Population PK Models as Individual Parameter Priors

From a Population Model to an Individual Prior

A population pharmacokinetic (PK) model describes both typical parameter values and how individuals vary around them. When data from a particular patient have not yet been used to update the model, this population description can provide a prior distribution for that patient’s PK parameters. It is a distribution, not a single “typical patient” estimate: it represents the model’s uncertainty about which parameter values are plausible for an individual drawn from the modeled population.

The earlier material on Bayesian updating of individual PK parameters and constructing individual likelihoods for nonlinear PK data provides the context for this step. Here the focus is the prior’s construction from population typical values, between-subject variability, and correlations. The prior is not yet conditioned on patient covariates, and it does not include the patient’s concentration observations.

This construction relies on assumptions that should be made explicit. The individual is treated as drawn from the same target population represented by the model; the typical values and variability estimates are treated as fixed; and the random-effects distribution and parameter scale are taken to be appropriate. These are modeling assumptions, not facts guaranteed by having fitted a population model. In particular, this tutorial’s normal model is on the parameter scale as written. A normal distribution can assign positive probability to values outside a parameter’s valid range, so its suitability must be assessed rather than presumed.

Between-subject variability in parameters is distinct from residual error in measured concentrations. The former describes differences among individuals’ PK parameters and supplies the parameter prior. Residual error describes discrepancies between model-predicted and observed concentrations; it belongs to the observation model used in the individual likelihood, not to \(\boldsymbol{\Omega}\).

Building the Covariance Matrix

For \(p\) parameters, let \(\omega_j\) be the between-subject standard deviation of parameter \(j\), and let \(\rho_{jk}\) be the correlation between the random effects for parameters \(j\) and \(k\). The corresponding covariance is \(\rho_{jk}\omega_j\omega_k\). The diagonal entries are variances, not standard deviations.

The marginal distribution of each parameter follows from the joint model: \(\theta_{ij}\sim N(\theta_{\mathrm{typ},j},\omega_j^2)\). A correlation does not change these marginal means or variances. It changes the joint distribution: it describes whether deviations in two parameters tend to occur together. A positive correlation associates above-typical deviations in one parameter with above-typical deviations in the other; a negative correlation associates deviations in opposite directions.

A valid covariance matrix must be symmetric and positive semidefinite. Consequently, every pairwise correlation must lie between \(-1\) and \(1\), but that condition alone is not enough for three or more parameters: the full matrix must also be mathematically valid. An invalid matrix cannot define the stated multivariate normal prior.

Why Correlation Changes the Joint Prior

The covariance matrix determines how the model weights combinations of parameter deviations. For a two-parameter normal prior with positive standard deviations \(\omega_1,\omega_2\) and correlation \(\rho\) satisfying \(|\rho|<1\), define standardized deviations \(z_1=(\theta_1-\theta_{\mathrm{typ},1})/\omega_1\) and \(z_2=(\theta_2-\theta_{\mathrm{typ},2})/\omega_2\). The squared Mahalanobis distance from the typical vector is

This expression is the quadratic form in the normal density. The density’s parameter-dependent factor is proportional to \(\exp(-d^2/2)\): a larger \(d^2\) means lower density at that point, all else equal. It is not itself a probability. The derivation follows by inverting the standardized correlation matrix \(R=\begin{pmatrix}1&\rho\\\rho&1\end{pmatrix}\), whose inverse is \((1-\rho^2)^{-1}\begin{pmatrix}1&-\rho\\-\rho&1\end{pmatrix}\), and evaluating \(\boldsymbol{z}^{T}R^{-1}\boldsymbol{z}\).

A Validity Check for Several Parameters

Pairwise correlations are not sufficient to establish that a multivariate prior is valid. The correlations must jointly form a positive semidefinite correlation matrix, and scaling that matrix by the parameter standard deviations must yield a positive semidefinite covariance matrix.

Assumptions, Diagnostics, and Alternatives

A practical construction should be checked against both the model specification and the parameter scales. Confirm which quantities are typical values, which entries are standard deviations or variances, the units and scales on which they were reported, and whether the correlations refer to between-subject random effects. If a model reports a covariance matrix directly, do not multiply its entries by standard deviations again. If it reports standard deviations and correlations, use the conversion above.

The parameter-scale normal model can be inappropriate when parameters must be positive or bounded. For example, a normal prior with mean 5.0 and standard deviation 3.0 assigns nonzero probability to negative clearance. A prior that gives appreciable probability to impossible values is a warning that the assumed scale or distribution may not suit the application. Alternative approaches can model random effects on transformed scales or use other distributions; selecting and deriving those constructions requires additional model specification. Do not silently truncate or transform a reported covariance matrix, since that changes the implied prior.

Another diagnostic is the reported correlation matrix. Correlations outside \([-1,1]\), asymmetry, or a matrix that is not positive semidefinite indicate a data-handling, rounding, or model-output problem. Rounding can occasionally make an otherwise valid matrix appear slightly invalid; investigate the unrounded estimates rather than treating a materially invalid matrix as usable. A near-singular matrix signals that some combinations of random effects have extremely little modeled variation and can make numerical calculations unstable.

This plug-in prior also treats the estimated typical values and covariance matrix as known. It therefore omits uncertainty in the population-model estimates themselves. That simplification may be reasonable when estimation uncertainty is small relative to between-subject variability, but can be consequential when the population data are limited. The prior is also only as relevant as the population from which it was estimated: it should not automatically be transferred to a patient population with materially different characteristics.

Compared with a single typical-value estimate, the distributional prior carries the modeled spread and dependence among parameters. Compared with a prior built from a different distribution or transformed scale, the parameter-scale multivariate normal is direct to construct and interpret but may have unsuitable support. The appropriate choice depends on the population model and parameter scales; the essential requirement is to use the model’s stated random-effects structure consistently, not to add or remove correlations for convenience.