Updating Pharmacokinetic States Across Dose Events

Why State Updates Must Follow the Events

When dose times are irregular, an amount cannot be advanced by repeatedly applying a formula that assumes one fixed dosing interval. Instead, divide the timeline at each dose or infusion event. Within each interval, the input and model parameters must be constant for the exact one-compartment update below to apply. At an event, update the state or input as appropriate, then propagate over the next interval, whose duration may be different.

This tutorial uses a one-compartment model with first-order elimination to make the bookkeeping explicit. The same event logic is useful in more complex NLME models, although their state equations may need numerical integration rather than the exact formula derived here. Earlier material on Event-Based Encoding of Irregular Dosing Histories motivates representing dose records as events; here the focus is how those events change the model state.

For an individual in an NLME model, \(CL\), \(V\), and therefore \(k\) may depend on that individual's parameters. Conditional on those parameter values, the update calculations below are deterministic. The state is the amount carried forward from one interval to the next; an infusion start or stop changes the input rate, not the amount itself.

The Exact Update Over One Interval

Suppose an interval begins at time \(t_j\) with amount \(A_j^+\), and ends after \(\Delta_j\) hours. The superscript \(+\) denotes the amount just after any dose applied at the beginning of the interval. Assume \(k\) and the infusion rate \(R_j\) remain constant during the interval. Solving the differential equation gives the amount just before the next event:

Here \(A_{j+1}^-\) is the amount immediately before the event at the interval end. The first term is the surviving amount present at the interval start. The second is the amount contributed by infusion during the interval, accounting for elimination while drug enters. If there is no infusion, \(R_j=0\) and the update reduces to exponential elimination.

The \(+\) and \(-\) superscripts distinguish the state on the two sides of an event; they do not indicate positive or negative amounts. For an interval with both a bolus at its start and an infusion running, first use the post-bolus amount as the initial state and include the infusion contribution in the interval formula.

Why the Formula Works

To justify the update, multiply the differential equation by the integrating factor \(e^{kt}\). The product rule gives \(\frac{d}{dt}(e^{kt}A(t))=Re^{kt}\), since \(R\) is constant on the interval. Integrating from \(t_j\) to \(t_j+\Delta_j\) yields:

Dividing by \(e^{k(t_j+\Delta_j)}\) and rearranging gives \(A(t_j+\Delta_j)=A(t_j)e^{-k\Delta_j}+\frac{R}{k}(1-e^{-k\Delta_j})\). This derivation requires constant \(R\) and \(k\) on the interval. A change in either means the interval must be split at that change, or the state equation must be solved using a method that accommodates time-varying inputs or parameters.

Worked Event-by-Event Calculations

Assumptions, Diagnostics, and Alternatives

The exact interval formula relies on a one-compartment amount state, first-order elimination, constant \(k\), and constant infusion rate within each interval. Bolus doses are treated as instantaneous intravenous inputs with full dose entering the compartment. These assumptions are appropriate for illustrating state updates, but are not universal pharmacokinetic assumptions. For oral dosing, for example, a dose generally enters through an absorption process rather than as an instantaneous central-compartment jump.

A useful audit is to check the state immediately before and after each event and verify that the interval duration is the difference between the actual event times. With nonnegative initial amount and infusion rate, the formula should give a nonnegative amount. When \(R=0\), it should yield a decline for \(k>0\); with \(R>0\), the amount during a sufficiently long infusion approaches \(R/k\), not an indefinitely increasing amount. These checks can reveal a wrong interval length, a missed event, or use of the wrong infusion rate.

A common mistake is to treat an infusion stop like a negative bolus, subtracting the amount infused or resetting the amount to zero. The correction is that the infusion stop changes only \(R\); drug in the compartment continues to be eliminated. Another mistake is to apply \(R\Delta\) as the full amount contributed by an infusion while ignoring elimination. The exact contribution is \((R/k)(1-e^{-k\Delta})\), which is less than \(R\Delta\) when \(k,\Delta>0\).

For this linear one-compartment case, exact event-by-event propagation avoids numerical integration error within each constant-input interval. Numerical ODE integration is the main alternative when the state equations are nonlinear, compartments interact in ways without a closed-form update, or parameters and inputs vary continuously. Integration must still respect discontinuous dose events by carrying the state across them and applying the event-specific change. A steady-state or repeated equal-interval formula is not a substitute for this calculation when intervals differ. The limitation of exact propagation is that its convenience depends on the model and the within-interval constancy assumptions.