Saturable Fractal Pharmacokinetics and Its Applications - Mathematical Methods and Models in Biomedicine - page 350

Several experiments provide support for this theory. Using data from an in vitro
study looking at the cytotoxicity of paclitaxel and docetaxel in three cell lines [ 55 ],
a power law relationship was found between the IC 50 and the exposure time.
Interestingly, for one cell line, the exponent was found to be 62 ( R 2

9957).
In addition, in another in vitro study of tumor cells in a supportive medium [ 36 ], the
concentration of paclitaxel in the medium after 4 h was related through a power law
to the initial concentration with an exponent of

− 1 . = 0 .

15 ( R 2 998).

The results from our study and a more in-depth analysis have been published
earlier where details can be found [ 46 , 47 ].

− 1 . = 0 .

4 Conclusion and Discussion

Biological systems are complex, but the advantage of such systems is the relatively
simple mathematics that seems to consistently emerge from their behavior. It is
well known that the liver has a complex geometry. The blood vessels supplying
it are arranged as a fractal tree [ 29 ], its cellular network has fractal properties [ 18 ],
and the perfusion of blood at the terminal branches is heterogeneous [ 63 ]. Both
transient and steady state reactions occurring within such spaces can exhibit
anomalous behavior. For transient reactions, it is assumed that there is a random
distribution of reactants [ 34 ]. Therefore, anomalous kinetics in the transient case
strictly results from the decreased efficiency of random walkers in exploring their
irregular space (quantified by d s ). In the steady state regime, however, there is
an influx of molecules. In regular geometries, this influx can cause a net stirring
effect [ 34 ]; however, in fractal and confined geometries, self-stirring is inefficient.
The spaces are instead characterized by large fluctuations in the local concentration
and an increasing segregation of molecules. This effect has been reported for
reaction-diffusion phenomena in physical systems [ 12 ]. As a result, under steady
state conditions, the distribution of molecules is partially ordered due to the
influx of molecules, and the non-randomness reduces the reaction probabilities and
consequently the reaction rate. To summarize, transient fractal kinetics occur in
well-stirred heterogeneous media while steady state kinetics occur in poorly stirred
heterogeneous media. Here, the term heterogeneity refers to the geometry of the
environment.
In the case of steady state fractal kinetics, Klymko and Kopelman [ 32 ] interpreted
non-integer values of X as characteristic of a microscopically heterogeneous
medium that is best described as a collection of kinetically independent clusters. The
kinetic rate coefficients are then kinetic averages taken over domains of different
sizes and local concentration. This interpretation is consistent with the studies that
reported high X values for reactions occurring on clusters and islands [ 50 ]. A similar
model can be developed for the liver, the organ predominantly responsible for the
elimination of drugs from the body. The metabolic enzymes are located in the liver
cells, called hepatocytes, which are organized around the terminal supply vessels.
Each set of vessels and their surrounding cells are called a sinusoid. Not only does