Robust Sampling for One-Compartment PK

Why a PK sampling schedule should be tested across scenarios

A sampling schedule that is informative at one assumed parameter vector can perform poorly if clearance, volume, or absorption differs from that assumption. In oral pharmacokinetics, these parameters shape different parts of the concentration-time curve, but their effects can overlap: a change in absorption rate may partly resemble a change in elimination rate over a limited time window. Robust sampling design makes that dependence visible by evaluating candidate schedules across plausible parameter scenarios rather than only at a single nominal value.

This tutorial compares three choices for a one-compartment oral model: optimize information at the nominal parameter vector, optimize its probability-weighted average across scenarios, or maximize the worst scenario's information. These criteria answer different design questions; none is universally best. A numerical comparison is meaningful only after the model, error assumptions, candidate times, scenario set, and utility have been specified.

Model, sensitivities, and information

Assume a single oral dose with first-order absorption and elimination, one compartment, fixed bioavailability \(F\), and no lag time. Let \(CL\) be clearance, \(V\) be volume, \(k=CL/V\) the elimination rate constant, and \(k_a\) the absorption rate constant. For \(k_a\ne k\), the predicted concentration at time \(t>0\) is

The formula assumes a linear one-compartment disposition model and known dose and bioavailability. For the worked calculations, observations have independent, constant-variance error on the log-concentration scale: \(\log Y_i=\log C(t_i;\theta)+\epsilon_i\), where \(\epsilon_i\sim N(0,\sigma^2)\). This is a local information calculation: it describes curvature near a specified parameter vector, not guaranteed finite-sample precision under model misspecification.

The sensitivities can be written directly in terms of \(k\), \(k_a\), and \(q=e^{-kt}-e^{-k_a t}\):

The volume derivative includes both the direct \(1/V\) contribution and the effect of changing \(k=CL/V\) while holding \(CL\) fixed. The absorption derivative includes the change to both the prefactor and the absorbed fraction term. These details matter: using a derivative with respect to \(k\) as though it were a derivative with respect to \(CL\) or \(V\) gives the wrong information matrix.

A sensitivity calculation at an early sample

From a nominal schedule to robust criteria

For three parameters and exactly three sampling times, \(G\) is square. If it is nonsingular, then \(\det I=\sigma^{-6}(\det G)^2\). More generally, when there are more observations than parameters, \(G^{\mathsf T}G\) is used directly. The determinant measures the volume of local information across parameter directions; a large value is desirable for D-optimal design. In the square case, \(\det G\) near zero indicates that the three sensitivity rows are nearly linearly dependent, so at least one parameter combination is weakly identified.

For scenario-based comparison, let scenario \(s\) have parameter vector \(\theta_s\) and weight \(w_s\), with \(w_s\ge0\) and \(\sum_s w_s=1\). Evaluate the same candidate schedule at each scenario. The nominal criterion uses a designated reference vector \(\theta_0\); expected utility maximizes the weighted average \( \sum_s w_s U(d;\theta_s)\); maximin design maximizes the worst scenario utility \( \min_s U(d;\theta_s)\). In each case the utility and scenario weights encode design priorities, not facts forced by the PK model.

Justification, diagnostics, and failure modes

The expected-utility and maximin calculations follow directly from evaluating one common utility function at each scenario and then applying the selected summary. For expected utility, a schedule \(d^*\) is preferred to every candidate \(d\) exactly when \(\sum_s w_s U(d^*;\theta_s)\ge\sum_s w_s U(d;\theta_s)\). For maximin, it is preferred when \(\min_s U(d^*;\theta_s)\ge\min_s U(d;\theta_s)\) for every candidate. Thus expected utility can tolerate a poor result in one low-weight scenario if stronger performance elsewhere compensates; maximin cannot. Both criteria depend on the candidate set and the scenarios supplied.

A useful diagnostic is to inspect the information matrix or the singular values of \(G\), not just the final utility. A very small singular value signals a poorly informed parameter direction. Also examine each scenario separately: an attractive average may conceal a disastrous scenario, while a maximin result may be driven by one extreme scenario. Check whether candidate times are feasible, whether the error variance is credible over the concentration range, and whether all scenarios remain within the model's domain. Near \(k_a=k\), the displayed concentration formula has a removable limit; numerical calculations should use that limit or a stable equivalent rather than subtracting nearly equal exponentials.

A common mistake is to treat the nominally best schedule as robust because its nominal determinant is large. In the example, nominal optimization selects B, but that choice ignores its weaker performance under faster elimination (Scenario 3). The correction is to state the intended criterion first, evaluate the chosen utility at every relevant scenario, and then compare schedules under that criterion. Another mistake is to interpret log-determinant utilities as ordinary percentages or as directly comparable across different parameterizations without care. A consistent nonsingular linear reparameterization adds the same constant to all schedule utilities for a fixed parameter dimension, but changing the parameterization can still change how results are interpreted.

For the example's common \(\sigma\) and three-parameter dimension, omitting \(\sigma^{-6}\) shifts every log-determinant utility by the same constant and does not change the schedule ranking. It does change the reported utility values. Rankings can change if schedules are evaluated under different error assumptions, because then the information scaling is not common. Likewise, changing the parameter dimension, scenario weights, plausible scenario set, or utility can change the decision. Scenario-based robustness is only as credible as those inputs.

Compared with nominal optimization, expected utility represents a weighted average and requires defensible scenario probabilities or weights. Maximin requires no probabilities, but can be dominated by an implausible extreme case and may sacrifice substantial average performance. A Bayesian expected-utility analysis can integrate over a parameter distribution rather than a finite scenario list; a scenario set is a transparent approximation when a small set of parameter combinations captures the design concerns. These choices do not replace checking model adequacy, operational constraints, or whether the parameter combinations are plausible jointly.

Limitations and interpretation

The local Fisher information uses derivatives at fixed parameter values and assumes the stated likelihood is adequate. It does not establish that the oral one-compartment model is correct, that the nominal error variance holds at every time, or that the selected schedule will perform well with sparse or censored observations. The log-determinant criterion rewards joint information but does not guarantee precise estimation of a particular parameter if another direction dominates. If the scientific objective is specifically clearance or exposure, a criterion targeted to that objective may be more appropriate.

Robust schedules are also limited by scenario construction. A finite set can miss important combinations, and arbitrary weights can make expected utility look more objective than it is. Report the scenarios and weights, show scenario-specific performance, and perform sensitivity checks when the ranking is close. The aim is not to eliminate uncertainty by design, but to make explicit which uncertainty the schedule is designed to tolerate.