Exposure Metrics Under Different Sampling Schedules

Why Sampling Time Is Part of an Exposure Comparison

An exposure metric is calculated from concentration observations, but the observations are only available at selected times. When two profiles are measured on different time grids, a difference between their calculated metrics can reflect a difference in the underlying concentration-time profiles, a difference in sampling, or both. This matters especially for peak measures and for area estimates near sharp changes or window boundaries.

The time grid is therefore part of the measurement procedure, not just a convenient way to organize the data. A numerical comparison is easiest to interpret when profiles use comparable sampling times and the same calculation rules. If schedules differ, the analyst should assess how that difference could affect each metric before attributing a contrast to exposure.

Definition: A sampling schedule is the set of times at which concentrations are planned or observed. For a profile with concentrations \(C(t_i)\) at times \(t_1,\ldots,t_k\), observed \(C_{\max}\) is the largest recorded concentration, and observed \(T_{\max}\) is the time of that recorded maximum, subject to any prespecified rule for ties. The trapezoidal AUC estimate sums the areas of trapezoids between successive observations. A partial AUC is an area estimate restricted to a specified time window, using a prespecified rule for handling window boundaries.

These are sampling-based quantities. In particular, observed \(C_{\max}\) is not automatically the continuous-time peak concentration, and observed \(T_{\max}\) is not automatically the time of that peak. The earlier material on sparse sampling and peak misspecification establishes why a peak between observations can be missed; here the focus is on what happens when schedules differ across profiles or study conditions.

How the Grid Enters the Calculations

For adjacent observations at \(t_i\) and \(t_{i+1}\), the linear trapezoidal contribution to AUC is the interval width multiplied by the average of the endpoint concentrations. Adding these contributions gives the estimate over the observed range:

\widehat{\mathrm{AUC}}_{t_1-t_k} = \sum_{i=1}^{k-1} \frac{C(t_i)+C(t_{i+1})}{2}(t_{i+1}-t_i).

Thus, changing the grid changes both the interval widths and the concentrations that serve as endpoints. The formula does not recover unobserved curvature. It defines an estimate based on straight-line segments between the available observations.

The same issue is especially direct for \(C_{\max}\): the maximum is selected from the recorded values, so a schedule that omits times near the peak can record a smaller maximum. The \(T_{\max}\) value is likewise selected from the observed times, so its possible values are restricted to that schedule. Without additional assumptions about profile shape, there is no general correction that recovers the unsampled peak time from the observed maximum alone.

Justification: If the underlying profile is \(C(t)\) and the observed times are a subset of the times over which its maximum is defined, then each recorded value satisfies \(C(t_i)\leq \max_t C(t)\). Taking the maximum over the recorded values gives \(C_{\max,\mathrm{observed}}=\max_i C(t_i)\leq \max_t C(t)\). Equality holds only if a sampled time attains the continuous-time maximum. This comparison assumes exact concentrations; measurement error can also make an observed value exceed the underlying profile.

For a partial AUC over \([a,b]\), the calculation must also define what happens if \(a\) or \(b\) is not an observation time. A common approach is linear interpolation between the observations bracketing a boundary, followed by trapezoidal integration within the window. The resulting estimate then depends on which observations bracket the boundary and how far apart they are.

Worked Example: Different Grids, Different Peak Summaries

Suppose two hypothetical profiles are measured on different schedules. Profile A has observations at 0, 1, 2, 4, and 8 hours, with concentrations 0, 6, 10, 5, and 1 mg/L. Profile B has observations at 0, 0.5, 1.5, 3, 6, and 8 hours, with concentrations 0, 4, 9, 8, 3, and 1 mg/L. Compare their observed peak summaries and trapezoidal AUC estimates.

Observed peaks. For A, the largest recorded concentration is 10 mg/L at 2 hours, so observed \(C_{\max}=10\) mg/L and observed \(T_{\max}=2\) hours. For B, the largest recorded concentration is 9 mg/L at 1.5 hours, so observed \(C_{\max}=9\) mg/L and observed \(T_{\max}=1.5\) hours. These summaries differ, but the schedules do not sample the same times. The observed values alone do not show whether the underlying profiles truly have different peaks.

AUC for A. Applying the trapezoidal formula interval by interval gives

\widehat{\mathrm{AUC}}_A = \frac{0+6}{2}(1-0) +\frac{6+10}{2}(2-1) +\frac{10+5}{2}(4-2) +\frac{5+1}{2}(8-4) =3+8+15+12=38\ \mathrm{mg\cdot h/L}.

AUC for B. Its intervals give

\widehat{\mathrm{AUC}}_B = \frac{0+4}{2}(0.5) +\frac{4+9}{2}(1) +\frac{9+8}{2}(1.5) +\frac{8+3}{2}(3) +\frac{3+1}{2}(2) =1+6.5+12.75+16.5+4 =40.75\ \mathrm{mg\cdot h/L}.

Both estimates cover 0 to 8 hours, yet they are 2.75 mg·h/L apart. Different grids provide different trapezoids and can produce different estimates even when the integration window is the same. The calculation does not establish that the underlying exposure differs by 2.75 mg·h/L; interpretation requires considering schedule and profile shape as well as the observed values.

Partial-AUC Boundaries and Comparability

A partial AUC comparison requires more than naming the window. Both profiles need the same start and end times, the same concentration and time units, and a consistent boundary-handling and integration rule. If a boundary falls between samples, interpolation makes the boundary concentration an estimate. A wide interval around that boundary can make the estimate sensitive to curvature that the observations do not reveal.

Even with the same window, unequal schedules can generate different estimates. Conversely, matching the window does not make two estimates equally reliable if one profile has dense observations around a rapid rise and fall while the other has a long interval across that region. The next example isolates the effect of interpolation and interval width.

Worked Example: A Partial-AUC Window Between Samples

Two hypothetical profiles are evaluated over the same 0-to-2-hour window. Profile A has observations \((0,0)\), \((1,8)\), and \((3,4)\), where each pair is time in hours and concentration in mg/L. Profile B has observations \((0,0)\), \((0.5,5)\), \((1.5,9)\), and \((3,4)\). Estimate each partial AUC using linear interpolation at 2 hours and trapezoidal integration within the window.

For A, 2 hours lies between 1 and 3 hours. Linear interpolation gives

\widehat{C}_A(2) = 8+\frac{2-1}{3-1}(4-8) =8-2=6\ \mathrm{mg/L}.

The area from 0 to 1 hour is \(\frac{0+8}{2}(1)=4\) mg·h/L. From 1 to 2 hours it is \(\frac{8+6}{2}(1)=7\) mg·h/L. Therefore, the estimated partial AUC is \(4+7=11\) mg·h/L.

For B, interpolate between 1.5 and 3 hours to estimate the concentration at 2 hours:

\widehat{C}_B(2) = 9+\frac{2-1.5}{3-1.5}(4-9) = 9+\frac{1}{3}(-5) =7.333\ \mathrm{mg/L}.

The areas are \(\frac{0+5}{2}(0.5)=1.25\) mg·h/L from 0 to 0.5 hours, \(\frac{5+9}{2}(1)=7\) mg·h/L from 0.5 to 1.5 hours, and \(\frac{9+7.333}{2}(0.5)=4.083\) mg·h/L from 1.5 to 2 hours. The estimated partial AUC is \(1.25+7+4.083=12.333\) mg·h/L.

The estimates differ by 1.333 mg·h/L. The profiles have different observed values as well as different grids, so this example cannot attribute the difference to schedule alone. It does show why the same nominal window is not sufficient to guarantee the same interpolation accuracy: the endpoint at 2 hours is inferred from different bracketing observations.

Ways to Make Comparisons More Defensible

The strongest practical safeguard is to use comparable schedules when planning the measurements. This is especially important around expected peaks, rapid changes, and partial-AUC boundaries. When data have already been collected on unequal schedules, the analysis should make the difference visible rather than silently treating all estimates as equally supported.

Use the same prespecified calculation rules. Apply consistent time units, integration conventions, analysis windows, and boundary interpolation rules. A shared algorithm removes avoidable procedural differences, though it cannot restore information that was never sampled.
Inspect the actual grids and profiles. Report observation times and plot concentrations against time. Mark the analysis window and its boundaries. Long gaps near a peak or window boundary deserve particular attention.
Use a common set of times cautiously. Interpolation to a common grid can make sampled values easier to compare, but interpolated values are not new observations. Results depend on the interpolation rule and can miss curvature within wide gaps.
Consider schedule-aware sensitivity checks. Recalculate estimates under defensible alternative treatments of uncertain intervals or boundaries, and show whether the substantive comparison changes. Such checks describe sensitivity; they do not determine the unobserved profile uniquely.
Use a model-based approach only with suitable support. A validated concentration-time model can estimate features between observation times, but its conclusions depend on model form and assumptions. It is not a neutral substitute for matching schedules.

These approaches have different purposes. Matched sampling directly improves design comparability. A common-grid calculation standardizes presentation but adds interpolation dependence. Model-based estimation can use a specified curve to represent between-sample behavior, but introduces structural assumptions. No approach can guarantee recovery of an unobserved sharp peak when the data and assumptions do not constrain it.

Diagnostics, Limitations, and a Common Mistake

Before interpreting a contrast, check whether the schedules cover the same time range, whether peak-region sampling is similarly dense, and whether partial-AUC boundaries are observed or interpolated. Confirm that each estimate uses the intended window and consistent units. Plotting both profiles against the actual observation times can reveal gaps that a single summary number hides. If an apparent difference is concentrated in a region where one profile is sparsely sampled, that is a reason to qualify the comparison, not proof that the difference is an artifact.

The limits are important: a denser grid can improve resolution but does not eliminate assay error or guarantee that the true peak was captured; interpolation assumes a straight line over each interval; and observed summaries cannot, without additional information, distinguish every possible underlying curve. Schedule differences should therefore be reported alongside the metric and considered when assessing how much confidence to place in its comparison.

Common mistake: “Both profiles use the same 0-to-8-hour window, so their AUC estimates are directly comparable and any difference is a real exposure difference.” The window alone does not make the estimates equally informative. Unequal interior times change trapezoidal segments and may miss different features. Use a consistent method, examine the schedules and profiles, and qualify conclusions when the sampling grids differ.

Worked Example: A Missed Peak Can Change the Observed Contrast

Suppose a hypothetical concentration profile reaches 10 mg/L at 2 hours. One schedule samples at 1, 2, and 3 hours and records 8, 10, and 8 mg/L. A second schedule samples only at 1 and 3 hours and records 8 and 8 mg/L. Compare the observed peak summaries.

With the first schedule, observed \(C_{\max}=10\) mg/L at observed \(T_{\max}=2\) hours. With the second, observed \(C_{\max}=8\) mg/L, and both sampled times tie for the maximum. If a prespecified tie rule selects the earliest time, observed \(T_{\max}=1\) hour for that schedule. The second schedule's observed maximum is 2 mg/L, or \(100(2/10)=20\%\), below the peak recorded by the first schedule.

The second schedule gives no observation at the known peak time in this hypothetical setup. Connecting its two observations by a straight line would also yield 8 mg/L at 2 hours, not 10 mg/L. Thus, interpolation cannot recover the peak from those endpoints without additional information about the profile's shape. In a real comparison, the underlying peak would not be known in advance; the example shows why a lower observed \(C_{\max}\) on a sparser schedule is not, by itself, evidence of lower underlying peak exposure.

Check Your Understanding

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

A profile is sampled at 0, 1, and 4 hours, with concentrations 0, 6, and 3 mg/L. What is the trapezoidal AUC from 0 to 4 hours? What feature of the schedule could affect the estimate?
One schedule samples at 2 hours and another samples only at 1 and 3 hours. Can the second schedule's observed \(C_{\max}\) be assumed to equal the first schedule's? Explain.
A partial-AUC window ends at 3 hours, but the bracketing observations are at 2 and 5 hours. What must be specified to calculate the partial AUC, and what is one limitation?
Two profiles have the same window but different interior sampling times. Give two checks that should precede interpreting their AUC difference.

Solutions. (1) The area is \(\frac{0+6}{2}(1-0)+\frac{6+3}{2}(4-1)=3+13.5=16.5\) mg·h/L. The long 1-to-4-hour interval may conceal curvature or a peak between observations. (2) No. The second schedule can miss a peak at 2 hours; its maximum is only the largest recorded value. (3) Specify a boundary rule, such as linear interpolation, and the integration rule. Interpolation over a wide interval assumes a straight-line segment and may not represent curvature. (4) Inspect the actual time grids and concentration-time plots; check peak-region spacing, window boundaries, and whether the same integration and interpolation rules were used. Answers should explain how the checks affect confidence in the comparison.