Choosing Sampling Times for Patient-Level Precision

Why Sampling Time Affects Patient-Level Precision

A concentration measurement is informative about a patient’s parameters only to the extent that the predicted concentration changes when those parameters change. That sensitivity depends on when the sample is collected. A late sample may help distinguish clearance values, while an early sample after an oral dose may be more responsive to absorption rate. A sample can also be precise analytically yet add little information about the forecast of interest.

The design question is therefore not simply “How many samples should be collected?” It is “At which times will the measurements most reduce uncertainty in the parameters or forecast that matter?” The answer depends on the dosing history, the model, residual error, the population-informed individual prior, and the target prediction. The calculations below are local: they describe sensitivity near a specified set of parameter values.

From Sensitivity to Information

Suppose the observed log concentrations \(y_i=\log C_i\) are described locally by a model \(f(t_i;\eta)=\log C(t_i;\eta)\), where \(\eta\) is a vector of transformed individual parameters. Assume that residual errors are independent, normally distributed, have mean zero, and share known variance \(\sigma^2\). Also assume the model and dosing times are correctly specified for this calculation. The sensitivity matrix \(J\) has one row per sample and one column per parameter:

Under the local linear approximation, the data contribute the information matrix \(J^\mathsf{T}J/\sigma^2\). If the individual’s prior distribution for \(\eta\) is approximately normal with covariance matrix \(\Omega\), its precision matrix is \(\Omega^{-1}\). The approximate posterior covariance is then

This expression assumes a locally adequate linearization, a correctly specified error model, and a positive-definite combined precision matrix. It is an approximation, not a replacement for fitting the full individual model. It makes the design principle explicit: sampling times matter through the rows of \(J\), while residual error and prior uncertainty determine how much those sensitivities can improve precision.

Worked Example: Clearance and Volume Sensitivity at Two Times

Turning Parameter Precision Into Forecast Precision

A useful design criterion is the uncertainty in a specified future prediction. If the future log concentration is \(g(\eta)=\log C(T;\eta)\), its sensitivity vector at the fitted or nominal parameter value is \(q=\nabla_\eta g\). A first-order Taylor approximation gives forecast variance due to parameter uncertainty as

This is uncertainty in the predicted mean log concentration due to uncertainty in the individual parameters. It does not include future residual error or assay error. For a predictive distribution of a future observation, those sources would also need to be included under their appropriate model assumptions.

Absorption-Phase Samples and Parameter Sensitivity

For first-order oral absorption in a one-compartment model, ignoring factors constant with respect to time and \(k_a\), the concentration is proportional to

where \(k_a\) is the absorption rate constant. An early sample can be sensitive to \(k_a\), while a sample after absorption is largely complete may carry little local information about it. This does not mean that an early measurement uniquely identifies absorption: parameter sensitivities can point in similar directions, and other parameters also affect the concentration.

Choosing a Design Criterion and Checking Its Limits

Several criteria can compare candidate schedules, and they answer different questions. A determinant criterion, such as the determinant of the information matrix, rewards joint information across parameters and is often interpreted through the local volume of the parameter uncertainty region. A criterion based on the trace of the covariance matrix summarizes overall parameter variance, but depends on parameter scaling. For a clinical forecast, directly comparing \(q^\mathsf{T}\Sigma_{\text{post}}q\) for the target times or outcomes is often more relevant. These criteria can rank the same candidate schedules differently.

A practical comparison should also account for feasibility: sample timing relative to doses, clinic attendance, assay limits, and whether a proposed time is tolerable for the patient. A theoretically informative time may not be usable. Model structure matters as well: if the absorption model, dose history, or residual-error assumptions are wrong, calculated precision can be misleading. Sensitivity-based designs are local, so a design optimized at one nominal parameter value may perform differently for a patient with substantially different parameters.

Diagnostics include checking whether sensitivity columns are nearly proportional, whether the information matrix is ill-conditioned, and whether small changes in nominal parameters or sample times greatly change the ranking. Such patterns indicate that parameters may remain difficult to distinguish or that the local result is fragile. Compare candidate schedules across plausible individual parameter values and inspect the forecast variance for the actual prediction target. The earlier material on parameter correlation in individual Bayesian fits is relevant here: strong posterior correlation can remain even when marginal standard errors appear moderate.

No single time is best for every parameter and every forecast. In the intravenous example, early and late samples supplied different directions of information about clearance and volume. In the oral example, an early absorption-phase sample was more sensitive to \(k_a\) than a late sample. In practice, candidate times should be evaluated jointly with the individual prior, measurement-error model, plausible parameter range, and intended prediction. The resulting calculation guides design; it does not guarantee precise predictions if the model or assumptions fail.