Singular-Value Analysis of Sensitivity Matrices

Why Analyze a Sensitivity Matrix?

A local sensitivity ranking can identify parameters that strongly affect one PBPK output at a time. It does not, by itself, show whether two parameters produce nearly the same pattern across several outputs. If their effects are similar, changing one parameter may partly compensate for changing the other. Singular-value analysis gives a systematic way to examine these combinations across a chosen collection of outputs.

The analysis begins with a matrix whose rows represent outputs and whose columns represent parameters. Its entries quantify how each output changes when each parameter changes near a specified baseline. The singular-value decomposition (SVD) then describes the matrix in terms of orthogonal directions in parameter-change space and the corresponding output-change directions. The results are local: they describe the model near the chosen parameter values and for the outputs included in the matrix.

Constructing and Scaling the Matrix

Let \(y_i(\boldsymbol{\theta})\) be output \(i\) from a PBPK model, where \(\boldsymbol{\theta}\) is the parameter vector. Outputs may be concentrations at selected times, AUCs, or other defined endpoints. A matrix can include different types of outputs, but its rows must have an explicit and defensible scaling.

For positive parameters and outputs, log scaling makes the entries elasticities: approximately, a 1% change in parameter \(\theta_j\) produces an \(A_{ij}\)% change in output \(y_i\), locally, when other parameters are held fixed. This scaling is not appropriate for outputs that may be zero or negative. A more general local scaling is \(A_{ij}=(r_j/q_i)(\partial y_i/\partial\theta_j)\), where \(r_j\) is a stated parameter-change scale and \(q_i\) is a stated output scale. These scales make entries comparable; they do not automatically represent measurement uncertainty.

A numerical matrix can be estimated by finite differences. With log scaling, for a small positive \(h\), a central difference is \([\,\log y_i(\theta_j e^h)-\log y_i(\theta_j e^{-h})\,]/(2h)\), holding other parameters fixed. A suitable \(h\) should be large enough to exceed numerical solver noise and small enough for the local approximation to be reasonable. Sensitivity calculations and their finite-difference checks are covered in the earlier material on finite-difference sensitivity calculations.

Before interpreting an SVD, specify the baseline parameter vector, the output rows, the parameter columns, and the scaling. If outputs differ greatly in units or magnitude, an unscaled matrix can be dominated by whichever output happens to have the largest numerical values. The choice of scales is part of the scientific question, not a cosmetic step.

What the Singular Values and Vectors Mean

For an \(m\)-by-\(p\) scaled sensitivity matrix \(A\), the SVD writes \(A=U\Sigma V^\mathsf{T}\). The columns of \(V\) are orthonormal directions in the \(p\)-dimensional scaled parameter-change space; the columns of \(U\) are orthonormal directions in the \(m\)-dimensional scaled output space. The nonnegative diagonal entries of \(\Sigma\), ordered from largest to smallest, are the singular values \(\sigma_1,\sigma_2,\ldots\).

For a small scaled parameter perturbation \(\delta z\), the local output change is \(\delta w\approx A\delta z\). In particular, if \(\delta z=v_k\), the \(k\)th right singular vector, then \(Av_k=\sigma_k u_k\). Thus, a unit perturbation in that parameter direction produces a local output-change vector of length \(\sigma_k\), using the stated scaling and ordinary Euclidean length. A small singular value means the corresponding combination of parameter changes has a small net effect on the selected outputs near the baseline. A zero singular value means the linearized matrix has an exact null direction.

This result follows directly from the matrix product:

because \(v_k\) has unit length. Since a vector's length is nonnegative, \(\|Av_k\|=\sqrt{\lambda_k}\). This derivation also explains why examining the eigenvectors of \(A^\mathsf{T}A\) is mathematically related to the SVD, although direct SVD algorithms are generally preferred for numerical stability.

A right singular vector is a combination, not a ranking of individual parameters. Its entries give the relative signed changes in the scaled parameters for that direction. Large entries with opposite signs can indicate that increasing one parameter and decreasing another produce offsetting effects across the selected outputs. Such a pattern is evidence of locally similar or compensating effects, not proof that the parameters are interchangeable under all doses, times, or physiological scenarios.

Exact Redundancy and the Role of the Selected Outputs

An exact zero singular value corresponds to a nonzero parameter direction \(v\) for which \(Av=0\). In the linear approximation, moving in that direction produces no change in any selected output. Such a direction can arise when there are more parameters than independent output-response patterns, or when columns are exactly dependent. In practical PBPK models, small rather than exactly zero singular values are more common, owing to near-similar effects and numerical approximations.

Scaling, Diagnostics, and Alternatives

A singular value has meaning only relative to the chosen row and column scales. If one output is rescaled by a factor of 100, its row in \(A\) is also rescaled under a non-dimensionless convention, which can change the singular values and vectors. Likewise, changing parameter scales changes the meaning of a unit direction in parameter space. For log elasticities, the entries are dimensionless, but the selected outputs and the use of equal row weighting still determine which response patterns are emphasized.

Useful checks include recalculating the matrix with different finite-difference step sizes, confirming that the PBPK solver tolerances are tight enough, and inspecting whether a small singular value persists under reasonable scaling choices. Also inspect the singular vectors themselves: a very small singular value is less informative if it is driven by an output row with unstable numerical calculations or by a scaling choice that was not justified. If stochastic simulation noise is present, repeat or otherwise quantify that noise before interpreting small matrix entries.

Singular-value analysis complements, rather than replaces, simpler summaries. Column norms summarize the overall size of each parameter's sensitivity vector but can miss similarity between columns. Pairwise column correlations can reveal similarity between two parameter effects but do not summarize combinations involving several parameters. SVD considers orthogonal combinations across all columns at once. These are different views of the same chosen local matrix, not universal rankings independent of outputs and scales.

The method is local and first-order. It does not show how sensitivities change far from the baseline, capture nonlinear interactions over a broad parameter range, or establish whether a parameter can be estimated from a particular dataset. Data-based identifiability requires considering the observations and their uncertainty, not only model sensitivities. A small singular value is therefore a diagnostic prompt to examine a weakly expressed parameter combination, not a standalone decision to fix or combine parameters.