Why Dose-Relative Time Is Not the Whole Story
A blood sample collected six hours after a dose has a clear dose-relative time, but that timing alone does not determine its interpretation. During a multiple-dose regimen, the measured concentration can include contributions from several earlier doses. A sample six hours after the first dose and one six hours after a dose in a repeated regimen can therefore differ even when the dose and the time since the most recent dose are identical.
This matters when planning sampling schedules. A single-dose profile can show how concentrations change after one administration. Repeated-dose samples describe concentrations under a continuing regimen, including accumulation and the changing profile across a dosing interval. To interpret either type of sample, the analyst needs the dose timing and enough information about the preceding dosing history.
A Simple Model for Accumulation
To make the effect of repeated dosing explicit, consider a hypothetical one-compartment model with instantaneous intravenous bolus doses, first-order elimination, a constant dose \(D\), a constant dosing interval \(\tau\), and linear pharmacokinetics. Let \(V\) be the apparent volume of distribution and \(k\) the elimination rate constant. Between doses, concentration declines exponentially. After a single dose, at elapsed time \(t\), the model predicts \(C_1(t)\):
Here \(D/V\) has concentration units and \(k\) has units of inverse time. For repeated doses, the contributions from prior doses add under the model's linearity assumption. At time \(t\) after a dose, where \(0\leq t<\tau\), a regimen that has reached steady state has contributions from the current dose and all preceding doses:
The factor \(e^{-k\tau}\) is the fraction of a dose's concentration contribution that remains after one dosing interval. Since \(k>0\) and \(\tau>0\), it lies between zero and one, so the geometric series converges. The ratio of the steady-state concentration to the single-dose concentration at the same dose-relative time is therefore:
This ratio is the model-specific accumulation factor. It does not depend on \(t\) in this simple model, but the absolute concentration does. This result assumes identical doses at regular intervals, linear superposition, constant \(k\) and \(V\), and a regimen continued long enough to approach steady state. It is not a universal multiplier for every drug or formulation.
Worked Example: Comparing Matched Times After One Dose and at Steady State
Suppose a hypothetical drug is given as an 8 mg intravenous bolus every 12 hours. Let its elimination half-life be 12 hours and let \(D/V=8\) mg/L. Compare concentrations six hours after a single dose and six hours after a dose in a steady-state regimen.
The elimination rate constant is \(k=\ln(2)/12=0.05776\ \mathrm{h}^{-1}\). Over one interval, \(e^{-k\tau}=e^{-0.05776(12)}=0.5000\). At six hours, \(e^{-k(6)}=e^{-0.05776(6)}=0.7071\).
For the single dose:
For the repeated regimen at steady state:
At the same six-hour dose-relative time, the steady-state prediction is \(5.656854\ldots\) mg/L higher using unrounded model values (5.6569 mg/L to four decimal places, or about 5.657 mg/L). The difference arises from residual contributions from earlier doses, not from a different elapsed time since the most recent dose.
Covering the Interval, Not Just One Convenient Time
A repeated-dose concentration changes throughout each dosing interval. Under the bolus model, it is highest immediately after a dose and declines until just before the next dose. A schedule that samples only near one part of the interval may describe that part well while providing little direct information about the rest. Sampling near the beginning, middle, and end of the interval gives a more interpretable picture of the within-interval profile.
The precise times should suit the question and the expected concentration-time shape. Early samples may be useful for characterizing the post-dose profile; samples later in the interval help describe the decline and the pre-dose concentration. The earlier material on peak and trough samples for exposure characterization discusses those sampling roles. Here the key point is that repeated-dose samples must be located within the interval and interpreted in relation to the accumulated concentration profile.
Worked Example: Interpreting Samples Across a Dosing Interval
Use the hypothetical regimen from the preceding example: an 8 mg bolus every 12 hours, a 12-hour half-life, and \(D/V=8\) mg/L. At steady state, calculate the predicted concentration immediately after a dose, six hours later, and immediately before the next dose.
The accumulation factor is \(1/(1-0.5)=2\). Immediately after the dose, \(t=0\), so:
At six hours, the remaining fraction from each contribution is \(e^{-k(6)}=0.7071\):
Immediately before the next dose, \(t\) approaches 12 hours from below. The fraction remaining over the interval is 0.5000:
The post-dose and pre-dose predictions are not interchangeable: they are 16.00 and 8.000 mg/L, respectively. A sample taken just before the next dose is a trough sample for this regimen; a sample soon after dosing is near the post-dose peak in this instantaneous-bolus model. In an actual study, sample collection times and dose times would need to be known accurately to classify samples this way.
Accumulation Develops Over Successive Doses
Steady state is approached rather than reached immediately. After a finite number of doses, the concentration includes only a finite number of prior-dose contributions. If \(m\) doses have been given at regular intervals and the time \(t\) is measured after the most recent dose, then:
This follows from the finite geometric-series identity. As \(m\) increases, \(e^{-mk\tau}\) approaches zero, leaving the steady-state expression. The number of elapsed half-lives is a useful rough guide to the extent of accumulation, but the exact fraction also depends on the dosing interval.
Worked Example: How Close Is the Profile to Steady State?
Continue with the hypothetical 12-hour half-life and 12-hour dosing interval. At six hours after a dose, the single-dose contribution is 5.657 mg/L. Since \(e^{-k\tau}=0.5\), the concentration after three doses is:
The steady-state concentration at this time is 11.31 mg/L. Thus the exact model ratio is \(1.75/2=0.875\), or 87.50% of the steady-state prediction. After four doses:
This is 93.75% of the steady-state value, because the four-dose sum is \(1-0.5^4=0.9375\) of the infinite sum. The example shows why the number and timing of prior doses matter: a sample at six hours is not necessarily a steady-state sample simply because it was collected six hours after a dose.
Design Checks, Alternatives, and Limits
Before interpreting repeated-dose samples, check that the dose-relative times are based on actual dosing times and that the preceding regimen is understood. A schedule described only as “six hours after dose” can be misleading if doses were delayed, omitted, or changed, or if participants had received different numbers of doses. Likewise, a nominal steady-state designation should be supported by a regimen long enough for the relevant elimination rate and interval, rather than assumed from the calendar alone.
For a sparse design, sampling can be distributed across participants so that the combined data cover different parts of the dosing interval. This uses the principle of staggered sampling across the dosing interval, but does not make each participant's samples a complete individual profile. The schedule should still include adequate representation of important interval regions; a large number of samples clustered at one dose-relative time does not provide equivalent coverage.
| Sampling approach | What it can show | Main interpretive caution |
|---|---|---|
| Single-dose sampling | Concentration change after one dose without prior-dose accumulation | May not represent exposure under the continuing regimen |
| Repeated-dose sampling at one interval time | Concentrations at a specified point in the regimen | Does not by itself describe the full interval profile |
| Repeated-dose sampling across the interval | Variation from post-dose concentrations toward trough | Requires dose history and interval timing to be interpretable |
The one-compartment bolus equations are a teaching model, not a general formula for every multiple-dose study. Oral absorption, infusions, nonlinear pharmacokinetics, changing clearance, and variable dosing intervals can alter the profile and the accumulation pattern. In those settings, dose-relative sampling remains useful, but predictions require a model that represents the relevant input and elimination processes. A concentration difference between single-dose and repeated-dose samples should not be attributed to accumulation alone if the formulation, dose, subject characteristics, or sampling conditions also differ.
Common Pitfalls
- Assuming equal time since the last dose means equal concentration. This ignores residual contributions from earlier doses. Compare samples only after considering the dose history and regimen state.
- Calling every sample after repeated dosing a steady-state sample. Steady state is approached over repeated intervals; the number of doses and elimination rate matter.
- Sampling at one time and inferring the whole interval. A single interval location cannot directly establish the shape or extremes elsewhere in the interval.
- Using the bolus accumulation factor for a different input model without justification. The formula depends on instantaneous bolus input, linear superposition, and regular dosing; other input patterns require appropriate equations.
Check Your Understanding
Answer each question, then compare with the solutions below.
- In the simple bolus model, if \(e^{-k\tau}=0.25\), what is the steady-state accumulation factor?
- For a regimen with a 12-hour half-life and a 12-hour interval, how does the steady-state concentration six hours after a dose compare with the single-dose concentration at six hours?
- In that same regimen, the predicted steady-state concentration immediately after a dose is 16 mg/L. What is the predicted concentration immediately before the next dose?
- Why is a design that collects every sample at the same dose-relative time limited, even if it includes many participants?
Solutions. (1) The factor is \(1/(1-0.25)=1.333\). (2) The factor is \(1/(1-0.5)=2\), so the steady-state concentration is twice the single-dose concentration at six hours. (3) Half the post-dose concentration remains over one half-life, so \(16(0.5)=8.000\) mg/L. (4) It estimates concentrations at that interval location, but does not directly cover the profile at other times, such as near the post-dose peak or pre-dose trough.