Optimal Sparse Sampling Designs for Pediatric PK

Why Sparse Sampling Requires Deliberate Design

In pediatric pharmacokinetic (PK) studies, a rich concentration-time profile from every participant may be impractical or ethically unacceptable. Yet a small number of samples can still be informative if their timing is chosen to distinguish the parameters and age-related patterns the study needs to estimate. The design problem is therefore not simply to collect fewer samples; it is to place limited samples where the expected profiles carry useful information, subject to clinical, analytical, and ethical constraints.

A sampling design specifies which participants are sampled, at what times, and in which age strata. It is judged against a stated objective, such as estimating clearance precisely, distinguishing absorption from elimination, or estimating population parameters without leaving a stratum poorly represented. Those objectives can compete. A time that is informative for one parameter may add little for another, while the most statistically attractive schedule may be infeasible or impose too much burden.

From Expected Profiles to Information

Suppose a model predicts the log concentration at time \(t\) as \(f(t,\boldsymbol{\theta})\), where \(\boldsymbol{\theta}\) is the vector of parameters to be estimated. For a local design calculation, assume observations satisfy \(Y(t)=f(t,\boldsymbol{\theta})+\epsilon\), with independent residual errors \(\epsilon\sim N(0,\sigma^2)\), and treat \(\sigma^2\) as known. The model must be differentiable with respect to the parameters near the values used for design. These assumptions give a tractable approximation; a population PK model with random effects, correlated residuals, or substantial parameter uncertainty may require a fuller calculation or simulation.

The sensitivity vector at time \(t\) is the gradient of the expected log concentration with respect to the parameters. A time is informative when changes in the parameters cause distinguishable changes in the expected observation. Under the stated independent normal-error model, the expected Fisher information for observations at times \(t_1,\ldots,t_N\) is the sum of the sensitivity contributions:

This matrix summarizes local information about the parameters jointly. If it is singular, the selected times cannot distinguish at least one parameter combination under the model. If it is invertible and the approximation is adequate, the inverse provides an approximate covariance matrix for parameter estimates. Thus, an informative schedule should not merely produce measurable concentrations: its sensitivity vectors must collectively support the quantities the study intends to estimate.

A Short Derivation: Why Separated Times Can Help

Consider the simple one-compartment intravenous bolus model after dosing, written on the log scale as \(f(t)=\log A-kt\). Here \(A\) is the concentration intercept and \(k\) is the elimination rate constant. The model is illustrative: it does not represent absorption, distribution beyond one compartment, or all features of pediatric PK. Its sensitivities are \((1,-t)^\mathsf{T}\), so the information matrix for \(N\) samples is:

The determinant is \(\bigl(N\sum_i t_i^2-(\sum_i t_i)^2\bigr)/\sigma^4\). Since \(N\sum_i t_i^2-(\sum_i t_i)^2=\sum_{i<j}(t_i-t_j)^2\), the determinant increases with the separation among sampling times in this model. With \(n_1\) samples at one time and \(n_2\) at another, it is \(n_1n_2(t_2-t_1)^2/\sigma^4\). The result explains why sampling at only one time cannot separate \(A\) from \(k\), and why time separation helps estimate both. It does not imply that the latest feasible time is always best: other parameters, concentrations below quantification limits, safety, and model departures can change the practical choice.

Designing Across Pediatric Age Strata

A design that performs well for one expected profile may perform poorly in another age stratum. Differences in body size, maturation, or other age-related factors can change the concentration-time profile and therefore the sensitivity of a measurement to model parameters. Earlier material on body-size scaling, maturation, and age-stratified diagnostics provides relevant model structure and checks; here the design question is how to ensure the proposed times remain informative across the intended strata.

A practical approach is to calculate or simulate expected profiles by stratum, then evaluate candidate schedules against the parameters of interest in each stratum. If every stratum must support estimation, one option is to impose a minimum information or precision target separately for each. Another is to optimize a weighted criterion across strata. A weighted average can hide a weakly informed stratum, so the design should also report stratum-specific performance. Weights should reflect the study objective and not merely the number of participants expected in each age group.

Simulation and Practical Constraints

Local information calculations depend on the assumed parameter values and simplified observation model. Simulation can compare candidate designs under a fuller population model, including between-participant variability, residual error, dosing and visit constraints, and plausible uncertainty in model parameters. For each candidate schedule, simulated data are generated and analyzed using the intended estimation approach. Repeating this process reveals how often estimation succeeds, how precise estimates are, and whether intervals achieve their intended coverage under the scenarios examined.

Before ranking schedules, specify what is fixed and what may vary. Relevant constraints can include the maximum number of research draws per participant, allowable visit windows, total blood volume limits, clinical-care timing, assay requirements, and the need to avoid unnecessary procedures. Ethical review, informed parental permission, and age-appropriate assent where applicable remain essential; statistical efficiency does not override participant protection. When clinically indicated blood draws can supply a sample, their timing and availability must be represented realistically rather than assumed.

A design should also account for uncertainty about the expected profiles. One strategy is to evaluate a candidate across a set of plausible parameter values or model scenarios, rather than optimizing at a single best estimate. A robust design may sacrifice some apparent efficiency under one scenario to avoid very poor performance under another. Sampling windows, rather than exact clock times, should be evaluated when actual visits cannot reliably occur at exact times.

Diagnostics, Alternatives, and Limitations

Several warning signs indicate that a proposed design may be failing. A singular or nearly singular information matrix signals weak separation of parameter effects. Very large standard errors or strong parameter correlations in simulated fits can reveal the same weakness. Estimates repeatedly at parameter boundaries, poor convergence, substantial bias, or inadequate interval coverage indicate that a nominally informative design may not work with the intended estimation method. Also inspect the predicted concentration range at each time: a theoretically sensitive time contributes little if measurements are frequently below quantification limits or unusable in practice.

Compare more than one design criterion. D-optimality is useful for joint parameter precision, but it can favor a schedule that improves already well-informed directions while leaving one important parameter weak. A-optimality focuses on the sum of approximate variances, and E-optimality on the weakest direction. A targeted criterion can instead prioritize a specific parameter or prediction. Simulation is more flexible than local information calculations but depends on the realism of its simulated models, scenarios, and analysis procedure. A pragmatic schedule built around clinical visits may be less statistically efficient in an idealized calculation yet more robust and achievable in practice.

No design criterion can protect against every misspecified structural model, unexpected age-related mechanism, or operational deviation. Nor does an optimal sampling schedule establish that the model’s resulting exposure estimates meet a clinical target; that is a separate interpretation question. Design conclusions should therefore state the model and scenarios assessed, the objective optimized, the constraints applied, and the performance that remained uncertain.