Initial Conditions Under Repeated Dosing

Why the Starting Effect-Site Concentration Matters

An effect compartment is a dynamic state: its concentration changes in response to plasma concentration and does not generally jump when plasma concentration changes abruptly. Under repeated dosing, that state can persist between doses. Beginning a simulation at a dose time with the effect-site concentration set to zero therefore makes a substantive assumption: it says there was no residual effect-site concentration at that time. If earlier doses have contributed drug to the effect site, the assumption can understate concentrations and distort predicted responses.

The appropriate initial value depends on the regimen and on the history before the simulation begins. It may be zero after sufficient drug-free time, a value carried forward from a previous interval, or the value implied by a periodic steady-state regimen. The calculations below use the first-order effect-compartment model and explicitly track that starting state.

Carry the State Across a Dosing Interval

For any known plasma concentration profile over an interval of length \(\tau\), the effect-site concentration at its end is

The first term is the surviving contribution from the starting effect-site concentration. The integral is the contribution from plasma exposure during the interval, weighted so that more recent exposure has had less time to equilibrate. This formula is the interval update: the end value becomes the starting value for the next interval. At a bolus, plasma concentration may jump, but the effect-site concentration remains continuous under this model.

A repeated regimen reaches a periodic effect-site state when the concentration at the same point in successive dosing intervals is equal. This is a fixed-point condition on the interval update, not an assumption that concentration is constant during the interval. For a simulation that has not yet reached that state, use the value inherited from the preceding interval instead.

Accumulation Across Infusion and Washout

The same state-carrying principle applies when plasma concentration changes in stages. For a constant plasma concentration \(C\) lasting \(T\) hours, solving the effect-compartment equation gives \(C_e(T)=C+[C_e(0)-C]e^{-k_{e0}T}\). During a subsequent period of zero plasma concentration lasting \(W\) hours, the effect-site concentration decays by the factor \(e^{-k_{e0}W}\). These expressions can be combined into a cycle update.

When Zero Is and Is Not Appropriate

A zero initial effect-site concentration is appropriate when the model’s assumed history supports it—for example, before any drug has been administered, or after a sufficiently long period with negligible prior input. It is not automatically appropriate just because the recorded data begin at a dose time. If earlier doses occurred, their contribution may remain even when plasma concentration has fallen substantially.

When plasma concentration is zero after a prior exposure, the effect-site state decays exponentially rather than disappearing instantly. If the state at the beginning of a drug-free period is \(C_{e,\mathrm{prior}}\), then after a washout of length \(w\),

Derivation and Practical Checks

The interval formula follows from a simple integrating-factor argument. Rearrange the model as \(\frac{dC_e}{dt}+k_{e0}C_e=k_{e0}C_p\), multiply by \(e^{k_{e0}t}\), and apply the product rule:

Integrate from \(t_0\) to \(t_0+\tau\), then multiply by \(e^{-k_{e0}(t_0+\tau)}\). The result is the state-update equation given earlier. For a constant input \(C_p=C\), the integral reduces to \(C(1-e^{-k_{e0}\tau})\), yielding \(C_e(t_0+\tau)=C+[C_e(t_0)-C]e^{-k_{e0}\tau}\). Thus the constant-input expression is a special case of the same update, not a separate accumulation rule.

In practice, check that the assumed concentration history includes doses before the observation window, that the starting state is defined at the same instant as the input profile, and that time units match those of \(k_{e0}\). Under a repeated, identical interval update \(C_{e,n+1}=rC_{e,n}+q\), where \(r=e^{-k_{e0}\tau}\) and \(q\) is the input contribution, the fixed point is \(C_e^*=q/(1-r)\) when \(0\le r<1\). The difference from that fixed point shrinks by \(r\) each interval. This fixed-point argument requires the same input contribution \(q\) each cycle; if plasma concentrations are still accumulating, the interval input is not yet identical.

A useful diagnostic is to compare the model’s predicted state at a cycle boundary with the state obtained by applying the interval update. A mismatch may indicate a calculation or implementation error, inconsistent boundary definitions, or an input profile that is not truly repeating. Large changes in predictions when the initial state is altered also signal that the observation window may be too short to make predictions insensitive to earlier exposure.

Alternatives, Assumptions, and Common Pitfalls

There are three common ways to specify the starting state. One can set \(C_e=0\) when no relevant earlier exposure is assumed; propagate the state forward from an earlier known or modeled value; or solve the repeated-cycle fixed-point equation when the regimen has reached periodic steady state. The fixed-point approach is convenient but is inappropriate if the regimen has just started, doses or intervals have changed, or plasma concentrations are not repeating. In those cases, propagate from the actual history instead.

The calculations assume a first-order, linear effect compartment with constant \(k_{e0}\), a specified plasma input, and no additional state changes outside the model. The effect-site concentration is often latent rather than directly observed, so its initial value may need to be based on dosing history or estimated with the model. Uncertain history or an uncertain initial state can therefore affect predictions. A more elaborate PK or PD model may be needed if the assumed plasma profile or effect-site dynamics do not adequately represent the system; changing the initial value alone does not repair a misspecified model.

• Mistake: reset \(C_e\) to zero at every dose. A bolus changes plasma concentration, not the continuous effect-site state in this model. Carry \(C_e\) forward across the dose.
• Mistake: assume a repeated dose automatically means effect-site steady state. A regimen may repeat before the state has converged. Use the actual prior intervals or verify the fixed-point condition.
• Mistake: equate low plasma concentration with zero effect-site concentration. The effect-site state has its own decay, so residual concentration can remain after plasma concentration becomes small.