Propagating PBPK Uncertainty to Exposure Ratios

Why Propagate Uncertainty to the Ratio?

A PBPK prediction of a drug-drug interaction is often summarized by an exposure ratio: exposure under the interaction condition divided by exposure under the control condition. A single predicted ratio can conceal how strongly the result depends on uncertain physiology, drug properties, or interaction parameters. Propagating those uncertainties through the model shows the range and shape of ratios implied by the specified inputs and model.

The key operation is to calculate a ratio for each common set of inputs, rather than to combine unrelated control and interaction predictions. A shared draw preserves the fact that uncertain physiology—such as hepatic blood flow or enzyme abundance—affects both simulations. This tutorial focuses on that propagation and the resulting model-based distributions for AUC and Cmax, not on assigning separate uncertainty sources to different levels of a clinical study.

The input distribution should reflect the uncertainty representation chosen for the prediction. It must specify plausible ranges and, where relevant, dependencies among inputs. For example, two physiological quantities may be correlated, or drug parameters may have been estimated jointly. Drawing each input independently when dependence is part of the available information changes the assumed joint distribution and therefore can change the ratio distribution.

Other simulation conditions must also be defined consistently: dose, dosing interval, victim-drug regimen, perpetrator regimen, and endpoint definition. The two scenarios should differ only in the intended interaction condition unless the study question explicitly requires another difference. The PBPK model must be applicable to both scenarios, and each simulation must use an adequate duration, sampling schedule, and numerical resolution for the endpoint being calculated. In particular, Cmax can be missed if concentration peaks are poorly resolved in time.

Paired Monte Carlo Propagation

Monte Carlo propagation repeatedly samples the uncertain inputs, runs the model under both conditions, and calculates the ratio within each paired draw. The resulting collection of ratios approximates the distribution induced by the specified input uncertainty and PBPK model. The ratio distribution can be asymmetric even when some input distributions are symmetric, because the model and the division used to form the ratio can be nonlinear.

The pair is essential: both outputs in a ratio correspond to the same sampled physiology and drug properties. If a parameter draw produces a high baseline exposure, for example, it may also produce a high interaction exposure. That shared movement can cancel in the ratio. Breaking the pairing discards this dependence and can distort the distribution of ratios.

What the Summaries Mean

A useful summary may include the median, selected quantiles, and a measure of spread. A central 90% model-based uncertainty interval, for example, is defined by the 5th and 95th percentiles of the simulated ratios. It describes the central range of ratio values under the specified input distribution and model, subject to Monte Carlo error. It is not automatically a frequentist confidence interval for a clinical-study estimand, nor does it account for model inadequacy unless that uncertainty has been represented.

A ratio is dimensionless when numerator and denominator use the same exposure units and endpoint definition. A ratio above 1 indicates a larger predicted endpoint under the interaction condition; a ratio below 1 indicates a smaller one. The distribution's spread communicates sensitivity to the specified uncertain inputs, not the probability that a particular clinical outcome will occur.

A Short Derivation: Why Pairing Changes Ratio Variability

For positive endpoints, write \(A=\mathrm{AUC}_{\mathrm{int}}\) and \(B=\mathrm{AUC}_{\mathrm{ctl}}\). The log ratio is \(\log R_{\mathrm{AUC}}=\log A-\log B\). Therefore, under any joint distribution for which these variances exist,

When the two log exposures tend to increase together, their covariance is positive, reducing the variance of the log ratio relative to the sum of the separate variances. Pairing retains this covariance. Independently combining control and interaction outputs generally removes or alters it, so it does not estimate the same ratio distribution. The same argument applies to Cmax when both endpoints are positive.

Diagnostics, Alternatives, and Limitations

The most direct diagnostic is to examine the simulated endpoints and ratios, not only a final interval. Check for failed simulations, nonpositive or implausible concentrations, inconsistent dose or time settings between scenarios, and outlying ratios. Determine whether unusual ratios reflect plausible parameter combinations and real model behavior or numerical problems, such as a poorly resolved Cmax. Repeat the calculation with more draws and compare key summaries; if quantiles move materially, Monte Carlo error remains important.

Also inspect whether the sampled inputs respect parameter bounds and dependencies. A distribution that assigns appreciable probability to physiologically impossible values can create an apparently broad uncertainty distribution that is an artifact of the input specification. Conversely, truncating a distribution without justification can conceal meaningful uncertainty. Sensitivity checks on the chosen distributions and dependencies help reveal how much the result depends on these choices.

Paired Monte Carlo propagation is flexible for nonlinear models and can represent skewness, interactions among inputs, and nonlinear transformations. A local sensitivity or delta-method approximation is faster and can clarify which small input perturbations drive local output variation, but it relies on a sufficiently smooth response near a reference parameter set. One-at-a-time analysis is useful for identifying directional effects while holding other inputs fixed; it does not, by itself, characterize a joint uncertainty distribution or preserve input dependence. These approaches can complement Monte Carlo propagation but answer different questions.

The resulting distribution is conditional on the PBPK model structure, the input distributions, and the chosen simulation and endpoint definitions. It does not establish that the model is correct, and it does not include omitted uncertainty merely because many simulations were run. A narrow distribution can therefore be misleading if important inputs were treated as known or if model discrepancy is substantial. Conversely, a wide distribution can arise from uncertain inputs, nonlinear amplification, or both.

Common Mistakes

• Using independent input draws for the two scenarios. This discards shared parameter dependence. Draw once, then use that same vector in the control and interaction runs.
• Calling a simulated quantile interval a clinical confidence interval. The quantiles summarize the specified model-based uncertainty distribution; they are not automatically a confidence interval for a clinical estimand.
• Taking the ratio of average endpoints as the average ratio. In general, \(\operatorname{mean}(A/B)\) differs from \(\operatorname{mean}(A)/\operatorname{mean}(B)\). Choose the summary that matches the question and state it explicitly.
• Assuming many draws fix a flawed model. More simulations reduce Monte Carlo error, but cannot repair implausible input distributions, incorrect model structure, or inadequate time resolution.
• Interpreting an AUC ratio as a Cmax ratio. These are distinct endpoints, and time-varying interactions can affect peak and total exposure differently. Propagate and summarize each endpoint separately.