Encoding Dose Histories and Accumulation

Why Dose History Is Part of the Model

A simulated concentration at a given time depends not only on the dose administered at that time, but also on doses that came before it. Those earlier doses may still contribute drug to the body. If a simulation begins with a concentration of zero when the person has actually received prior doses, the resulting profile can underestimate exposure. A duplicated dose or an incorrect administration time can instead inflate or shift the predicted profile.

The introductory material on single-dose and steady-state exposure establishes the relevant concentration calculations for those settings. Here the emphasis is on encoding the sequence of doses as simulation events and ensuring that the concentration history and exposure window match the intended regimen.

For the worked calculations, assume a one-compartment model with instantaneous intravenous bolus doses, volume of distribution \(V\), and first-order elimination rate constant \(k\). The model is linear: each dose contributes independently, and the total concentration is the sum of the contributions from all doses already administered. Assume \(V\) and \(k\) are constant over time, doses are administered as recorded, and no other drug input or elimination process is present.

Here \(D_j\) is the amount in dose event \(j\), administered at time \(t_j\), and \(C(t)\) is the concentration at time \(t\). The sum includes only doses administered by time \(t\). At a bolus time, it is useful to distinguish the concentration immediately before the dose, \(C(t^-)\), from the concentration immediately after it, \(C(t^+)\). The bolus produces the jump \(C(t^+)=C(t^-)+D/V\). Between events, concentration declines exponentially under the stated assumptions.

Encoding Events and Accumulation

A simulation event record should preserve the actual sequence of administration, rather than just the planned regimen or the dose at the start of the reporting window. Set a time origin explicitly—for example, the first scheduled dose or the start of a treatment cycle—and express every event time relative to it. Include relevant doses before the analysis window, including doses in a prior treatment period if their drug contribution has not become negligible.

For an oral dose, infusion, or another input type, the event record must also contain the information required to describe that input, such as route, bioavailability, or infusion duration when applicable. The bolus equations below are not formulas for those input processes; the same bookkeeping principle still applies: represent the input at its correct time and retain the prior history required by the model. Do not replace an input history with an assumed steady-state concentration unless that is the intended initial condition and is consistent with the regimen.

Under repeated equal bolus doses \(D\) every \(\tau\) time units, the concentration contribution at time \(u\) after a dose, for \(0\leq u<\tau\), is

This expression assumes the repeated regimen has continued long enough to reach steady state. It represents the sum of the current dose's remaining contribution and the contributions of all earlier doses at intervals \(\tau,2\tau,\ldots\) before it. A finite dose list approximates this infinite history; whether that is adequate depends on how many prior doses are included and how quickly drug is eliminated.

Why the Summation Formula Holds

For one bolus at time \(t_j\), the model gives a concentration increment \(D_j/V\), followed by exponential decline. At time \(t\geq t_j\), that dose contributes \((D_j/V)e^{-k(t-t_j)}\). In a linear one-compartment model, concentration responses to separate doses add. Summing those individual contributions over all events administered by time \(t\) gives the dose-history equation above.

For the repeated-dose case, let the time since the most recent dose be \(u\). The most recent dose contributes \((D/V)e^{-ku}\); the dose one interval earlier contributes \((D/V)e^{-k(u+\tau)}\), and the dose \(m\) intervals earlier contributes \((D/V)e^{-k(u+m\tau)}\). Thus:

The last equality uses the geometric-series result \(\sum_{m=0}^{\infty}r^m=1/(1-r)\), which requires \(|r|<1\). Here \(r=e^{-k\tau}\), and \(0<e^{-k\tau}<1\) when \(k>0\) and \(\tau>0\). If the elimination rate is very small relative to the dose interval, contributions from remote doses decline slowly; a short history may then be inadequate to approximate the steady-state sum.

How an Incorrect History Distorts Exposure

Exposure calculations also depend on the state at the beginning of the integration window. For a bolus contribution \(C_0\) already present at the beginning of an interval of length \(T\), its AUC contribution under first-order elimination is \(C_0(1-e^{-kT})/k\). A dose given during the interval contributes only from its administration time to the end of that interval. Therefore, omitting a prior dose can bias AUC even when all doses scheduled inside the reporting window are recorded correctly.

Diagnostics, Alternatives, and Limitations

Before interpreting simulated exposure, check that the event history is internally consistent. A practical review includes the following:

• Time origin and units: Confirm that every event time uses the same origin and time unit as \(k\), the dosing interval, and the exposure window.
• Event ordering: Sort events by time, verify that doses at a shared time are intentional, and distinguish pre-dose from post-dose concentrations at bolus times.
• Completeness: Compare the simulated history with the intended and actual regimen, including missed, delayed, held, or changed doses. Do not silently substitute planned doses for administered doses.
• Initial condition: Verify that the starting concentration reflects the prior history or an explicitly justified reset. A zero initial concentration is not equivalent to steady state.
• Concentration checks: For boluses, each event should create the expected jump \(D/V\). Between events under this model, concentration should decline exponentially; unexplained jumps or discontinuities suggest an event or implementation error.
• Exposure window: Check which event contributions fall within the AUC interval and whether residual concentration at its start has been retained.

A finite event history and an initialized steady-state profile are two ways to represent prior dosing. A finite history makes individual events explicit and is useful when doses vary or are missed, but it may omit contributions from sufficiently remote doses. A steady-state initial condition is efficient for a stable repeated regimen, but it is inappropriate when the regimen has not reached steady state or when prior doses differ from the assumed schedule. The choice should match the simulated clinical history, not merely simplify the computation.

The calculations here rely on linear superposition, constant \(V\) and \(k\), and bolus input. They do not cover nonlinear concentration-dependent disposition, time-varying parameters, or the full input functions for non-bolus routes. In those situations, the event history still needs correct times and amounts, but the simple exponential contributions cannot necessarily be added. The model must be evaluated using its appropriate input and disposition equations.

Common mistake: “The AUC window starts at time 0, so set the concentration at time 0 to zero and include only doses given after time 0.” This is wrong whenever earlier doses still contribute drug at time 0. The integration window limits the times over which AUC is calculated; it does not erase the pharmacologic state accumulated before the window. Use the correct concentration at the window start, or simulate enough prior history to generate it.