The trapezoid method of obtaining AUC
This method, which has intuitive appeal, consists of connecting points on the concentration-time curve with straight line segments and then using the area under this polygon to approximate the actual area under the curve. It is called the trapezoid method because each segment of the polygon is a trapezoid. The area of a trapezoid is:
One disadvantage is the systematic error that occurs when the curve isn't straight. As you can see in the figures, when the curve is concave upward (positive second derivative) the trapezoid method over estimates, while if the curve is convex upward (negative second derivative) the method under estimates.
If you are on a desert island and your laptop battery is completely discharged, this method is attractive since it only requires data points, and not a curve. If you have time and motivation, even a hand drawn curve can be used to get a better value than the trapezoid method. If your computer is running, and you have software to fit a curve to the data, you will get a better answer if you compute the real area under a curve, any curve. The best curve to fit pharmacokinetic data is usually a sum of exponentials, but there are exceptions.
There are some that might argue that the trapezoid method has the advantage of being objective, in the sense that it consists of a set of rules; if you do it ten times you get the same answer each time. If you draw a curve by hand, you will probably get ten different answers, but each will be better than the trapezoid method. I don't see the advantage of reproducibly obtaining incorrect answers, but do what you think best.
The trapezoid method seems reasonable, if not clever. However, in many cases it is neither. Consider an apparently dumb alternative. You have a large number of time points:
a. take every other time point and use the trapezoid method to find the area (green line in the figure).
b. take the time points left over (the ones in between the ones used for the trapezoid method) and just multiply the concentration by the time separation (the horizontal blue line in the figure). Call this the rectangle method.
To compare the methods, note that if the blue line is rotated about its C value (the point half way between a and b), the area in the resulting trapezoid is the same as in the original rectangle. Thus, rotate the blue line to be parallel to the green line.
Now it is clear that the blue method under estimates, and the green method over estimates the area. The difference between the two methods is the parallelogram with a green top and blue bottom. To see which one is better, divide the difference between the blue and green methods into two equal parts by the black dashed lines. Since the real curve, the red curve, is on the blue side of the dashed lines, the blue error is less than the green error.
How can the dumb rectangle method be better than the clever trapezoid method? We think the trapezoid method is better only because it appears to follow the curve better. A deeper explanation is beyond the scope of this essay.