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Pharmacokinetics · Noncompartmental Analysis

Noncompartmental Analysis: AUC and AUMC

Learn how noncompartmental analysis uses the area under the concentration-time curve and the area under the first moment curve to quantify drug exposure, characterize drug persistence, and calculate mean residence time.

Intermediate Noncompartmental Analysis AUC AUMC Clinical Pharmacology
01 · The big picture

1. What Is Noncompartmental Analysis?

Noncompartmental analysis (NCA) is a model-independent approach for summarizing pharmacokinetic concentration-time data. Unlike compartmental analysis, NCA does not require the investigator to specify a one-, two-, or multi-compartment structural model for the entire concentration-time profile.

Instead, NCA uses quantities calculated directly from the observed concentration-time data and, where necessary, a limited description of the terminal phase. Important NCA quantities include Cmax, Tmax, AUC, AUMC, clearance, volume-related parameters, terminal half-life, and mean residence time.

Core idea: AUC summarizes the amount of exposure over time, while AUMC gives greater weight to observations occurring later in time. Together, AUC and AUMC provide the foundation for several important NCA parameters, including mean residence time.
02 · AUC

2. What Is AUC?

Area under the concentration-time curve (AUC) is the integral of drug concentration over time:

\[ AUC_{0-t}=\int_0^t C(\tau)\,d\tau \]

Conceptually, AUC measures the total concentration-time exposure over a specified interval. For concentration expressed in mg/L and time in hours, AUC has units of mg·h/L.

AUC can be calculated over a finite observation interval, such as AUC0-t, or extrapolated to infinite time to obtain AUC0-∞.

QuantityMeaningTypical interpretation
AUC0-tArea from time zero through the last relevant observed timeObserved concentration-time exposure over the measured interval
AUC0-∞Area from time zero extrapolated to infinite timeTotal exposure when the terminal portion is appropriately characterized
AUCt-∞Extrapolated terminal areaExposure after the last observed concentration

AUC is one of the most important exposure measures in pharmacokinetics because it captures both concentration and duration of exposure.

03 · AUMC

3. What Is AUMC?

Area under the first moment curve (AUMC) is the integral of concentration multiplied by time:

\[ AUMC_{0-t}=\int_0^t \tau C(\tau)\,d\tau \]

The term first moment comes from the additional weighting by time. Whereas AUC integrates concentration alone, AUMC integrates \(tC(t)\). Concentrations occurring later therefore contribute more strongly to AUMC than concentrations occurring at the same magnitude earlier in time.

Think of the difference this way: AUC asks, “How much concentration-time exposure occurred?” AUMC asks, “How much exposure occurred, with later exposure receiving greater weight because it occurred later?”

For concentration in mg/L and time in hours, AUMC has units of mg·h²/L.

04 · Comparing the two

4. AUC vs. AUMC

FeatureAUCAUMC
Mathematical quantity\(\int C(t)\,dt\)\(\int tC(t)\,dt\)
Time weightingNo additional time weightingEach concentration is weighted by time
Typical unitsConcentration × timeConcentration × time²
Primary interpretationTotal concentration-time exposureTime-weighted exposure
Important derived quantityClearance and exposure comparisonsMean residence time when combined with AUC

The distinction becomes particularly important when calculating mean residence time (MRT). Because AUMC weights exposure according to when it occurs, dividing AUMC by AUC produces a time scale describing the average time molecules remain in the system under the relevant assumptions.

05 · Numerical integration

5. How Are AUC and AUMC Calculated From Data?

In a clinical PK study, concentrations are usually observed at discrete sampling times rather than continuously. Numerical integration is therefore used to approximate the areas.

5.1 AUC using the linear trapezoidal rule

Between two observations \((t_i,C_i)\) and \((t_{i+1},C_{i+1})\), the linear trapezoidal contribution is:

\[ AUC_i=\frac{C_i+C_{i+1}}{2}(t_{i+1}-t_i) \]

The total observed AUC is the sum of the interval contributions:

\[ AUC_{0-t}=\sum_{i=1}^{n-1}\frac{C_i+C_{i+1}}{2}(t_{i+1}-t_i) \]

5.2 AUMC using the linear trapezoidal rule

For AUMC, the quantity being integrated is \(tC(t)\). Define:

$$ M_i=t_iC_i $$

The corresponding trapezoidal contribution is:

\[ AUMC_i=\frac{t_iC_i+t_{i+1}C_{i+1}}{2}(t_{i+1}-t_i) \]

Therefore:

\[ AUMC_{0-t}= \sum_{i=1}^{n-1} \frac{t_iC_i+t_{i+1}C_{i+1}}{2}(t_{i+1}-t_i) \]
Practical point: the AUMC calculation is not simply “AUC multiplied by time.” Each observed concentration is first multiplied by its corresponding sampling time, and the resulting first-moment curve is then integrated.
06 · Numerical methods

6. Why Can the Trapezoidal Method Change in the Elimination Phase?

During a declining concentration profile, the choice of numerical integration method can influence the estimated area. A common approach in PK software is the linear-up/log-down method.

For an increasing concentration interval, linear interpolation is generally used. For a declining interval, logarithmic interpolation can better represent a concentration profile that is approximately exponential during elimination.

For two concentrations where \(C_{i+1}

\[ AUC_i= \frac{C_i-C_{i+1}} {\ln(C_i)-\ln(C_{i+1})} (t_{i+1}-t_i) \]

The method used for an NCA should therefore be specified when reporting PK results, particularly when reproducing or comparing analyses from different software implementations.

07 · Extrapolation

7. From AUC0-t to AUC0-∞

The observed concentration-time data usually end before the drug has been completely eliminated. NCA can therefore estimate the remaining terminal exposure using the terminal elimination rate constant, commonly denoted \(\lambda_z\).

If the terminal phase is adequately described by first-order decline:

\[ C(t)=C_{\text{last}}e^{-\lambda_z(t-t_{\text{last}})} \]

The extrapolated AUC from the last measured concentration to infinity is:

\[ AUC_{t-\infty}=\frac{C_{\text{last}}}{\lambda_z} \]

Thus:

\[ AUC_{0-\infty}=AUC_{0-t}+ \frac{C_{\text{last}}}{\lambda_z} \]

The terminal rate constant is typically estimated from the slope of the log concentration-time profile during the terminal phase:

\[ \lambda_z=-\text{slope}\left[\ln(C)\text{ versus }t\right] \]
Important: AUC extrapolation depends on the quality of the terminal-phase data. If the terminal phase is poorly sampled or incorrectly identified, the extrapolated portion can be unreliable.
08 · AUMC to infinity

8. From AUMC0-t to AUMC0-∞

AUMC can also be extrapolated beyond the final observed concentration. Under a first-order terminal decline, the extrapolated first moment is:

\[ AUMC_{t-\infty} = \frac{t_{\text{last}}C_{\text{last}}}{\lambda_z} + \frac{C_{\text{last}}}{\lambda_z^2} \]

Therefore:

\[ AUMC_{0-\infty} = AUMC_{0-t} + \frac{t_{\text{last}}C_{\text{last}}}{\lambda_z} + \frac{C_{\text{last}}}{\lambda_z^2} \]

The additional \(1/\lambda_z^2\) term is important. AUMC is more sensitive to the terminal portion of the concentration-time curve than AUC because late concentrations receive increasing time weight.

Why this matters: uncertainty in the terminal phase can have a particularly noticeable effect on AUMC and therefore on MRT. This is one reason adequate terminal sampling is important when residence-time parameters are of interest.
09 · Mean residence time

9. Mean Residence Time: AUMC Divided by AUC

One of the most important quantities derived from AUMC is mean residence time (MRT).

For an IV administration in the simplest NCA setting:

\[ MRT_{0-\infty} = \frac{AUMC_{0-\infty}} {AUC_{0-\infty}} \]

The units work out naturally:

\[ \frac{\text{mg}\cdot\text{h}^2/\text{L}} {\text{mg}\cdot\text{h}/\text{L}} = \text{h} \]

MRT can be interpreted as the average residence time of drug molecules in the system, under the assumptions and definitions associated with the particular PK analysis.

For an IV bolus in a simple linear one-compartment model:

\[ MRT=\frac{V}{CL} \]

Since \(k=CL/V\), this is also:

\[ MRT=\frac{1}{k} \]

More complex models and routes of administration require corresponding interpretation. In particular, an extravascular MRT includes the effects of the input process as well as disposition.

10 · Worked example

10. Worked Example: Calculating AUC, AUMC, and MRT

Consider the following hypothetical concentration-time data after an IV dose. Concentration is measured in mg/L and time in hours.

Time (h)Concentration (mg/L)Time × Concentration
000
188
21224
4936
8432

Step 1: Calculate AUC0-8

Using the linear trapezoidal rule:

\[ \begin{aligned} AUC_{0-8} &= \frac{0+8}{2}(1-0) +\frac{8+12}{2}(2-1)\\ &\quad+ \frac{12+9}{2}(4-2) +\frac{9+4}{2}(8-4)\\ &=4+10+21+26\\ &=\mathbf{61\ mg\cdot h/L} \end{aligned} \]

Step 2: Calculate AUMC0-8

First calculate \(tC(t)\) at each time point, then integrate that curve:

\[ \begin{aligned} AUMC_{0-8} &= \frac{0+8}{2}(1-0) +\frac{8+24}{2}(2-1)\\ &\quad+ \frac{24+36}{2}(4-2) +\frac{36+32}{2}(8-4)\\ &=4+16+60+136\\ &=\mathbf{216\ mg\cdot h^2/L} \end{aligned} \]

Step 3: Estimate the terminal elimination rate constant

For illustration, use the concentrations at 4 and 8 hours to characterize the terminal decline:

\[ \lambda_z = \frac{\ln(9)-\ln(4)}{8-4} \approx \mathbf{0.203\ h^{-1}} \]

The corresponding terminal half-life is:

\[ t_{1/2,z} = \frac{\ln(2)}{\lambda_z} \approx \mathbf{3.42\ h} \]

Step 4: Estimate AUC0-∞

\[ AUC_{8-\infty} = \frac{4}{0.2027} \approx19.73\text{ mg·h/L} \] $$ AUC_{0-\infty} = 61+19.73 = \mathbf{80.73\ mg\cdot h/L} $$

Step 5: Estimate AUMC0-∞

\[ AUMC_{8-\infty} = \frac{(8)(4)}{0.2027} + \frac{4}{(0.2027)^2} \approx255.17\text{ mg·h}^2\text{/L} \] $$ AUMC_{0-\infty} = 216+255.17 = \mathbf{471.17\ mg\cdot h^2/L} $$

Step 6: Calculate MRT0-∞

\[ MRT_{0-\infty} = \frac{471.17}{80.73} \approx \mathbf{5.84\ h} \]

In this example, the estimated mean residence time is approximately 5.84 hours. Notice that more than simply the observed AUMC is needed: extrapolation of both AUC and AUMC depends on the terminal rate constant.

Worked-example takeaway: AUC summarizes total exposure, AUMC adds time weighting, and their ratio produces a time-based summary—MRT. The terminal extrapolation becomes especially important for AUMC because late observations receive greater weight.
11 · Terminal contribution

11. Why the Extrapolated Portion Matters

AUC0-∞ and AUMC0-∞ combine the observed portion of the profile with an extrapolated terminal component.

A useful diagnostic is the fraction of AUC that is extrapolated:

\[ \%AUC_{\text{extrapolated}} = 100\times \frac{AUC_{t-\infty}} {AUC_{0-\infty}} \]

A large extrapolated fraction indicates that a substantial part of the estimated total exposure occurs after the last observed concentration. This does not automatically invalidate an analysis, but it makes the total exposure estimate more dependent on assumptions about the terminal phase.

The issue is even more important for AUMC because its time weighting magnifies the influence of late concentrations.

Interpretation principle: always distinguish what was directly observed from what was extrapolated. An AUC or AUMC estimate extending to infinity is partly data-driven and partly dependent on the terminal-phase model used for extrapolation.
12 · IV administration

12. AUC, AUMC, and IV Dosing

For a linear IV dose, clearance can be estimated from dose and AUC:

\[ CL=\frac{Dose}{AUC_{0-\infty}} \]

Thus, AUC is directly connected to a fundamental disposition parameter.

For IV administration, MRT is based on the ratio of AUMC to AUC:

\[ MRT_{IV} = \frac{AUMC_{0-\infty}} {AUC_{0-\infty}} \]

For an IV bolus in a simple linear system, MRT can also be related to clearance and volume:

\[ MRT_{IV}=\frac{V}{CL} \]

This provides a useful connection between model-independent NCA quantities and familiar compartmental PK parameters.

13 · Extravascular dosing

13. What Changes After Oral or Other Extravascular Dosing?

After oral administration, the observed concentration-time profile reflects both drug input and drug disposition. Absorption can therefore affect AUC, AUMC, and MRT.

For a linear system with complete information and appropriate assumptions, AUC remains an exposure measure. However, MRT after extravascular administration incorporates the time associated with the input process as well as the residence time associated with disposition.

This distinction is important when comparing IV and oral administration. The difference between an IV MRT and an extravascular MRT is not necessarily evidence of a different disposition process; part of the difference can arise because drug must first be absorbed.

Key distinction: AUC is primarily an exposure measure, whereas MRT is a time-based summary influenced by both the input and disposition processes appropriate to the route of administration.
14 · Interpretation

14. What AUC and AUMC Do—and Do Not—Tell You

QuantityWhat it tells youWhat it does not automatically tell you
AUCOverall concentration-time exposureThe unique physiological mechanism producing that exposure
AUMCTime-weighted exposureA unique mechanistic description of distribution or elimination
MRTA summary of residence timeA complete description of the concentration-time profile
AUC0-∞Estimated total exposure under the terminal extrapolationExposure that was entirely observed experimentally
AUMC0-∞Estimated total first momentA quantity independent of terminal-phase assumptions

NCA is therefore powerful for summarizing exposure and disposition, but it should not be interpreted as if it were a mechanistic compartmental model.

15 · Sampling

15. Why Sampling Design Matters

The reliability of AUC and AUMC depends heavily on the concentration-time data collected during the study.

  • Early samples help characterize absorption or the initial disposition phase.
  • Samples around Cmax help define the peak concentration and timing.
  • Multiple terminal samples help establish whether a meaningful terminal phase can be identified.
  • Appropriate spacing helps capture the shape of the concentration-time profile.
  • Late sampling becomes particularly important when AUC0-∞, AUMC, or MRT is a major objective.

Poor sampling can affect not only the observed AUC and AUMC but also the estimated terminal rate constant and, consequently, all quantities that depend on extrapolation.

Sampling principle: an NCA calculation can be mathematically correct while still being scientifically limited by inadequate concentration-time sampling.
16 · Practical workflow

16. A Practical NCA Workflow for AUC and AUMC

  1. Review the dosing and sampling design. Confirm dose, route, formulation, sampling times, and concentration units.
  2. Inspect the concentration-time profile. Identify absorption, peak, distribution, and apparent terminal phases where applicable.
  3. Calculate observed AUC. Integrate concentration over time using the prespecified numerical method.
  4. Calculate observed AUMC. Integrate \(tC(t)\) over the same observed interval.
  5. Identify the terminal phase. Select an appropriate set of terminal observations according to the analysis plan and data characteristics.
  6. Estimate \(\lambda_z\). Use the terminal log-linear decline to estimate the terminal rate constant.
  7. Calculate AUC0-∞. Add the estimated terminal AUC to the observed AUC.
  8. Calculate AUMC0-∞. Add the appropriate terminal first-moment contribution.
  9. Calculate derived parameters. For example, MRT can be calculated as AUMC/AUC.
  10. Assess extrapolation. Report the extrapolated contribution and consider whether the terminal characterization is adequate for the intended interpretation.

17. Key Takeaways

  • AUC is the area under the concentration-time curve and summarizes drug exposure over time.
  • AUMC is the area under the first moment curve, obtained by integrating \(tC(t)\), so later concentrations receive greater weight.
  • AUC is typically expressed in concentration × time units, while AUMC has concentration × time² units.
  • For discrete concentration-time data, AUC and AUMC are calculated using numerical integration such as the trapezoidal method.
  • AUC0-∞ requires extrapolation beyond the last observed concentration when drug remains in the system at the end of sampling.
  • AUMC0-∞ requires a corresponding terminal extrapolation and is particularly sensitive to the terminal phase because of its time weighting.
  • The terminal elimination rate constant \(\lambda_z\) is central to extrapolating AUC and AUMC to infinity.
  • Mean residence time is calculated as \(MRT=AUMC/AUC\) using the corresponding integration limits.
  • For IV administration in a simple linear system, MRT can be related to \(V/CL\).
  • After extravascular administration, MRT reflects both input and disposition processes, so route-specific interpretation is important.
  • NCA provides useful model-independent summaries but does not by itself provide a mechanistic compartmental description of drug disposition.
  • The quality of AUC, AUMC, and MRT estimates depends strongly on the concentration-time sampling design and characterization of the terminal phase.
Next step

Where to Go Next

A natural progression is to study Cmax, Tmax, and exposure metrics, followed by clearance and volume of distribution, half-life and elimination rate constants, and then more advanced NCA topics such as terminal-phase selection, partial AUC, dose proportionality, bioavailability, and bioequivalence.

Once AUC and AUMC are understood, the connection between observed concentration-time data and clinically meaningful PK parameters becomes much clearer: exposure is summarized by AUC, time-weighted exposure by AUMC, and residence time by their ratio.

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