From a Dose Diary to a Concentration Prediction
A dose diary is not itself a pharmacokinetic history until each entry has been translated into a dose amount and a time. A dose taken late changes the time available for absorption and elimination before a sample; a partial dose changes the amount entering the system. When several such events occur, their effects accumulate. Treating the diary as a regular schedule can therefore produce a different predicted concentration, even when the intended regimen was regular.
The calculations below use a deliberately simple model so the effect of event timing is visible. They are conditional predictions for one hypothetical individual, using fixed pharmacokinetic parameters. They do not estimate population parameters or account for between-individual variability.
The dose history is represented by the event list \((t_i,D_i)\), not by the intended regimen alone. The time difference \(t-t_i\) is the age of dose \(i\) at the prediction time. When an oral dose is administered at the prediction time, its contribution is zero at that instant in this model: drug first enters the absorption depot, not the central compartment. If \(k_a=k\), the limiting single-dose expression is \(F D_i k(t-t_i)e^{-k(t-t_i)}/V\).
Why Contributions Add
For one dose at time \(t_i\), let \(u=t-t_i\) be elapsed time and let \(A_g(u)=D_i e^{-k_a u}\) be the amount remaining in the absorption depot. The input rate into the central compartment is \(F k_a A_g(u)\). If \(A_c(u)\) is the central amount, its balance equation is \(dA_c/du=F k_aD_i e^{-k_a u}-kA_c\), with \(A_c(0)=0\). Multiplying by the integrating factor \(e^{ku}\) and integrating gives the single-dose result:
For multiple doses, linearity means the balance equation's input is the sum of the dose-specific input rates. The resulting central amount, and therefore concentration, is the sum of the single-dose contributions. This superposition is why all recorded doses still contributing at the sample time must be included, including doses administered over unequal intervals.
Worked Reconstructions
Worked Example: Three Irregular Doses Versus a Regular-Schedule Assumption
Suppose a hypothetical patient's diary records 100 mg at 0 hours, 100 mg at 8 hours, and 100 mg at 21 hours. Predict concentration at 24 hours. For comparison, consider a regular-schedule assumption with doses at 12 and 24 hours in the modeled window. Use \(F=1\), \(D_i=100\) mg, \(V=50\) L, \(k_a=1\) h\(^{-1}\), and \(k=0.1\) h\(^{-1}\). These are illustrative values, not estimates from a study.
The diary intervals are \(8-0=8\) hours and \(21-8=13\) hours. At 24 hours, the three recorded doses have been in the system for 24, 16, and 3 hours. The regular-schedule assumption contributes doses aged 12 hours and 0 hours; the dose at 24 hours contributes zero at that instant.
For these parameters, the shared coefficient is \(F D_i k_a/[V(k_a-k)]=100/(50\times0.9)=2.222\) mg/L. Thus the diary-based prediction is:
The regular-schedule prediction is:
Subtracting the unrounded predictions gives \(C_{\text{diary}}-C_{\text{regular}}=1.517\) mg/L. Under the specified assumptions, the irregular diary predicts a higher concentration at 24 hours than this regular-schedule reconstruction. The contrast reflects the different event histories in the modeled window, not a general claim that irregular dosing always raises concentration.
Worked Example: A Second Irregular History
A second hypothetical diary records 100 mg at 0 hours and 100 mg at 10 hours. At 24 hours, compare its predicted concentration with the same regular-schedule assumption used above, which has one contributing dose aged 12 hours. Keep the same parameter values, so the coefficient remains 2.222 mg/L.
The two diary doses are aged 24 and 14 hours. Therefore:
The regular prediction is still \(0.6693\) mg/L. Using unrounded values, the difference is:
For this particular pair of histories, the diary-based prediction is 0.08028 mg/L higher at 24 hours. The sign and size of a difference depend on the actual times, amounts, parameters, and prediction time; they should be computed rather than inferred just from the label “irregular.”
Worked Example: Unequal Intervals and a Partial Dose
A third hypothetical diary records 100 mg at 0 hours, 50 mg at 9 hours, and 100 mg at 23 hours. The 50 mg entry is the amount actually taken. Predict the concentration immediately before the dose recorded at 23 hours, using the same model parameters as above.
The intervals are 9 hours and 14 hours. Immediately before the final dose, the first dose is aged 23 hours and the partial dose is aged 14 hours. The dose at 23 hours has not yet been administered at the prediction time, so it is not included. Because the 50 mg dose is half the reference dose, its concentration contribution is also halved under the linear model:
This prediction includes both earlier events and uses their separate elapsed times. Replacing the two unequal intervals with a single nominal interval, or treating the partial dose as 100 mg, would represent a different history and yield a different prediction.
Practical Reconstruction and Checks
A reliable calculation begins by establishing a consistent time origin and units. Convert every diary entry to elapsed time from that origin, record the amount actually taken, and identify which events occurred before the concentration prediction time. If a dose is reported as delayed, use its actual time rather than its scheduled time. If only part was taken, use that amount. Events after the prediction time cannot contribute to the prediction.
A compact event table helps expose errors before calculation:
| Event time | Recorded amount | Elapsed time at prediction | Included? |
|---|---|---|---|
| 0 h | 100 mg | 24 h | Yes |
| 8 h | 100 mg | 16 h | Yes |
| 21 h | 100 mg | 3 h | Yes |
| 24 h | 100 mg, under comparison schedule | 0 h | Zero contribution at that instant |
The table is not a substitute for checking the diary's meaning. Confirm whether times are actual administration times, whether amounts are total dose or amount swallowed, and whether the record's time zone or date boundary has been handled consistently. A missing entry is not automatically evidence of a zero dose, and a scheduled dose is not evidence that it was taken.
Diagnostics, Alternatives, and Limits
Before interpreting a reconstructed exposure, check that all dose times and amounts are plausible and that the prediction time is later than the events being included. Verify units: with \(D_i\) in mg and \(V\) in L, the resulting concentration is mg/L; rate constants must use the reciprocal of the time unit used for dose times. A calculated negative concentration, a concentration contribution from a future event, or a nonzero oral contribution exactly at administration time signals a coding or formula error under this model.
The event-based calculation is appropriate when the diary is informative and the assumed linear pharmacokinetic model is adequate. A regular-schedule calculation is a useful comparator when the question is specifically how much predictions change under that simplification, but it is not a correction for uncertain diary entries. If timing or dose amounts are unknown, reporting a single reconstructed history can imply more certainty than the records support. If absorption, clearance, or bioavailability changes over time, or pharmacokinetics are nonlinear, the simple formula and superposition used here may not apply.
In a nonlinear mixed-effects analysis, these calculations can be understood as individual predictions conditional on specified parameters and event history. Population inference additionally requires a model for parameter variability and observations; those components are deliberately outside these numerical reconstructions. The key comparison here is between two explicit event histories under the same stated conditional model.
- Wrong: “The patient was prescribed 100 mg every 12 hours, so I will use doses at 0, 12, and 24 hours.” Correction: A prescription describes the intended schedule. Reconstruct the administered events from the diary, and label any regular-schedule calculation as an assumption for comparison.
- Wrong: “The concentration after the last dose depends only on that dose.” Correction: Earlier doses can still contribute. Sum all doses administered before the prediction time that have nonzero model contributions.
- Wrong: “A missed diary entry means the patient took zero.” Correction: Distinguish a documented zero dose from a missing or ambiguous record; do not silently convert uncertainty into an observed event history.
Check Your Understanding
Answer each question from the stated model and distinguish actual diary events from assumed schedules.
- A 100 mg dose was taken at 6 hours. At what elapsed time should its contribution be evaluated for a prediction at 20 hours?
- Under the linear model, how does the concentration contribution of a 40 mg dose compare with that of a 100 mg dose at the same time and with the same parameters?
- Why does a dose administered exactly at the prediction time contribute zero concentration at that instant in the first-order absorption model used here?
- Name two checks that can reveal a dose-history reconstruction or calculation error.
Solutions. (1) The elapsed time is \(20-6=14\) hours. (2) It is \(40/100=0.4\) times the 100 mg contribution, because the model is linear in dose amount. (3) At elapsed time zero, \(e^{-k(0)}-e^{-k_a(0)}=1-1=0\); oral absorption has not yet produced central-compartment concentration at that instant. (4) Examples include checking that only doses at or before the prediction time are included, verifying time and rate-constant units, confirming that the amounts match the diary, and checking that each included event has a nonnegative elapsed time.