Combining Nonoverlapping Partial-AUC Windows

Why Combining Windows Requires a Coverage Check

A partial AUC describes exposure within a specified time window. When several windows are reported, adding their partial AUCs is meaningful only if we are clear about which times are covered, which are omitted, and whether any time is covered more than once. Adjacent windows can form one continuous interval; separated windows leave a gap; overlapping windows count some exposure twice if their AUCs are simply added.

These distinctions matter when a combined quantity is intended to represent exposure over selected periods rather than over the entire elapsed time. A sum across separated windows can be correct for the selected periods while being smaller than AUC across the full span. Conversely, a sum across overlapping windows can exceed AUC over their union. The comparison concerns time coverage, not a change in the concentration profile.

The integral definition assumes that \(C(t)\) is integrable over the finite windows. In pharmacokinetic applications the concentration profile is ordinarily nonnegative, and AUC has units of concentration multiplied by time, such as mg·h/L. An endpoint shared by two adjacent windows has zero duration, so that single point contributes no additional area to the integral. In a numerical calculation, however, one must still avoid accidentally treating a finite interval as if it were only a shared endpoint.

Adjacent Windows and Separated Windows

Suppose \(W_1=[a,b]\) and \(W_2=[b,c]\), with \(a\leq b\leq c\). They meet at \(b\) but have no overlap of positive duration. Additivity of the integral gives

Thus, adjacent windows can be added to obtain AUC across the combined continuous interval. The same principle applies to a collection of windows whose interiors do not overlap. If those windows are separated, their sum gives AUC over the union of the selected windows, not automatically AUC across the span from the earliest start to the latest end.

The gap distinction can be written directly. For \(a<b<c<d\), the selected windows \([a,b]\) and \([c,d]\) omit \((b,c)\). Therefore

The right-hand side shows that the selected-window sum equals the AUC across the full span only when the omitted gap has zero AUC. A gap with positive duration does not, by itself, prove that its AUC is positive; the concentration could be zero throughout that gap. In ordinary profiles with positive concentrations over the gap, the selected-window sum is less than the full-span AUC.

Overlapping Windows: Count the Intersection Once

For two windows \(A\) and \(B\) that overlap, adding their individual AUCs counts the overlap in both windows. The general identity is

One justification is to consider each time \(t\). If it belongs to neither window, both sides contribute zero. If it belongs to exactly one window, both sides count \(C(t)\) once. If it belongs to both, the left side counts \(C(t)\) twice, while the right side counts it once in the union and once in the intersection. Integrating these equal contributions over time proves the identity. Rearranging gives

For a nonnegative concentration profile, the intersection AUC is nonnegative. It is generally necessary to subtract the intersection AUC once to obtain the union AUC. However, positive overlap duration alone does not guarantee a positive intersection AUC: if concentration is zero almost everywhere in the overlap, that integral is zero, and the sum already equals the union AUC. More generally, the sum equals the union AUC exactly when the intersection AUC is zero.

Practical Calculation and Checks

When AUCs are calculated from sampled concentrations, the accounting principle applies to the curve represented by the numerical method. If a window boundary falls between sampling times, use the specified boundary-interpolation method before calculating the window AUC. For a combined calculation, use consistent boundary concentrations and a consistent integration rule on the shared profile. If a sampled interval must be split at a window boundary, calculate the pieces so that each part of the interval is represented once in the selected union.

For more than two windows, define \(m(t)\) as the number of windows containing time \(t\). The sum of the individual AUCs weights concentration by this coverage count:

The union AUC instead counts each time in the union once. This makes a useful audit possible: list the windows in chronological order, mark every start and end, and partition the time axis into pieces on which coverage is constant. A piece covered twice contributes twice to the uncorrected sum; a piece covered zero times inside the overall span is a gap.

A common mistake is to assume that adding two partial AUCs always gives the AUC from the earliest start to the latest end. That is wrong if the windows are separated, because the gap is omitted, and it is wrong if they overlap, because the overlap is duplicated. The correction is to state the target first: for separated windows, add the gap only if the target includes it; for overlapping windows, subtract the intersection once if the target is the union.

An alternative to summing precomputed window AUCs is to partition the requested union into nonoverlapping pieces and integrate each piece once. This is often the clearest method when there are several overlaps or when boundaries cut through sampled intervals. Inclusion-exclusion is convenient for two windows; direct partitioning is usually easier to audit for more complicated collections. Neither approach removes numerical integration error: both depend on the available observations, boundary interpolation, and chosen trapezoidal rule. The arithmetic can be correct even when the profile is sparsely sampled or the numerical approximation is imperfect.

Keep the interpretation narrow. A sum over disjoint windows is the exposure accumulated in those windows under the chosen integration method. It does not imply that the intervening exposure is zero, and it is not the full-span AUC unless the gaps have zero AUC. Likewise, a union AUC from overlapping windows is not the uncorrected sum unless the intersection AUC is zero. Reporting the windows, units, and whether gaps or overlaps were handled makes the quantity interpretable and reproducible.