Integrated Case Workshop: Missed Doses and Changing Renal Function

Why an Integrated Event History Matters

A concentration prediction depends on more than the prescribed regimen. It depends on the administrations that actually occurred and on how the patient’s physiology changed between them. When a dose may have been skipped and renal function has been measured only intermittently, the analyst must make both histories explicit before calculating exposure. Otherwise, a mismatch between observations and predictions can be incorrectly attributed to the pharmacokinetic model.

This case uses hypothetical data and a deliberately simple one-compartment model. Its purpose is to show how to construct and compare internally consistent event histories, not to prescribe a dose or establish an individual patient’s true history from limited observations. The population model parameters are treated as fixed for the calculations; no parameter estimation is performed.

For a dose at time \(t_j\), the concentration contribution remaining at time \(t\geq t_j\) is the dose divided by volume, multiplied by the exponential of the negative cumulative elimination since administration. Contributions from all doses are added. The cumulative elimination is the integral of the time-varying rate constant; it is not generally equal to one constant rate multiplied by elapsed time.

This expression follows by solving \(dC/dt=-k(t)C\) between events: separating variables gives \(dC/C=-k(t)\,dt\), and integration from \(t_j\) to \(t\) gives the exponential survival factor. At a bolus event, concentration increases by \(D_j/V\). Thus the concentration immediately after the dose is the concentration immediately before it plus \(D_j/V\). An observation recorded at exactly the dose time must be identified as pre-dose or post-dose; the two values differ by this increment.

Worked Case: Missed Dose With Stepwise Renal Function

Suppose the hypothetical patient is prescribed 100 mg intravenous boluses every 12 hours, beginning at hour 0. The recorded administrations are 100 mg at hours 0, 12, 36, and 48. The dose scheduled for hour 24 is confirmed to have been missed. eGFR measurements are 60 mL/min/1.73 m² at hour 0, 40 at hour 24, and 30 at hour 48. Use the stepwise convention in the model definition: 60 applies from hour 0 up to 24, 40 from hour 24 up to 48, and 30 from hour 48 onward. The resulting elimination rates are 0.100, 0.06667, and 0.05000 h\(^{-1}\), respectively.

The calculation makes the event history visible: there is no hour-24 dose term. A missing record and a confirmed missed administration are not interchangeable facts. If the record is uncertain rather than confirmed, it is more defensible to calculate a separate history that includes the possible administration than to silently insert a dose into the primary history.

Comparing Alternative Histories

An alternative history should change only the assumption under investigation. Here, one plausible alternative is that the hour-24 dose was actually administered but was absent from the record. Keep the same renal-function history and all other doses. At hour 60, this extra dose has experienced 24 hours at \(k=0.06667\) and 12 hours at \(k=0.05000\), for cumulative elimination \(1.60+0.6000=2.20\). Its predicted contribution is \(5.00e^{-2.20}=0.5540\) mg/L. The alternative-history prediction is therefore \(4.194+0.5540=4.748\) mg/L. Relative to the missed-dose prediction, this is about \(0.5540/4.194=13.21\%\) higher.

That difference is conditional on the stipulated dose, volume, and renal-effect model. It is not a probability that the dose was taken, nor does a higher prediction establish which history occurred. The value of the comparison is that it quantifies how much the uncertain event matters under explicit assumptions.

Covariate History Is Also an Assumption

The preceding calculations use a stepwise eGFR history. A different, also explicit, choice is linear interpolation between the recorded measurements. Interpolation assumes the covariate changed smoothly between measurement times; it does not assert that the unobserved renal function is known. Since clearance is proportional to eGFR in this example, interpolation changes the integrated elimination and therefore changes the concentration contributions from earlier doses.

Exposure Over an Interval

Concentration at one time is not the only useful comparison. For an interval with constant elimination rate \(k\), concentration declines exponentially, so the area under the concentration-time curve (AUC) from \(t_0\) to \(t_1\) is \(C(t_0)\{1-e^{-k(t_1-t_0)}\}/k\). This follows by integrating \(C(t_0)e^{-k(t-t_0)}\) over the interval. The units are concentration multiplied by time, here mg·h/L.

Under the missed-dose, stepwise history, the post-dose concentration at hour 48 is approximately 7.642 mg/L. The rate from hour 48 to 60 is constant at 0.05000 h\(^{-1}\), so AUC over that interval is approximately \(7.642273(1-e^{-0.6000})/0.05000=68.96\) mg·h/L. If the hour-24 dose had instead been administered, it would add \(5.00e^{-1.60}=1.009483\) mg/L to the hour-48 pre-dose concentration and therefore to the post-dose concentration. The corresponding AUC increment from hour 48 to 60 is \(1.009483(1-e^{-0.6000})/0.05000=9.109\) mg·h/L. The alternative history’s AUC is thus about 78.07 mg·h/L over this interval. This AUC comparison, like the concentration comparison, is conditional on the same structural assumptions.

Diagnostics, Limitations, and Common Pitfalls

A reliable reconstruction begins with a timeline containing both administrations and observations. Check the event time, dose amount, units, and whether each concentration is pre-dose or post-dose. Confirm that each covariate value is applied over the intended interval and that cumulative elimination includes every portion of the interval since the relevant dose. A simple numerical check is that a dose’s contribution must decline as its cumulative elimination increases, and contributions from doses at the same time add linearly in this model.

• Do not convert an absent record automatically into a missed dose. Distinguish confirmed omission, uncertain administration, and missing documentation. Calculate alternative histories when the record cannot resolve the event.
• Do not apply a new covariate value before its measurement time without stating an interpolation rule. Stepwise carry-forward and interpolation are different assumptions and can produce different predictions.
• Do not compare a pre-dose observation with a post-dose prediction. At a bolus time, the model has a jump of \(D/V\); align observation and prediction on the same side of the event.
• Do not treat a closer prediction as proof of history. Residuals also reflect measurement error and model inadequacy. Repeated residual patterns, event-record review, and sensitivity to plausible histories are more informative than a single comparison.
• Do not overinterpret the renal relationship. The proportional eGFR-clearance relation and parameter values here are hypothetical. Real applications require an appropriate model, relevant units, and evaluation of whether renal measurements adequately represent the physiology affecting clearance.

An alternative to a single assumed history is a sensitivity analysis that reports predictions over plausible event and covariate histories. A richer model could represent gradual renal change, uncertainty in parameters, or random effects, but those additions do not remove the need to establish event ordering and define how covariates are handled between measurements. This workshop’s calculations isolate the impact of those histories while holding the structural parameters fixed.