Sensitivity-Based Identifiability Diagnostics

Why Sensitivity Is Relevant to Identifiability

A PBPK model can respond strongly to changes in several parameters and still fail to distinguish those parameters from one another. For example, increasing intrinsic clearance while also changing blood flow may leave a specified plasma concentration profile nearly unchanged. The individual parameters influence the output, but their effects may be difficult to separate using that output and design.

Sensitivity-based identifiability diagnostics examine this local distinguishability. They ask how the specified model outputs change when parameters are perturbed near a particular parameter vector. They do not establish that parameters are uniquely determined across the full parameter space, or even that a fitted model has only one acceptable solution. Those are stronger questions than a local derivative calculation can answer.

The diagnostic depends on what is observed: which outputs, at which times, under which dose and physiological conditions, and with what measurement precision. It also depends on how parameters and outputs are scaled. A statement that two parameters are “distinguishable” is therefore incomplete unless it is tied to a particular model, parameter region, output set, design, and scale.

Constructing a Local Diagnostic

Let \(\boldsymbol{\theta}\) denote the parameter vector, and let \(\mathbf{y}(\boldsymbol{\theta})\) collect the model predictions for the specified outputs. The local sensitivity matrix \(J\) has one row for each output and one column for each parameter. Entry \(J_{ij}\) is the derivative of output \(i\) with respect to parameter \(j\), evaluated at a chosen parameter vector. Sensitivities may be calculated by the finite-difference, forward-sensitivity, or adjoint approaches covered earlier.

Raw derivatives have units, and their size depends on the units used for parameters and outputs. For interpretation across parameters, a common local scale is the derivative of a log output with respect to a log parameter. For positive outputs and parameters, this is an elasticity: approximately, the percent change in an output for a one-percent change in a parameter. More generally, define dimensionless parameter coordinates \(\boldsymbol{\phi}\) and scaled outputs \(\mathbf{z}\), and form \(S=\partial\mathbf{z}/\partial\boldsymbol{\phi}\). The chosen scales should represent meaningful parameter changes and output changes.

If output errors have known or specified covariance matrix \(\Sigma\), a precision-weighted sensitivity matrix is \(S_w=\Sigma^{-1/2}S\). This weighting expresses output changes in units of their error scale. When errors are independent with standard deviations \(\sigma_i\), this amounts to dividing each sensitivity row by \(\sigma_i\). If error covariance is not available, unweighted diagnostics can still be descriptive, but they do not account for the relative information in measurements of different precision.

The singular values of the scaled matrix summarize how much it changes different parameter directions. A very small singular value relative to the largest indicates a direction that has little effect on the specified scaled outputs. The condition number, the ratio of the largest to the smallest singular value when the matrix has full column rank, summarizes the spread in these effects. A large condition number warns that distinguishing some parameter combinations may be numerically unstable. The singular-value analysis of sensitivity matrices gives the underlying decomposition; here the emphasis is on interpreting it as an identifiability diagnostic.

Why the Local Diagnostic Works—and Where It Stops

The justification follows from a first-order expansion. Assume the model output map is differentiable in a neighborhood of the parameter vector, the selected output and parameter scales are fixed, and the local derivatives are calculated accurately. For a small perturbation, the Taylor expansion is

Write a singular-value decomposition \(S=U D V^\mathsf{T}\), where the columns of \(V\) are parameter-space directions and the diagonal entries of \(D\) are singular values. For a unit direction \(\mathbf{v}_j\) associated with singular value \(d_j\), \(S\mathbf{v}_j=d_j\mathbf{u}_j\). To first order, a perturbation of size \(a\) along that direction changes the scaled output by approximately \(a d_j\). A small \(d_j\) therefore means a small predicted output change for that perturbation. If the change is small relative to output error or numerical uncertainty, the data and model outputs may provide little local ability to distinguish that direction.

A zero singular value means the derivative matrix has a null direction: infinitesimal changes in that direction have zero first-order effect on the selected outputs. This is a local, first-order statement. Nonlinear curvature can still change outputs for larger movements, and a derivative can vanish at a point even when the output map is one-to-one nearby. Conversely, a well-conditioned local matrix does not rule out distant parameter vectors that give similar outputs. Sensitivity conditioning is thus a diagnostic of local behavior, not a general identifiability proof.

Profiles, Conditioning, and Added Outputs

A useful diagnostic combines visual or numerical profile comparison with singular values. Pairwise profile cosines reveal near-parallel columns; column norms indicate their scaled magnitudes; singular values assess combinations of all columns simultaneously. Pairwise comparisons alone can miss a weak combination involving several parameters, while the smallest singular value can be affected by scaling and by any nearly redundant parameter direction.

An output can improve distinguishability when its sensitivity pattern adds information in a direction not already represented by the existing outputs. However, simply adding more time points does not guarantee improvement: closely spaced or redundant observations may contribute nearly repeated sensitivity patterns. A new endpoint should be assessed on its own precision and scale, and the resulting matrix should be recomputed for the complete specified output set.

Assumptions, Diagnostics, and Common Pitfalls

Interpret a sensitivity-based diagnostic only after checking its construction. The local approximation assumes differentiability near the evaluation point and sufficiently small perturbations. Numerical derivatives should be checked against reasonable step-size changes or a second sensitivity calculation when feasible. Sensitivities that vary sharply across the plausible parameter region may make a single-point diagnostic unrepresentative. The parameter scaling should reflect meaningful changes; changing from a parameter to its logarithm can substantially change the diagnostic because it changes the perturbation being assessed.

• Check the outputs and design. State which concentrations, times, endpoints, doses, or physiological scenarios enter the matrix. The result applies to that selection, not automatically to every possible PBPK prediction.
• Check scaling and error weights. Confirm that parameters are perturbed on interpretable scales and outputs are scaled consistently. If precision weighting is used, document the assumed covariance or standard deviations.
• Check derivative stability. Compare calculations under smaller and larger finite-difference steps, or compare with available forward or adjoint sensitivities. Solver tolerances and noisy numerical derivatives can create misleading small singular values.
• Check more than one parameter vector. A local diagnostic can change across the parameter space. Where scientifically relevant, evaluate it at multiple plausible parameter vectors rather than treating one nominal point as universal.

A common mistake is to conclude that a small singular value proves the parameters are structurally non-identifiable. The reasoning is that a near-null direction produces little output change, so no data could distinguish it. The correction is that the calculation is local, first-order, and tied to a specified output set, scaling, and error model. Nonlinear behavior away from the evaluation point, other outputs, or additional experimental conditions may provide distinction. The reverse mistake is also possible: a well-conditioned matrix at one point does not establish global uniqueness.

These approaches answer related but different questions. Sensitivity conditioning is efficient for locating local weak directions and comparing specified outputs. It is not a substitute for structural analysis when the question concerns ideal uniqueness, or for fit-based uncertainty assessment when the question concerns the information in observed data.