Multiple Peaks and Cmax Selection

Why Multiple Peaks Require Careful Selection

A concentration–time profile does not always rise to one peak and then decline smoothly. A sampled profile may have several rises and falls, and therefore several local peaks. The pharmacokinetic summary \(C_{\max}\), however, is not the first peak, the last peak, or an average of the peaks: it is the largest concentration actually observed in the profile. Its associated time, \(T_{\max}\), must be selected from the observation time or times at which that largest value occurs.

This distinction matters whenever a profile has more than one apparent peak. First identify local maxima to describe the sampled pattern; then compare their concentrations across the full profile to select the global observed maximum. These operations concern the recorded observations. They do not establish the precise maximum of an unobserved continuous concentration–time curve.

Definition: Suppose a profile contains \(n\) observations at strictly increasing times \(t_1<t_2<\cdots<t_n\), with corresponding concentrations \(C_1,C_2,\ldots,C_n\). For \(n\geq 2\), an observation is a sampled local maximum if its concentration is at least as large as the concentration at each observed immediate neighbor. An endpoint has just one immediate neighbor, so it is compared with that neighbor only. A global observed maximum is the largest concentration among all \(n\) observations. \(C_{\max}\) denotes that largest observed concentration, and \(T_{\max}\) denotes the time or times at which it was observed.

In symbols, an interior observation \(C_i\), where \(1<i<n\), is a sampled local maximum when \(C_i\geq C_{i-1}\) and \(C_i\geq C_{i+1}\). The first observation is a local maximum when \(C_1\geq C_2\), and the last when \(C_n\geq C_{n-1}\). This endpoint definition uses the one neighbor that exists; it does not extend to a one-observation profile. When \(n=1\), a global observed maximum exists, but a local maximum is not defined by these neighbor comparisons.

The inequalities are non-strict so that equal adjacent concentrations can be handled. A run of equal observations at the top of a peak can contain more than one observation satisfying the local-maximum rule. Treat such a run as one flat peak region when describing the profile, rather than counting each equal observation as a separate peak. The earlier material on flat peaks and tied \(T_{\max}\) values addresses how to report ties.

From Local Peaks to the Global Observed Maximum

The selection rule is direct: compare every observed concentration and retain the largest value. Local-maximum identification helps describe where the sampled profile turns or levels off, but it is not a substitute for checking the full data set. In particular, a smaller local peak does not become \(C_{\max}\) merely because it occurs earlier, looks sharper, or is the first peak in the profile.

C_{\max}=\max_{1\leq i\leq n} C_i, \qquad T_{\max}=\{t_i:C_i=C_{\max}\}.

Here \(T_{\max}\) is written as a set to make clear that more than one observation time may correspond to the largest observed concentration. If a reporting system requires a single \(T_{\max}\), apply the prespecified tie convention and state it. Do not silently choose one tied time or imply that the data identified a unique time when they did not.

Result: For a profile with \(n\geq 2\), every global observed maximum is also a sampled local maximum under the neighbor-comparison definition above. A sampled local maximum need not be global. If the global maximum occurs at several tied observations, each corresponding time belongs to \(T_{\max}\).

The first statement follows because a global maximum is at least as large as every observation, including each immediate neighbor. The endpoint cases require only one such comparison; an interior observation requires two.

C_i=\max_{1\leq j\leq n} C_j \quad\Longrightarrow\quad C_i\geq C_{i-1}\ \text{and}\ C_i\geq C_{i+1} \quad (1<i<n).

At the first or last observation, the same argument gives the required inequality against its sole neighbor. The condition \(n\geq2\) matters: with one observation there is no neighbor to compare against, so the stated definition does not classify it as a local maximum even though it is the global observed maximum.

Worked Examples

Worked Example: Selecting the Larger of Two Peaks

Consider hypothetical concentrations measured in mg/L at the following times. Identify the sampled local maxima and determine \(C_{\max}\) and \(T_{\max}\).

Time (h)	Concentration (mg/L)
0	0
1	5
2	3
3	8
4	4
6	2

At \(t=1\) h, the concentration is a local maximum because \(5\geq0\) and \(5\geq3\). At \(t=3\) h, it is also a local maximum because \(8\geq3\) and \(8\geq4\). The first and last observations are not local maxima: \(0<5\) at the first endpoint, and \(2<4\) at the last.

Compare the local-maximum concentrations: \(5\) mg/L and \(8\) mg/L. The full profile has no observation greater than \(8\) mg/L, so \(C_{\max}=8\) mg/L and \(T_{\max}=3\) h. The earlier local peak is part of the profile description, but it is not the global observed maximum.

Worked Example: A Global Maximum at an Endpoint

For a second hypothetical profile, concentrations at times \(0,1,2,3,4\) h are \(7,4,3,2,1\) mg/L. Find all local maxima and select \(C_{\max}\).

The first observation is an endpoint, so compare it with its only neighbor: \(7\geq4\). It is a sampled local maximum. The observation at \(1\) h is not one because \(4\geq7\) is false. Each later interior observation is smaller than its preceding neighbor, and the last observation is smaller than its sole neighbor. Thus, there is one local maximum, at \(t=0\) h.

The largest value in the complete set is \(7\) mg/L. Therefore, \(C_{\max}=7\) mg/L and \(T_{\max}=0\) h. An endpoint can be the global observed maximum; the rule does not require the profile to rise from a sampled baseline before a maximum can be selected.

Worked Example: Tied Global Peaks

A hypothetical profile has concentrations, in mg/L, of \(1,4,9,6,3,7,9,5\) at times \(0,1,2,3,4,5,6,7\) h. Identify the local maxima, the global observed maximum, and the associated times.

At \(1\) h, \(4\) mg/L is not a local maximum because its next neighbor is \(9\) mg/L. At \(2\) h, \(9\geq4\) and \(9\geq6\), so \(9\) mg/L is a local maximum. At \(5\) h, \(7\geq3\) but \(7\geq9\) is false, so \(7\) mg/L is not a local maximum. At \(6\) h, \(9\geq7\) and \(9\geq5\), so it is another local maximum.

The largest observed concentration is \(9\) mg/L, and it occurs at both \(2\) h and \(6\) h. Thus, \(C_{\max}=9\) mg/L and \(T_{\max}=\{2,6\}\) h. If a prespecified convention reports the earliest time in a tie, the reported scalar \(T_{\max}\) would be \(2\) h, but the tie should still be recognized when interpreting the profile.

Practical Checks and Common Failure Modes

A reliable selection begins by keeping times and concentrations paired, then comparing observations in chronological order. A concentration value cannot be compared with the wrong neighbor because rows were sorted independently, and a local-maximum check must use the immediately preceding and following observations—not whichever values happen to look closest in magnitude.

Checking only the first peak. The first local maximum may be lower than a later one. Compare concentrations across the entire observed profile before assigning \(C_{\max}\).
Calling every rise a peak. An observation is not an interior local maximum merely because it exceeds the preceding value; it must also be at least as large as the following value.
Ignoring the endpoint rule. At an endpoint, compare with the sole observed neighbor. Do not require a nonexistent second neighbor.
Reporting a unique time for a tie without a convention. The maximum concentration can occur at multiple times. Report all such times or clearly state the rule used to select one.
Calling a sampled peak the true continuous peak. \(C_{\max}\) is computed from observed concentrations. It does not locate an unobserved value between samples or prove that the continuous profile attained its maximum at \(T_{\max}\).

If the data contain missing, nonquantifiable, or otherwise specially handled values, the analysis must follow the applicable data-handling rules before the maximum is selected. Those rules can affect which observations are eligible for comparison. Likewise, a suspicious isolated value or an unexpected sequence may warrant checking the data record, units, and time ordering. Such checks do not authorize changing or omitting an observation without a stated, justified data-handling decision.

How This Selection Differs From Other Peak Descriptions

The observed-maximum rule is deliberately simple and reproducible: take the maximum among the eligible recorded concentrations. Listing local maxima adds a description of the sampled shape, especially when the profile rises again after an earlier decline. Neither operation estimates a smooth curve. A fitted or smoothed curve can produce a peak between observation times, but that is a model-based estimate requiring additional assumptions; it is not the observed \(C_{\max}\) defined here.

This distinction also limits interpretation. The selected value and time summarize the available observations and their sampling times. They do not, by themselves, explain why the profile has multiple peaks or establish the timing of an underlying biological process. Those questions require additional evidence and methods. For the observed metric, the essential discipline is to distinguish a local feature from the largest recorded concentration and to preserve any genuine tie.

Check Your Understanding

For each question, identify the relevant comparisons before deciding.

At times \(0,1,2,3\) h, concentrations are \(2,6,4,5\) mg/L. Which observations are local maxima, and what are \(C_{\max}\) and \(T_{\max}\)?
Why is an interior observation that is greater than its preceding observation not necessarily a local maximum?
A profile contains one observation only. Does it have a global observed maximum under the definition? Is it a local maximum under the neighbor-comparison rule in this tutorial?
Two observations at different times both equal the largest concentration. What should be reported about \(T_{\max}\)?

Solutions. (1) At \(1\) h, \(6\geq2\) and \(6\geq4\), so it is a local maximum. At \(3\) h, the endpoint concentration \(5\geq4\), so it is also a local maximum. The largest value is \(6\) mg/L, hence \(C_{\max}=6\) mg/L and \(T_{\max}=1\) h. (2) It must also be at least as large as its following observation; otherwise it does not satisfy both neighbor comparisons. (3) Yes. The sole concentration is the global observed maximum because it is the only value. No: with \(n=1\), there is no neighbor comparison, and the local-maximum definition here requires \(n\geq2\). (4) Report both times as \(T_{\max}\), or state the prespecified convention if a single time must be reported; do not imply the observed maximum occurred uniquely.