A First-Order Fisher Matrix for a One-Compartment PK Model

Why Linearize a Pharmacokinetic Model?

Population design requires accounting for differences between individuals as well as measurement error. In a one-compartment model, an individual’s clearance and volume affect every predicted concentration, and random variation in those parameters induces correlation among that individual’s repeated measurements. A first-order linearization turns this nonlinear dependence into an approximate mean and covariance that can be used to construct population Fisher information.

The approximation is local: it describes concentration predictions near the nominal population parameters and near zero random effects. It does not make the underlying pharmacokinetic model linear. The distinction matters because the resulting covariance and Fisher information are useful design quantities, but their accuracy depends on how well the local approximation represents the distribution of predictions.

One-Compartment Model and Its Random Effects

Consider an idealized one-compartment model after an instantaneous intravenous bolus. For subject \(i\), concentration at time \(t\) after dosing is

where \(D\) is dose, \(CL_i\) is clearance, and \(V_i\) is volume of distribution. Suppose clearance and volume have log-normal interindividual variation:

where \(CL\) and \(V\) are the typical population values. Let \(\eta_i=(\eta_{CL,i},\eta_{V,i})^\top\) have mean zero and covariance matrix

Assume random effects are independent across subjects. For the derivation below, measurement errors are independent of the random effects and have mean zero and covariance \(R_i\). A common special case is additive, independent residual error with \(R_i=\sigma^2 I\). These assumptions specify the error structure; they do not imply that the exact marginal distribution of concentrations is normal.

Linearizing the Concentration Predictions

Write the nominal prediction, evaluated at zero random effects, as

The random effects enter through log clearance and log volume. Differentiating the log concentration first makes their effects transparent:

At \(\eta_i=0\), the derivatives of concentration are the nominal concentration multiplied by the derivatives of its logarithm. Thus

For \(m_i\) sampling times, collect the nominal predictions in \(f_i(0;\theta)\) and the derivatives in the \(m_i\times2\) Jacobian \(J_i\). The first-order expansion is

Taking expectations under the stated zero-mean random-effects assumption gives the approximate population mean. Taking the covariance gives the random-effect contribution to within-subject covariance; independent residual error adds \(R_i\).

The superscript indicates a first-order approximation. In particular, the approximate mean is the prediction at zero random effects, not a claim that the exact mean of the nonlinear concentration model must equal that prediction.

From the Approximate Distribution to Fisher Information

For a fixed design, the first-order approximation treats \(y_i\) as multivariate normal with mean \(\mu_i^{(1)}\) and covariance \(\Sigma_i^{(1)}\). This working likelihood follows exactly from the linearized model if the random effects and residual errors are normal. With nonnormal random effects, it is a Gaussian approximation based on the first two moments. Assume subjects are independent, the covariance matrices are positive definite, and the model is differentiable in the population parameters of interest.

Let \(\lambda=(\lambda_1,\ldots,\lambda_q)^\top\) collect the parameters for which information is required. Both the approximate mean and covariance may depend on \(\lambda\). For one subject, the Fisher information under the multivariate normal model is

The first term is information carried by changes in the mean relative to its uncertainty. The second is information carried by changes in the covariance. Omitting the second term is justified only when covariance is parameter-independent for the parameters being estimated, or when one deliberately reports the mean contribution alone. For independent subjects, information adds: \(\mathcal I=\sum_i\mathcal I_i\). If subjects have different fixed covariates or sampling schedules, their corresponding means, Jacobians, and covariance matrices must be evaluated separately before summing.

Correlated Random Effects and Design Interpretation

When clearance and volume random effects are correlated, the off-diagonal elements of \(\Omega\) contribute to the covariance between measurements. For two times \(t_r\) and \(t_s\), write their derivative rows as \(j_r=(a_r,b_r)\) and \(j_s=(a_s,b_s)\). Their random-effect covariance is

This shows why it is not generally valid to add independent clearance and volume variance contributions while ignoring their covariance. The sign and size of the cross-effect depend on the sensitivities at both times as well as on the random-effect correlation. Residual error contributes separately through \(R_i\), as specified by the observation model.

A design can therefore change population information in two linked ways: changing sampling times changes the sensitivity matrix \(J_i\), and it changes sensitivity of the nominal mean to parameters. Those same changes also alter \(\Sigma_i^{(1)}\), which weights mean information and may itself carry information through its parameter dependence. This is the population-level counterpart of distinguishing repeated-measurement residual noise from between-subject variability.

Assumptions, Diagnostics, and Limitations

The calculation rests on several conditions: the concentration model is differentiable at the nominal values; the random effects are centered at zero with finite covariance \(\Omega\); the chosen residual covariance \(R_i\) is appropriate; and the local linear expansion is adequate over the random-effect values that have substantial probability. The Gaussian Fisher formula additionally uses a nonsingular covariance matrix and the normal working likelihood. These assumptions should be checked against the model and design rather than treated as automatic consequences of writing random effects on a log scale.

Useful checks include verifying derivative signs and units, confirming that \(\Omega\) is positive semidefinite, and checking that \(\Sigma_i^{(1)}\) is positive definite. A computed negative variance or a non-positive-definite covariance signals an algebraic or model-specification error. For approximation adequacy, compare the nominal prediction and its first-order changes with direct predictions at plausible positive and negative random-effect values. Large curvature, strong multiplicative variability, or predictions far from the local tangent plane are warnings that the first-order moments may be poor.

A common mistake is to treat the first-order population mean as the exact mean of the nonlinear model, or to set the population covariance equal to residual variance alone. The correction is to distinguish the nominal mean approximation from the exact marginal mean, and to include \(J_i\Omega J_i^\top\) as well as \(R_i\). Another mistake is to compute only the mean-sensitivity term and call it the full Fisher information even when the covariance depends on estimated parameters. Use the covariance term in the normal Fisher formula whenever that dependence is part of the model being estimated.

The approximation is most useful when local sensitivities summarize the relevant random-effect range and when a tractable population information calculation is needed. It can be less reliable when variability is large or the prediction is strongly curved as a function of random effects. It also depends on the residual model: additive error, proportional error, and correlated residual error produce different \(R_i\) and may change the information substantially. Unlike a conditional individual Fisher calculation evaluated at a particular random effect, the population calculation represents between-subject variation through the approximate covariance.