Repeated Doses and Turnover-Rate Identifiability

Why Repeated Doses Can Reveal Turnover

A single response measurement after a dose can be compatible with several combinations of turnover speed and drug-effect magnitude. Repeated dosing adds a time pattern: the response rises or falls during exposure, then may recover between doses. The accumulation of successive response changes and the shape of each cycle can help distinguish a slowly turning-over response with a moderate drug effect from a rapidly turning-over response with a stronger effect.

The distinction is not automatic. Repeated doses are informative only when the exposure history and sampling times are known well enough, the response model is appropriate, and the data contain measurable changes across the dosing cycle. The calculations below use an inhibitory production model to make these conditions explicit. The same design logic applies more broadly, but the equations depend on the chosen indirect-response mechanism.

The model says that the deficit moves toward the current inhibition level at a rate set by \(k_{out}\). When inhibition is constant, the deficit approaches that level exponentially. When inhibition ends, the deficit decays toward zero. This simple relationship is useful for identifiability reasoning even when the actual inhibition varies with concentration; in that case, the exposure-driven effect must be represented rather than assumed constant.

Assumptions Behind the Cycle Calculations

For the closed-form repeated-cycle results, suppose each dosing cycle has an exposure period of length \(\tau_{on}\) with constant inhibition \(I\), followed by an effect-free period of length \(\tau_{off}\). Assume \(k_{out}\) is constant, the response is observed without measurement error for the calculations, and the system has reached a repeating steady cycle: the response at a given phase is the same from one cycle to the next. In practical data, these are modeling assumptions to assess, not facts guaranteed by repeated dosing.

Constant inhibition is an approximation to a concentration-varying drug effect. It may be reasonable if concentration changes little over the period being represented or if the model uses an appropriate average effect. If drug concentrations remain pharmacologically active during the nominal off-period, that period is not effect-free, and the simple off-period decay formula does not apply. The effect compartment or concentration-effect relationship must then be accounted for in the model.

Here, “peak” and “trough” refer to the deficit, not necessarily the response: a larger deficit corresponds to a lower response. With inhibitory production, the deficit generally rises during exposure and falls during recovery. A different indirect mechanism can reverse or alter this pattern, so the sign and interpretation of the measured response must match the model.

Deriving the Repeating Peak and Trough

The periodic formulas follow by solving the two within-cycle equations and imposing the repeating-cycle condition. Define \(a=e^{-k_{out}\tau_{on}}\) and \(b=e^{-k_{out}\tau_{off}}\). At periodic steady state, the trough before a dose equals the trough after the preceding cycle, so \(z_T^{next}=z_T\). The off-period equation gives \(z_T=bz_P\). Substituting this into the on-period equation gives \(z_P=I+(bz_P-I)a\), or \(z_P(1-ab)=I(1-a)\). Therefore:

These expressions show two different roles. The turnover rate governs how quickly the response moves during both phases, while the inhibition magnitude sets the level toward which the deficit moves during the exposure phase. Their effects are intertwined in a single peak measurement, but the peak-to-trough pattern and the timing between samples add information about the rate.

Using Repeated Cycles to Estimate Both Parameters

If the trough and peak are measured in a periodic cycle, the effect-free interval gives \(b=z_T/z_P\). Therefore \(k_{out}=-\ln(b)/\tau_{off}\). Once \(k_{out}\) is estimated, the exposure-period equation can be rearranged to estimate \(I\). The trough-to-peak decay identifies the rate independently of the inhibition magnitude under the stated assumptions; the rise during exposure then informs the magnitude.

Why Dosing Interval Changes the Information

The dosing interval affects both accumulation and the visibility of recovery. If doses arrive before much turnover occurs, the deficit may accumulate from cycle to cycle. At periodic steady state, the trough remains elevated, and the peak-to-trough contrast can be small. A longer effect-free interval can reveal more of the decay and make the rate easier to assess, provided the deficit remains measurable. Conversely, an interval much longer than the response time may allow near-complete recovery and produce little accumulation.

Identifiability, Diagnostics, and Limitations

Structural identifiability asks whether distinct parameter values can generate exactly the same ideal, noise-free observations under a specified design and model. Practical identifiability asks whether the available, noisy observations constrain the parameters precisely enough to be useful. Repeated cycles can improve practical identifiability by supplying multiple observations at informative phases, but collecting more doses at the same uninformative phase does not necessarily resolve a parameter confounding.

A single early response point can illustrate the ambiguity. Starting at baseline with constant inhibition, \(z(t)=I(1-e^{-k_{out}t})\). If a hypothetical deficit of \(0.120\) is observed two hours after exposure begins, then \(k_{out}=0.100\ \mathrm{h}^{-1}\) implies \(I=0.120/(1-e^{-0.200})\approx0.662\). But \(k_{out}=0.200\ \mathrm{h}^{-1}\) implies \(I=0.120/(1-e^{-0.400})\approx0.364\). Both parameter pairs reproduce that single observation. Measurements across the cycle are needed to tell whether the response changes slowly or the drug effect is simply smaller.

Several diagnostics indicate when the simple repeated-cycle interpretation may be failing:

• No stable cycle: successive peaks or troughs continue to drift, so the periodic steady-state equations are not appropriate. Transient accumulation may still contain information, but it must be modeled from the actual initial condition.
• Unexpected off-period shape: the deficit does not follow an approximately exponential decay when effect is assumed absent. Residual drug effect, concentration variation, a changing baseline, or a misspecified response model may be responsible.
• Weak peak–trough contrast: if the change is small relative to measurement variability, the ratio \(z_T/z_P\) is unstable. The resulting rate estimate can be imprecise even when the equations are structurally informative.
• Near-zero deficit measurements: taking a logarithm of a ratio requires positive measurements. Values near zero, or values that cross the baseline, make the ratio calculation unstable or incompatible with the assumed inhibitory-deficit model.
• Inconsistent estimates across cycles: materially different off-period rate estimates may indicate noise, changing exposure, incomplete periodicity, or failure of the constant-rate assumption.

A common mistake is to treat a large accumulated deficit under a short dosing interval as proof of a large drug effect. Accumulation depends on both effect magnitude and turnover: deficits can remain high because the response recovers slowly, even if the inhibition is moderate. The correction is to use timed changes within the cycle, especially a well-characterized recovery interval, to constrain \(k_{out}\) before interpreting the exposure-phase change as an estimate of effect magnitude.

An alternative is to estimate turnover from a separate washout or single-dose profile, then use repeated-dose data mainly to estimate the effect. That can be useful when the washout period is well observed and truly effect-free. Repeated-dose designs have the advantage of testing whether the inferred rate and effect also describe accumulation and recurring response patterns. Their limitations include dependence on exposure characterization, the assumed indirect-response mechanism, and sufficient observations at both exposure and recovery phases. If inhibition is concentration-dependent rather than constant, the full time-varying effect should be used; the formulas here then serve as design intuition, not a substitute for the appropriate model.