Encoding Delayed, Partial, and Interrupted Doses

Why Actual Administration Events Matter

A dosing schedule describes what was intended; an event history describes what entered the dosing process and when. When an administration is late, reduced, or interrupted, encoding the schedule as if it were followed gives the model a different input history from the one recorded. That difference can matter even when the total amount is eventually the same, because a pharmacokinetic system responds to the timing and pattern of input as well as its total amount.

The event-based representation used for irregular dosing histories applies here too. Each administration record needs to identify the individual, the actual time, the amount or rate, and the route or model compartment receiving the dose. An interrupted infusion requires enough information to mark when input starts, stops, and resumes. All times and durations must use consistent units.

Definition: A dose event records an actual administration or a change in administration, rather than merely a planned dose. For an instantaneous dose into a model compartment, the event specifies its actual time \(t_j\) and amount \(A_j\). For an infusion, the input rate \(u(t)\) is specified over the intervals during which drug is delivered; a pause has \(u(t)=0\). A delayed dose is placed at its actual administration time, and a partial dose is represented by the amount actually administered.

This encoding assumes the recorded administration times and amounts are the best available account of what was delivered, that the route and receiving compartment are correctly identified, and that units are consistent. For a constant-rate infusion segment, the delivered amount is rate multiplied by duration. If a pump record or administration record does not establish the actual amount, do not silently substitute the planned amount: represent the known information and flag the uncertainty for the analysis.

Encoding a Delayed or Partial Dose

For an instantaneous dose into a depot compartment, the dose event produces a jump in the amount in that compartment. Between dose events, the compartment evolves according to the model. For example, under first-order absorption with absorption rate constant \(k_a\), the depot amount \(A_g(t)\) satisfies:

\frac{dA_g(t)}{dt}=-k_aA_g(t), \qquad A_g(t_j^+)=A_g(t_j^-)+A_j.

Here \(t_j\) is the actual administration time, \(A_j\) is the administered amount, and \(t_j^-\) and \(t_j^+\) denote the instants immediately before and after the event. For a delayed, partial oral dose, use the delayed actual time and the reduced actual amount in the jump. Do not create an additional event at the scheduled time unless an administration actually occurred then.

Worked Example: A Delayed, Partial Oral Dose

Suppose a hypothetical patient is scheduled to receive 100 mg orally at 08:00 but actually takes 60 mg at 10:00. Assume the model uses a depot compartment and \(k_a=0.80\ \text{h}^{-1}\). There are no other doses in the interval.

The event is a 60 mg dose at 10:00. There is no dose event at 08:00. Thus the depot amount does not jump at 08:00; it jumps by 60 mg at 10:00. At 12:00, two hours after the actual dose, the depot amount is:

A_g(12{:}00)=60\exp[-0.80(2)] =60\exp(-1.6) \approx 12.12\ \text{mg}.

The units are consistent because \(k_a\) is in inverse hours and the elapsed time is in hours. Using 100 mg at 08:00 would encode both the wrong amount and the wrong time, and would therefore generate a different modeled absorption history.

The relevant amount is the amount administered by the recorded route, not automatically the planned amount or a later estimate of systemic availability. If the model includes bioavailability, it acts according to the model's definition and should not be applied a second time when constructing the dose record.

Interrupted Infusions as Piecewise Input

An infusion provides input over time rather than as an instantaneous jump. On a segment from \(s\) to \(e\) with constant rate \(R\), the input is \(u(t)=R\) for \(s\leq t<e\), and the delivered amount is \(R(e-s)\). An interruption divides the administration into separate segments. During the gap, input is zero; on resumption, input begins again at the documented rate and time.

u(t)= \begin{cases} R_1, & s_1\leq t<e_1,\\ 0, & e_1\leq t<s_2,\\ R_2, & s_2\leq t<e_2. \end{cases} \qquad A_{\mathrm{delivered}}=\sum_j R_j(e_j-s_j).

The rate may be the same on each active segment or may change. A useful equivalent event representation records a start and stop for each segment, or records each segment's start, rate, and duration. The representation must make the pause explicit so the model does not continue delivering drug through it.

Worked Example: Encoding an Interrupted Infusion

A hypothetical 600 mg infusion is administered at a constant rate of 300 mg/h. It starts at 08:00, stops at 08:40, resumes at 09:10, and finishes at 10:30. Assume the recorded rate applies throughout both active segments.

The first active segment lasts 40 minutes, or \(40/60=2/3\) hour. Its delivered amount is:

300\ \text{mg/h}\times\frac{2}{3}\ \text{h}=200\ \text{mg}.

The pause lasts from 08:40 to 09:10, so the input is zero for 30 minutes. The second active segment lasts 80 minutes, or \(80/60=4/3\) hours. Its delivered amount is:

300\ \text{mg/h}\times\frac{4}{3}\ \text{h}=400\ \text{mg}.

The total is \(200+400=600\) mg, consistent with the recorded total. The input history is 300 mg/h from 08:00 to 08:40, zero from 08:40 to 09:10, and 300 mg/h from 09:10 to 10:30. Encoding one uninterrupted infusion from 08:00 to 10:00 would deliver the same total at the same rate but at the wrong times; extending it to 10:30 would incorrectly deliver 750 mg.

For an amount-based event system, the same history can be represented by two infusion segments: 200 mg over \(2/3\) hour beginning at 08:00, and 400 mg over \(4/3\) hours beginning at 09:10. The gap must remain a gap, not an additional segment. Whether an implementation uses rates, segment amounts and durations, or paired start-and-stop records, its resulting \(u(t)\) must be the same.

Checking Amounts, Durations, and Input

A basic consistency check for a constant-rate segment is \(A=RD\), where \(A\) is delivered amount, \(R\) is rate, and \(D\) is duration. If two of these are known, the third can be checked or calculated, but that arithmetic does not establish which value is correct when records conflict. Resolve discrepancies using the underlying administration information rather than altering an event merely to make the planned dose appear complete.

Worked Example: A Reduced Infusion Stopped Early

In a hypothetical case, a 200 mg infusion is planned to run for one hour starting at 14:00, corresponding to a planned rate of 200 mg/h. The actual infusion starts late, at 14:20, and is stopped at 14:50. The administration record confirms a constant rate of 200 mg/h while it is running and no restart.

The actual duration is 30 minutes, or \(0.5\) hour. Therefore the delivered amount is:

A_{\mathrm{actual}}=200\ \text{mg/h}\times0.5\ \text{h}=100\ \text{mg}.

Encode input at 200 mg/h from 14:20 to 14:50, with zero input outside that interval for this administration. Do not encode the planned 200 mg as an instantaneous dose at 14:00 or as an infusion continuing to 15:00. The planned dose is useful context for checking the record, but it is not the actual input history.

A model may express the amount in a central compartment through an input term, for example \(dA_c(t)/dt=\text{other model terms}+u(t)\). In that representation, \(u(t)\) has units of amount per time. The event construction specifies \(u(t)\); it does not replace the model's disposition, absorption, or other differential equations.

Derivation: Why Splitting the Infusion Preserves Its Recorded Input

For any infusion input \(u(t)\), the amount delivered between times \(a\) and \(b\) is the integral of its rate over that interval. For a piecewise constant rate, divide the interval at every start, stop, or rate change. Integrating each constant segment gives its rate times its duration; summing the segment amounts gives the total input:

\int_a^b u(t)\,dt =\sum_j\int_{s_j}^{e_j}R_j\,dt =\sum_j R_j(e_j-s_j).

For the interrupted-infusion example, this calculation gives 200 mg in the first segment and 400 mg in the second. The pause contributes \(\int u(t)\,dt=0\). This justifies the segment representation as a faithful accounting of the recorded input. It does not imply that any alternative timing with the same total amount produces the same concentration profile: the model evolves during and between input intervals.

The argument assumes the infusion rate is constant within each recorded segment. If the rate varies continuously, the segment amount is instead \(\int_s^e u(t)\,dt\); a single rate-times-duration calculation is appropriate only if the rate is constant or if the rate used is a correctly calculated average over that segment.

Diagnostics, Alternatives, and Limitations

Before fitting or interpreting an NLME model, inspect the constructed event history for errors that change the input. Check that actual times follow the administration record; each partial dose uses the actual amount; every infusion's units agree; and segment amounts sum to the documented total when the total is known. For interrupted infusions, confirm that each pause has zero input and that resumed segments have their own correct start and stop times. Compare planned and actual records to find unusual deviations, but do not replace the actual history with the plan.

Several shortcuts are tempting but generally answer a different question:

Encoding choice	What it represents	Why it can be wrong for the actual history
Use the scheduled time and amount	The intended administration	It shifts or enlarges input when the dose was late or partial.
Use the actual total amount at one time	A bolus at a chosen time	It removes the duration and pauses of an infusion.
Average an interrupted infusion across one continuous interval	A smoothed input over the full span	It places input during the pause and changes when drug enters the model.
Enter a planned remainder as a dose	A hypothetical additional administration	It adds drug that was not recorded as delivered.

Some data sources provide only a scheduled dose or an uncertain report, rather than verified actual administration. In that case, the true history is not identified by the available record. Distinguishing an uncertain or absent record from a confirmed zero administration is essential: an uncertain record should not be converted to a zero event or to a full planned dose without justification. The event history can document what is known, but cannot recover unrecorded administration details by itself.

Common signs of faulty encoding include an infusion amount that fails \(A=RD\), overlapping segments that imply unintended simultaneous input, a nonzero rate during a documented pause, or a dose event at the scheduled time despite a documented late administration. These checks identify data and coding inconsistencies; they do not prove that the recorded history is correct. When records conflict, retain a transparent audit trail and resolve or represent the uncertainty explicitly rather than silently choosing a convenient value.

Key takeaway: Encode what was actually administered: the actual time and amount for an instantaneous dose, and the actual active intervals and rates for an infusion. Keep the planned schedule separate from the model's input history.

Check Your Understanding

Answer each question using the event-history principles above. Full solutions follow the questions.

A hypothetical patient is scheduled for 80 mg orally at 09:00 but takes 50 mg at 11:00. What dose event should be encoded?
A constant-rate infusion runs at 120 mg/h from 13:00 to 13:30, pauses until 14:00, and resumes at 120 mg/h until 14:45. How much is delivered in each active segment, and what is the total?
A planned 90 mg infusion is scheduled to begin at 16:00. The record confirms that 60 mg was administered from 16:20 to 16:50 at a constant rate. What rate and event interval are consistent with the record?
Why is it generally incorrect to replace the interrupted infusion in question 2 with one constant-rate infusion over 13:00 to 14:45, even if an average rate is chosen to preserve the total amount?

Solutions. (1) Encode one 50 mg oral dose at 11:00, not an 80 mg dose at 09:00. (2) The first segment lasts \(0.5\) h and delivers \(120\times0.5=60\) mg. The second lasts \(0.75\) h and delivers \(120\times0.75=90\) mg. The total is \(60+90=150\) mg; input is zero during the 30-minute pause. (3) The actual duration is \(0.5\) h, so the consistent rate is \(60/0.5=120\) mg/h. Encode input at that rate from 16:20 to 16:50. (4) A continuous averaged input places drug into the model during the recorded pause and at different times during the active periods. Preserving the integrated amount does not preserve the time-varying input history.