Repeated Dosing and Target Recovery

Why Repeated Dosing Requires Two Clocks

After a dose, drug concentration changes as free drug is eliminated, binds target, and may be released from or removed with drug-target complex. Meanwhile, target is synthesized, turns over, and may be depleted through complex formation and internalization. Repeated dosing therefore involves at least two timescales: the time over which drug remains in the body and the time over which the target pool recovers. A concentration that changes from one dose to the next does not, by itself, reveal which process is responsible.

The useful distinction is between drug accumulation, meaning residual drug from earlier doses contributes to concentrations after later doses, and target depletion or recovery, meaning the abundance of available target changes over time. These processes can occur together, and either can affect observed pharmacodynamic response. The equations below use the full TMDD state variables: free drug \(D\), free target \(R\), and drug-target complex \(C\).

Repeated-Dose Full TMDD Model

The equations assume a well-mixed compartment, constant rate parameters, and the stated processes as the relevant routes of drug and target turnover. They describe concentrations between doses; for an intravenous bolus of amount \(Q\) into a compartment of volume \(V\), the dose produces the jump \(D(t^+)=D(t^-)+Q/V\), while \(R\) and \(C\) are continuous at the instant of dosing. A repeated schedule with interval \(\tau\) applies this jump at each dosing time \(t_j=j\tau\). Other routes require an absorption or input model rather than this instantaneous jump.

Without drug, the baseline target concentration is \(R_0=k_{\mathrm{syn}}/k_{\mathrm{deg}}\). During dosing, free target alone is not the whole target pool: total target is \(R_{\mathrm{tot}}=R+C\). Adding the target and complex equations gives

Binding and dissociation cancel in this total-pool balance because they redistribute target between free and complexed forms. Synthesis adds target, while degradation of free target and internalization of complex remove it. Consequently, a low free-target measurement may indicate substantial binding rather than depletion of the total target pool. Target depletion should be interpreted using the measured or modeled pool that matches the biological question.

Recovery Between Doses: A Useful Limiting Calculation

When drug and complex have become negligible after a dose, the target equation reduces to \(dR/dt=k_{\mathrm{syn}}-k_{\mathrm{deg}}R\). Solving this linear equation from a post-dose concentration \(R_{\mathrm{post}}\) gives

Thus recovery is governed by target turnover, not simply by the drug elimination half-life. If the target is at a fraction \(r\) of baseline immediately after the dose, its fraction at time \(\Delta\) is \(1-(1-r)e^{-k_{\mathrm{deg}}\Delta}\). This is a limiting approximation: while appreciable drug or complex remains, binding, dissociation, and complex internalization continue to affect the target trajectory.

Repeated Target Depletion and Interval Effects

A simple recurrence illustrates how incomplete recovery can accumulate across a schedule. Suppose, as an explicit approximation, each dose instantaneously removes the same fraction \(f\) of the free target present immediately before dosing. Between doses, assume the drug and complex become negligible and target recovers according to the preceding exponential solution. Let \(R_n\) denote free target immediately before dose \(n\), and let \(a=e^{-k_{\mathrm{deg}}\tau}\). The post-dose value is \((1-f)R_n\), so the next pre-dose value is

This recurrence is not a replacement for the full TMDD model. It is a transparent way to isolate the effect of target turnover and interval under the stated pulse-depletion approximation. If it reaches a stable repeating value \(R_*\), set \(R_{n+1}=R_n=R_*\) and solve:

For \(0\leq a<1\) and \(0\leq f\leq1\), the denominator is positive. The recurrence contracts deviations from its stable value by the factor \(a(1-f)\), whose magnitude is less than one, so it approaches that value under these assumptions. A shorter interval makes \(a\) larger: less recovery occurs before the next dose. This does not establish that a shorter interval always produces a particular exposure or response in a full TMDD system, because dose size and ongoing drug-target dynamics also matter.

Drug Accumulation Is a Separate Calculation

A common first approximation to drug accumulation is the linear one-compartment model with instantaneous doses, first-order elimination, constant volume, and no target-mediated disposition. Under those assumptions, the fraction of a concentration increment remaining one interval later is \(q=e^{-k_{\mathrm{el}}\tau}\). At periodic steady state, the concentration immediately after a dose is the sum of the current increment and all surviving earlier increments:

where \(\Delta C=Q/V\). The accumulation ratio relative to the concentration increment from one dose is therefore \(1/(1-e^{-k_{\mathrm{el}}\tau})\). This result is a benchmark, not a general repeated-dose TMDD formula. In TMDD, target binding and complex disposition can make drug clearance dependent on concentration and on the evolving target pool.

Reading Exposure and Response Together

Interpreting repeated-dose profiles requires tracking the states relevant to the observation. Free drug concentration, total drug concentration, free target, total target, and occupancy are not interchangeable. A response linked to occupancy can change because drug exposure changed, because target abundance changed, or because both changed. The introductory material on linking target occupancy to pharmacodynamic response establishes why response interpretation must follow the specified exposure-response relationship; the repeated-dose question adds time-varying target and drug states.

For example, lower free drug after a later dose does not prove that drug accumulation is absent: higher target availability can increase binding and target-mediated removal, while residual drug can still be present. Conversely, a higher trough concentration does not prove target recovery. The concentration profile and target trajectory must be evaluated jointly, with the observation model stated clearly. When the full model is used, numerical integration across every dose and the intervals between them preserves the changing state carried into the next dose.

Diagnostics, Alternatives, and Limitations

Useful checks start with model predictions at matched times in the dosing cycle. Compare pre-dose and post-dose free drug, free target, complex, and total target across successive cycles. If these states have not become periodic, a steady repeating profile has not yet been reached. Check whether the observed changes are consistent with the specified dose jumps, synthesis and turnover, and the model’s concentration definitions. Residual patterns or mismatches between free and total measurements may point to an inadequate observation model or missing disposition process, rather than proving target depletion.

The limiting exponential recovery calculation is appropriate only after drug-mediated processes are negligible and the free target returns toward a stable baseline under constant synthesis and degradation. The pulse-depletion recurrence adds the stronger assumption that every dose removes the same fraction of the target available before that dose. The linear accumulation ratio assumes no target-mediated clearance and first-order elimination. These are useful comparisons because each isolates a process, but none should be substituted for a full TMDD simulation when binding and complex persist through the interval.

A model can also be numerically plausible yet weakly supported by data. Sparse sampling may fail to distinguish rapid changes in drug concentration from slower target recovery; concentration assays may measure different combinations of free and bound species. Parameter uncertainty in target turnover or complex internalization can substantially alter predicted recovery. Treat numerical simulations as conditional on the model, parameters, dosing history, and measured species—not as direct evidence that one mechanism has been uniquely identified.