Full and Partial AUC Estimation

Why Estimate Both Full and Partial AUC?

The area under a plasma concentration-time curve (AUC) summarizes exposure over a defined time window. In a drug-drug interaction (DDI) study, the full dosing-interval AUC can show how total exposure over that interval differs between conditions. But the same full-interval area can conceal meaningful differences in when exposure occurs. A prespecified partial AUC focuses on an interval that is relevant to a particular clinical question, such as early exposure after dosing or a later period in which metabolism or inhibition may be important.

An AUC is not defined by the concentration data alone: the start and end times are part of the quantity being estimated. A partial AUC is therefore not simply an approximate version of the full AUC. It answers a different question about exposure during a particular interval. Interpreting a comparison requires the same window, units, and integration approach in each condition.

Definition: For concentration \(C(t)\), the AUC over \([t_a,t_b]\) is the integral \(\int_{t_a}^{t_b} C(t)\,dt\). A full dosing-interval AUC covers a complete dosing interval, commonly \([0,\tau]\), where \(\tau\) is the interval between doses. A partial AUC covers a prespecified subinterval, such as \([t_a,t_b]\) with \(0 \leq t_a < t_b \leq \tau\). If concentration is measured in mg/L and time in hours, AUC is in mg·h/L.

For a steady-state dosing interval, the profile from one dose time to the next repeats under the steady-state assumption; AUC over that interval is often written \(AUC_{0-\tau}\). An interval can also be specified for a non-steady-state profile, but it should not be described as a steady-state interval AUC unless steady state is supported. The calculations below estimate area from observed concentration-time samples; they do not establish that the sampling schedule adequately represents the underlying profile.

Numerical Integration From Samples

Suppose measured concentrations are \(C_i\) at times \(t_i\), ordered from earliest to latest. Between adjacent observations, the simplest estimate connects the points by a straight line. The area for that segment is the area of a trapezoid:

AUC_{i,i+1} = \frac{C_i+C_{i+1}}{2}(t_{i+1}-t_i)

Summing the segment areas whose time ranges lie within the requested interval gives the estimated AUC. The linear trapezoidal rule assumes concentration changes linearly between adjacent samples. It is exact for a truly linear segment, but real concentration-time curves need not be linear between observations.

Worked Example: Full AUC Over One Dosing Interval

Consider hypothetical concentrations measured over an 8-hour dosing interval:

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

Using linear trapezoids, the four segment areas are:

\begin{aligned} AUC_{0-1} &= \frac{0+4}{2}(1-0)=2 \\ AUC_{1-2} &= \frac{4+6}{2}(2-1)=5 \\ AUC_{2-4} &= \frac{6+3}{2}(4-2)=9 \\ AUC_{4-8} &= \frac{3+1}{2}(8-4)=8 \end{aligned}

Adding them gives \(AUC_{0-8}=2+5+9+8=24\) mg·h/L. This is the estimated area over the specified 8-hour interval using the linear rule. It is not an estimate of area after 8 hours.

When concentrations are declining, a straight line can overestimate the area if the decline is closer to exponential. If concentration decreases from \(C_i\) to \(C_{i+1}\), with both values positive, a log trapezoid assumes that log concentration changes linearly over the segment. Equivalently, concentration follows an exponential curve within that segment. Its area is:

AUC_{i,i+1} = (t_{i+1}-t_i)\frac{C_i-C_{i+1}}{\ln(C_i/C_{i+1})}

This expression requires positive, unequal concentrations. If the two positive concentrations are equal, the limiting area is \(C_i(t_{i+1}-t_i)\). A common hybrid approach uses linear trapezoids for rising segments and log trapezoids for declining segments. The selected rule should be applied consistently and reported; changing the rule between conditions can make an apparent exposure difference reflect calculation choices rather than profiles.

Worked Example: Comparing Linear and Log Trapezoids

Suppose a hypothetical concentration falls from 8 mg/L at 2 hours to 4 mg/L at 4 hours. The interval length is 2 hours. The linear estimate is:

AUC_{\mathrm{linear}}=\frac{8+4}{2}(4-2)=12\ \text{mg·h/L}

For a log trapezoid, both concentrations are positive and unequal, so the formula applies:

AUC_{\mathrm{log}} = 2\frac{8-4}{\ln(8/4)} = \frac{8}{\ln(2)} \approx 11.54\ \text{mg·h/L}

The log estimate is smaller because it assumes exponential decline rather than a straight line between the endpoints. Neither estimate can be declared universally correct from the two samples alone; the choice is an assumption about the unobserved segment. Additional samples or a well-supported model could help characterize the curve, but cannot make an unsupported integration assumption disappear.

Partial AUCs and Boundary Times

Partial intervals should be selected before examining the results when they are intended to answer a clinical question. For example, a protocol may define an early post-dose window or a late interval in advance. The boundary times must be stated clearly, and the same boundaries must be used for all conditions being compared. The partial area is calculated by summing the areas only within that window.

If a boundary falls between sampling times, its concentration must be estimated before the partial area can be calculated. With linear interpolation between samples, for \(t_i \leq t \leq t_{i+1}\):

\widehat{C}(t) = C_i+\frac{C_{i+1}-C_i}{t_{i+1}-t_i}(t-t_i)

This interpolation uses the same straight-line assumption as the linear trapezoidal rule. If a log-linear rule is being used for a declining segment, interpolation should instead follow the corresponding exponential assumption. Whichever method is used, the boundary estimate and the partial AUC depend on that assumption.

Worked Example: Partial AUC With Interpolated Boundaries

A hypothetical profile has concentrations of 2 mg/L at 0 hours, 6 mg/L at 2 hours, 3 mg/L at 4 hours, and 1 mg/L at 6 hours. Estimate the partial AUC from 1 to 5 hours using linear interpolation and linear trapezoids.

At 1 hour, interpolate between 0 and 2 hours:

\widehat{C}(1)=2+\frac{6-2}{2-0}(1-0)=4\ \text{mg/L}

At 5 hours, interpolate between 4 and 6 hours:

\widehat{C}(5)=3+\frac{1-3}{6-4}(5-4)=2\ \text{mg/L}

The partial area from 1 to 2 hours is \((4+6)(2-1)/2=5\) mg·h/L. From 2 to 4 hours it is \((6+3)(4-2)/2=9\) mg·h/L. From 4 to 5 hours it is \((3+2)(5-4)/2=2.5\) mg·h/L. Therefore, the estimated \(AUC_{1-5}\) is \(5+9+2.5=16.5\) mg·h/L.

What Full and Partial Areas Can Show

A full-interval AUC is useful when the question concerns total exposure over the complete dosing interval. A partial AUC is useful when the timing of exposure matters and the interval has a prior clinical or mechanistic rationale. For instance, a change concentrated in an early window may be diluted in the full-interval total. Conversely, an early partial area alone may miss a change occurring later in the interval. Neither area, by itself, identifies the biological mechanism responsible for the profile.

Worked Example: Same Full AUC, Different Partial AUC

Compare two hypothetical profiles sampled at 0, 1, 2, 4, and 8 hours. Profile A has concentrations 0, 4, 6, 3, and 1 mg/L. Profile B has concentrations 0, 8, 4, 2, and 2 mg/L. Use linear trapezoids. For Profile A, the segment areas are 2, 5, 9, and 8 mg·h/L, so the full \(AUC_{0-8}\) is 24 mg·h/L.

For Profile B, the segment areas are:

\begin{aligned} AUC_{0-1} &= \frac{0+8}{2}(1)=4 \\ AUC_{1-2} &= \frac{8+4}{2}(1)=6 \\ AUC_{2-4} &= \frac{4+2}{2}(2)=6 \\ AUC_{4-8} &= \frac{2+2}{2}(4)=8 \end{aligned}

Thus, Profile B also has a full \(AUC_{0-8}=4+6+6+8=24\) mg·h/L. But the early partial \(AUC_{0-2}\) is \(2+5=7\) mg·h/L for Profile A and \(4+6=10\) mg·h/L for Profile B. The full areas are equal while the early areas differ by 3 mg·h/L. This demonstrates why a prespecified partial AUC can reveal a time-localized difference that a full-interval total does not convey; it does not establish why the profiles differ.

For a DDI comparison, calculate the same AUC endpoint in the interacting and reference conditions. A ratio for a given window can be expressed as the interacting-condition AUC divided by the reference-condition AUC. The numerator and denominator must refer to the same time window and use compatible units and calculation rules. A full-interval ratio and a partial-interval ratio are distinct exposure comparisons and need not be equal. A partial AUC should not be selected after seeing which window produces the largest condition difference, because that makes the result difficult to interpret as a prespecified endpoint.

Assumptions, Diagnostics, and Alternatives

The numerical estimate depends on the measurements and on what is assumed between them. Before interpreting an AUC, check that time points are correctly ordered, concentration and time units are consistent, and the requested boundaries are covered by observations or a declared interpolation method. Inspect the profile for unexpectedly large gaps, abrupt changes, or implausible values. A long gap across a steep rise or decline can produce substantial integration error even when the arithmetic is correct. Sparse sampling can also miss a localized feature entirely; reducing the calculation to a single number does not repair that loss of information.

Log trapezoids cannot be applied directly to a segment with a zero or nonpositive concentration, and a value below the assay’s quantification limit is not automatically an observed zero. The handling of such values should follow the analysis plan and measurement context. Do not quietly replace them with a convenient positive value just to permit a logarithm. For partial AUCs, inspect whether boundaries are directly observed or interpolated and whether the result is sensitive to the assumptions at those boundaries.

A noncompartmental trapezoidal estimate is relatively direct but depends on sampling density and numerical interpolation. A model-based approach can integrate a continuous fitted concentration-time curve, which may help when observations are sparse or a time window requires interpolation. Its estimate then depends on the model’s structure and parameters. When a PBPK simulation provides a concentration-time trajectory, its integrated area is a model prediction; it is not the same thing as directly integrating observed concentrations. The choice between approaches should reflect the question, available data, and assumptions that can be defended.

Common pitfall: “The full AUC is unchanged, so exposure cannot have changed in a clinically relevant way.” This reasoning treats the total area as if it described the entire shape of the profile. Two profiles can have the same full AUC but different partial AUCs, as the worked example shows. The correction is to define the relevant exposure window in advance, calculate it consistently in each condition, and interpret it alongside—not as a replacement for—the full-interval endpoint when both questions matter.

Calculation Workflow

Define the endpoint.
State whether the target is a full dosing-interval AUC or a partial AUC, and give its start and end times.

Check the inputs.
Confirm time and concentration units, the ordering of observations, and whether each endpoint is observed or must be interpolated.

Choose and apply an integration rule.
Use a stated rule, such as linear trapezoids or linear-up/log-down trapezoids, consistently across the relevant profiles.

Sum and inspect.
Add only the segment areas inside the defined window, check the units and arithmetic, and assess whether sampling gaps or unsupported interpolation could materially affect the result.

Check Your Understanding

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

A hypothetical profile has concentrations 2 mg/L at 0 hours, 6 mg/L at 1 hour, and 2 mg/L at 3 hours. Using linear trapezoids, calculate \(AUC_{0-3}\).
For positive concentrations of 10 mg/L at 2 hours and 5 mg/L at 4 hours, calculate the log-trapezoidal area for that segment.
Why must the boundaries of a partial AUC be specified consistently for the two conditions in a DDI comparison?
A report gives equal full-interval AUCs for two conditions. Does this establish that their partial AUCs are equal? Explain.

Solutions. (1) The 0–1 hour area is \((2+6)(1)/2=4\) mg·h/L. The 1–3 hour area is \((6+2)(3-1)/2=8\) mg·h/L. Thus \(AUC_{0-3}=4+8=12\) mg·h/L. (2) The log area is \(2(10-5)/\ln(10/5)=10/\ln(2)\approx14.43\) mg·h/L. (3) The two estimates must represent exposure over the same time window; otherwise, their difference could reflect different boundaries rather than a condition effect. (4) No. Equal total areas do not require equal areas within a subinterval, because one profile can have relatively more area in that subinterval and less elsewhere.