Calculating AUC, Cmax, Cmin, And Trough Exposure

Why Sampling Time Matters

A concentration-time data set can be summarized in several ways, but the summaries answer different questions. The area under the concentration-time curve (AUC) accumulates concentration over a specified time window. Cmax and Cmin summarize the largest and smallest observed concentrations in the sampled window. A scheduled trough is a concentration tied to a specific dosing time, usually just before the next dose. These measures cannot be substituted for one another: their meanings, units, and sampling requirements differ.

The calculations below use hypothetical concentration measurements. Before calculating, verify that each concentration has the same units and refers to the same analyte and sample matrix, that each time is recorded in a common time unit, and that the values belong to the intended observation window. If the times or dosing history do not correspond to that window, the resulting summary may not describe the exposure of interest.

Definition: For ordered concentration measurements \(C_0,\ldots,C_m\) at times \(t_0<\cdots<t_m\), the linear trapezoidal AUC over the observed interval \([t_0,t_m]\) is the sum of the areas of trapezoids formed between successive measurements. The observed \(C_{\max}\) is the largest measured concentration in the specified window, and the observed \(C_{\min}\) is the smallest measured concentration in that window. A scheduled trough concentration is the concentration measured at the prespecified trough time, commonly immediately before a scheduled dose.

Calculating AUC by the Linear Trapezoidal Rule

For two consecutive samples, the linear trapezoidal rule approximates the concentration between samples as changing linearly. The area for that interval is the time difference multiplied by the average of the endpoint concentrations. Summing the interval areas gives AUC over the observed sampling interval.

\mathrm{AUC}_{t_0\text{ to }t_m} = \sum_{i=0}^{m-1} \frac{C_i+C_{i+1}}{2}(t_{i+1}-t_i)

Here, \(C_i\) is the concentration measured at time \(t_i\). Each interval can have a different duration; equal spacing is not required. If concentration is measured in mg/L and time in hours, each interval area and the total AUC have units of mg·h/L. The AUC is not itself a concentration.

The approximation has a simple geometric justification. If concentration is assumed to vary linearly from \(C_i\) to \(C_{i+1}\), the line’s area over the interval is the area of a trapezoid. Equivalently, integrating that straight line gives the endpoint average multiplied by the interval duration:

\int_{t_i}^{t_{i+1}} \left[ C_i+\frac{C_{i+1}-C_i}{t_{i+1}-t_i}(t-t_i) \right]dt = \frac{C_i+C_{i+1}}{2}(t_{i+1}-t_i)

Thus the trapezoid is exact for a truly linear concentration change between those two times. When the actual curve bends between samples, it is a numerical approximation, and sparse sampling can make the approximation less representative. The calculated AUC covers only the interval bounded by the first and last included samples; it does not automatically include time before the first sample or after the last.

Worked Example: AUC and Concentration Summaries Across One Dosing Interval

In a hypothetical study, a participant receives a dose at time 0 h. Concentrations over the 8 h dosing interval are 0 mg/L at 0 h, 6 mg/L at 2 h, 4 mg/L at 6 h, and 1 mg/L at 8 h. Calculate the linear trapezoidal AUC, observed Cmax and Cmin, and scheduled trough concentration if the next dose is due at 8 h.

The ordered times are 0, 2, 6, and 8 h. The three interval durations are 2, 4, and 2 h, so the samples are not all equally spaced. Calculate each trapezoid:

From 0 to 2 h: \(\frac{0+6}{2}(2-0)=3(2)=6\) mg·h/L.

From 2 to 6 h: \(\frac{6+4}{2}(6-2)=5(4)=20\) mg·h/L.

From 6 to 8 h: \(\frac{4+1}{2}(8-6)=2.5(2)=5\) mg·h/L.

Therefore, \(\mathrm{AUC}_{0\text{ to }8}=6+20+5=31\) mg·h/L. The largest sampled value is 6 mg/L, so observed \(C_{\max}=6\) mg/L at 2 h. The smallest sampled value is 0 mg/L, so observed \(C_{\min}=0\) mg/L at 0 h. The scheduled trough is the measurement at the next dose time, 8 h, and is 1 mg/L. In this window, the sampled minimum and scheduled trough are different metrics and have different values.

Observed Maximum, Minimum, and Scheduled Trough

For a specified set of samples, the calculations of observed extrema are direct:

C_{\max,\mathrm{obs}}=\max(C_0,\ldots,C_m), \qquad C_{\min,\mathrm{obs}}=\min(C_0,\ldots,C_m)

The subscript “obs” emphasizes that these are extrema among the recorded samples. Unless sampling captures the actual peak or nadir, they need not equal the true maximum or minimum of the continuous concentration curve. A reported observed Cmax therefore depends on the sampling schedule: a peak between samples may be missed. Likewise, a low concentration occurring between samples may not be represented by observed Cmin.

A scheduled trough is determined by the dosing schedule, not by ranking all observed values. For a dose due at time \(t_{\mathrm{dose}}\), the trough is conventionally the concentration at \(t_{\mathrm{dose}}^{-}\), meaning immediately before administration. The sample time should be close enough to that scheduled time, and collected before the dose, to support that interpretation. A sample collected after the dose is not a predose trough. Nor should a nearby sample be silently treated as an exact scheduled trough without stating the timing difference and the rule used to handle it.

Worked Example: Equally Spaced Sampling and AUC

A hypothetical concentration profile is sampled at 0, 4, 8, and 12 h, with concentrations 2, 8, 5, and 3 mg/L, respectively. The next scheduled dose is at 12 h. Calculate AUC over the observed interval and the three concentration summaries.

The sampling intervals are 4 h, 4 h, and 4 h; they are equally spaced. The trapezoidal calculation is:

From 0 to 4 h: \(\frac{2+8}{2}(4)=5(4)=20\) mg·h/L.

From 4 to 8 h: \(\frac{8+5}{2}(4)=6.5(4)=26\) mg·h/L.

From 8 to 12 h: \(\frac{5+3}{2}(4)=4(4)=16\) mg·h/L.

Thus, \(\mathrm{AUC}_{0\text{ to }12}=20+26+16=62\) mg·h/L. The observed \(C_{\max}\) is 8 mg/L, and the observed \(C_{\min}\) is 2 mg/L. Because the 12 h sample was collected immediately before the scheduled dose, the scheduled trough is 3 mg/L. These results describe the measured window and sample set; they do not establish that the continuous curve’s true maximum occurred exactly at 4 h.

Sampling Requirements and Data Checks

AUC can be calculated for irregularly spaced samples as long as each time and concentration are known and the measurements are ordered correctly. The interval durations must be calculated from the actual times, not replaced by a typical or average interval. For Cmax and Cmin, the sampled values must belong to the stated window, and the report should make clear that they are observed extrema. For a scheduled trough, the data must include a valid measurement at the relevant predose time. AUC and observed extrema can still be calculated when that trough sample is absent, but the trough itself cannot be reported as measured.

Worked Example: Irregular Sampling With No Scheduled Trough Sample

In a hypothetical dosing interval ending at 8 h, concentrations are measured at 0, 1.5, 5, and 7.5 h as 1, 7, 4, and 2 mg/L. Calculate the AUC over the observed interval and observed extrema. Determine whether the scheduled trough at 8 h is available.

The actual interval durations are 1.5, 3.5, and 2.5 h. The trapezoid areas are:

From 0 to 1.5 h: \(\frac{1+7}{2}(1.5)=4(1.5)=6\) mg·h/L.

From 1.5 to 5 h: \(\frac{7+4}{2}(5-1.5)=5.5(3.5)=19.25\) mg·h/L.

From 5 to 7.5 h: \(\frac{4+2}{2}(7.5-5)=3(2.5)=7.5\) mg·h/L.

The AUC from 0 to 7.5 h is \(6+19.25+7.5=32.75\) mg·h/L. The observed \(C_{\max}\) is 7 mg/L and observed \(C_{\min}\) is 1 mg/L. There is no concentration measurement at 8 h, immediately before the scheduled dose. Therefore, the scheduled trough is not available as a measured value. The 7.5 h measurement is not automatically the 8 h trough, and the AUC calculated here ends at 7.5 h rather than 8 h.

Common Pitfalls, Diagnostics, and Alternatives

A useful quality check is to calculate each interval duration and area separately, then verify that their sum uses every consecutive pair exactly once. Confirm that the units agree: if time is in minutes but the intended report is in hours, convert durations before calculating an AUC in concentration·hours. Check for duplicate or out-of-order times, unexpected concentration units, and samples recorded on the wrong side of a dose. A negative concentration is usually incompatible with an ordinary measured drug concentration and should trigger a data review rather than be accepted without explanation.

Values below the assay’s quantification limit require a prespecified handling rule. Substituting zero, a fraction of the quantification limit, or another value can change AUC and observed extrema; there is no universal substitution justified by the trapezoidal formula itself. Document the rule used. Also inspect the sampling density around a likely peak: a large gap across a rapidly changing part of the profile makes observed Cmax particularly vulnerable to missing the true peak.

Common mistake: “The lowest concentration in my data is the trough, because trough means the minimum.” This is incorrect when the lowest sample was taken at another time. The correction is to define Cmin as the lowest observed value in the specified window and define trough by the prespecified predose sampling time. They may happen to be equal, but the definitions are different.

The linear trapezoidal rule is a transparent noncompartmental calculation from measured concentrations. A model-based approach can instead use a fitted concentration-time function to estimate values between samples or integrate a modeled curve. That alternative depends on the adequacy of the model and its assumptions; it does not make sparse or mistimed measurements irrelevant. For straightforward summaries, report the trapezoidal interval and observed sampling schedule so readers can see exactly what the calculation covers.

These summaries have practical limitations. AUC based only on observed endpoints is a partial-window AUC if the intended window extends beyond those endpoints. Observed extrema depend on when samples were collected, and a trough depends on an accurately timed predose sample. Report the time window, concentration and time units, whether extrema are observed, and whether the trough was measured at the scheduled time. That information makes the numerical result interpretable and helps prevent comparisons of metrics that were obtained under different sampling conditions.

Check Your Understanding

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

Concentrations are 2, 6, and 4 mg/L at 0, 2, and 5 h. Calculate linear trapezoidal AUC from 0 to 5 h, including units.
For the samples in question 1, state observed Cmax and Cmin. Can either value be asserted to be the true maximum or minimum between samples?
A scheduled dose is due at 10 h. The last concentration sample is at 9.5 h, before dosing. Is that sample a measured trough at 10 h?
Why can a sample set support AUC and observed Cmax calculations but not a measured scheduled trough?

Solutions. (1) From 0 to 2 h, the area is \(\frac{2+6}{2}(2)=8\) mg·h/L. From 2 to 5 h, it is \(\frac{6+4}{2}(3)=15\) mg·h/L. Total AUC is \(8+15=23\) mg·h/L. (2) Observed Cmax is 6 mg/L and observed Cmin is 2 mg/L. Neither is necessarily the true continuous-curve extremum, because a peak or minimum may occur between samples. (3) No. It is a pre-dose sample at 9.5 h, not a measurement at the scheduled 10 h trough time; report its actual time rather than labeling it the 10 h trough. (4) AUC and observed Cmax can be calculated from the available ordered samples, but a measured trough requires a concentration collected at the prespecified predose time. Without that sample, the trough is unavailable as a measured value.