Saturable Fractal Pharmacokinetics and Its Applications - Mathematical Methods and Models in Biomedicine - page 351

each sinusoid have a different number and distribution of hepatocytes, it receives
a different portion of the blood flow. Therefore, the liver can be considered as a
network of clusters of sinusoids. Because X increases as the size of the clusters
decreases [ 50 ], X

= 1 means that the liver acts as a homogeneous, well-mixed

compartment and X 1 indicates segmentation and a lack of mixing.

This interpretation is consistent with a model proposed by Weiss [ 63 ], who
described the transit times in the liver as being determined by both the micro-
mixing and macro-mixing processes. He suggested two models at different ends
of the spectrum: (a) a distributed model in which the sinusoids are parallel and
there is complete segregation of the pathways, and (b) a dispersion model in which
the sinusoids are interconnected and there is perfect micro-mixing. Considering our
results in this framework, X provides a quantitative measure of the degree of micro-
mixing between sections of the liver and locates the model somewhere between
Weiss's two extreme models.
This chapter describes applications of fractal kinetics under steady state con-
ditions to pharmacokinetics. We have demonstrated that a steady state fractal
Michaelis-Menten equation best describes the elimination of the drug mibefradil
from dogs. Furthermore, it accounts for the long-time power law behavior of the
concentration through the inclusion of a fractal reaction order, X . This anomalous
reaction order suggests that the liver, the organ of elimination for mibefradil, is
best treated as a collection of clusters of sinusoids. The higher the value of X ,the
less mixing that occurs between adjacent sinusoid clusters. Mibefradil is an orally
administered calcium antagonist with a relatively short elimination time, while
paclitaxel is an intravenously administered anticancer drug that can remain in the
body for over three days. Nevertheless, the elimination curves of both drugs can be
accurately described using a similar power law relationship.
We conclude that transient fractal kinetics is appropriate for describing reactions
that occur within well-mixed heterogeneous environments, while steady state fractal
kinetics better describes reactions that occur in under-stirred heterogeneous spaces.
The latter can occur due to the continuous influx of drug molecules through
recycling in the circulatory system.

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