Trapezoid rule sampling

Blood samples were taken at 0.25, 0.5, 1.5, 2.5, 4.5, 6.5, 8.5, 10.5 and 12.5 h and the concentration of theophylline in serum was assayed by a spectrophotometric method [7]. The samples were analysed in duplicate. Bioavailability was calculated from the area under the concentration curve following the trapezoidal rule. 

Let Q be the smallest experimentally determined concentration so that and let Q be determined at a fixed sampling distance r for various values of t. Then one point is known on each curve in addition to the common point y = 1, (> = 0. Such a set of points is illustrated by the black circles in Figure 8.2. To each point corresponds a known value of obtained from equation (8.4). Further, the area included between each curve, the concentration axis and the ordinates = 0 and is equal to f Xd i). Thus F d i) is approximated by the trapezoidal rule for, first of all. 

Obviously Cmax and r ax are observational and determined directly from experimental results or the curve. Generally, AUC is determined using the trapezoidal rule, as the arithmetic sum of average plasma concentration at two sampling times divided by the... 

Fig. 2.3 Application of the trapezoidal rule for estimating area under the curve (AUC). The observed (measured) plasma drug concentration-time data are plotted on arithmetic coordinates. Total area under the curve is obtained by adding together areas of the trapezoids, the triangle from time zero to the first measured datum point and the calculated area under the extrapolated (terminal) portion of the curve, Cp yka, where Cp(n is the last measured datum point and kd is the apparent first-order disposition rate constant. The sample collection times and duration of sampling determine how well and/or completely the curve is defined.

The AUC should generally be estimated by applying the linear trapezoidal rule, but in cases where sampling times are separated by long intervals,... 

A true area under the curve (TRUE AUC) was calculated for each subject using the full concentration-time profile generated from the simulation. Sampling was performed at 0, 0.25, 0.5, 0.75,1, 2, 4, 6, 8,10,12,16, and 24 hours. All areas under the curve were calculated using the noncompartmental analysis module in WinNonlin Version 4.0, using the log/linear trapezoidal rule. 

To assess whether food impacted the pharmacokinetics of a new drug, such a study was conducted in 12 healthy, male subjects. Plasma samples were collected and analyzed for drug concentrations. AUC(0—oo) was calculated using the linear trapezoidal rule and Cn3. was determined from direct observation of the data. 

Following the administration of a drug by an intravenous injection, if it is necessary to use a two-compartment model, the area under the plasma concentration-time curve from f = 0 to f = f (the last sampling time) may be estimated by using trapezoidal rules, as mentioned earlier. Additionally, the area under plasma concentration-time curve from f = f to f = may be computed using the following equation ... 

Some pharmacokinetic software packages perform noncompartmental analysis without fitting the entire response curve. These programs compute the elimination rate constant (k) for the terminal elimination phase of the data, and then use a trapezoidal rule with this elimination rate constant to compute AUC and AUMC. With these terms, the total body clearance, the steady-state volume of distribution, and the mean residence time in the body can be calculated. Without C , it is not possible to calculate the volume of distribution of the central compartment or the mean residence time of the sampling compartment. The latter term is therefore critical in accurately determining these parameter values and depends on an unbiased and close fit of the data to Equations 13.2 and 13.6. 

Equations 3.122 and 3.124 for the gas phase serve only to compute distributions of Y and along bed height, which is necessary to calculate q and Wd. They can easily be integrated numerically, e.g., by the Euler method, at each time step. The integrals in Equations 3.121 and 3.123 can be numerically calculated, e.g., by the trapezoidal rule. This allows Equations 3.121 and 3.123 to be solved by any ODE solver. The model has been solved for a sample case and the results are shown in Eigure 3.20. 

In summary, the trapezoidal rule is easy to implement and usually accurate enough. Furthermore, it can be used with nonequally spaced data (such as real experimental data). When there is a limit to the number of sample points, but they can be placed at will, then Gauss integration is often the best choice. For example, suppose a restriction is that only four samples can be obtained on a process during a test whose duration is 1 h. When during the hour should the samples be collected The answer to this is left as an exercise. 

---