Reparameterizing Informative Parameter Combinations

Why Reparameterize?

A PBPK model may reproduce observed concentrations well even when two or more of its parameters are poorly separated by the available data. For example, oral exposure can depend on bioavailability and clearance mainly through their ratio. Many different bioavailability–clearance pairs can therefore produce the same exposure. In that situation, reporting both parameters as if each were independently well determined gives a misleading impression of what the data establish.

Reparameterization expresses the model in coordinates aligned with the directions to which the observations are most and least sensitive. The more informative coordinates may correspond to physiologically meaningful ratios, products, or effective rates. This does not create information or prove that an individual parameter is known. It makes the information—and the remaining ambiguity—clearer.

Definition: An informative parameter combination is a function of model parameters that changes along a direction to which the selected model outputs are sensitive. A weakly informed combination changes along a direction that produces little change in those outputs, relative to the output scaling and observation precision used in the analysis.

Local Sensitivity Directions and New Coordinates

Let $\boldsymbol{\theta}$ denote the parameters of interest, and let $\mathbf{r}$ be a vector of model outputs represented on suitable scales. The outputs might be log concentrations at selected times or standardized endpoints. Choose a reference parameter vector $\boldsymbol{\theta}_*$. For positive parameters, define dimensionless log deviations by $\eta_j=\log(\theta_j/\theta_{*,j})$. Near the reference, changes in the outputs are approximated by

\Delta\mathbf{r}\approx J\Delta\boldsymbol{\eta}, \qquad J_{ij}=\left.\frac{\partial r_i}{\partial\eta_j}\right|_{\boldsymbol{\theta}_*}.

If the outputs have different scales or precisions, their rows should be scaled accordingly; for example, a residual may be divided by its observation standard deviation. Thus the sensitivity directions depend on the outputs included and their scaling. The singular-value decomposition $J=U\Sigma V^\mathsf{T}$ identifies orthogonal directions in parameter-coordinate space. The columns of $V$ are those directions, and the corresponding diagonal entries of $\Sigma$ give their local sensitivity magnitudes.

Writing $\mathbf{c}=V^\mathsf{T}\boldsymbol{\eta}$ expresses deviations in those directions. A large singular value means a unit change in that coordinate has a relatively large local effect on the scaled outputs; a small one means its effect is relatively small. These are local statements around $\boldsymbol{\theta}_*$, not guarantees about all parameter values or all possible observations.

Justification: Under the local linear approximation, the squared change in the scaled outputs for a parameter displacement $\Delta\boldsymbol{\eta}$ is $$ \|J\Delta\boldsymbol{\eta}\|^2 =\Delta\boldsymbol{\eta}^{\mathsf{T}}J^{\mathsf{T}}J\Delta\boldsymbol{\eta}. $$ For a right singular vector $\mathbf{v}_k$, $J^{\mathsf{T}}J\mathbf{v}_k=\sigma_k^2\mathbf{v}_k$. Therefore a displacement of size $a$ along that direction produces a linearized output change of size $|a|\sigma_k$. Small $\sigma_k$ identifies a locally weakly informed direction; it does not by itself identify the cause of the weakness or establish global non-identifiability.

Worked Examples of Informative Combinations

Worked Example: Oral Exposure Identifies a Clearance-to-Bioavailability Ratio

Suppose a hypothetical one-dose oral analysis uses the relationship $AUC=D F/CL$, where $D$ is dose, $F$ is bioavailability, and $CL$ is systemic clearance. Let $D=100$ mg, $F=0.60$, and $CL=3.0$ L/h. Calculate the exposure and identify the combination it informs.

Substitution gives

AUC=\frac{(100\ \mathrm{mg})(0.60)}{3.0\ \mathrm{L/h}} =20\ \mathrm{mg\cdot h/L}.

In this relationship, exposure depends on $F/CL$, or equivalently on apparent oral clearance $CL/F$. The parameter values give $CL/F=3.0/0.60=5.0$ L/h. Now take a different pair, $F=0.80$ and $CL=4.0$ L/h. It has the same $CL/F=5.0$ L/h, so it gives $AUC=(100)(0.80)/4.0=20$ mg·h/L as well.

Thus this AUC alone distinguishes neither pair: it informs $CL/F$, not $F$ and $CL$ separately. Other data or assumptions may help separate them, but the algebraic relationship by itself does not. The combination $CL/F$ has physiological and pharmacokinetic meaning as apparent clearance after oral dosing.

Worked Example: Reading Strong and Weak Directions from a Sensitivity Matrix

Consider a hypothetical local sensitivity matrix for log-scale bioavailability and clearance coordinates, $(\eta_F,\eta_{CL})$, and two scaled outputs:

J= \begin{pmatrix} 1&-1\\ 0.1&0.1 \end{pmatrix}.

The first output responds to the log ratio $F/CL$; the second responds more weakly to a joint change in both parameters. Calculate the singular directions and interpret them.

First calculate

J^\mathsf{T}J= \begin{pmatrix} 1.01&-0.99\\ -0.99&1.01 \end{pmatrix}.

Its eigenvectors are $\mathbf{v}_1=(1,-1)^\mathsf{T}/\sqrt{2}$ and $\mathbf{v}_2=(1,1)^\mathsf{T}/\sqrt{2}$. Their eigenvalues are $2.00$ and $0.0200$, so the singular values are $\sqrt{2}=1.414$ and $\sqrt{0.0200}=0.1414$. The log-coordinate combinations are therefore $c_1=(\eta_F-\eta_{CL})/\sqrt{2}$, the relatively strong direction, and $c_2=(\eta_F+\eta_{CL})/\sqrt{2}$, the relatively weak direction.

For a displacement of $0.10$ along $\mathbf{v}_1$, the linearized output-change magnitude is $0.10(1.414)=0.1414$. Along $\mathbf{v}_2$, the same-sized displacement gives $0.10(0.1414)=0.01414$. The first direction changes the ratio $F/CL$ while keeping the log product approximately unchanged; the second changes the product while keeping the ratio approximately unchanged. The sensitivity matrix therefore supports a clearer local interpretation in terms of ratio and product than in terms of two separately ranked parameters. The numeric comparison applies only to these outputs, scales, and reference point.

Worked Example: An Effective Tissue Exchange Rate

Consider a hypothetical tissue compartment with volume $V_t=10$ L, permeability-surface-area product $PS$, partition coefficient $K_p$, arterial concentration $C_a$, and tissue concentration $C_t$. Suppose its exchange equation is

V_t\frac{dC_t}{dt}=PS\left(C_a-\frac{C_t}{K_p}\right).

During a period with $C_a=0$, the equation reduces to

\frac{dC_t}{dt}=-\frac{PS}{V_tK_p}C_t.

For $PS=2.0$ L/h and $K_p=4.0$, the decay rate is $2.0/(10\times4.0)=0.0500$ h$^{-1}$. The hypothetical alternative $PS=3.0$ L/h and $K_p=6.0$ gives the same rate, $3.0/(10\times6.0)=0.0500$ h$^{-1}$. Under this zero-input equation, the concentration time course is $C_t(t)=C_t(0)\exp[-PS\,t/(V_tK_p)]$. Data from that phase inform the effective rate $PS/(V_tK_p)$, but do not separate $PS$ from $K_p$.

This effective rate has units of inverse time and describes the local washout rate in this model. Data during nonzero arterial input can also inform the source coefficient $PS/V_t$; with sufficiently informative input and concentration observations, that additional behavior may help distinguish $PS$ and $K_p$. Whether it does so depends on the data and model, not merely on rewriting the equation.

Constructing and Interpreting Combinations

For positive parameters, log coordinates are often convenient because relative rather than absolute changes are compared. If a right singular vector has two entries with opposite signs, movement along it increases one parameter while decreasing the other, often suggesting a ratio. Entries with the same sign suggest a joint increase or decrease, often suggesting a product. For example, $\eta_1-\eta_2=\log[(\theta_1/\theta_{*,1})/(\theta_2/\theta_{*,2})]$ describes a relative change in a ratio. Similarly, $\eta_1+\eta_2$ describes a relative change in a product. The reference values make the log arguments dimensionless.

These interpretations are guides, not automatic naming rules. With three or more parameters, a singular vector may mix several parameters with unequal weights. A combination such as $a\eta_1+b\eta_2+c\eta_3$ may not reduce to a familiar physiological quantity. In that case, report the direction and its coefficients, inspect the model equations for a meaningful derived quantity, and state when no simple physiological interpretation is available.

A coordinate transformation should preserve the parameter domain and be invertible if the aim is only to express the same model in different coordinates. If only a subset of combinations is retained, information in discarded directions is being set aside; that is a dimension-reduction decision, not a harmless relabeling. A useful combination should also have units and a physiological interpretation that are explicit. Ratios of like quantities may be dimensionless, while effective clearances and rates retain their respective units.

Assumptions, Diagnostics, and Alternatives

The local sensitivity analysis assumes the model is differentiable near the reference parameter vector and that the selected outputs and their scales are appropriate to the question. The local linear approximation is most useful for sufficiently small parameter changes. It can fail when the model is strongly nonlinear over the relevant range, when parameters are near a boundary, or when changes alter the qualitative behavior of the model. Log coordinates require positive parameters; other parameter domains need coordinates that respect their constraints.

Before assigning physiological meaning to a direction, check the sensitivity calculation and the analysis choices. Derivatives should be stable to reasonable changes in finite-difference step size or numerical solver tolerance. Compare the direction across relevant outputs, time points, doses, and physiological scenarios: a direction weak for one endpoint may be informative for another. Inspect whether a small singular value persists under reasonable output scaling and reference values. A very small singular value can also result from redundant outputs, inadequate observation coverage, or numerical error. It is a diagnostic, not a stand-alone proof.

Reparameterization differs from two nearby strategies. Fixing a parameter removes it from estimation or variation; this may be appropriate for a low-influence parameter under a justified assumption, but it does not describe a trade-off between parameters. Lumping parameters replaces them with a shared or combined representation when their effects are similar; a useful reparameterization instead seeks coordinates aligned with informative and weak directions. Direct estimation in the original parameters retains their familiar names, but can obscure a narrow trade-off and lead to unstable individual estimates. None of these approaches supplies information absent from the observations.

A Common Pitfall

Common mistake: “The sensitivity analysis shows $F$ and $CL$ are both influential, so each is separately estimated well.” This does not follow. Two parameters can each have large individual sensitivities while changing together in a way that leaves outputs nearly unchanged. The correction is to examine joint directions: a weak singular direction or a near-constant ratio may reveal that the data constrain a combination rather than the individual parameters.

A second caution is to avoid interpreting a singular vector as a universal physiological law. Its meaning depends on the local reference point, the chosen outputs, and their scaling. Present the combination as a property of the model-data analysis being considered, and state what additional observations or assumptions would be needed to separate its components.

Check Your Understanding

Answer each question, then compare with the short solutions below.

If an oral AUC is proportional to $F/CL$, which combination remains constant when $F$ and $CL$ are both multiplied by the same positive factor?
For a sensitivity matrix, what does a small singular value indicate under the local linear approximation?
In the tissue washout example, what effective rate is informed by a zero arterial concentration phase, and what are its units?
Why should sensitivity directions be recalculated or checked when the output set or its scaling changes?

Solutions. (1) The ratio $F/CL$ is unchanged, so this AUC relationship predicts the same exposure. (2) A displacement along its corresponding parameter direction produces a relatively small change in the selected scaled outputs near the reference point; it does not prove global non-identifiability. (3) The rate is $PS/(V_tK_p)$, with units h$^{-1}$ when $PS$ is in L/h, $V_t$ in L, and $K_p$ is dimensionless. (4) The sensitivity matrix describes how those particular outputs, on those particular scales, respond to parameters. Changing the outputs or their weights changes the matrix and can change which directions appear informative.