Why Timing Can Change the Comparison
A concentration is not determined by the prescribed dose alone. It also depends on when the sample is taken relative to the latest dose, how many doses have already been given, and what regimen preceded the current one. A comparison that ignores any of these features can attribute a timing or accumulation difference to the regimen itself.
Two distinct questions are often confused. A steady-state comparison asks how regimens differ after each has reached its repeating concentration-time pattern. A transient comparison asks how concentrations differ at a specified time during treatment, including the effects of each regimen’s start-up or switch history. Both questions can be useful, but they require different comparison conditions. Comparing one regimen at steady state with another shortly after it starts answers neither question cleanly.
A Simple Model for Seeing the Timing Effects
The following derivations use a one-compartment model with instantaneous intravenous bolus doses, volume of distribution \(V\), and first-order elimination rate constant \(k\). Assume \(V\) and \(k\) are constant, doses are given at fixed interval \(\tau\), and concentrations add linearly across doses. The elimination half-life is \(t_{1/2}=\ln(2)/k\). These assumptions make the timing effects transparent; they are not guaranteed to hold for every drug or route.
For a dose of amount \(D\), the immediate concentration increment is \(D/V\). If the dose interval is \(\tau\), define \(r=e^{-k\tau}\), the fraction of a concentration increment remaining one interval later. With repeated equal doses, each earlier increment is multiplied by another factor of \(r\).
Here \(C_n^{+}\) is the concentration immediately after the \(n\)th dose increment, when dosing began from zero and the first dose was given at time zero. The steady-state post-dose concentration is the limit as \(n\) increases:
The dose-count convention matters. After n dose increments, the fraction of steady-state post-dose concentration is \(1-r^n\). If the first dose is at time zero, the \(n\)th dose occurs after \(n-1\) elapsed intervals. Thus, immediately after the dose at time \(n\tau\), there have been \(n+1\) increments and the fraction is \(1-r^{n+1}\), not \(1-r^n\).
Within an interval, concentration continues to decline. At \(s\) hours after a post-dose concentration, where \(0\leq s<\tau\), it is multiplied by \(e^{-ks}\). Therefore, at steady state, the profile at phase \(s\) is \(C_{\mathrm{ss}}(s)=C_{\mathrm{ss}}^{+}e^{-ks}\). A peak-phase value and a later post-dose sample are not interchangeable even when the regimen is unchanged.
Worked Example: A Sampling-Time Difference Is Not a Regimen Effect
Suppose a hypothetical patient receives 100 mg by intravenous bolus every 8 hours. Let \(V=20\) L and the half-life be 8 hours. Compare a sample taken immediately after a dose with one taken 2 hours after a dose, assuming the regimen is at steady state.
The immediate concentration increment is \(D/V=100/20=5\) mg/L. Since the interval equals one half-life, \(r=e^{-k\tau}=0.5\). Thus the steady-state post-dose concentration is:
Two hours after the dose, the remaining fraction is \(e^{-k(2)}=2^{-2/8}=2^{-0.25}=0.8409\). The concentration is therefore:
The 1.591 mg/L difference is caused by sampling at different phases of the same dosing interval. It is not evidence that the regimen changed or that drug exposure differs between two treatment groups. A comparison intended to assess regimens must compare concentrations at the same post-dose phase, or compare a prespecified summary of each complete profile.
Before Steady State: Count Doses, Not Just Intervals
A common error is to label a concentration “after three intervals” and then use the accumulation fraction for three dose increments. The number of elapsed intervals and the number of doses administered are different when a dose is given at time zero. The geometric sum above counts dose increments, so the dose schedule must be checked before applying it.
Worked Example: Correctly Counting Accumulation
Use the same hypothetical model: \(D=100\) mg, \(V=20\) L, half-life 8 hours, and \(\tau=8\) hours. Doses are given at 0, 8, 16, and 24 hours. Calculate the post-dose concentrations at 16 and 24 hours and compare them with steady state.
The increment is 5 mg/L and \(r=0.5\). At 16 hours, the dose at time zero and the doses at 8 and 16 hours have given three increments:
Steady state is \(5/(1-0.5)=10.00\) mg/L, so the fraction reached after these three increments is \(8.750/10.00=0.8750\), or 87.50%. At 24 hours, the scheduled fourth dose is administered. There are then four increments:
This is \(9.375/10.00=0.9375\), or 93.75% of steady state. The 24-hour value is not 87.50%: the dose at 24 hours contributes a fourth increment. Equivalently, after \(m\) elapsed intervals from a dose at time zero, the immediate post-dose value after the dose at \(m\tau\) includes \(m+1\) increments.
For a regimen comparison, suppose a 120 mg regimen is also started from zero on the same schedule. Its increment is 6 mg/L. At 16 hours its post-dose concentration is \(6(1+0.5+0.25)=10.50\) mg/L; its steady-state post-dose concentration is \(6/(1-0.5)=12.00\) mg/L. The transient comparison is 8.750 versus 10.50 mg/L at the same phase after the same number of doses. The steady-state comparison is 10.00 versus 12.00 mg/L. Both are internally aligned, but they answer different questions.
Regimen Changes Carry the Previous Concentration Forward
After a dose change, the concentration does not reset to zero or instantly jump to the new regimen’s steady state. Drug remaining from the previous regimen contributes to subsequent concentrations. Under linear elimination, the concentration at time \(t\) can be calculated by adding the contributions from all doses administered by that time:
Here \(D_j\) is the amount of dose \(j\), administered at time \(t_j\). This superposition formula includes doses from before a regimen change as well as doses given afterward. It is valid under the stated linear, one-compartment bolus assumptions. For other models, the corresponding model-predicted profile must account for the relevant absorption and disposition processes.
Worked Example: A Dose Increase Does Not Immediately Produce Its New Steady State
Suppose the hypothetical 100 mg every-8-hour regimen above has reached steady state. Immediately after a scheduled dose, the concentration is 10.00 mg/L. At time zero, the next dose is increased to 150 mg, while the interval and pharmacokinetic parameters remain unchanged. Find the concentration immediately after this new dose and after the next 150 mg dose at 8 hours.
Just before the dose at time zero, the old steady-state concentration is the old post-dose concentration after one interval of elimination: \(10.00(0.5)=5.00\) mg/L. The new 150 mg dose adds \(150/20=7.500\) mg/L. Immediately after the new dose, concentration is therefore:
At 8 hours, the 12.50 mg/L decays by a factor of 0.5 to 6.250 mg/L before the next dose. The second 150 mg dose adds another 7.500 mg/L:
The new regimen’s steady-state post-dose concentration is \(7.500/(1-0.5)=15.00\) mg/L. The first post-change concentration is 12.50 mg/L, not 15.00 mg/L, because the concentration history is still transitioning. This example also shows why a comparison based only on the current dose amount can omit the residual contribution from earlier doses.
How to Make the Comparison Fair
First decide whether the question concerns steady-state performance or a transient treatment course. For a steady-state question, simulate or observe each regimen at its repeating pattern and compare the same phase of the interval, or compare the same prespecified profile metric. For a transient question, specify the starting concentrations, prior regimens, dose times, and comparison time; then compare both regimens at the same elapsed time and the same phase relative to dosing.
Specify whether the target is a steady-state concentration, a transient concentration at a defined time, or a summary such as the area under the concentration-time curve over a specified interval.
Record time since the latest dose and time since regimen initiation or change. Match both when the comparison is intended to isolate regimen differences at a transient time point.
Include prior doses and residual concentrations, or explicitly assume a common starting condition such as zero concentration. Do not silently give one regimen a different accumulation history.
Use the same post-dose phase, the same interval summary, or the same defined clinical outcome for each regimen. Label transient and steady-state results separately.
A useful diagnostic is to plot concentration against both clock time and time since the most recent dose. Check dose timestamps, interval lengths, treatment-start times, and any regimen changes. If two supposedly aligned profiles show an unexpected offset, verify that the dose counts and residual concentrations agree with the recorded histories before interpreting the difference pharmacologically.
Alternatives, Failure Modes, and Limits
A single matched-time concentration is easy to interpret when the question is explicitly about that phase, but it may not represent overall exposure or the entire concentration-time shape. Comparing full profiles can reveal differences in peaks, troughs, and timing, while a prespecified interval summary can be more compact. The summary must match the scientific question: two profiles can have similar averages but different peaks or troughs. A steady-state calculation is efficient under stable linear assumptions; explicit transient simulation is preferable when starting conditions or dose changes matter.
| Approach | What it compares | Main caution |
|---|---|---|
| Matched-time concentration | Concentrations at the same post-dose phase and, for transient questions, same treatment time | May not represent the rest of the profile |
| Steady-state profile | Repeating concentration patterns under unchanged regimens | Misleading if a regimen has not reached its repeating pattern |
| Transient model simulation | Concentrations along specified dose histories and times | Depends on correct dose records and an adequate PK model |
| Interval summary | A prespecified measure over a defined interval | Different summaries answer different questions and can hide profile-shape differences |
Warning signs include a large apparent regimen difference that shrinks when samples are aligned by post-dose time; a reported “steady-state” value obtained after only a few doses without checking accumulation; a discontinuity at a dose change that incorrectly resets concentration; and comparisons that use different numbers of administered doses. In real data, missed or delayed doses, changing clearance, nonlinear pharmacokinetics, absorption delays, and measurement error can also make simple bolus-model calculations inadequate. In those cases, use a model appropriate to the route and drug, preserve the actual dosing history, and describe the remaining assumptions.
Common Mistake
Correction: Calendar day does not establish either matching post-dose phase or steady state. Check the exact dose times and count the administered doses. If the comparison is transient, align treatment time and dose phase and include prior concentrations. If it is intended as a steady-state comparison, establish that each regimen’s profile has reached its repeating pattern before comparing the same phase or interval summary.
The geometric accumulation result is a useful check, not a universal rule for every dosing situation. It follows from fixed intervals, a constant fraction \(r\) remaining per interval, and linear superposition. When these assumptions fail, calculate profiles using a suitable pharmacokinetic model rather than applying the geometric formula mechanically.
Check Your Understanding
Answer each question, then check the solutions below.
- For the 100 mg every-8-hour example, why is a sample at 2 hours not directly comparable with one immediately after a dose if the intended comparison is a regimen effect?
- With a dose at time zero, how many dose increments contribute immediately after the dose at \(3\tau\)? What fraction of steady-state post-dose concentration is reached if \(r=0.5\)?
- A regimen is changed while concentration is nonzero. Should the new-regimen steady-state concentration be assigned immediately after the first new dose? Explain.
- Name two timing or history details that should be aligned for a transient comparison.
Solutions. 1. The 2-hour sample has undergone 2 hours of elimination; in the example it is 8.409 mg/L versus 10.00 mg/L immediately post-dose, despite the unchanged regimen. 2. Four increments contribute: the doses at \(0,\tau,2\tau,3\tau\). The fraction is \(1-r^4=1-0.5^4=0.9375\), or 93.75%. 3. No. Residual drug and incomplete accumulation mean the post-change concentration generally differs from the new regimen’s steady state; use the dose history to calculate the transient. 4. Align time since the most recent dose and elapsed time since regimen initiation or change, and account for prior doses or starting concentration.