Why a Missed Administration Changes the Event History
A pharmacokinetic model responds to what was administered, not merely to what was scheduled. When an administration is documented as missed, the model’s dose-event history should therefore contain no administration event at that scheduled time. The state still evolves through the interval: drug already in the body continues to be distributed and eliminated according to the model. Removing a dose event does not reset concentration to zero.
This tutorial focuses on documented missed administrations. It does not treat an unknown administration status as a confirmed miss. The dose history used for prediction must describe the administrations that are known to have occurred, while retaining all other relevant events and the correct state at the start of the modeled interval.
State Evolution and Dose-Event Encoding
For a precise calculation, consider a one-compartment model with instantaneous intravenous bolus administrations and first-order elimination. For individual \(i\), let \(C_i(t)\) be concentration, \(V_i\) volume of distribution, and \(k_i\) the elimination rate constant. Between dose events, the model is
At an administered bolus of amount \(D_{ij}\), concentration increases by \(D_{ij}/V_i\). If the modeled interval begins at time \(s\), with the state \(C_i(s+)\) specified immediately after any events at \(s\), then at time \(t>s\) the post-event concentration is
The initial-state term is essential: it carries forward drug present at the beginning of the interval. The sum includes only administrations that occurred. A dose at time \(t\) is included for a post-dose prediction \(C_i(t+)\); for a pre-dose prediction \(C_i(t-)\), that event is excluded. This convention makes the prediction at a shared observation and dose time unambiguous.
The equation follows by solving the elimination equation between events and applying the concentration jump at each administered dose. If there is no event at a scheduled time, there is no jump there; the concentration continues along the between-event solution. In a general nonlinear NLME model, superposition of dose contributions need not hold. The same event-encoding principle still applies, but the state equations must be propagated with the actual administered events and the correct initial state.
Carry forward the model-predicted state immediately before or after the starting-time events, using one stated convention consistently.
Include administrations that occurred at their recorded times and amounts. Do not create an event for a documented miss.
Apply the model’s state equations over each interval, then apply a dose input only at an event that occurred.
Use the event ordering to decide whether a dose at the same time is included in a post-dose or pre-dose prediction.
Worked Predictions for Missed-Dose Patterns
Worked Example: One Skipped Dose in a Linear Model
Suppose an individual receives 100 mg intravenous boluses at time 0 and, if adherent to the planned schedule, every 24 hours. For this hypothetical calculation, let \(V_i=20\) L and \(k_i=0.1000\ \text{h}^{-1}\). The dose increment is
Assume the time-0 dose occurred and the concentration immediately after it was \(C_i(0+)=5.000\) mg/L. The dose scheduled at 24 hours was documented as missed, and the next dose occurred at 48 hours.
At 24 hours, before any hypothetical 24-hour dose, the remaining concentration is
Because the scheduled 24-hour administration did not occur, the predicted post-time concentration remains 0.4536 mg/L; there is no 5.000 mg/L jump. At 48 hours, just before the administered dose, the time-0 contribution is
The dose at 48 hours adds 5.000 mg/L, giving \(C_i(48+)=5.041\) mg/L. Under the planned schedule with the 24-hour dose administered, the 48-hour post-dose prediction would instead be \(5.000e^{-4.8}+5.000e^{-2.4}+5.000=5.495\) mg/L. The difference, 0.4536 mg/L, is exactly the 24-hour dose’s contribution after 24 hours of elimination. This subtraction works here because the model is linear.
Worked Example: Several Consecutive Skipped Doses
Consider a second hypothetical individual with 80 mg boluses, \(V_i=20\) L, and \(k_i=0.08000\ \text{h}^{-1}\). A dose occurs at time 0; the scheduled doses at 24 and 48 hours are both documented as missed; and the dose at 72 hours occurs. Thus each administered dose adds \(80/20=4.000\) mg/L.
Starting from \(C_i(0+)=4.000\) mg/L, the concentration just before the 72-hour dose under the actual history is
The 72-hour dose raises the post-dose prediction to \(4.0126\) mg/L. Under the planned schedule, the 24- and 48-hour doses would also have contributed. The planned post-dose prediction at 72 hours would be
The actual-history prediction is lower by \(4.000(e^{-3.84}+e^{-1.92})=0.6724\) mg/L. The two consecutive missed doses leave no jumps at 24 or 48 hours, but the concentration from the time-0 dose continues to decay throughout. The calculation assumes the same \(k_i\) and \(V_i\) across the interval and bolus input at the recorded 72-hour time.
Worked Example: Applying the History to an Individual NLME Prediction
Suppose a third hypothetical individual has an individual elimination rate \(k_i=0.1200\ \text{h}^{-1}\), \(V_i=25.00\) L, and 100 mg scheduled boluses at 0, 12, and 24 hours. The dose at 0 occurred, the 12-hour dose was documented as missed, and the 24-hour dose occurred. The 0- and 24-hour administrations each produce an increment of \(100/25=4.000\) mg/L. Let \(C_i(0+)=4.000\) mg/L.
The post-dose prediction at 24 hours using the actual history is
If all three scheduled doses had occurred, the 12-hour dose would add \(4.000e^{-1.44}=0.9477\) mg/L by hour 24. The planned-history prediction would therefore be \(4.2245+0.9477=5.172\) mg/L. In an NLME analysis, the individual parameter values may reflect the model’s population parameters and that individual’s random effects; the documented event history is applied to those parameters to obtain the individual prediction. The missed dose does not itself imply a change in \(k_i\), \(V_i\), or the random effects.
Assumptions, Checks, and Failure Modes
The numerical examples rely on an instantaneous intravenous bolus, a single well-mixed compartment, constant \(V_i\) and \(k_i\) over the displayed interval, and first-order elimination. The event times and administered amounts are treated as known, and the starting concentration is specified. These assumptions justify the exponential decay and dose increments used in the arithmetic. A model with absorption, multiple compartments, time-varying parameters, or nonlinear elimination requires its own state equations; the missed administration is still omitted, but the displayed closed-form concentration calculations may no longer apply.
In practice, inspect both the event records and model predictions. Useful checks include:
- Confirm that each documented miss produces no dose input at its scheduled time, while subsequent administered doses retain their actual times.
- Check pre-dose and post-dose predictions around every event. A prediction should have a jump only when the model and event type imply an input at that time.
- Verify that the interval-start state is carried forward. Reinitializing concentration to zero at a missed dose incorrectly erases drug from earlier administrations.
- Compare predictions generated from the documented history with predictions from the planned schedule as a diagnostic contrast, not as evidence that either history is correct.
- Review implausible abrupt changes, unexpected peaks, or systematic residual patterns near missed-dose times. Such patterns can indicate an event-time, event-ordering, or history-encoding error, although they can also reflect model misspecification or inaccurate records.
A common error is to shift every later dose forward when one scheduled dose is missed. That represents a delayed regimen, not the event history described here, unless later administrations actually occurred at shifted times. Another error is to subtract a dose contribution from a nonlinear model as though dose effects were additive. For a linear model such as the examples, contribution subtraction is valid; for a nonlinear state model, rebuild the event list and propagate the state equations.
The main alternative is to calculate predictions under a planned schedule rather than the documented actual schedule. Planned-schedule predictions can be useful as a reference or counterfactual, but they are not predictions conditional on the administration history when doses were missed. Event-based encoding is especially useful for irregular histories because it preserves each observed administration time without forcing a uniform interval. Its limitation is equally important: predictions are conditional on the accuracy and completeness of the recorded event history and on the adequacy of the pharmacokinetic model.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- In the one-compartment bolus model, what happens to concentration at a scheduled time when the dose is documented as missed?
- An individual has \(C_i(s+)=2.0\) mg/L, \(k_i=0.10\ \text{h}^{-1}\), and no administered doses during the next 10 hours. What is the predicted concentration at hour 10?
- For 50 mg boluses with \(V_i=10\) L and \(k_i=0.05\ \text{h}^{-1}\), a dose at time 0 occurs and the scheduled 8-hour dose is missed. Calculate the concentration at hour 8 before any subsequent dose.
- Why should later administrations not automatically be shifted when an earlier scheduled administration was missed?
- Why is it generally unsafe to calculate a missed-dose prediction in a nonlinear model simply by subtracting a dose contribution from the planned-schedule prediction?
Solutions. (1) There is no dose-related jump; the state continues to evolve according to the model between events. (2) The interval contains no dose events, so \(C_i(10)=2.0e^{-(0.10)(10)}=2.0e^{-1}=0.7358\) mg/L. This carries forward the initial state rather than resetting it. (3) The administered dose adds \(50/10=5.0\) mg/L at time 0. Thus \(C_i(8-)=5.0e^{-(0.05)(8)}=5.0e^{-0.4}=3.352\) mg/L. The missed 8-hour dose contributes no jump. (4) A later dose is placed at its actual recorded administration time; shifting it would encode a different history unless it was in fact delayed. (5) In a nonlinear model, the effect of a dose may depend on the state, so dose contributions need not add independently. Propagate the model using the actual event list instead.