Propagating Pharmacokinetic Uncertainty into Exposure-Response

Why Pharmacokinetic Uncertainty Matters

An exposure–response analysis often begins with individual exposure estimates from a population pharmacokinetic (PK) model. A common workflow estimates PK parameters, calculates an exposure metric for each participant, and then treats those calculated exposures as known when fitting the response model. That is a plug-in analysis: the PK estimates are inserted into the exposure calculation, but their uncertainty is not carried into the exposure–response slope or its interval.

This simplification can make an interval too narrow. If plausible PK parameter values produce meaningfully different individual exposures, the slope estimated from those exposures can also change. The direction and size of the change depend on how the exposure metric responds to the PK parameters and how the outcome data identify the slope. Propagation does not necessarily make the point estimate move in one predictable direction; its clearest consequence is often greater uncertainty.

The target here is specifically uncertainty in the PK parameters used to calculate exposure. It is distinct from describing exposure as measured with error or fitting a separate model for an exposure measurement process. Those are related issues, but they require additional assumptions. A useful first question is whether the reported exposure–response result is conditional on one fitted PK model, or whether it reflects uncertainty across plausible PK parameter values.

Model and Assumptions

Let \(\theta\) denote the PK parameter vector, and let \(C_i(\theta)\) be the chosen exposure metric for participant \(i\) under that parameter vector. For example, \(C_i(\theta)\) could be an area-under-the-curve measure or a concentration summary; the appropriate metric depends on the exposure–response estimand and scientific question. Suppose the response model has a slope \(\beta\) for exposure. In a simple linear illustration, its conditional mean is

The example is deliberately simple. Its purpose is to show uncertainty propagation, not to prescribe a response distribution or functional form for every clinical endpoint. A valid analysis must specify a response model suitable for the outcome and the estimand. For the linear model, usual slope inference additionally relies on an appropriate mean specification and, for the familiar standard error, independent errors with constant variance; a normal-theory interval also relies on normal errors or a suitable large-sample approximation.

The PK uncertainty calculation also needs a defensible distribution or approximation for \(\theta\), such as a covariance matrix from parameter estimation, a bootstrap distribution, or a posterior distribution. The PK model, data, and estimation procedure must support that uncertainty representation. If individual PK parameters, such as individual clearances, are estimated, their relevant uncertainty and dependence on population parameters may also need to be retained. Treating estimated exposures as fixed is not justified merely because the exposure values have been calculated.

Worked Example: The Plug-In Slope

First-Order Propagation and Its Justification

A useful approximation follows by representing the slope as a function of uncertain PK parameters. If \(\hat{\theta}\) is an estimate of \(\theta\), and \(g(\hat{\theta})\) is the slope estimate produced using that estimate, a first-order Taylor expansion around \(\hat{\theta}\) gives

If the PK parameter estimate has covariance matrix \(V_{\theta}\), this approximation gives the PK contribution to the slope variance:

The justification is direct: the approximate change in the slope is the gradient multiplied by the parameter change. The variance of that linear combination is the gradient, the parameter covariance matrix, and the transposed gradient in the displayed order. This approximation works best when parameter uncertainty is modest and the slope changes approximately linearly over the region of plausible parameter values. Strong nonlinearity, skewed parameter distributions, or parameters near boundaries can make it unreliable.

For a simple combination of response-model and PK uncertainty, suppose the conditional slope estimate has variance \(V_{\mathrm{response}}\), and the PK contribution has variance \(V_{\mathrm{PK}}\). If the two contributions can reasonably be treated as independent, then

Independence should not be assumed automatically. When the response estimate and PK estimates are dependent, their covariance contributes to the total variance. More generally, form a joint vector of estimated quantities, use its covariance matrix, and apply the gradient of the final slope with respect to that whole vector. This retains covariance rather than adding variance components as if they were unrelated.

Propagating Uncertainty With Draws

When the mapping from PK parameters to exposure is nonlinear or the parameter uncertainty is not well summarized by a covariance matrix, propagate uncertainty with draws. In a frequentist bootstrap, resample or simulate PK data under the fitted model, refit the PK model for each replicate, recalculate individual exposures, and refit the response model. In a Bayesian analysis, draw PK parameters from their joint posterior distribution, calculate exposures for each draw, and carry those exposures into the response analysis. In either case, the draws should retain relevant dependence among PK parameters; varying each parameter independently when the fitted parameters are correlated can produce implausible combinations.

For a nested calculation, it can also be important to represent response-model uncertainty conditional on each PK draw. One practical procedure is to draw PK parameters, calculate exposures, fit the response model, and retain its estimate and uncertainty for each draw. The resulting distribution includes variation across PK draws and within-draw response uncertainty. If the response and PK analyses share data or parameters, a joint procedure is preferable to an artificial independence assumption.

Choosing an Approach and Checking Its Performance

The main approaches answer related but not identical questions. A plug-in analysis is straightforward and can be useful as a conditional reference, but its ordinary interval does not account for uncertainty in the PK estimates. A delta method is efficient when the mapping is smooth and uncertainty is modest, but depends on a local linear approximation and a suitable covariance matrix. Bootstrap or posterior draws can preserve nonlinearities and asymmetric uncertainty, provided the PK refitting or posterior distribution is implemented correctly. A joint PK–response model can represent dependence and estimate shared quantities together, but its assumptions, parameterization, and computation require careful evaluation.

Useful checks include comparing the propagated and plug-in results, examining the range and shape of the exposure and slope distributions across PK draws, and verifying that parameter correlations and individual exposure calculations are retained. In a bootstrap, inspect whether fits converge and whether a small number of failed or extreme replicates dominate the interval. In a draw-based analysis, check that the number of draws is sufficient for stable interval endpoints and that the draws represent the intended PK uncertainty. If exposure barely changes across plausible PK draws, a large change in the slope interval calls for investigation; if exposure changes substantially but the interval does not, determine whether the response data or the model structure genuinely constrain the slope.

No propagation method repairs a misspecified PK model, an unsuitable exposure metric, or a response model that does not address the intended estimand. Nor does a wider interval itself establish that the exposure–response relationship is causal. Interpretation still depends on the estimand and causal assumptions for the analysis. Propagated uncertainty describes what the specified models and data leave uncertain; it does not by itself resolve structural uncertainty about whether those models are appropriate.