Why the Dosing History Is Part of the Model
A prescribed regimen describes what a patient was instructed to take. It does not establish what entered the body. A patient’s report describes what they recall or choose to report, while a device record may document an event such as a container opening rather than ingestion. These histories can agree, but they are not interchangeable. When the PK model uses the wrong one, a predicted concentration can be wrong even if the structural PK model is otherwise adequate. That discrepancy can then affect predicted response and estimated exposure-response relationships.
The aim is not to assume that one source is always correct. It is to identify what each source measures, decide which history is defensible for the analysis, and assess whether conclusions depend on that choice. A dose history must specify both amount and time: the same amount administered at different times generally produces different concentrations at an assessment time.
For subject \(i\), represent an administration history as dose-time pairs \((D_{ik},t_{ik})\), where \(D_{ik}\) is the amount administered at time \(t_{ik}\). Prescriptions can also be represented as dose-time pairs, but they encode intended rather than necessarily actual input. A missed dose contributes no dose at its scheduled time; a delayed dose contributes its amount at a later time; and a partial dose contributes a smaller amount than prescribed. For infusions or other non-instantaneous inputs, the input record must also represent duration or rate.
How Different Histories Propagate Through a PK/PD Model
The following calculations use a deliberately simple model to isolate the effect of the input history. Assume an intravenous bolus, one well-mixed compartment, fixed volume \(V\), and first-order elimination with rate constant \(k\). Doses are completely delivered into the compartment, and the parameters do not change between histories. Under these assumptions, concentration from one dose \(D\), given at time \(t_d\), is \(D/V\) at administration and then declines exponentially.
This expression follows from superposition. Between administrations, the concentration satisfies \(dC/dt=-kC\), whose solution over an interval of length \(u\) is \(C(t+u)=C(t)\exp(-ku)\). A bolus of amount \(D_{ik}\) increases concentration by \(D_{ik}/V\). Applying that increase at each administration time and allowing each contribution to decay independently gives the sum above. The result requires linear disposition: elimination and distribution are not saturated, and one dose does not change the disposition of another.
If PD is described by a concentration-effect model, the history affects response through the predicted concentration or another exposure summary. For example, the instantaneous Emax model is \(R=E_{\max}C/(EC_{50}+C)\), where \(E_{\max}\) is the maximum drug-related response and \(EC_{50}\) is the concentration giving half that maximum. The model assumes the response at the assessment time depends on the current concentration, with no additional delay or effect compartment.
Worked Example: A Missed Dose Changes Predicted Concentration
Hypothetical data. A patient is prescribed 100 mg intravenous boluses at 0, 12, and 24 hours. Let \(V=20.0\) L and \(k=0.100\ \mathrm{h}^{-1}\). A verified administration record shows that the 12-hour dose was not given, while the 0- and 24-hour doses were given. Calculate each history’s predicted concentration at 30 hours.
The model assumptions above apply: the doses are boluses into the same compartment, disposition is linear, and the stated administration record correctly identifies the doses given. Under the prescribed history, the elapsed times at 30 hours are 30, 18, and 6 hours. Each dose contributes \(100/20.0=5.00\ \mathrm{mg/L}\) immediately after administration. Thus
Using \(\exp(-3)=0.04979\), \(\exp(-1.8)=0.1653\), and \(\exp(-0.6)=0.5488\),
For the administered history, omit the 12-hour dose. The 0-hour and 24-hour contributions remain:
The prescribed-history prediction is higher by \(3.819-2.993=0.826\ \mathrm{mg/L}\). Relative to the prescribed-history prediction, the administered-history concentration is about \(0.826/3.819=21.6\%\) lower. This difference is caused by the input history in this calculation; it does not demonstrate that the PK parameter values are wrong.
From Input Differences to Response and Estimation
In an analysis, competing histories can be used to generate competing model predictions while holding the PK/PD structure and parameter values fixed. That comparison isolates the consequences of the history choice. In a fitted model, however, parameters may also change to compensate for a mismatched input. A lower concentration inferred from a recorded history, for example, may be offset by a larger estimated potency or exposure-response slope. Thus the history can affect both predictions and parameter estimates.
Worked Example: A Concentration Difference Changes an Emax Prediction
Hypothetical data. At a response assessment, a PK calculation using the prescribed history gives \(C=1.061\ \mathrm{mg/L}\), while a calculation using a verified administered-dose history gives \(C=0.8581\ \mathrm{mg/L}\). These are model-predicted concentrations at the same assessment time. Assume \(E_{\max}=8.00\) response units and \(EC_{50}=0.5000\ \mathrm{mg/L}\), and use the instantaneous Emax model.
The Emax assumptions are that the stated parameter values apply to both predictions and that response depends on the current concentration as specified, without hysteresis or a delay term. Under the prescribed history,
Under the administered history,
The difference is \(5.438-5.055=0.383\) response units, with the administered-history prediction about \(7.04\%\) lower than the prescribed-history prediction. The response difference is smaller in percentage terms than the concentration difference because the Emax curve is nonlinear and begins to flatten as concentration rises. These are conditional model predictions, not observed treatment effects.
Worked Example: A History Choice Changes an Estimated Exposure-Response Slope
Hypothetical data. Suppose one observed response is \(4.00\) units, and the analysis uses the zero-intercept model \(R=\beta C\), where \(\beta\) has units of response units per \(\mathrm{mg/L}\). The two candidate histories yield concentrations of \(1.061\ \mathrm{mg/L}\) (prescribed) and \(0.8581\ \mathrm{mg/L}\) (administered). For this illustration, assume the model form is appropriate, the observed response is treated as exact, and the only unknown is \(\beta\). With one observation, the fitted slope is the response divided by its model concentration.
Using the prescribed history gives
Using the administered history gives
The second estimate is \(4.661/3.770-1=0.236\), or about \(23.6\%\), higher. In this simple example the observed response is unchanged, so the lower concentration under the administered history requires a steeper fitted slope. With multiple observations and a richer model, parameters may shift in more complicated ways. This calculation demonstrates sensitivity to the input history; it does not establish which history is correct or imply that one slope is biologically true.
Choosing and Checking a History
The history used in an analysis should be selected according to the question and the evidence available. A prescribed history is appropriate for predicting outcomes under the intended regimen, provided the question is explicitly about that regimen. It is not a substitute for actual administration when estimating exposure experienced by patients. Reported histories can capture doses and timing unavailable elsewhere, but may be affected by recall or reporting. Electronic records can provide precise event times, yet an opening event, dispensing event, or scheduled entry need not establish ingestion or delivery of the full dose. Directly observed administration or a medication-administration record may provide stronger evidence in a particular setting, but should still be checked for completeness and correct time conventions.
Useful checks include comparing the sources for each dose rather than relying only on a single percentage-adherence summary; checking that units, date, and time zone are consistent; confirming that recorded amounts and intervals are plausible; and plotting dose events against concentration and response observations. A predicted concentration pattern that systematically disagrees with measured concentrations may signal an incorrect history, an inadequate PK model, or both. Concentration data alone do not necessarily identify which explanation is responsible. Where the evidence cannot resolve a discrepancy, calculate predictions under defensible alternative histories and report whether the conclusion changes.
| History source | What it represents | Important limitation |
|---|---|---|
| Prescribed regimen | Intended doses and schedule | Does not establish that doses were taken or administered |
| Patient report | Recalled or reported dose-taking behavior | May omit, mis-time, or misstate administrations |
| Electronic or other recorded events | Logged events associated with medication use or delivery | An event may not prove the delivered or absorbed amount |
| Verified administration record | Doses documented as administered in the relevant setting | May still have incomplete entries, errors, or ambiguous time conventions |
A common alternative is to summarize adherence as the fraction of prescribed doses recorded as taken. Such a summary can be useful descriptively, but it discards timing and sometimes amount. For example, taking every dose but several hours late can yield a 100% dose-count summary while producing different concentrations at a sampling time. Similarly, recording that a dose was taken does not distinguish a full from a partial dose unless amount is captured. When PK/PD predictions depend on concentration over time, dose-time histories are generally more informative than a single adherence percentage.
Common Mistakes and Limitations
- Using the prescription as if it proved administration. A regimen is an instruction. Use it for intended-regimen predictions, not as an unexamined proxy for exposure actually received.
- Treating a recorded event as proof of ingestion. Interpret the event according to what the device or record actually measures.
- Ignoring timing because the total dose is unchanged. In the elimination model, an earlier dose contributes \(D/V\exp\{-k(t-t_d)\}\), which depends on the administration time \(t_d\).
- Attributing every prediction discrepancy to adherence. Incorrect dose history is one possible explanation; structural PK/PD misspecification, parameter uncertainty, or data problems can also contribute.
- Claiming the history caused a biological response difference from model predictions alone. The examples compare conditional predictions. They do not by themselves identify causal effects or establish actual response outcomes.
The simple calculations have important limits. The one-compartment bolus model may not describe oral absorption, multicompartment distribution, infusion inputs, nonlinear PK, or time-varying clearance. The instantaneous Emax model will not describe delayed or indirect responses when those processes matter. In real applications, the history should be entered into the appropriate PK input model and uncertainty in both model parameters and the history should be considered. Sensitivity analyses across plausible histories are especially valuable when the source records disagree and the estimated exposure-response relationship is consequential.
Check Your Understanding
Answer each question, then compare your reasoning with the brief solutions.
- In the one-compartment bolus model, what happens to a dose’s contribution to concentration when the time since administration increases?
- A prescribed 80 mg dose is recorded as a 40 mg administration at the same time. Under linear PK, what fraction of the prescribed dose’s concentration contribution does the recorded dose produce?
- Why can an electronic container-opening record not automatically be treated as proof that a full dose was absorbed?
- In the Emax model, why might a 20% decrease in concentration produce less than a 20% decrease in predicted response?
- What is the main interpretive limitation of estimating an exposure-response slope from the single-observation example?
Solutions. (1) Its contribution declines as \(\exp(-ku)\), where \(u\) is elapsed time, so it becomes smaller for \(k>0\). (2) The contribution is proportional to dose under the assumed linear PK, so \(40/80=0.500\), or 50% of the prescribed-dose contribution. (3) Opening documents an event, not necessarily ingestion, complete delivery, or absorption. (4) The Emax curve is nonlinear and flattens toward \(E_{\max}\); response changes depend on the starting concentration relative to \(EC_{50}\). (5) One observation under a zero-intercept model determines the slope mechanically, so it provides no assessment of model fit, residual variability, or whether the linear model is appropriate.