Presenting Sensitivity and Uncertainty for Regulatory Review

Why Sensitivity and Uncertainty Need Separate Reporting

A PBPK prediction is conditional on a model structure, parameter values, scenario assumptions, and a defined population or individual. A reviewer therefore needs more than a single predicted value. They need to see which inputs matter for the reported outcome, how uncertain inputs were represented, and how that uncertainty affects the prediction. The presentation should also make clear what the analysis does not represent: uncertainty in selected inputs is not the same as differences among people, and neither automatically captures limitations in model structure.

This reporting supports interpretation of the analysis for its stated context of use. It is not a substitute for reporting verification or validation evidence, and it should not imply that a prediction is more certain simply because it is displayed with a precise number of decimal places.

Before presenting any sensitivity ranking or interval, identify the output, units, population, dosing or exposure scenario, and calculation method. For example, “the predicted maximum plasma concentration in adults after the specified dose” is interpretable; “the model output” is not. State whether results are conditional on a fixed set of parameters, propagated over uncertain parameter values, simulated over individuals, or generated by combining these sources.

Reporting Parameter Influence

A simple local sensitivity analysis changes one input around a reference value while holding other inputs fixed. For a positive parameter \(\theta_j\) and positive output \(Y\), a normalized local sensitivity, also called an elasticity, can be approximated by a central finite difference:

Here \(Y_0\) is the output at the reference parameter set, and \(\delta\) is the fractional perturbation. This quantity is dimensionless. Near the reference value, it estimates the percentage change in \(Y\) associated with a 1% change in \(\theta_j\). A positive value indicates that increasing the parameter tends to increase the output locally; a negative value indicates the reverse. The magnitude indicates local responsiveness, not the parameter’s uncertainty or its overall contribution to prediction uncertainty.

This interpretation relies on a sufficiently small perturbation and a locally smooth response. The selected perturbation should be reported: a ranking can change with perturbation size, especially near thresholds or nonlinear regions. For parameters with meaningful bounds, the perturbations must remain within those bounds. A one-at-a-time analysis also holds all other inputs fixed, so it cannot reveal interactions between inputs.

A useful regulatory table reports the parameter name, physiological or model role, unit, reference value, perturbation range, resulting output values, sensitivity measure, and evidence or rationale for the range. Include the output definition and reference scenario in the table caption or accompanying text. Report the direction as well as the magnitude; a list of absolute sensitivities alone hides whether an increase in an input raises or lowers the prediction.

Local one-at-a-time analysis is useful for transparent screening. If the model is nonlinear over plausible ranges, parameters interact, or the input distribution is broad, supplement it with a global analysis that varies multiple inputs over their stated distributions or ranges. A global ranking answers a different question: how much an input contributes to output variation across the chosen input space, under the specified sampling and dependence assumptions. State the method and assumptions rather than presenting a ranking as universal. A ranking is specific to the output, scenario, input ranges, and method used.

Propagating Input Uncertainty

Uncertainty propagation carries specified uncertainty in model inputs through the PBPK model to produce a distribution of predictions. A distribution should have a traceable basis: data, literature, expert elicitation, or an explicit assumption. Report its form and parameters, units, bounds, and any dependence with other inputs. If an input is fixed because no distribution was available, say so; a fixed value does not mean that the value is known without uncertainty.

The following linear calculation illustrates why uncertainty in an output depends both on sensitivity and on input uncertainty. Suppose a model output is \(Y=2\theta_1+\theta_2\), where \(Y\) is in mg/L and the two normalized inputs are dimensionless. Assume independent normal inputs, \(\theta_1\sim N(10,1^2)\) and \(\theta_2\sim N(4,2^2)\). Under these assumptions, the mean and variance follow from linearity and independence:

The variance formula follows because the variance of a linear combination includes covariance terms; independence makes the covariance zero. If the inputs were correlated, the additional term \(2(2)(1)\operatorname{Cov}(\theta_1,\theta_2)\) would be required. For the stated normal, linear, independent-input example, the output is normal with standard deviation \(\sqrt{8}=2.828\) mg/L. Its central 90% interval is:

Separate Uncertainty, Variability, and Model Limitations

In a PBPK analysis, input uncertainty might concern an uncertain enzyme abundance or a tissue partition parameter. Variability might concern differences in body weight or enzyme activity among individuals in the target population. A simulation that draws an uncertain population-level parameter once per simulation is not equivalent to a simulation that draws an individual-specific value for each simulated person. The sampling level and interpretation should be stated.

If both sources are relevant, show them separately where feasible: for example, a distribution of population predictions from parameter uncertainty, and a distribution across simulated individuals conditional on a given parameter set. A combined distribution can be useful, but label the included sources and explain how they were combined. Do not call the combined range an “uncertainty interval” if it also contains between-individual variability without qualification.

A model limitation is different again. It may include a biological process that is absent, a weakly supported mechanism, or a population outside the model’s well-supported scope. Unless a limitation is represented through explicit alternative model structures or other quantified scenarios, it is not captured by a parameter uncertainty interval. Describe such limitations alongside the numerical results, including the potential direction or consequence when it can be reasoned through. Do not imply that a wide interval automatically accounts for every model limitation.

Tables, Plots, and Diagnostics for Review

A compact table can make assumptions auditable. For each uncertain input, report its role, unit, reference or central value, distribution or range, evidence basis, dependencies, and whether it represents uncertainty or individual variability. A second table can summarize outputs by scenario, including units, median or mean as appropriate, interval definition, and the sources propagated. Keep scenario labels explicit and define abbreviations.

A tornado plot is useful for a local one-at-a-time analysis when it shows the reference prediction and the low- and high-input predictions, not only a ranked bar length. Label the perturbation and the output units. For propagated uncertainty, a distribution plot or quantile plot can show shape, skewness, multiple modes, and tail behavior that a mean and standard deviation would conceal. A scenario comparison plot should use a common scale and show the scenario-specific prediction summaries without visually conflating input uncertainty and population variability.

Diagnostics should check whether the reported summaries describe the simulation output adequately. Inspect distributions and tails, confirm that sampled inputs remain within intended bounds, examine parameter dependence, and verify that simulations converged or that Monte Carlo precision is adequate for the reported quantiles. For local sensitivity, repeat calculations at a smaller and larger perturbation when practical; substantial ranking changes signal nonlinearity or scale dependence. Check whether the result is driven by a small number of extreme draws. If it is, report that feature and investigate its source rather than silently truncating the distribution.

No single display is sufficient for every purpose. Local sensitivity is relatively easy to explain and trace, but it is tied to a reference point and may miss interactions. Global sensitivity can reflect broad ranges and interactions but depends on the selected joint input distribution and attribution method. Deterministic scenario analysis is transparent for discrete assumptions but does not, by itself, assign probabilities to scenarios. Choose the approach that answers the stated question and explain what it leaves unresolved.

Common Reporting Pitfalls

• Equating influence with uncertainty contribution. A parameter can have high sensitivity but a narrow plausible range. Report sensitivity and input uncertainty separately.
• Calling every range a prediction interval. State whether it comes from parameter uncertainty, individual variability, both, or a deterministic scenario range.
• Hiding assumptions behind a distribution. A probability distribution is not self-justifying. Give its evidence basis, bounds, and dependence assumptions.
• Overinterpreting precision. Many reported digits do not remove uncertainty in inputs, model structure, or simulation estimates.
• Treating model limitations as quantified uncertainty. State limitations that were not represented in the propagated inputs or alternative scenarios.