Model Free Pharmacokinetic Analysis

A mammillary system of pools is pictured on the left. It consists of a central pool connected to a number of peripheral pools. This is usually a good model for a drug given IV. In the most simple version, the non-blood tissues are modeled as one pool, which means the drug level in the blood is represented by two exponential terms.

Two pools will not enable a good fit, if you have more than a few time points (e.g. greater than 10), the data is fairly accurate (e.g. standard deviation less than 10 percent), and the sample times cover a fairly long time (e.g. out to 24 hours). This is actually a vindication of the idea that the pools in the theory have a representation in the physical world. Some tissues, e.g. those in the liver, have an open architecture, allowing drugs to be taken up and returned rapidly to the circulation rapidly. Other tissues, e.g. those in skeletal muscle, have are perfused more slowly by blood that flows through vessels with walls of greater integrity, and thus give rise to longer time constants.

This is not a real problem, especially if you are using a computer to fit the data to exponentials. Just use three pools.

Is there a simpler way ? If you don't have a computer or good software to fit exponential curves to the data, there are graphical methods to obtain the exponential terms. However, they are tedious, especially for three pools. Some people become uneasy with pool theory, and their apprehension increases with the complexity of the system. Are the three pools I find correct? What do they really mean? If some of the parameters are inaccurate, how will that affect my analysis? Finally, often you only want (or think you only want) two numbers to characterize the pharmacokinetics of the drug.

The elimination constant is often one of the two desired parameters (the other is the volume of distribution, which I won't discuss here). Drug is often eliminated only from the blood, by the kidneys.

In this case the area under the curve, AUC, as shown by the equations on the right, is all you need to find the elimination constant.

Eq. 1 is the definition of the elimination constant: the rate drug is eliminated equals a constant times its concentration.

Eq. 2 says that if you add up (integrate) all the drug eliminated, you get the total drug given, the dose.

Eq. 3 just moves the constant out from the integral, and Eq. 4 gives you Ke.

It is misleading to call this model free analysis, because it depends on a very specific model: all drug must be eliminated from the pool from which the concentrations are measured, and the elimination rate must be a constant times the concentration in that pool.

The AUC introduces an important concept used in many fields outside of pharmacokinetics. The top expression on the left defines n terms, where each term is characterized by the power of t inside the integral. The AUC that we have been talking about is just the term where n=0, i.e. it's the 0th term in this series. The terms of this series are called the moments of the curve. In pharmacokinetics the term where n=1 is also sometimes calculated. It represents the mean transit time of a drug molecule through the central pool.

Moments also have intuitive meaning in the field of physics, where the 0th moment is the mass, the 1st moment is the center of mass (or center of gravity), and the 2nd moment is the moment of inertia. You can evaluate an infinite number of integrals with increasing values for n. The more terms you know, the more you know about c. Thus you can consider the expression a way of transforming a curve into a series of parameters, M0, M1, M2, ...which characterize that curve.

The Laplace Transform of c(t), defined in the second equation, is different from the moment series in two ways. First, the function inside the integral (the kernel) is a negative exponential of t, instead of t raised to a power. Secondly, the equation defines a continuos function of s, not a series of discrete terms with integral values for n. This transform is a powerful tool in the analysis of a system of pools, and is used in many other fields, e.g. electronics. Other kernels define other integral transforms, for example the sine and cosine define the Fourier transform.