions than a single local calculation. In particular, population designs require accounting for between-individual variation, which is the subject of the later material on hierarchical models.

Common Mistake

Wrong reasoning: “The schedule has the largest determinant and the steepest concentration-time slopes, so it must estimate every parameter precisely.” This combines two distinct errors: steep slopes ignore residual variance, and the determinant summarizes joint uncertainty rather than each parameter's marginal precision.

Correction: Calculate information using the appropriate variance weights, confirm that parameter scales are consistent, and inspect the inverse information matrix or uncertainty for the specific parameters and functions of interest. Then check whether the proposed times contribute distinct sensitivity directions and whether the conclusions persist across plausible nominal values.