Diagnosing Uptake and Metabolism Parameter Identifiability

Why Identifiability Matters

A plasma concentration–time curve can be well described by a hepatic model without uniquely determining every clearance inside that model. In particular, uptake into hepatocytes and metabolism within hepatocytes can have compensating effects on the predicted plasma concentrations. A fitted curve may therefore support a precise prediction of overall disposition while providing weak evidence about the individual uptake and metabolic clearances.

The practical question is not simply whether an optimizer returns estimates for both parameters. It is whether materially different pairs of uptake and metabolism clearances also fit the available plasma data nearly as well. Profile likelihoods and parameter correlations help answer that question. They diagnose how much the data distinguish the two parameters; they do not by themselves establish which biological mechanism is correct.

Here, \(CL_u\) denotes hepatic uptake clearance and \(CL_{met}\) denotes intracellular metabolic clearance. Both are positive and are measured in L/h in the examples. Plasma concentrations are observations of a model output, not direct measurements of either clearance. The earlier material on hepatic transport and the well-stirred liver establishes how these processes can be represented in a PBPK model; this tutorial concerns whether plasma observations distinguish their parameter values.

Model and Likelihood Assumptions

Suppose a deterministic PBPK model predicts plasma concentrations \(C_i(\theta)\) at sampling times \(t_i\), where \(\theta\) includes \(CL_u\), \(CL_{met}\), and any other fitted parameters. For clarity, consider a lognormal observation model:

This likelihood assumes that the model predictions are appropriate for the observation times, that log-scale residuals are independent and normally distributed with a common variance, and that the residual variation is represented adequately by \(\sigma\). A real analysis may need a different error model, such as time-varying variance or correlated residuals. Identifiability conclusions depend on the model, the observation design, the error assumptions and which other parameters are estimated.

A profile likelihood for \(CL_u\) fixes \(CL_u\) at a chosen value and maximizes the likelihood over \(CL_{met}\) and all other nuisance parameters. This asks how well the data can fit at each fixed uptake clearance when the remaining parameters are allowed to adjust. The profile for \(CL_{met}\) is defined analogously. For a single parameter, a common approximate 95% likelihood-based interval contains values satisfying \(-2\log L_{prof}(\theta)+2\log L_{max}\leq 3.84\). This cutoff relies on regular likelihood conditions and a sufficiently informative fit; it is not a universal guarantee, especially near boundaries or under severe non-identifiability.

How Parameter Correlation and Sensitivity Reveal a Ridge

A local sensitivity describes how a model output changes for a small parameter change near a fitted value. For positive clearances, log parameters are often useful: changing \(\log CL_u\) by a small amount corresponds approximately to a proportional change in \(CL_u\). Let \(J_u\) and \(J_{met}\) be the vectors of derivatives of the log plasma predictions with respect to \(\log CL_u\) and \(\log CL_{met}\), respectively.

If these vectors are nearly opposite, a small increase in both log clearances can leave the sampled predictions almost unchanged. The two parameter estimates can then move together along a likelihood ridge. This means a positive correlation between their estimates is plausible. Conversely, nearly aligned sensitivity vectors may allow opposing parameter changes to compensate. The direction of compensation must be determined from the sensitivities; it cannot be inferred just from the names of the parameters.

In a local weighted least-squares or likelihood approximation, information about the parameters is related to \(J^\mathsf{T}WJ\), where \(W\) weights observations according to their precision. If sensitivity columns are nearly dependent, this matrix has a small eigenvalue: there is a parameter-change direction that has little effect on the fitted outputs. The resulting uncertainty is large in that direction. A correlation coefficient summarizes the local shape of the fitted uncertainty, but a single coefficient cannot describe a curved or asymmetric likelihood ridge.

Reading a Profile Likelihood

A likelihood profile is more informative than a point estimate alone. A narrow, well-defined minimum suggests that the data constrain the profiled parameter within the fitted model. A broad, flat region means many values can fit after the other parameters are re-optimized. An open profile that remains acceptable toward a parameter bound means the data may not provide a finite interval on that side. The nuisance parameter estimates along the profile are also important: they show whether changes in one clearance are being offset by changes in the other.

Diagnostics, Alternatives and Limitations

Profile likelihood, a local covariance matrix, and repeated fits from different starting values answer related but distinct questions. The local covariance approximation is efficient when the likelihood is well approximated by a quadratic near a clear interior maximum. It can be misleading for a long, curved ridge or a boundary estimate. A profile exposes asymmetry and re-optimizes nuisance parameters, but it can be computationally demanding and still depends on correct optimization at each fixed value. Multiple starting values help detect optimizer failures or competing maxima; they do not prove identifiability. Resampling can assess estimation variability under a chosen design and model, but it cannot repair a structural lack of information.

Useful warning signs include a nearly singular information matrix, large standard errors, strong parameter correlation, very different estimates from different initial values, and profiles that are flat or remain acceptable at imposed bounds. Check that each fixed-parameter fit has converged, inspect residuals for systematic patterns, and verify that parameter bounds are not creating an artificial profile endpoint. If an optimizer reports a single precise estimate while the profile is broad, the point estimate should not be interpreted as strong evidence about the individual clearance.

A common remedy for weak practical identifiability is to add observations that carry different information, such as an independent uptake assay or a distinct sampling design. The added information must be linked to the model with its own defensible measurement assumptions. A second output is not automatically informative: if it responds to the two clearances in the same relative way as the plasma data, it may leave the ridge largely unchanged. Identifiability is also conditional on the model structure. A narrow profile under a misspecified model does not establish that the clearance estimates are biologically correct.

Common Pitfall

A frequent error is to infer the compensating direction from verbal intuition, for example, “more uptake must be offset by less metabolism.” That conclusion is not generally justified: the signs and magnitudes of the model sensitivities determine how predictions change. In the sensitivity example above, uptake and metabolism sensitivities had opposite signs. Increasing one while decreasing the other reinforced prediction changes; increasing both produced near-cancellation. Always calculate or inspect the relevant sensitivity directions and verify them against profiled parameter pairs.