Common Interpretation Errors In Sparse Repeated-Dose Data

Why Sparse Repeated-Dose Profiles Are Easy to Misread

A concentration measured just before a scheduled dose can be useful, but it is not a compressed record of everything that happened during the preceding dosing interval. It is one value at one time. A peak, an interval average, and the area under the concentration-time curve (AUC) are different summaries, and a single trough does not generally determine them. The interpretation becomes still less secure if the actual administration time differs from the recorded or scheduled time.

The calculations here use a deliberately simple reference model: instantaneous intravenous boluses, one compartment, constant volume $V$, and first-order elimination with rate constant $k>0$. Between doses, concentration follows $dC/dt=-kC$. This model is a transparent way to expose interpretation errors; its assumptions are not claims that every sparse profile follows this model.

Definition: A trough is a concentration measured immediately before a dose. In repeated-dose data, a valid pre-dose trough is defined relative to the actual administration time. A sample taken just before the scheduled time is not necessarily a trough if administration was delayed, advanced, or otherwise different from the schedule.

Let doses occur at times $t_j$, with dose $D_j$, and let $A_j=D_j/V$ denote the concentration increment from dose $j$. A trough is therefore not defined by its concentration alone: its timing relative to the dose history matters. It also matters whether a reported time is a scheduled time, an administration time, or a sample-collection time.

Concentration After Repeated Boluses

For this model, each dose contributes a concentration that decays exponentially after administration. Adding the contributions gives the concentration at any time $t$ after the dose history has begun.

Result: If the concentration is initially zero and doses are instantaneous boluses at times $t_j$, then $$ C(t)=\sum_{j:t_j\leq t} A_j e^{-k(t-t_j)}. $$ This formula assumes the same linear elimination rate $k$ applies throughout and that concentration changes only by elimination between boluses.

To justify the formula, consider a time $t$ between consecutive doses. The concentration present at the previous dose time decays by the factor $e^{-k\Delta t}$, from the solution of $dC/dt=-kC$. At a dose time, the concentration increases by $A_j$. Thus each dose’s increment decays by that same factor from its own administration time. Adding these increments gives the stated sum. This derivation is also a reminder that the concentration at a trough reflects residual contributions from multiple doses, not just the most recent dose.

Worked Example: Calculating a Steady-State Trough

Suppose a one-compartment model has $k=\ln(2)/6$ per hour, so the elimination half-life is 6 hours. A bolus adds $A=3$ mg/L, and doses are given every 12 hours. Find the steady-state pre-dose trough and post-dose concentration.

At steady state, the residual fraction after one 12-hour interval is $$ r=e^{-k\tau}=e^{-(\ln(2)/6)(12)}=e^{-2\ln(2)}=\frac{1}{4}. $$

Immediately after a dose, the current increment $A$ is added to the residual from all earlier doses. The residuals form a geometric series: $$ C_{\text{post}}=A(1+r+r^2+\cdots)=\frac{A}{1-r} =\frac{3}{1-1/4}=4\ \text{mg/L}. $$

Just before the next dose, this concentration has decayed for 12 hours: $$ C_{\text{trough}}=C_{\text{post}}r=4\left(\frac{1}{4}\right)=1\ \text{mg/L}. $$ Equivalently, the pre-dose residual series is $A(r+r^2+\cdots)=Ar/(1-r)=1$ mg/L. Both calculations agree.

A Trough Is Not an Interval Summary

For constant bolus size $A$ and interval $\tau$, write $r=e^{-k\tau}$. At steady state, the post-dose concentration is $A/(1-r)$, and during the interval it decays exponentially. Integrating that curve gives the interval AUC:

Derived result: Under the stated one-compartment, repeated-bolus assumptions, the steady-state interval AUC and interval-average concentration are $$ \operatorname{AUC}_{0,\tau} =\int_0^\tau \frac{A}{1-e^{-k\tau}}e^{-kt}\,dt =\frac{A}{k}, \qquad C_{\text{average}}=\frac{A}{k\tau}. $$ The pre-dose trough is $C_{\text{trough}}=Ar/(1-r)$, which is a different quantity.

For the AUC derivation, integrate the exponential profile: $$ \int_0^\tau \frac{A}{1-e^{-k\tau}}e^{-kt}\,dt =\frac{A}{1-e^{-k\tau}}\frac{1-e^{-k\tau}}{k} =\frac{A}{k}. $$ Dividing by the interval length $\tau$ gives the average concentration. The trough is the endpoint value just before the next dose, not this integral or its average. The AUC expression is consistent with dose divided by clearance, since clearance in this model is $kV$ and $A/k=(D/V)/k=D/(kV)$.

Worked Example: The Same Trough, Different Interval Profiles

Compare two steady-state regimens with $\tau=12$ hours. In the first, $k=\ln(2)/6$ per hour and $A=3$ mg/L. In the second, $k=\ln(2)/12$ per hour and $A=1$ mg/L. Calculate each trough, post-dose concentration, and interval-average concentration.

First regimen. Here $r=e^{-k\tau}=1/4$. Therefore, $$ C_{\text{trough}}=\frac{Ar}{1-r} =\frac{3(1/4)}{3/4}=1\ \text{mg/L}, \qquad C_{\text{post}}=\frac{A}{1-r}=\frac{3}{3/4}=4\ \text{mg/L}. $$ Its interval average is $$ C_{\text{average}}=\frac{A}{k\tau} =\frac{3}{2\ln(2)} \approx 2.164\ \text{mg/L}. $$

Second regimen. Now $r=e^{-\ln(2)}=1/2$, so $$ C_{\text{trough}}=\frac{1(1/2)}{1/2}=1\ \text{mg/L}, \qquad C_{\text{post}}=\frac{1}{1/2}=2\ \text{mg/L}. $$ The interval average is $$ C_{\text{average}}=\frac{1}{\ln(2)} \approx 1.443\ \text{mg/L}. $$

Both troughs equal 1 mg/L, but the post-dose concentrations are 4 and 2 mg/L, and the interval averages differ. The trough alone cannot distinguish these profiles. This is a concrete counterexample to treating a trough as a complete summary of the interval.

Dose-Time Uncertainty Changes What a Sample Means

A recorded schedule and the actual dose history are not interchangeable. If a dose is administered later than expected, then a sample collected at the scheduled pre-dose time is closer to the actual previous dose than the schedule suggests. Under elimination, a shorter elapsed time means less decay and therefore a higher concentration, all else equal.

Worked Example: A One-Hour Administration Delay

Use the first regimen above: $k=\ln(2)/6$ per hour, $\tau=12$ hours, and $A=3$ mg/L. Under a regular schedule, the sample at hour 12 is immediately before the next dose, and the trough is 1 mg/L. Suppose instead that the whole regular dosing schedule is shifted one hour later, while the sample is still collected at hour 12. What concentration does the model predict?

With doses at hours $1,13,25,\ldots$, the doses contributing at hour 12 were given at hours $1,-11,-23,\ldots$. Their elapsed times are $11,23,35,\ldots$ hours. Thus $$ C(12)=3\left(e^{-11k}+e^{-23k}+e^{-35k}+\cdots\right) =\frac{3e^{-11k}}{1-e^{-12k}}. $$

Under the nominal schedule, the corresponding trough is $3e^{-12k}/(1-e^{-12k})=1$ mg/L. The ratio of the shifted-schedule concentration to that nominal trough is $$ \frac{3e^{-11k}/(1-e^{-12k})}{3e^{-12k}/(1-e^{-12k})} =e^k =e^{\ln(2)/6} \approx 1.122. $$ Therefore $C(12)\approx1.122$ mg/L, about 12.2% higher than the nominal trough. This difference is caused by the timing shift in this model, not by a change in dose size or elimination rate.

Real records may contain uncertainty in both dose times and sample times. A sample labelled “12 hours” could mean 12 hours after the scheduled dose, while the elapsed time since the actual dose is different. The distinction is especially important when concentrations change appreciably over the size of the timing discrepancy. A small clock-time error may have little impact for slow elimination and a larger impact for faster elimination; its effect should be evaluated relative to the model’s time scale, not assumed negligible.

Diagnostics, Alternatives, and Limitations

Before interpreting a sparse concentration as a trough or comparing it across occasions, check the records and the model assumptions. In practice, the most useful diagnostic is often a timeline that places actual administrations and sample collections on the same clock. If only scheduled dose times are available, state that limitation rather than silently treating them as actual times.

Check the event order. Confirm whether the sample was collected before or after the relevant administration, and calculate elapsed time from the actual dose if known.
Check interval consistency. A sample at the same nominal clock time may occur at different elapsed times after a dose when dosing intervals vary or doses are delayed.
Check what is being summarized. A trough is a point value; an interval average requires an interval profile or a model-based calculation; AUC is an integral.
Check model fit and context. A one-compartment exponential decline may be inadequate with absorption, infusion, multiple compartments, changing clearance, or other departures from the assumptions.

When the goal is to characterize a whole interval, alternatives include collecting samples at multiple informative times, estimating a model-based profile with suitable dose and sample timestamps, or reporting a quantity designed for the question, such as AUC or interval average. These approaches are not interchangeable: a model-based estimate relies on model assumptions, while a trough requires fewer measurements but answers a narrower question. Dense sampling can reveal more of a profile, but it does not repair inaccurate dose histories or automatically distinguish sources of variability.

A central limitation is identifiability. Sparse observations can be compatible with multiple combinations of dose timing, elimination rate, dose contribution, and other model features. The examples show that even under a very simple model, a trough does not uniquely determine peak or average concentration. In a more complex pharmacokinetic setting, additional uncertainty can make the ambiguity greater. Avoid making a stronger claim than the data and assumptions support.

Common mistake: “The sample was collected at the scheduled pre-dose time, so it is a trough and summarizes exposure over the interval.” This reasoning confuses the scheduled clock time with the actual dose history and confuses a point measurement with an interval summary. Check the actual administration and sample times; call the value a pre-dose trough only when it was collected immediately before the actual dose, and use an interval-based quantity when the question concerns the whole interval.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions below.

For $A=2$ mg/L, $k=\ln(2)/4$ per hour, and $\tau=8$ hours, calculate $r$, the steady-state trough, and the post-dose concentration.
For the regimen in Question 1, calculate the interval AUC and interval-average concentration. Explain why the trough is not either of those quantities.
A sample is collected at the scheduled pre-dose time, but the preceding dose was administered one hour later than scheduled. Is the sample necessarily a trough? Explain.
Under a regular repeated-bolus schedule, what additional information is needed to calculate the concentration at an arbitrary time from the dose-history formula?

Solutions. (1) $r=e^{-k\tau}=e^{-2\ln(2)}=1/4$. Hence $C_{\text{trough}}=Ar/(1-r)=2(1/4)/(3/4)=2/3$ mg/L, and $C_{\text{post}}=A/(1-r)=2/(3/4)=8/3$ mg/L. (2) The interval AUC is $A/k=2/(\ln(2)/4)=8/\ln(2)\approx11.54$ mg·h/L. The average is $(A/k)/\tau=1/\ln(2)\approx1.443$ mg/L. The trough, $2/3$ mg/L, is the concentration at one endpoint just before a dose; it is not an integral or an average. (3) No. The sample’s relation to the actual administration determines whether it is immediately pre-dose; a scheduled time alone does not establish that. (4) One needs the elimination rate $k$, the dose increments $A_j$, and the actual times $t_j$ of all doses that contribute, as well as the time $t$ at which concentration is sought. Without reliable timing or adequate model information, the calculated profile may not be identifiable from sparse concentrations alone.