Reconstructing Dosing Histories for Individual Forecasts

Why the Dosing History Is Part of the Forecast

An individual forecast is conditional not only on the pharmacokinetic parameters used, but also on the dosing history supplied to the model. A correct estimate of clearance cannot compensate for a dose entered at the wrong time, and an accurate concentration measurement may appear inconsistent with a forecast if the route or infusion duration has been recorded incorrectly. Reconstructing the history means translating the best available account of administrations into model inputs, not assuming that a prescribed regimen and the doses actually taken are identical.

The calculations below use simple linear pharmacokinetic models to make the consequences of history errors explicit. They are illustrations, not patient-specific dosing advice. A real forecast also depends on whether the structural model and individual parameter estimates are suitable for the patient and drug.

Definition: A dosing event is a model input describing an administered dose: its amount, route or delivery method, and time; for an infusion, its duration or rate must also be specified. A dosing history is the set of events supplied to the model over the relevant time period. An interval such as every 12 hours is a rule for generating scheduled times; it is not itself proof that each scheduled dose was administered.

Representing Events in a Pharmacokinetic Model

For a simple one-compartment model with linear elimination, let \(CL\) denote clearance, \(V\) the apparent volume of distribution, and \(k=CL/V\) the elimination rate constant. Assume \(CL\) and \(V\) are constant over the period considered. A model forecast sums the concentration contributions from the events in the history, provided the model is linear and each event is represented with the correct input function.

For an IV bolus of amount \(D\) given at time \(t_d\), the concentration contribution at a later time \(t\) is the initial concentration increment \(D/V\), multiplied by the fraction remaining after elimination:

C_{\mathrm{bolus}}(t)=\frac{D}{V}\exp[-k(t-t_d)],\qquad t\geq t_d.

For several boluses, add the contribution from every dose already given. A recorded interval is useful for constructing the event times: for example, a 500 mg dose every 12 hours starting at time zero generates event times 0, 12, 24, and so on, in hours. If actual administration times are available, those times are generally the appropriate inputs rather than a nominal schedule.

An infusion is not an instantaneous bolus. If amount \(D\) is infused at a constant rate over duration \(T\), the rate is \(R=D/T\). After the infusion ends at \(t_d+T\), its contribution is:

C_{\mathrm{infusion}}(t)=\frac{D}{CL\,T}\left(1-\exp[-kT]\right)\exp[-k(t-t_d-T)],\qquad t\geq t_d+T.

For an oral dose under a first-order absorption model, let \(F\) be bioavailability and \(k_a\) the absorption rate constant. If \(k_a\neq k\), the concentration contribution at time \(t\) after a dose at \(t_d\) is:

C_{\mathrm{oral}}(t)=\frac{F D k_a}{V(k_a-k)} \left\{\exp[-k(t-t_d)]-\exp[-k_a(t-t_d)]\right\},\qquad t\geq t_d.

These expressions use the stated model assumptions: one well-mixed compartment, linear pharmacokinetics, and the specified input and absorption forms. They do not imply that different routes are interchangeable. Route affects how and when drug enters the model; an oral dose must not be entered as an IV bolus simply because both records list an amount in milligrams.

Why Superposition Works

The event-summing calculation follows from the linear one-compartment equation. Between bolus doses, the concentration satisfies \(dC/dt=-kC\), whose solution over elapsed time \(u\) is \(C(u)=C(0)\exp(-ku)\). An IV bolus adds \(D/V\) instantaneously, so its later contribution is the bolus expression above. Because the differential equation is linear, the solution for a sum of bolus inputs is the sum of their individual solutions.

For a constant-rate infusion beginning at time zero, the equation is \(dC/dt=R/V-kC\), with \(C(0)=0\). Multiplying through by the integrating factor \(\exp(kt)\) and integrating from 0 to \(T\) gives:

\exp(kT)C(T)=\frac{R}{V}\int_0^T\exp(ku)\,du =\frac{R}{Vk}\left(\exp(kT)-1\right).

Since \(Vk=CL\) and \(R=D/T\), this becomes \(C(T)=D(1-\exp[-kT])/(CL\,T)\). After the infusion ends, the concentration declines exponentially, yielding the post-infusion expression. This derivation also shows why changing the duration changes both the rate during infusion and the time available for elimination before a later sample.

Worked Example: Reconstructing a Repeated-Dose Interval

Hypothetical data. A patient is documented as receiving 500 mg IV every 12 hours, with the first administration at time zero. A concentration is forecast at hour 30. Use \(V=50\) L and \(CL=5\) L/h, so \(k=5/50=0.1000\) h\(^{-1}\). Assume instantaneous IV boluses and linear one-compartment disposition.

The event times before hour 30 are 0, 12, and 24 hours. Each dose has \(D/V=500/50=10.00\) mg/L as its initial concentration contribution. Therefore:

\begin{aligned} C(30) &=10.00\left\{\exp[-0.1000(30)] +\exp[-0.1000(18)] +\exp[-0.1000(6)]\right\}\\ &=10.00(0.049787+0.165299+0.548812)\\ &=7.639\ \text{mg/L}. \end{aligned}

If the history were mistakenly entered as every 24 hours, the modeled events before hour 30 would instead be at 0 and 24 hours. The forecast would be \(10.00(0.049787+0.548812)=5.986\) mg/L. That is 1.653 mg/L lower than the forecast from the documented 12-hour interval. This comparison isolates the effect of changing the interval; it does not establish which schedule was actually followed.

How Common Recording Errors Change Forecasts

A history error changes the input to the prediction model. The direction and size of its effect depend on the model, the sample time, and the error itself. The next examples hold pharmacokinetic parameters fixed so that the numerical difference can be attributed to the history entry.

Worked Example: A Dose Entered at the Wrong Time

Hypothetical data. A 500 mg IV bolus is actually administered at hour 0, and a concentration is forecast at hour 8. Let \(V=50\) L and \(CL=5\) L/h, so \(k=0.1000\) h\(^{-1}\). Assume the bolus model applies and there are no other relevant doses.

With the correct administration time, the elapsed time is 8 hours:

C_{\mathrm{correct}}(8)=\frac{500}{50}\exp[-0.1000(8)] =10.00(0.449329)=4.493\ \text{mg/L}.

Suppose the recorded time is hour 2 instead of hour 0. The model now treats the dose as only 6 hours before the forecast:

C_{\mathrm{recorded}}(8)=10.00\exp[-0.1000(6)] =10.00(0.548812)=5.488\ \text{mg/L}.

The incorrect time raises the forecast by \(5.488-4.493=0.995\) mg/L, or about 22.1% relative to the correct-history forecast. The parameter values did not change: the discrepancy is caused by the two-hour shift in the event time.

Worked Example: Entering an Oral Dose as an IV Bolus

Hypothetical data. A 500 mg oral dose was given at time zero. At hour 2, forecast the concentration using \(F=0.80\), \(k_a=1.00\) h\(^{-1}\), \(V=50\) L, and \(CL=5\) L/h, giving \(k=0.1000\) h\(^{-1}\). Assume the first-order absorption model is appropriate and that \(k_a\neq k\).

Using the oral input model:

\begin{aligned} C_{\mathrm{oral}}(2) &=\frac{(0.80)(500)(1.00)}{50(1.00-0.1000)} \left\{\exp[-0.1000(2)]-\exp[-1.00(2)]\right\}\\ &=8.8889(0.818731-0.135335)\\ &=6.075\ \text{mg/L}. \end{aligned}

If the same 500 mg entry is incorrectly treated as an IV bolus, the prediction is:

C_{\mathrm{IV}}(2)=\frac{500}{50}\exp[-0.1000(2)] =10.00(0.818731)=8.187\ \text{mg/L}.

The route error increases this forecast by \(8.187-6.075=2.112\) mg/L, about 34.8% relative to the oral-model forecast. The difference arises because the oral model includes incomplete bioavailability and absorption over time, while the IV bolus model places the full amount directly into the central compartment at time zero.

Worked Example: Recording the Wrong Infusion Duration

Hypothetical data. A 600 mg IV infusion starts at hour 0 and actually runs for 2 hours. Forecast the concentration at hour 5 using \(CL=6\) L/h, \(V=60\) L, and \(k=0.1000\) h\(^{-1}\). Assume a constant infusion rate, linear one-compartment disposition, and no other doses.

The correct rate is \(600/2=300\) mg/h. The infusion ends at hour 2, so the sample is 3 hours after its end:

\begin{aligned} C_{\mathrm{correct}}(5) &=\frac{600}{(6)(2)}(1-\exp[-0.1000(2)])\exp[-0.1000(3)]\\ &=50(1-0.818731)(0.740818)\\ &=6.714\ \text{mg/L}. \end{aligned}

If the duration is incorrectly recorded as 1 hour, the model uses a rate of 600 mg/h, an infusion end time of hour 1, and 4 hours of post-infusion decline:

\begin{aligned} C_{\mathrm{recorded}}(5) &=\frac{600}{(6)(1)}(1-\exp[-0.1000(1)])\exp[-0.1000(4)]\\ &=100(1-0.904837)(0.670320)\\ &=6.379\ \text{mg/L}. \end{aligned}

The incorrect duration lowers the forecast by \(6.714-6.379=0.335\) mg/L, approximately 5.00% of the correct-history forecast. The intermediate value \(0.904837\) is retained here so that the displayed arithmetic supports the reported result.

Reconstruction, Diagnostics, and Limitations

A practical reconstruction starts by listing each relevant event with its amount, route, date and time, and, when applicable, infusion duration. Convert times to one consistent scale relative to the forecast or observation time. Use an interval to generate scheduled times only when that is the information available, and distinguish scheduled events from confirmed administrations. Check units as well: an amount in milligrams, an infusion rate in milligrams per hour, and a concentration in milligrams per litre are different quantities.

Before using a forecast, check whether the model-generated event sequence agrees with the available record. Look for impossible orderings, duplicate entries, omitted doses, inconsistent dates, implausible infusion durations, or a route that conflicts with the formulation. A useful sensitivity check is to calculate forecasts under a small number of plausible alternative records, such as the two possible administration times documented in separate sources. This shows how dependent the prediction is on the history without claiming that either alternative is known to be true.

A recurring-schedule shortcut can be convenient when doses are genuinely regular and the model assumptions support it. An explicit event-by-event history is more flexible when times vary or the record includes an interruption. Either approach can be wrong if it encodes a prescribed schedule as administered doses without evidence. Likewise, a simple bolus model may be inappropriate for an infusion, an oral formulation, nonlinear pharmacokinetics, or a drug requiring a more complex compartmental description. More elaborate history models may represent uncertainty in adherence or dose times, but a deterministic reconstruction is not itself such a model.

The forecast differences calculated here are conditional comparisons: they show what the specified model predicts under alternative inputs. They are not confidence intervals, and they do not measure all uncertainty in the patient’s parameters, residual error, or structural model. A precise numerical output can therefore still be misleading when the event record is uncertain or the input model is unsuitable.

Common Mistakes

Substituting the prescribed interval for actual administration times. A q12h instruction generates scheduled times; it does not verify that doses were taken on that schedule.
Treating amount as the only relevant input. The same amount can produce different concentration-time profiles when route, absorption, or infusion duration differs.
Using the sample time as the dose time. Concentration contributions depend on elapsed time since each administration, so dose and sample timestamps must be separately identified.
Assuming a changed history proves a changed pharmacokinetic parameter. A mismatch may result from the recorded input rather than clearance or volume; the history and model should be checked before interpreting the mismatch.
Reporting arithmetic that uses rounded intermediates as though they were exact. Retain extra digits through calculations, then round the final result consistently with the input precision.

Key takeaway: An individual forecast is conditional on a model-ready dosing history. Encode each event’s amount, route, timing, and infusion duration correctly, and check how plausible recording errors change the predicted concentration before relying on the result.

Check Your Understanding

Answer each question, then compare with the worked solutions below.

A 250 mg IV bolus is given at hour 0. With \(V=25\) L and \(k=0.0800\) h\(^{-1}\), what is its predicted concentration contribution at hour 5?
A 400 mg dose is documented every 8 hours, beginning at time zero. List the scheduled event times before hour 20, and explain why the list does not establish actual administrations.
For a 200 mg IV infusion over 2 hours with \(CL=4\) L/h and \(k=0.1000\) h\(^{-1}\), calculate the concentration at the end of infusion, assuming no other doses.
In the preceding infusion scenario, what modeling input changes if the record says the same dose was infused over 4 hours instead? Name two consequences for the forecast.

Solutions. (1) The initial contribution is \(250/25=10.00\) mg/L. At hour 5, \(C=10.00\exp[-0.0800(5)]=10.00(0.670320)=6.703\) mg/L. (2) The scheduled times are 0, 8, and 16 hours. They follow from the stated interval and start time, but actual administration requires confirmation from the dosing record or another reliable source. (3) The rate is \(R=200/2=100\) mg/h and \(V=CL/k=4/0.1000=40\) L. Thus \(C(2)=(R/CL)(1-\exp[-k(2)])=(100/4)(1-\exp[-0.2000])=25(0.181269)=4.532\) mg/L. (4) The duration input changes from 2 to 4 hours while the total amount remains 200 mg. The modeled rate falls from 100 to 50 mg/h, and the infusion ends later; both the concentration accumulated during infusion and the time available for elimination before a given later sample can therefore change.