Sampling Designs That Separate TMDD Processes

Why Sampling Design Matters in TMDD

A TMDD model can describe several processes that all change measured drug concentrations: drug binds target, drug–target complex is internalized, and target is synthesized and removed. A concentration decline by itself may not reveal which process produced it. Sampling design should therefore give these mechanisms opportunities to make distinguishable predictions, rather than concentrating measurements in a time window where their effects look similar.

The earlier material on identifiability of binding, internalization, and target turnover motivates this design problem: even a correctly specified model may not support separate parameter estimates if the observations contain little information to distinguish the parameters. Here the practical question is where in time, at what exposure levels, and for which analytes to collect data. The schedule should be based on plausible parameter ranges and the intended assay, and should be refined for the route of administration and study constraints.

Mechanisms and Their Informative Time Windows

For a simple shared-compartment model, let \(C\) be free drug concentration, \(R\) free target concentration, and \(B\) drug–target complex concentration. Let drug enter at amount-per-time rate \(u(t)\) into a compartment of volume \(V\). Assume linear elimination of free drug with rate constant \(k_{\mathrm{el}}\), reversible binding with association rate constant \(k_{\mathrm{on}}\) and dissociation rate constant \(k_{\mathrm{off}}\), target synthesis at rate \(k_{\mathrm{syn}}\), free-target degradation at rate \(k_{\mathrm{deg}}\), and irreversible complex internalization at rate \(k_{\mathrm{int}}\). For clarity, assume no target recycling and no complex-mediated return of drug.

When free drug is approximately constant over a short interval, the binding complex has an approximate relaxation rate \(k_{\mathrm{on}}C+k_{\mathrm{off}}+k_{\mathrm{int}}\), conditional on treating \(C\) as fixed. If internalization is negligible during binding equilibration, the corresponding binding rate is approximately \(k_{\mathrm{on}}C+k_{\mathrm{off}}\). These are local timescale guides: in a full model, \(C\), \(R\), and \(B\) change together. Binding information is generally concentrated early, when drug and target first encounter one another and concentrations can change rapidly.

Internalization acts on complex over a longer interval when \(B\) persists. Its importance depends not only on \(k_{\mathrm{int}}\), but also on competition with dissociation: in this simple model, the probability that a complex is internalized rather than dissociating during one complex episode is \(k_{\mathrm{int}}/(k_{\mathrm{off}}+k_{\mathrm{int}})\). Thus, a late fall in drug concentration alone may be hard to attribute to internalization without early information on binding and measurements that constrain target or complex behavior.

In the absence of drug and complex, the target approaches its baseline \(R_0=k_{\mathrm{syn}}/k_{\mathrm{deg}}\), with recovery governed by \(k_{\mathrm{deg}}\). Consequently, target measurements after drug and complex have become negligible can reveal turnover on its own characteristic timescale. Drug troughs also matter: concentrations near the end of a dosing interval show whether target-mediated loss remains important at low exposure and whether the target has had time to recover.

Deriving a Practical Sampling Principle

A useful design principle follows from the shape of a simple target-recovery trajectory. Suppose drug and complex are negligible after time zero, and target concentration at that time is \(R_i\). The model reduces to \(dR/dt=k_{\mathrm{syn}}-k_{\mathrm{deg}}R\). Since \(R_0=k_{\mathrm{syn}}/k_{\mathrm{deg}}\), its solution is

If \(R_0\) and \(R_i\) are known for this calculation, the sensitivity of the predicted concentration to \(k_{\mathrm{deg}}\) is

The absolute sensitivity is proportional to \(t e^{-k_{\mathrm{deg}}t}\). Differentiating this factor gives \(e^{-k_{\mathrm{deg}}t}(1-k_{\mathrm{deg}}t)\), which is zero at \(t=1/k_{\mathrm{deg}}\), positive before that time, and negative after it. Thus, under these simplifying assumptions, a target sample near one turnover time can be more informative about \(k_{\mathrm{deg}}\) than a sample immediately after drug removal. This does not prescribe a universal time point: uncertainty in the rate, measurement error, and lingering drug or complex all affect the useful schedule.

The same logic supports spreading measurements across processes. Very early drug concentrations can reveal rapid concentration changes; intermediate concentrations can inform complex formation and loss; troughs describe end-of-interval behavior; and later target measurements can capture recovery. A schedule should include multiple windows rather than trying to infer all rates from one dense cluster.

Dose Levels and What to Measure

Time sampling alone is not enough if every dose produces nearly the same degree of target occupancy. A useful dose-ranging design includes exposures below, near, and above the affinity scale. Under a rapid-equilibrium approximation with total target \(T=R+B\), negligible internalization during equilibration, and fixed free drug \(C\), \(K_D=k_{\mathrm{off}}/k_{\mathrm{on}}\), and

This relation is a design guide, not a substitute for fitting a full dynamic model. It shows why observations only at \(C\gg K_D\) may be uninformative about the exact affinity: occupancy is close to one across a broad range of \(K_D\). Measurements spanning the transition around \(K_D\), alongside lower exposures, can better reveal the concentration dependence. Because TMDD makes dose-to-concentration relationships nonlinear, dose levels should be selected using plausible predicted exposures, not by assuming that doubling dose doubles concentration.

The analyte matters as much as the time. Free drug, total drug, free target, and total target do not provide interchangeable observations. Early free-drug measurements can show the changing concentration available for binding. Total drug may include drug in complex, depending on the assay. A target assay that measures only free target can reflect occupancy as well as changes in target abundance; a total-target measure can help distinguish target binding from target depletion if the assay and model definitions align. Collecting paired drug and target observations at selected times can be more informative than collecting many drug concentrations alone.

Building and Comparing Candidate Designs

A practical design can be assembled in stages. First, specify plausible ranges for binding, internalization, turnover, elimination, and absorption parameters. Second, translate those ranges into early, intermediate, trough, and recovery windows. Third, select dose levels whose predicted exposures cover a useful range of target occupancy. Finally, match each measurement to the process it can inform and check feasibility, assay sensitivity, and burden on participants.

A sparse design may be appropriate when burden is severe, but it risks missing rapid binding and offering too few points to distinguish curvature. A dense early PK design is useful for fast changes but may still provide little turnover information if target is never measured later. A dose-ranging design adds exposure contrast, while a repeated-dose design adds trough and recovery information; neither replaces the need for early observations. The strongest feasible design usually combines complementary windows and analytes rather than maximizing sample count in one window.

Design evaluation can formalize this comparison. For example, local sensitivity analysis asks how much predicted observations change when one parameter changes near a nominal value. If two parameters produce nearly proportional changes across all scheduled observations, their effects may be difficult to separate. A Fisher information calculation can summarize this local information, but its conclusions depend on the assumed model, parameter values, and observation-error model. It cannot rescue a design that omits the time window or analyte needed to reveal a mechanism.

Common Pitfalls and Limitations

• Sampling only at the peak and trough. Those points can characterize exposure endpoints but may miss the early binding transition and intermediate complex dynamics. Add early and intermediate observations when feasible.
• Using only high doses. Saturating exposure can obscure the concentration range where occupancy changes with drug. Include exposures near and below the plausible affinity scale when justified.
• Calling free-target decline target turnover. A free-target assay can fall because target is occupied, because target abundance changes, or both. Interpret it with the assay definition and, where possible, paired total-target or drug measurements.
• Assuming a late sample is automatically informative. If drug or complex persists, the target-recovery equation does not apply. Confirm that the proposed recovery window corresponds to low exposure under plausible model predictions.
• Overinterpreting a single schedule. Informative times depend on uncertain parameters, route, assay error, and between-subject variability. Consider a range of plausible values and adapt the schedule when early data update those ranges.

A common mistake is to argue, “Binding is rapid, so the first postdose sample is enough to estimate binding.” Rapid binding can instead occur largely before that sample, leaving only an apparent equilibrium or an early concentration change that is confounded with other processes. The correction is to place multiple samples as early as practical, ensure assay timing and absorption are understood, and recognize that if the transition is faster than feasible sampling, binding-rate constants may not be separately estimable even with a well-executed study.