Assessing Identifiability of CYP DDI Parameters

Why Identifiability Matters

A CYP drug-drug interaction (DDI) model can reproduce victim-drug concentrations well while leaving important parameters uncertain. For example, a larger estimate of victim fraction metabolized (\(f_m\)) may be offset by a different CYP abundance scaling factor (\(A\)); a change in perpetrator potency may then compensate for either. The resulting predictions can look stable over the observations used for fitting, even though the individual parameter estimates are not uniquely determined.

Identifiability asks whether the available observations distinguish the values of the parameters in a model. It is not the same as asking whether the model fits well. A good fit describes agreement between predictions and data; identifiability concerns whether substantially different parameter values would also fit those data.

Definition: A parameter is structurally identifiable if ideal, noise-free observations under the specified model and design uniquely determine its value. It is practically identifiable if the actual observations, with their finite sampling and measurement error, constrain its value sufficiently precisely. Parameters can be structurally identifiable but practically weakly identified.

The diagnostics here apply to fitted PBPK models, but their interpretation depends on the model and observations. To make the logic explicit, consider a simplified clearance model. Let \(A\) be dimensionless CYP abundance relative to a reference value, \(f_m\) the fraction of baseline victim clearance assigned to the CYP pathway, \(I\) the perpetrator concentration at the enzyme site, and \(q\) an effective potency with units inverse to those of \(I\). Assume reversible inhibition, so the CYP pathway is multiplied by \(1/(1+qI)\). A normalized clearance prediction is

y(I)=1-f_m+\frac{A f_m}{1+qI}.

This illustration assumes that the non-CYP component is unchanged by the perpetrator, that \(I\) is known at the relevant enzyme site, and that the inhibition function is appropriate. It is not a substitute for the full PBPK model: in a fitted model, concentrations at enzyme sites and the functional form of CYP effects may themselves depend on model structure. The example isolates how abundance, pathway contribution, and potency can trade off.

Three Complementary Diagnostics

No single diagnostic establishes identifiability. Parameter correlations summarize local covariation near an optimum; profile likelihood examines fit as one parameter is varied and nuisance parameters are re-estimated; alternative parameter sets make possible trade-offs concrete. Agreement among these views is more informative than any one result.

Interpretation: A high absolute parameter correlation, a broad or irregular profile likelihood, or very different parameter sets with nearly equal fit are warnings that the data do not sharply distinguish those parameters. None alone proves that a parameter is structurally unidentifiable.

A local parameter-correlation matrix is usually derived from the curvature of the objective function near the fitted values. A correlation near \(+1\) or \(-1\) indicates that small changes in two estimates tend to occur together in the local approximation. The sign indicates direction: negative correlation means that increasing one parameter is locally associated with decreasing the other. This matrix is a diagnostic, not a direct measurement of truth or evidence that the model is wrong. It may also miss nonlinear trade-offs, multiple minima, or uncertainty created by parameter boundaries.

A profile likelihood focuses on one parameter while allowing the others to adjust. For each fixed value of the parameter of interest, optimize the remaining parameters and record the best achievable likelihood (or objective value). A flat profile means a range of values can fit nearly as well. An asymmetric profile indicates that a symmetric standard error may be misleading; a boundary or multiple minima requires particular care.

Worked Example: Finding a Confounded Parameter Ridge

Suppose hypothetical clearance observations are summarized by the simplified model above. At baseline, \(I=0\), and during one perpetrator condition, \(I=2\) concentration units. The observed normalized clearances are \(y(0)=1.2\) and \(y(2)=0.9\). Find two different parameter sets that reproduce both observations.

At baseline, the model gives \(y(0)=1-f_m+A f_m\). Thus the baseline observation requires

$$ 1+f_m(A-1)=1.2. $$

For \(f_m=0.5\), this gives \(A=1.4\). At \(I=2\), the CYP contribution must equal \(y(2)-(1-f_m)=0.9-0.5=0.4\). Therefore \(0.7/(1+2q)=0.4\), which gives \(q=0.375\) per concentration unit.

A second choice is \(f_m=0.8\). The baseline equation then gives \(A=1.25\). For the interaction observation, \(1-f_m=0.2\), so \(1/(1+2q)=0.7\); hence \(q=0.2143\) per concentration unit. Both sets predict \(y(0)=1.2\) and \(y(2)=0.9\), despite having different abundance, fraction metabolized, and potency estimates.

These are exact alternative fits to the two observations, not a claim that either parameter set is the true one. The observations constrain combinations of the parameters, rather than uniquely fixing all three. An additional interaction condition could help distinguish the sets, but if its prediction difference is smaller than the observation error, practical identifiability may still be weak.

Why the Trade-Off Occurs

The local sensitivity of a prediction to a parameter indicates how much the prediction changes for a small parameter change. For the illustrative model, let \(D=1+qI\). Its partial derivatives are

\frac{\partial y}{\partial f_m}=-1+\frac{A}{D}, \qquad \frac{\partial y}{\partial A}=\frac{f_m}{D}, \qquad \frac{\partial y}{\partial q}=-\frac{A f_m I}{D^2}.

At baseline, \(I=0\), so the prediction has no sensitivity to \(q\). Baseline data alone cannot estimate potency in this model. For one baseline and one interaction observation, the sensitivity matrix has two rows and three columns. Its rank cannot exceed two, so it cannot locally distinguish all three parameters. Additional observations may increase the rank, but only if they provide sufficiently different sensitivities.

This is a local justification for examining both the number and the information content of observations. Repeated measurements at nearly identical perpetrator concentrations can improve precision for the predictions at that condition without adding much ability to separate parameters. Conditions that change the model’s sensitivity to \(f_m\), \(A\), and \(q\) differently are more useful for distinguishing their effects.

Worked Example: Interpreting a Parameter Correlation

Suppose a hypothetical fit reports a local correlation of \(-0.98\) between \(f_m\) and \(q\). For illustration, assume their local standard errors are \(0.12\) and \(0.10\) per concentration unit, respectively. The local covariance is

\operatorname{Cov}(f_m,q) =(-0.98)(0.12)(0.10)=-0.01176.

The negative sign means that nearby fits tend to pair a larger \(f_m\) with a smaller \(q\), or the reverse. It does not mean that increasing \(f_m\) causes potency to decrease. The magnitude, close to one, signals a strong local trade-off and cautions against interpreting either estimate in isolation.

A correlation matrix alone does not show how far the trade-off extends or whether it bends. A profile likelihood or explicit alternative fits can test that. Also check whether the correlation is calculated on the scale used for estimation: correlations can look different after parameter transformations, particularly for positive parameters such as potency.

Profiling Potency

For a likelihood-based fit, a one-parameter profile fixes \(q\) at a chosen value and maximizes the likelihood over the nuisance parameters, here including \(A\) and \(f_m\). If independent observations have normally distributed errors with known standard deviation \(\sigma\), maximizing the likelihood is equivalent to minimizing the sum of squared residuals, \(SSE\). The likelihood-ratio statistic at a fixed \(q\), compared with the best fit, is

\Delta \chi^2(q)=\frac{SSE(q)-SSE_{\min}}{\sigma^2}.

Under regular conditions and for a single profiled parameter, an approximate 95% likelihood interval includes values for which \(\Delta\chi^2\leq 3.84\). This cutoff is a large-sample likelihood-ratio result, not a guarantee that the approximation is accurate in every PBPK fit. Boundaries, strong nonlinearity, sparse data, and model misspecification can affect its calibration.

Worked Example: Calculating a Profile-Likelihood Interval

Consider hypothetical profile results for potency \(q\), with known residual standard deviation \(\sigma=0.02\) normalized-clearance units. After re-optimizing the other parameters at each fixed \(q\), the best fit has \(SSE_{\min}=0.0016\). At \(q=0.45\), suppose the profiled \(SSE\) is \(0.0024\). Then

\Delta \chi^2(0.45) =\frac{0.0024-0.0016}{0.02^2} =\frac{0.0008}{0.0004}=2.00.

Because \(2.00<3.84\), this value is inside the approximate 95% profile interval. If the profiled \(SSE\) at \(q=0.40\) is \(0.0032\), then \(\Delta\chi^2(0.40)=4.00\), placing that value just outside the interval by this criterion. A scan across \(q\), with nuisance parameters re-optimized at every point, is needed to locate both interval limits; two evaluated points do not determine the complete interval.

For example, if the full scan crosses the cutoff near \(q=0.41\) and \(q=0.74\), the approximate interval is \(0.41\) to \(0.74\) per concentration unit. This relatively broad range indicates that the data constrain potency only to that range under the assumed model and error specification. It is not a 95% probability statement about a fixed parameter’s location.

Diagnostics, Alternatives, and Limitations

When a parameter appears weakly identified, inspect the fitted predictions as well as the parameter summaries. A model may predict observed victim profiles closely across the available conditions while giving unstable \(A\), \(f_m\), or \(q\) estimates. Look for a nearly singular covariance or sensitivity matrix, large parameter correlations, profiles that remain shallow over a wide range, multiple profile minima, or estimates that repeatedly reach imposed bounds. These patterns can also arise from numerical scaling or optimizer failure, so assess convergence and repeat optimization from different starting values before interpreting them as scientific evidence.

The three diagnostics answer related but different questions. A correlation matrix is fast and useful for spotting local trade-offs, but relies on a local approximation. Profile likelihood accommodates nonlinear adjustment of nuisance parameters and can reveal asymmetry, but requires repeated stable optimization and depends on the assumed likelihood. Alternative parameter sets give a direct demonstration that predictions can be similar under different estimates, but a few examples do not establish the full shape of uncertainty. Comparing all three is more robust than treating any one as conclusive.

Identifiability is conditional on the model, observations, and error assumptions. Incorrectly fixed inputs, a misspecified inhibition function, uncertain enzyme-site concentrations, or unrepresented variability can make an apparently precise estimate misleading. Conversely, weak identification of individual parameters does not automatically invalidate predictions for the conditions observed: the poorly identified combinations may still yield similar predictions there. Predictions outside those conditions may be more sensitive to the unresolved trade-off.

Common pitfall: “The optimizer converged, the standard errors are finite, and the fit is good, so each CYP parameter is identified.” This reasoning confuses numerical convergence and fit quality with information in the data. Check correlations, profiles, and competing fits; report uncertainty and avoid interpreting a single point estimate as uniquely supported when substantially different parameter sets fit nearly as well.

Check Your Understanding

Answer each question using the definitions and simplified model above. Brief solutions follow the questions.

In \(y(I)=1-f_m+A f_m/(1+qI)\), which parameter has no effect on the baseline prediction when \(I=0\)?
Why can two observations not locally identify all three parameters \(A\), \(f_m\), and \(q\) in this model?
A profile point has \(SSE(q)=0.0040\), \(SSE_{\min}=0.0030\), and known \(\sigma=0.01\). Calculate \(\Delta\chi^2(q)\) and determine whether it is inside the approximate 95% profile interval.
What does a correlation of \(-0.95\) between two estimates indicate, and what does it not establish?
Name one warning sign of practical non-identifiability and one reason to verify an apparent warning before drawing a scientific conclusion.

Solutions. (1) \(q\), because its derivative is zero at \(I=0\). (2) The sensitivity matrix has two rows and three columns, so its rank is at most two and cannot locally distinguish three parameters. (3) \(\Delta\chi^2=(0.0040-0.0030)/0.01^2=0.0010/0.0001=10.0\), which exceeds 3.84 and is outside the approximate interval. (4) It indicates strong local negative covariation: nearby fits tend to increase one estimate as the other decreases. It does not establish causation, structural non-identifiability, or that either point estimate is wrong. (5) A broad profile, high absolute correlation, or materially different parameter sets with similar fit are warnings. Check optimization stability and scaling, because numerical problems can imitate weak information in the data.