Why Absorption Events Need Explicit Records
A dose history is more than a list of dose amounts. For each dose, the model may also need to know when it was administered, which absorption pathway receives it, whether input is delayed, and whether input is spread over a duration or distributed through transit. These details determine when drug enters the absorption model. A schedule that records only “400 mg every 12 hours” is incomplete if the input is lagged or extended.
A useful event record keeps the administered dose time distinct from the time an absorption input begins. This prevents a lag from being mistakenly added to the dose time in one part of a model and added again in another. It also makes it possible to check repeated doses on one absolute time scale, including occasions when the input from one dose continues as the next dose is given.
For event \(j\) and pathway \(p\), let \(A_{jp}=D_jw_{jp}\) be the amount assigned to that pathway, where \(w_{jp}\) is its dose fraction. This definition assumes that the fraction is applied to the dose amount in the record. If a model applies bioavailability elsewhere, do not multiply by it again when constructing \(A_{jp}\). The record should state where any such factor is applied.
The equations below describe the input contributed by an event to an absorption model. They require the event time, lag, duration, and distribution parameters to use compatible time units. For example, if \(t_j\) and \(L_{jp}\) are in hours, \(T_{jp}\) and transit-time parameters must also be in hours.
Encoding Lagged, Extended, and Transit Inputs
A lag shifts the absorption input in time; it does not change the recorded administration time. For a zero-order input, the amount assigned to the pathway is delivered at a constant rate over a specified duration after the lag. The interval starts at \(t_j+L_{jp}\) and ends at \(t_j+L_{jp}+T_{jp}\).
Here \(R_{jp}(t)\) is the input rate, in amount per unit time. The half-open interval is a bookkeeping convention: the choice of rate at a single endpoint does not change the total amount delivered. A transit input is recorded differently. It begins from the dose time plus any lag, then follows the specified transit-time distribution rather than a constant rate over a fixed interval.
The function \(g_{jp}\) is the transit input-rate distribution for that pathway, with units of inverse time and total area one. Its recorded specification must include the parameters required by the chosen transit model, such as mean transit time and transit-compartment number in the parameterization used by the model. The dose time is the origin for the transit process; the mean transit time is not an additional fixed lag. A transit distribution generally spreads input over time, so recording only its mean as if it were a zero-order duration would change the model.
For a dose divided among pathways, use separate pathway records or an equivalent representation that makes every allocation explicit. If fractions allocate the entire dose, they sum to one. If some dose is intentionally unallocated to the modeled pathways, the sum can be less than one, but that remainder must be understood rather than accidentally omitted. A repeated dose creates new event records; it does not reset or erase earlier events.
Worked Schedules
Worked Example: Repeated Lagged Zero-Order Doses
Suppose a hypothetical subject receives 400 mg orally at 08:00 and 20:00 on the same day. Each dose is assigned to one zero-order absorption pathway with a lag of 1.5 hours and an input duration of 4 hours. Use hours after 08:00 as the time scale, so the dose times are \(t_1=0\) h and \(t_2=12\) h.
For each event, the assigned amount is \(A=400\) mg. The input rate is \(400/4=100\) mg/h. For the first dose, input begins at \(0+1.5=1.5\) h and ends at \(1.5+4=5.5\) h. For the second, it begins at \(12+1.5=13.5\) h and ends at \(13.5+4=17.5\) h.
| Event | Dose time (h) | Assigned amount (mg) | Lag (h) | Input interval (h) | Rate (mg/h) |
|---|---|---|---|---|---|
| 1 | 0 | 400 | 1.5 | [1.5, 5.5) | 100 |
| 2 | 12 | 400 | 1.5 | [13.5, 17.5) | 100 |
Each event delivers \(100\text{ mg/h}\times4\text{ h}=400\) mg. The intervals do not overlap, and the 12-hour dose spacing remains 12 hours in the recorded administration times. The lag shifts both input intervals equally; it does not change the dose spacing or the duration.
Worked Example: Recording a Dose Split Across Two Pathways
Consider hypothetical 500 mg doses given every 24 hours, at \(t=0\) h and \(t=24\) h. Suppose 60% of each dose is assigned to an immediate transit pathway and 40% to a lagged zero-order pathway. The transit pathway has a mean transit time of 3 hours and uses \(N=2\) in the model's stated transit parameterization. The zero-order pathway has a 2-hour lag and a 5-hour duration.
For either dose, the transit-pathway amount is \(500(0.60)=300\) mg, and the zero-order amount is \(500(0.40)=200\) mg. The fractions account for the full dose because \(0.60+0.40=1.00\). Record a separate pathway entry for each dose-pathway combination:
| Dose time (h) | Pathway | Amount (mg) | Lag (h) | Additional timing specification |
|---|---|---|---|---|
| 0 | Transit | 300 | 0 | Mean transit time 3 h; \(N=2\) |
| 0 | Zero-order | 200 | 2 | Duration 5 h; input interval [2, 7) |
| 24 | Transit | 300 | 0 | Mean transit time 3 h; \(N=2\) |
| 24 | Zero-order | 200 | 2 | Duration 5 h; input interval [26, 31) |
The zero-order rate for each entry is \(200/5=40\) mg/h. The transit entries do not have a five-hour input interval: their input follows the transit distribution. Under the stated mean-transit-time definition, the mean input time for the first transit entry is \(0+3=3\) h, and for the second it is \(24+3=27\) h. These are means, not fixed input times. The amount accounting checks because each dose contributes \(300+200=500\) mg across its two pathways.
Worked Example: Checking Overlap in an Irregular Schedule
A hypothetical 300 mg dose is given at \(t=0\) h and a second 300 mg dose at \(t=8\) h. Both are assigned to a zero-order pathway with a lag of 1 hour and duration of 10 hours. Determine the input intervals, rates, and whether their inputs overlap.
For each dose, the rate is \(300/10=30\) mg/h. The first input interval is \([0+1,0+1+10)=[1,11)\) h. The second is \([8+1,8+1+10)=[9,19)\) h. The overlap is \([9,11)\), which lasts \(11-9=2\) hours. During that period, both event rates contribute, so the total input rate is \(30+30=60\) mg/h. Before 9 h, only the first event contributes; from 11 h until 19 h, only the second contributes.
The total input over the complete schedule is \(30(10)+30(10)=600\) mg, equal to the two assigned amounts. Overlap does not mean that either event should be deleted or its duration shortened. It means the model receives the sum of the active event rates at those times.
Justification and Timing Checks
The event equations follow from two basic bookkeeping requirements: shifting an event by a lag moves its input in time, and the total amount assigned to the event must be preserved. For zero-order input, the rate is constant over an interval of length \(T_{jp}\). Integrating that rate gives the assigned amount:
For a transit input, shifting the distribution by \(t_j+L_{jp}\) changes where it occurs, not its area. Substituting \(u=t-t_j-L_{jp}\) gives \(\int R_{jp}(t)\,dt=A_{jp}\int_0^\infty g_{jp}(u)\,du=A_{jp}\). Thus both representations preserve the assigned amount, while describing different timing patterns.
When multiple events contribute to the same modeled input, the total input rate is the sum of their rates:
This is an accounting identity for the specified input rates. It does not imply that downstream concentrations or other model outputs can always be added event by event; those outputs depend on the model's disposition and state equations. The event-level checks below concern the input schedule itself.
Use one clearly defined scale for every dose time, lag, duration, and transit-time parameter. Convert clock times to elapsed times before checking intervals.
Record the amount assigned to each pathway and verify that the pathway fractions account for the intended dose without accidental duplication.
For a lagged zero-order entry, calculate the start as \(t_j+L_{jp}\) and the end as \(t_j+L_{jp}+T_{jp}\). For transit, preserve the event origin and record the distribution and its parameters.
Check that each event contributes its assigned amount and inspect intervals from all events together. Overlapping inputs are summed, not silently shifted or truncated.
Diagnostics, Alternatives, and Limitations
A reliable event table should allow someone to reconstruct when each modeled input occurs and how much amount it carries. Common warning signs include a lag stored both in the event time and in a lag field, a duration entered in minutes when the schedule uses hours, pathway fractions that sum to more than one without justification, or an input rate whose rate-times-duration does not equal the assigned amount. For transit entries, a duration field should not be used as a substitute for the transit-distribution parameters.
Event records are especially useful when doses have different times, lags, pathway allocations, or durations. An alternative is to provide a model with a precomputed input function over time. That can be appropriate when the input is already available on a time grid, but a coarse grid may obscure event boundaries or shift an onset through interpolation. In either representation, retain enough information to verify the original dose times and amounts.
A lagged event and an event recorded at a later administration time can produce the same nominal start time for one input component, but they are not the same dose history. The latter changes the administration record and can misalign other pathways or repeated doses. Similarly, a zero-order input and a transit input can share the same mean timing while having different time profiles. Event records make those distinctions explicit; they do not establish that a particular absorption model is scientifically appropriate.
The equations assume that the event's assigned amount is fully represented by its input function, that the transit density is normalized, and that event times and parameters are correctly specified. If the model uses a different accounting convention for bioavailability or pathway fractions, document it and apply it consistently. A correctly formed event table cannot correct an incorrect model structure or inaccurate source dose history.
Check Your Understanding
For each question, identify the event timing and amount consequences. Concise solutions are included below.
- A 240 mg dose at \(t=5\) h is assigned to a zero-order pathway with a lag of 0.5 h and duration of 3 h. What are its input interval and rate?
- Two 200 mg doses are given at \(t=0\) h and \(t=6\) h. Each has a lag of 1 h and duration of 8 h on the same zero-order pathway. When do the inputs overlap, and what is the total rate during overlap?
- A 600 mg dose is split between two pathways with fractions 0.25 and 0.75. What amount is assigned to each pathway, and do the fractions account for the dose?
- A transit event has a dose time of 10 h, a lag of 2 h, and a mean transit time of 4 h. What is its shifted transit origin and mean input time? Is input concentrated at that mean time?
- Explain why a transit input with mean transit time 4 h should not be encoded as a zero-order input lasting 4 h.
Solutions. (1) The interval is \([5+0.5,\ 5+0.5+3)=[5.5,8.5)\) h. The rate is \(240/3=80\) mg/h. (2) The intervals are \([1,9)\) h and \([7,15)\) h, so they overlap from 7 h to 9 h. Each rate is \(200/8=25\) mg/h, giving a total rate of 50 mg/h during the 2-hour overlap. (3) The assigned amounts are \(600(0.25)=150\) mg and \(600(0.75)=450\) mg; the fractions sum to 1.00 and the amounts sum to 600 mg. (4) The shifted transit origin is \(10+2=12\) h, and the mean input time is \(12+4=16\) h. The mean is a summary of the transit distribution, not a claim that all input occurs at 16 h. (5) A transit input follows a normalized time distribution whose shape and spread depend on its specified parameters. A zero-order input instead has a constant rate over a fixed interval. Matching one mean timing value does not make those event profiles equivalent.