Interpreting Local Sensitivity Rankings

What a Local Sensitivity Ranking Tells You

A PBPK model can contain many parameters, but a sensitivity ranking is meaningful only after specifying which output is being examined and where in parameter space the model is evaluated. For example, the ranking for plasma concentration at 2 hours may differ from the ranking for area under the concentration-time curve, even for the same simulated subject and dose. A local ranking describes the model’s response to small changes near a stated reference parameter set; it is not a universal ordering of parameter importance.

The earlier material on local sensitivity defines sensitivity as an output derivative, and dimensionless sensitivity expresses response to relative parameter changes. Here the focus is how to compare and interpret those quantities. In particular, a steep derivative does not by itself show that a parameter contributes greatly to prediction uncertainty: that also depends on how uncertain the parameter is.

The sign of \(s_j\) indicates the local direction of change: a positive derivative means a small increase in \(\theta_j\) increases the output near the reference point; a negative derivative means it decreases the output. Ranking by absolute magnitude focuses on response size and does not preserve this direction, so signs should be reported separately. Raw derivatives have units of output divided by parameter units. Their magnitudes can therefore depend on the units used to express a parameter.

Dimensionless coefficients make relative changes comparable across parameters with different units, but they do not incorporate uncertainty in those parameters. They also require care when an output is zero or near zero, or when a parameter is zero or can take negative values. In those cases, a relative change or the coefficient above may be undefined or unstable. A scientifically chosen scale for the parameter and output, or a raw derivative with explicit units, may be more suitable.

Why the Local Approximation Works

A local ranking relies on a first-order approximation around the reference parameter values. If the output is differentiable there and the parameter changes are small, the first-order Taylor expansion is

This expression explains the interpretation of the derivative: holding other parameters fixed, the output change associated with parameter \(j\) is approximately \(s_j\Delta\theta_j\). It also shows why a derivative alone cannot establish prediction influence. The change depends on both the derivative and the size of the parameter change under consideration.

From Derivatives to Prediction Uncertainty

To assess how parameter uncertainty may affect an output, the first-order approximation can be combined with a covariance matrix for the parameters. Let \(\boldsymbol{\Sigma}\) be the covariance matrix of parameter uncertainty near \(\boldsymbol{\theta}_0\), and let \(\mathbf{s}\) be the vector of local derivatives. Under the first-order approximation, the output variance is approximately

This follows by writing the output deviation as approximately \(\mathbf{s}^{\mathsf T}\Delta\boldsymbol{\theta}\). The variance of a linear combination is the sum of the products of its coefficients and the corresponding covariances, giving the expression above. If parameter uncertainties are independent, the covariance matrix is diagonal and the expression reduces to

In that independent case, \(|s_j|\operatorname{SD}(\theta_j)\) is the approximate output-scale change associated with one parameter standard deviation. Its square is that parameter’s term in the first-order output variance. With correlated parameters, cross terms also contribute. Consequently, the independent-parameter terms cannot generally be interpreted as a complete, additive allocation of prediction variance.

Diagnostics, Comparisons, and Limits

A local ranking is most useful when it is reproducible and clearly tied to its conditions. Check that the reference parameter set is stated, derivatives are evaluated for the same output and scenario, and the ranking scale is consistent. If finite differences were used to obtain derivatives, the perturbation should be small enough to represent local behaviour but large enough to avoid numerical error; repeating the calculation with a smaller and a larger perturbation can reveal instability. A ranking that changes greatly with perturbation size, solver tolerances, or small changes to the reference point should not be presented as a robust ordering.

The main alternatives answer related but distinct questions. Raw derivatives rank response to equal absolute changes in stated parameter units. Dimensionless coefficients rank response to equal proportional changes, when relative scaling is meaningful. Uncertainty-weighted local calculations ask how parameter uncertainty may propagate to the output, requiring parameter standard deviations and, when relevant, covariances. These approaches should not be collapsed into a single label of “importance”: the appropriate one depends on the scientific question and available uncertainty information.

All these calculations are local and first-order. Large parameter changes, strong curvature, or a reference point near a sharp transition can make the linear approximation inaccurate. A local derivative ranking also applies only to its chosen output, time, dose, and physiological scenario; it does not establish a global ranking or show that parameters are identifiable from data. Those limitations matter when using rankings to guide model investigation or parameter reduction: a low local derivative for one output is not proof that a parameter can be ignored for other predictions.