Identifiability of Binding and Target-Turnover Parameters

Why identifiability matters in TMDD

A TMDD model can fit a drug concentration profile closely while leaving its mechanistic parameters poorly determined. For example, a rapid decline in free drug may reflect systemic elimination, target-mediated internalization, or a combination of binding and target abundance. A numerical estimate for each parameter does not by itself show that the data distinguish those explanations.

Identifiability asks what information the model outputs contain about its parameters. It is useful to separate two cases. Structural identifiability concerns whether ideal, continuous, noise-free observations under a specified model uniquely determine parameters. Practical identifiability concerns whether the actual finite, noisy observations constrain them precisely enough to estimate reliably. A parameter combination may be structurally identifiable in principle but practically weakly identified in a particular data set.

State equations and observable balances

Let \(D\) denote free drug concentration, \(R\) free target concentration, and \(C\) drug–target complex concentration. Use \(k_{\mathrm{on}}\) for the association rate constant, \(k_{\mathrm{off}}\) for dissociation, \(k_{\mathrm{int}}\) for complex internalization, \(k_{\mathrm{deg}}\) for free-target degradation, and \(k_{\mathrm{syn}}\) for target synthesis. Let \(k_{\mathrm{el}}\) be the first-order systemic elimination rate for free drug, and \(u(t)\) the drug input rate expressed as concentration per time. In this formulation, internalization removes both drug and target in the complex; free-target degradation acts on \(R\).

Here \(D,R,C\) have concentration units, \(k_{\mathrm{on}}\) has units concentration\(^{-1}\) time\(^{-1}\), the rate constants have units time\(^{-1}\), and \(k_{\mathrm{syn}}\) and \(u(t)\) have concentration-per-time units. The drug and target balances follow by adding the corresponding state equations. For total drug \(D+C\), the binding terms cancel, leaving systemic elimination of free drug and internalization of complex. For total target \(T=R+C\), complex formation and dissociation cancel, but free-target degradation and complex internalization remain.

These balances are central to identifiability. They also make clear why the free-target equation contains \(+k_{\mathrm{off}}C\), not \(+(k_{\mathrm{off}}+k_{\mathrm{int}})C\): dissociation returns target to the free state, whereas internalization removes the complex and its target. When there is no drug and no complex, the baseline target concentration is \(R_0=k_{\mathrm{syn}}/k_{\mathrm{deg}}\). Baseline alone therefore identifies that ratio, not necessarily the synthesis and degradation rates separately.

How sensitivities expose confounding

Suppose the observed model output at observation times is \(g(t_i;\theta)\), where \(\theta\) collects the unknown parameters. A local sensitivity matrix has entries \(S_{ij}=\partial g(t_i;\theta)/\partial\log(\theta_j)\). Using log parameters expresses a perturbation as a relative change. If two columns of this matrix are proportional, the corresponding parameter changes have nearly the same effect on the outputs. A direction \(v\) with \(Sv\) close to zero describes a parameter combination that can change substantially while changing predictions little.

The local justification is a first-order expansion: for a small vector \(\delta\) of changes in log parameters, the output change is approximately \(S\delta\). If the columns are linearly dependent, some nonzero \(\delta\) gives \(S\delta=0\); that parameter direction is invisible to first order. If the columns are merely close to dependent, the direction is weakly informed and estimates are sensitive to noise or modeling assumptions. This is a diagnostic for local identifiability, not proof of global uniqueness: separated parameter values can sometimes produce the same outputs even when local sensitivities have full rank.

Drug-only data: identifiable combinations at low dose

At low drug concentrations, suppose target is approximately at baseline, \(R\approx R_0\), and complex adjusts rapidly enough to be treated as quasi-steady. Then \(dC/dt\approx0\), so the complex equation gives \(C\approx qD\), where \(q=k_{\mathrm{on}}R_0/(k_{\mathrm{off}}+k_{\mathrm{int}})\). With constant input stopped after a dose, the total-drug balance implies \((1+q)dD/dt\approx-(k_{\mathrm{el}}+k_{\mathrm{int}}q)D\). Thus the free-drug decline rate is approximately \(\lambda=(k_{\mathrm{el}}+k_{\mathrm{int}}q)/(1+q)\). A terminal slope in this regime informs \(\lambda\), not each of its component parameters.

Drug-only profiles can offer more information than a single low-dose slope, particularly when the profile includes conditions where target depletion or saturation matters. Even then, parameter separation depends on how distinctly each parameter changes the observed drug trajectory. At high concentrations, target-mediated removal can be small relative to systemic elimination, making \(k_{\mathrm{el}}\) easier to distinguish while providing little information about binding. Conversely, a low-dose decline can be strongly influenced by a combined target-mediated contribution. The precise information depends on the measured drug quantity and the model assumptions; an apparent fit at one dose does not establish that all mechanisms are separately identified.

Joint target measurements can separate rates

Joint observations can add information that is absent from free drug alone. If free target \(R\) and total target \(T=R+C\) are both observed, then \(C=T-R\) is determined from those outputs. In ideal continuous data, the complex equation can be written as \(dC/dt=k_{\mathrm{on}}(DR)-(k_{\mathrm{off}}+k_{\mathrm{int}})C\). With enough variation in \(DR\) and \(C\), this equation can distinguish \(k_{\mathrm{on}}\) and the sum \(k_{\mathrm{off}}+k_{\mathrm{int}}\). It does not by itself divide that sum into its two components.

The total-target balance supplies a second relationship. If \(k_{\mathrm{syn}}\) and \(k_{\mathrm{deg}}\) are known, and \(C>0\), then \(k_{\mathrm{int}}=(k_{\mathrm{syn}}-k_{\mathrm{deg}}R-dT/dt)/C\). The complex equation can then yield \(k_{\mathrm{off}}\) by subtraction from the identified sum. If synthesis and degradation are unknown, target observations must also contain enough independent variation to distinguish their contributions from internalization. For example, the balance is a regression relationship with intercept \(k_{\mathrm{syn}}\) and terms in \(R\) and \(C\); those terms cannot be separated if their observed patterns are exactly dependent.

Measurements, alternatives, and failure modes

The data type determines which balance is available. Free-drug-only data do not directly reveal \(R\) or \(C\); joint free- and total-target measurements make \(C=T-R\) available, while a direct complex measurement could supply similar information. Total target alone does not generally reveal the split between free target and complex. These distinctions concern information in the outputs, not the choice of an estimation algorithm. A full TMDD model preserves the separate states and can use their different time courses; reduced approximations are useful when their assumptions hold, but often expose only composite quantities such as \(q\) or \(\lambda\). Neither representation guarantees that parameters are identifiable from the measurements at hand.

Several checks help reveal weak identification or model-data mismatch:

• Inspect sensitivity patterns or parameter profiles for broad, flat directions and strongly correlated parameter changes.
• Check whether the state combinations needed for separation actually vary. Nearly constant \(R\), for example, can make changes in binding and target abundance difficult to distinguish from drug-only observations.
• Compare estimates across plausible starting values and reasonable perturbations of the data. Large shifts with nearly unchanged predictions indicate weak information, not a uniquely determined mechanism.
• Check whether the inferred rates and predicted state trajectories satisfy the drug, complex, and total-target balances. A small residual in one observed output cannot compensate for an inconsistent balance elsewhere.

A common mistake is to treat a precise-looking optimizer estimate as evidence that every mechanistic parameter is identified. The correction is to ask which output changes when each parameter changes, and whether those changes differ from the effects of the other parameters. Another mistake is to infer \(k_{\mathrm{int}}\) from the complex equation alone: that equation identifies the sum \(k_{\mathrm{off}}+k_{\mathrm{int}}\) under suitable variation, not its components. The total-target balance or other independent information is needed to separate them.

Identifiability conclusions are conditional on the state definitions, observation model, and assumed turnover structure. For example, changing which target pool is degraded changes the target balance and therefore changes what can be inferred from target trajectories. Finite, noisy measurements also make derivatives such as \(dT/dt\) difficult to estimate directly; the balance derivations explain information content, but do not guarantee precise rate estimates in practice. In NLME applications, individual variability and residual error add further estimation considerations, but they do not remove a structural confounding built into the model-output relationship.