Pooled Concentration-Time Curves Can Hide Individual Trajectories

Why a Pooled Curve Can Be Misleading

A concentration-time curve formed by averaging subjects at each scheduled time is easy to read as though it described one typical patient. But the average at one time need not come from the same subject as the average at the next time. Subjects may peak at different times, have different exposure, or change in opposite directions. The resulting curve can therefore conceal clinically relevant variation in the individual trajectories.

Here, a subject-level profile is the set of concentrations recorded for one subject across scheduled times. A pooled pointwise mean curve is formed by averaging the observed concentrations across subjects separately at each time. The distinction matters: the first preserves which measurements belong together, while the second summarizes each time point without displaying those subject identities.

Definition: Suppose $n$ subjects are measured at the same scheduled times $t_1,\ldots,t_K$, and let $C_{ik}$ be the concentration for subject $i$ at time $t_k$. The pooled pointwise mean at $t_k$ is $$ \bar C(t_k)=\frac{1}{n}\sum_{i=1}^{n} C_{ik}. $$ It is the arithmetic mean of the $n$ concentrations at that time. Connecting the values $\bar C(t_k)$ gives a pooled curve; it does not, by itself, show any one subject’s trajectory.

The definition assumes the same subjects contribute at every listed time and that each subject receives equal weight at each time. Those conditions make comparisons across times especially clear. The examples below use complete, balanced, hypothetical data, with concentration in mg/L and time in hours. They are illustrations of averaging, not observations from a real study.

Different Peak Times Can Flatten or Shift the Pooled Peak

If subjects reach their maxima at different times, averaging at each time can lower the maximum and produce a pooled shape that no individual follows. The pooled curve may look smoother, flatter, or more prolonged than the separate subject profiles.

Worked Example: Two Subjects Peak at Different Times

Two hypothetical subjects have concentrations at 0, 1, and 2 hours as shown below.

Subject	1 h (mg/L)	2 h (mg/L)
A	10	0
B	0	10
Pooled mean	5	5

At 0 hours, the pooled mean is $(0+0)/2=0$ mg/L. At 1 hour it is $(10+0)/2=5$ mg/L, and at 2 hours it is $(0+10)/2=5$ mg/L. Subject A peaks at 1 hour and subject B peaks at 2 hours, each at 10 mg/L. The pooled curve instead has the same value, 5 mg/L, at both 1 and 2 hours. Its maximum is only half of either subject’s maximum, and its flat top is not the trajectory of either subject.

Exposure also differs between these profiles. Using straight-line interpolation between scheduled observations and the trapezoidal rule, subject A’s area from 0 to 2 hours is $(0+10)(1)/2+(10+0)(1)/2=10$ mg·h/L. Subject B’s area is $(0+0)(1)/2+(0+10)(1)/2=5$ mg·h/L. The pooled curve’s corresponding trapezoidal area is $(0+5)(1)/2+(5+5)(1)/2=7.5$ mg·h/L.

Thus the pooled curve conceals both the differing peak times and the differing individual areas. In this balanced example, its area is the average of the two subject-level areas: $(10+5)/2=7.5$ mg·h/L. That equality is not evidence that either individual had the pooled curve’s shape.

Opposite Changes Can Cancel in the Mean

A pooled curve can also hide the direction of individual change. If one subject’s concentration rises while another’s falls by the same amount, their pointwise mean can remain unchanged. The flat mean does not imply that the concentrations were stable within subjects.

Worked Example: A Flat Mean Hides Rising and Falling Profiles

Suppose two hypothetical subjects have concentrations at 0 and 1 hour. Subject A changes from 8 to 2 mg/L, while subject B changes from 2 to 8 mg/L.

At 0 hours, their mean is $(8+2)/2=5$ mg/L. At 1 hour, it is $(2+8)/2=5$ mg/L as well. The pooled mean curve is flat at 5 mg/L over these two observations.

The subject-level changes tell a different story. Subject A changes by $2-8=-6$ mg/L, a decrease of 6 mg/L. Subject B changes by $8-2=+6$ mg/L, an increase of 6 mg/L. Their changes average to $(-6+6)/2=0$ mg/L, which is why the pooled mean does not change. It would be incorrect to conclude that both subjects, or a representative subject, had a stable concentration.

What Averaging Preserves—and What It Does Not

For complete, balanced data, the change in the pooled mean between two scheduled times equals the mean of the subjects’ changes over that interval. This follows directly from the definition. If all $n$ subjects are observed at both $t_k$ and $t_{k+1}$, then

\bar C(t_{k+1})-\bar C(t_k) =\frac{1}{n}\sum_{i=1}^{n}C_{i,k+1} -\frac{1}{n}\sum_{i=1}^{n}C_{ik} =\frac{1}{n}\sum_{i=1}^{n}(C_{i,k+1}-C_{ik}).

So a pooled increase means that the average subject-level change is positive; it does not mean every subject increased. A zero average change can arise because all subjects were stable, or because increases and decreases offset one another. The pooled curve alone does not distinguish those patterns.

There is a related result for trapezoidal exposure. It clarifies why averaging can preserve an average area while still concealing each subject’s profile. Let the common time intervals be $\Delta t_k=t_{k+1}-t_k$. Define subject $i$’s trapezoidal area as

A_i=\sum_{k=1}^{K-1}\frac{\Delta t_k}{2}(C_{ik}+C_{i,k+1}).

The trapezoidal area calculated from the pooled means is

A_{\mathrm{pool}} =\sum_{k=1}^{K-1}\frac{\Delta t_k}{2}\{\bar C(t_k)+\bar C(t_{k+1})\}.

Substitute the definition of each pooled mean and rearrange the finite sums:

\begin{aligned} A_{\mathrm{pool}} &=\sum_{k=1}^{K-1}\frac{\Delta t_k}{2} \left\{\frac{1}{n}\sum_{i=1}^{n}C_{ik} +\frac{1}{n}\sum_{i=1}^{n}C_{i,k+1}\right\}\\ &=\frac{1}{n}\sum_{i=1}^{n} \sum_{k=1}^{K-1}\frac{\Delta t_k}{2}(C_{ik}+C_{i,k+1})\\ &=\frac{1}{n}\sum_{i=1}^{n}A_i. \end{aligned}

The equality requires the same subjects at every time, common scheduled times, equal subject weights, and the same linear trapezoidal calculation for each profile and the pooled curve. It says the pooled area equals the arithmetic average of individual trapezoidal areas under these conditions. It does not recover individual areas, peak times, maxima, or the trajectories between observations. Nor does the trapezoidal rule guarantee that the calculated area equals the unknown true area under a continuously observed concentration curve.

Worked Example: Two Different Sets of Profiles, One Pooled Curve

Consider again the pooled means $0,5,5$ mg/L at 0, 1, and 2 hours. The first possible set of profiles is A: $(0,10,0)$ and B: $(0,0,10)$ mg/L. A different possible set is C: $(0,5,5)$ and D: $(0,5,5)$ mg/L. In the second set, the pooled means are also 0, 5, and 5 mg/L, since the two subjects have those same values at each time.

But the individual patterns differ. In the first set, both subjects reach 10 mg/L, with peaks at different times; their trapezoidal areas are 10 and 5 mg·h/L. In the second set, both subjects reach 5 mg/L, and each has a trapezoidal area of $(0+5)(1)/2+(5+5)(1)/2=7.5$ mg·h/L. Both sets have an average area of 7.5 mg·h/L and exactly the same pooled curve, but their subject-level maxima, peak patterns, and areas are not the same.

This demonstrates a basic limit of the pooled curve: knowing its time-specific means does not tell us how the concentrations were distributed among subjects or which values belonged to the same subject.

How to Inspect a Pooled Curve Responsibly

A pooled curve is useful for displaying average concentration at each scheduled time, but it should be read alongside information that retains subject identity. For balanced data, plotting each subject’s observed profile, or tabulating each subject’s values, makes differences in timing and direction visible. Subject-level summaries such as observed maximum concentration, time of observed maximum, and trapezoidal area can also show whether a modest-looking pooled curve combines similar profiles or quite different ones.

These displays are complementary, not interchangeable. The pointwise mean answers, “What is the average observed concentration at this time?” A collection of subject profiles answers, “How do the observed concentrations vary and change within subjects?” A mean curve can be accurate for the first question while being inadequate for the second.

Check the peak: compare the pooled maximum with each subject’s observed maximum and time of maximum. A pooled peak may be lower or occur at a time when no subject reaches their own maximum.
Check direction: compare changes between consecutive times within subjects. A flat mean can conceal increases and decreases that cancel.
Check exposure summaries: if areas are relevant, inspect the individual areas as well as the pooled curve. The average-area equality above does not reveal the range or pattern of individual areas.
Check the data behind each time point: the mean is only interpretable as a common-subject comparison when the relevant observations and weighting are understood. The examples and derivation here assume a complete, equally weighted schedule.

Common Mistake

A common mistake is to treat a pooled curve as though it were a measured trajectory for a typical subject: “The mean concentration rose to 5 mg/L and then stayed there, so subjects generally had a 5 mg/L plateau.” In the first worked example, neither subject followed that curve: one peaked at 1 hour and the other at 2 hours, and both reached 10 mg/L. The correction is to describe the curve precisely as the pointwise mean, then examine subject-level profiles before making claims about individual change, peak timing, or exposure.

Pooling is not inherently wrong; it answers a specific summary question. The limitation is that averaging across subjects removes information about the joint pattern of measurements within each subject. A smooth or stable mean should therefore not be mistaken for evidence of smooth or stable individual profiles.

Check Your Understanding

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

At two times, subject A has concentrations 3 and 9 mg/L, and subject B has 7 and 1 mg/L. Calculate the pooled means and each subject’s change. What does the pooled change conceal?
In the two-subject, three-time example with profiles $(0,10,0)$ and $(0,0,10)$ mg/L, what are the two individual trapezoidal areas from 0 to 2 hours, and what is the pooled-curve area?
State the conditions that allow the trapezoidal area of a pooled curve to equal the average of individual trapezoidal areas.
Two different subject-level data sets have identical pooled means at every scheduled time. What can be concluded about their pooled curves, and what cannot be concluded about their individual profiles?

Solutions. (1) The pooled means are $(3+7)/2=5$ mg/L at the first time and $(9+1)/2=5$ mg/L at the second. Subject A increases by $9-3=6$ mg/L, while subject B decreases by $1-7=-6$ mg/L. The pooled change is zero, concealing the opposite individual directions. (2) Subject A’s area is $(0+10)(1)/2+(10+0)(1)/2=10$ mg·h/L. Subject B’s area is $(0+0)(1)/2+(0+10)(1)/2=5$ mg·h/L. The pooled curve is $0,5,5$ mg/L, with area $(0+5)(1)/2+(5+5)(1)/2=7.5$ mg·h/L. (3) The same subjects must contribute at every time, scheduled times must be common, weights must be equal across subjects, and the same trapezoidal rule must be applied. (4) Their pooled curves match at the scheduled times, but the subjects’ peak times, maxima, changes, and individual areas may differ. The pooled means alone do not identify those individual patterns.