Saturable Fractal Pharmacokinetics and Its Applications - Mathematical Methods and Models in Biomedicine - page 335

clearance curve. While traditionally clearance curves are fit by a sum of exponential
terms, some groups have shown that they can also be fit by a single negative power
according to y at − α e − β t ,orbytwo

sequential negative powers [ 52 , 66 ]. Within a given range, while the same curves
can be fit within experimental error by a sum of negative exponentials, the converse
is not true [ 66 ]. Different explanations for these power-law and gamma-function fits
have been proposed. Because the drugs examined by Norwich and Siu [ 52 ]were
predominately eliminated through the liver, they developed a model based on the
anatomy of the liver and the flow of blood through it. Their convection-diffusion
equations for the functional unit of the liver, the acinus, generated approximate
solutions with gamma and power functions. Wise [ 66 ] explained the power law fits
in terms of a heterogeneous distribution of drug particles, where each particle cycles
through the system a number of times by series of random walks with drift.
Dose-proportionality can be identified using a simple plot of the pharmacokinetic
parameter as a function of the dose. The graph will be a straight line with a zero
intercept if the parameter is linearly proportional to dose. As a better diagnostic
tool, the “power model” was proposed [ 24 ]

at − γ , by a gamma function according to y ( t )= =

D β ,

y = α (17)

where y is a pharmacokinetic parameter and D is the dose. When log y is plotted
as a function of log D , the slope of the line will be equal to the parameter β .

Two scenarios are discussed in this chapter: 1

(dose-dependence). The study [ 24 ] found that the slopes analyzed were internally
consistent, and they were compared to an expected value of one. We propose to
build on this model by expanding the definition of the exponent to include fractional
values.

β = 0 (dose-independence) and β =

2.3 Compartmental Models

To understand and predict drug behavior through the body, compartmental models
are commonly used in pharmacokinetics [ 28 ]. In principle, each organ should be
represented by a separate compartment (see Fig. 3 for a schematic illustration)
thatisassumedtocontaina homogenously distributed concentration of drug
molecules undergoing a set of chemical kinetic reactions. In practice, models with
a few hypothetical compartments have been used with variable success rates. The
compartment's input/output is typically assumed to be governed by linear kinetic
processes with constant rate coefficients.
A compartment is defined by the number of drug molecules having the same
probability of undergoing a set of chemical kinetic processes. The exchange of
drug molecules between compartments is described by kinetic rate coefficients.
The classical compartmental model is based on two main assumptions: (a) each
compartment is homogeneous (i.e. there is instantaneous mixing) and (b) the kinetic