Testing Alternative TMDD Mechanisms With Predictive Checks

Why Compare TMDD Mechanisms Predictively?

Two TMDD models can fit the same free-drug concentration curve while implying different fates for drug-bound target. In one mechanism, the drug-target complex is degraded and the target is lost with it. In another, the complex is internalized but the target is returned to the free-target pool. These mechanisms can have distinct implications for target recovery and pharmacology, even when the observed drug profiles are similar.

A predictive check asks whether each fitted model can generate observations resembling the data, at the actual sampling times and for the actual assays. It is not enough to compare smooth curves for free drug alone. The model should be checked against every measured quantity that bears on the mechanism: free drug, total drug, free or total target, and drug-target complex. A failure concentrated in target measurements can be more informative about target recycling than a small improvement in a drug-only fit.

Define the Competing Structures

Let \(D\) be free drug concentration, \(R\) free target concentration, and \(C\) drug-target complex concentration. Let \(I_D(t)\) denote drug input expressed as concentration per unit time. The association and dissociation rate constants are \(k_{\mathrm{on}}\) and \(k_{\mathrm{off}}\); \(k_{\mathrm{el}}\) is nonspecific free-drug elimination, \(k_{\mathrm{in}}\) is target production, \(k_{\mathrm{out}}\) is free-target loss, and \(k_{\mathrm{int}}\) is complex internalization or removal. Concentrations use consistent units, and rate constants have units consistent with those concentrations and time.

For the complex-degradation model, one minimal TMDD system is:

For the target-recycling model, the drug and complex equations have the same form, while the target equation includes return of target at internalization:

The equations make the structural difference explicit: the recycling model adds \(k_{\mathrm{int}}C\) to free target. In both systems, drug bound in complex is removed when complex is internalized. In practice, the assay observation model must also be specified. For example, if an assay reports total drug as drug equivalents and detects both free drug and circulating complex, its model prediction is \(D_{\mathrm{tot}}=D+C\). If an assay reports total target and detects free target plus circulating complex, the prediction is \(R_{\mathrm{tot}}=R+C\). Assays that detect only one form require the corresponding prediction, not an assumed total.

A fair comparison holds the dose history, covariates, sampling times, parameterization where appropriate, and observation models constant. Each mechanism must be fitted or otherwise assigned a defensible parameter distribution; predictive differences should not arise merely because one model was given more favorable measurement-error assumptions. Predictions should include relevant uncertainty in parameters, between-subject variation, and residual error.

Why Target Measurements Can Separate the Mechanisms

A short mass-balance derivation shows why target observations are especially useful. Define the circulating total target as \(T=R+C\). Add the free-target and complex equations. In the degradation model, the binding and dissociation terms cancel:

Thus, complex internalization is a loss route for circulating target. In the recycling model, the additional \(k_{\mathrm{int}}C\) in the free-target equation cancels the complex loss term:

The models therefore predict different target recovery when appreciable complex has formed. They need not predict equally distinct free-drug trajectories. Binding terms cancel from the derivative of circulating total drug \(D+C\) in both models, giving \(d(D+C)/dt=I_D(t)-k_{\mathrm{el}}D-k_{\mathrm{int}}C\). The drug profile can still differ indirectly because the target trajectory changes binding, but a drug-only comparison may provide weak evidence about target fate. These identities assume the stated compartments and complete recycling; delayed recycling, partial recycling, or additional target pools require different equations.

Run Checks on Observations, Not Just Curves

For each fitted mechanism, simulate replicate data at the same dose, covariates, sampling times, and assay types as the observed data. Each replicate should pass through the observation model, including measurement error and any relevant between-subject variability. Compare simulated and observed values using displays and summaries appropriate to the analyte: time-course overlays, predictive intervals, and patterns in the discrepancy across time and assay.

A simple descriptive discrepancy at one observation is \(z=(y-\mu_{\mathrm{pred}})/s_{\mathrm{pred}}\), where \(y\) is the observed value, and \(\mu_{\mathrm{pred}}\) and \(s_{\mathrm{pred}}\) are the mean and standard deviation of its predictive distribution. A large absolute value indicates that the observation is unusual under that predictive distribution if its location and scale are adequately summarized by the mean and standard deviation. It is not a universal pass/fail threshold: the distribution may be skewed, and many observations are being checked. Prefer empirical predictive intervals or simulation-based ranks when the predictive distribution is nonnormal.

Look for Joint, Time-Structured Evidence

A mechanism is more convincing when it accounts for a coherent pattern across related observations, not merely one favorable point. For example, a degradation model might predict adequate free-drug concentrations but repeatedly underpredict total target during recovery. A recycling model might reproduce the target profile but systematically miss complex concentrations. The second pattern is evidence against that model's adequacy even if its average target prediction is closer.

Before attributing a mismatch to mechanism, check whether it could arise from a different source: an assay that does not measure the assumed molecular forms, an inaccurate assay error model, an incorrect baseline target level, or a misspecified sampling or dosing record. Also check that the numerical solution is stable enough for the comparison. A predictive check cannot distinguish structural error from these other causes without additional information.

Alternatives, Limitations, and Practical Decisions

Predictive checks answer whether a fitted model reproduces relevant features of the observations. They complement, rather than replace, other comparisons. Likelihood-based criteria such as AIC summarize fit while penalizing model complexity under their assumptions; they do not show which analyte or time drives a mismatch. A residual-based diagnostic can expose patterns in fitted observations, while simulation-based checks make the variability expected under the full model more explicit. Mechanistic experiments or additional sampling can provide stronger discrimination when the existing data support similar predictions from both structures.

Predictive checks have important limits. If the target assay is absent, noisy, or unable to distinguish free from complexed target, the key mechanistic prediction may be weakly observed. If both models generate nearly identical predictive distributions over the observed dose range and times, no diagnostic can reliably choose between them from those data. Conversely, a better predictive fit does not prove that a model's internal biological story is correct; different mechanisms can be observationally equivalent.

When predictions overlap, report the lack of discrimination rather than selecting a mechanism from a small numerical advantage. Consider whether target and complex assays, later recovery samples, or dose conditions that create different amounts of complex would make the predictions separate. The useful design is the one that targets a predicted contrast while keeping assay measurement and observation models credible.

Common Pitfall

• Wrong reasoning: “The recycling model fits free drug slightly better, so target recycling has been demonstrated.”
• Correction: A small drug-only improvement is not direct evidence about target fate. Compare predictive distributions for the target and complex measurements as well, account for uncertainty and assay definitions, and state when the available observations do not discriminate the mechanisms.