The Trapezoidal Approximation

The integrals in equations (C) and (F) can also be evaluated numerically using the E-Z Solve software (file exl5-7.msp). The calculated value of t is 15.8 s, slightly less than that obtained using the trapezoidal approximation. 

The area under the curve, between the limits x = 50 and x = 200, was calculated by using each of the preceding equations (9-5, 9-6 and 9-7) the rectangular approximation, the trapezoidal approximation and Simpson s rule. In each case a constant x increment of 10 was used. A portion of the spreadsheet is shown in Figure 9-18. 

As expected, the trapezoidal approximation gives more nearly correct values than does the bar-graph approximation, for the same number of panels. For 10 panels, the trapezoidal approximation gives a result of 0.135810 for the integral in Eq. (5.54). For 100 panels, the trapezoidal approximation is correct to five significant digits. 

Figure 5.6 Figure to illustrate the trapezoidal approximation (enlarged view of one panel shown). 

Using the trapezoidal approximation with five panels,... 

Specifically, the pharmacokinetic curve involved in this case is the curve representing the plasma drug concentration versus time function and, even when this function is not explicitly known, the area under the curve (AUC) can be estimated by a technique called the trapezoidal approximation. Briefly stated, this divides the... 

This first moment (or, more strictly speaking, according to Yamaoka et al., the unnormalized first moment) is called the AUMC (area under the [first] moment curve). It is estimated by the trapezoidal approximation of the area under the curve having the product of plasma drug concentration multiplied by time on the ordinate and time on the abscissa. AUMC is rarely used per se in pharmacokinetics. However, the ratio of AUMC/AUC is widely used in non-compartmen-tal pharmacokinetic analysis. This ratio, the MRT, is described in considerable detail below. 

Note that this formula provides quite good approximation if the function to be integrated is smoothly enough and the number of intervals is enough large as well. However, the error in the trapezoid approximation accoimt for the fact that the interpolation is based on a single interval (so with n= I the above definition) and therefore reads for a single interval as ... 

In developing this approximation, we still have left some area under the f x) out of the sum (near the maximum of f x), for example), or included some area of trapezoids that lies above f x). Although we will not derive limits on this error in the trapezoidal approximation here, analysis shows that the error can be expressed in the following form ... 

In the trapezoidal approximation the height of the bar is taken as the average of the values of the function at the two sides of the panel. This gives an area for the panel that is the same as that of a trapezoid whose upper comers match the integrand tunction at the sides of the panel, as shown in Figure 7.6. 

Example 7.16. Using the trapezoidal approximation with five panels, calculate the value of the integral... 

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