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# Area under the plasma concentration time curve (AUC)

The area under the plasma (serum, or blood) concentration versus time curve (AUC) has an number of important uses in toxicology, biopharmaceutics and pharmacokinetics.

Toxicology AUC can be used as a measure of drug exposure. It is derived from drug concentration and time so it gives a measure how much - how long a drug stays in a body. A long, low concentration exposure may be as important as shorter but higher concentration. Some drugs are dosed using AUC to quantitate the maximum tolerated exposure (AUC Dosing).

Biopharmaceutics The AUC measured after administration of a drug product is an important parameter in the comparison of drug products. Studies can be performed whereby different drug products may be given to a panel of subject on separate equations. These bioequivalency or bioavailability studies can be analysed by comparing AUC values.

Pharmacokinetics Drug AUC values can be used to determine other pharmacokinetic parameters, such as clearance or bioavailability, F. Similar techniques can be used to calculate area under the first moment curve (AUMC) and thus mean resident times (MRT).

## Calculation of AUC using the Trapezoidal Rule

Figure 2.10.1. Linear Plot of Cp versus Time showing AUC and AUC segment

The area under the plasma concentration time curve (AUC) is very useful for calculating the relative efficiency of different drug products (We'll talk about this later, see Chapter 18). It can used to calculate the total body clearance (CL) and the apparent volume of distribution.

If we have a smooth line for concentration versus time or an equation for Cp versus time from a pharmacokinetic model we could slice the area into vertical segments. Each segment would be very thin, Δt and in extreme dt, in width (much smaller than the segment in Fig 2.10.1). The total AUC is calculated by adding these segments together. In calculus this would be the integral. Each very narrow segment has an area = Cp*dt. Thus the total area (AUC) is given by Equation 2.10.1:

Equation 2.10.1. Total AUC calculated from Very Narrow Segment

Moving ahead a little to Chapter 5 the integrated equation for plasma concentration as a function of time is:

Equation 2.10.2. Cp versus Time after an IV Bolus dose

This is essentially the same as the integrated equation we derived for amount of drug remaining using Laplace transforms. The difference here is the use of V, apparent volume of distribution, to convert amount into concentration. We can substitute this equation for Cpt into Equation 2.10.1 to derive an analytical equation for AUC:

Equation 2.10.3. AUC calculated as the integral of Cp versus time

From Math Table we would find that:

With a = kel (in Equation 2.10.3), t1 = 0, and t2 = ∞,

At t = 0, e-kel*t = 1 and at t = ∞, e-kel*t = 0

Therefore:

Equation 2.10.4. AUC Calculated from Concentration and kel

This is analytical integration (exact solution, given exact values for V and kel)

Note: For t = 0 to ∞, AUC = Cp0/kel. With t = t to ∞, AUC is calculated as Cpt/kel. We will use this result further down on this page
That is:

Rearranging Equation 2.10.4 to solve for V gives:

V = Dose/(AUC * kel)

Equation IV-18. Volume of Distribution Calculated from Dose, AUC and kel

We could use Equation 2.10.4 to calculate the AUC value if we knew DOSE, kel, and V but usually we don't do this. We can calculate AUC directly from the Cp versus time data. We need to use a different approach. The simplest, most common approach is a numerical approximation method called the trapezoidal rule.

Other methods have been described by Yeh and Kwan (Yeh, K.C. and Kwan, K.C. 1978 "A comparison of numerical integrating algorithms by trapezoidal Lagrange and spline approximation", J. Pharmacokin. Biopharm., 6, 79-98) and Purves (Purves, R.D. 1992 "Optimum numerical integration methods for estimation of area under the curve (AUC) and area under the moment curve (AUMC)", J. Pharmacokin. Biopharm., 20, 211-226).

Figure 2.10.2. Linear Plot of Cp versus Time showing Typical Data Points

We can calculate the AUC of each segment if we consider the segments to be trapezoids. [Four sided figure with two parallel sides].

The area of each segment can be calculated by multiplying the average concentration by the segment width. For the segment from Cp2 to Cp3:

This segment is illustrated in Fig 2.10.3 below.

Figure 2.10.3. Linear Plot of Cp versus Time showing One Trapezoid

The area from the first to last data point can then be calculated by adding the areas together.

Note: Summation of data point information (non-calculus)

This gives:

Figure 2.10.4. Linear plot of Cp versus time showing areas from data 1 to data n

To finish this calculation we have two more areas to consider. The first and the last segments.

After a rapid IV bolus, the first segment can be calculated after determining the zero plasma concentration Cp0 by extrapolation.

Thus

If we assume that the last data points follow a single exponential decline (a straight line on semi-log graph paper) the final segment can be calculated from the equation above from tlast to infinity:

Thus the total AUC can be calculated as:

 Table 2.10.1. Example Calculation of AUC Time (hr) Concentration (mg/L) Δ AUC AUC (mg.hr/L) 0 100* 1 71 85.5 85.5 2 50 60.5 146.0 3 35 42.5 188.5 4 25 30.0 218.5 6 12 37.0 255.5 8 6.2 18.2 273.7 10 3.1 9.3 283.0 Total 8.9** 291.9
* Extrapolated value ** Calculated as Cplast/kel

AUC Practice Problems
References

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Copyright 2001-3 David W. A. Bourne (david@boomer.org)

This file was last modified: Wednesday 26 May 2010 at 07:39 PM