Saturable Fractal Pharmacokinetics and Its Applications - Mathematical Methods and Models in Biomedicine - page 339

at − α e − β t [ 61 ].
Different explanations for these fits have been proposed, including a stochastic ran-
dom walk model based on the cycling of molecules in and out of the plasma [ 48 ], a
set of convection-diffusion equations for transit in the liver, and gamma-distributed
drug residence times [ 65 ]. The authors explain the power law behavior by hepatic
processes although tissue distribution may be playing an equally important role as
shown in [ 62 ].
In this chapter, we introduce a model based on fractal kinetics with an anomalous
reaction order as a physiologically based mechanism that generates power law tails
using only one compartment. This model, which may be a gross oversimplification
of the actual situation in the human body, is naturally interpreted in terms of the
anatomy and physiology of the liver, the organ of drug elimination.

power law, two sequential power laws, or the gamma function y ( t )=

2.6 Transient Fractal Kinetics

Anacker and Kopelman [ 2 ] found that reactions that occur on or within fractal
media exhibit anomalous kinetics that do not follow the classical mass-action form.
Specifically, the kinetic rate coefficient is time-dependent [ 32 ]:

k 0 t − h

k = , (30)

where

d s
2 .

= −

h 1 (31)

The quantity d s is the spectral dimension that describes the path of a random
walker within the medium [ 50 ]. The classical case corresponds to d s = 2. Equations

( 30 )and( 31 ) have been supported by experiments of trapping and binary reactions
on the Sierpinski gasket, percolation clusters, and lattices with disordered transition
rates [ 4 , 41 , 44 ] to name but a few examples. While Eq. ( 30 ) applies to diffusion-
limited reactions on fractals, it also applies to any situation for which h

0.
Equation ( 30 ) has been incorporated into pharmacokinetics through both non-
compartmental and compartmental models. The former includes the homogeneous-
heterogeneous distribution model introduced by Macheras [ 42 ] to quantify the
global and regional characteristics of blood flow to organs. The latter includes the
fractal compartmental model [ 16 ] in which a classical compartment was used to
represent the plasma while a fractal compartment was used to represent the liver.
In this formalism, the rate of elimination from the liver is given by

>

k 0 t − h C v = . (32)

Simulations of the model showed that h plays a significant role in determining
the shape of the concentration-time curve [ 9 ].