Saturable Fractal Pharmacokinetics and Its Applications - Mathematical Methods and Models in Biomedicine - page 343

1, since this value corresponds to
the classical model with an exponential tail, which is inconsistent with a power law.
The fact that the proposed steady state model predicts long-time power law
behavior provides a point of comparison with other models. For example, the
Michaelis-Menten model predicts an exponential tail, and the transient fractal and
fractal Michaelis-Menten equations predict stretched exponential tails of the form
C

Note that Eqs. ( 45 )-( 47 ) are undefined for X =

at 1 − h ( t ) ∼ exp ( ) .

3 Applications

3.1 Mibefradil: Spatially Induced Nonlinearity

Mibefradil is a calcium antagonist that was developed to reduce ventricular fib-
rillation [ 58 ]. It is orally administered, and its major site of elimination is the
liver. Studies done on chronically instrumented dogs [ 58 ] concluded that observed
nonlinear pharmacokinetics are due to dose- and time-dependent reduction of
hepatic clearance of the drug. Fuite et al. [ 16 ] proposed that the source of this
reduction is the fractal geometry of the liver. The circulation in the liver can be
divided into the macro-circulation (including the hepatic artery and the hepatic
and portal veins) and the micro-circulation (consisting of the portal vein, hepatic
arterioles, and the sinusoids) [ 8 ]. The microvasculature of the liver consists of
vessels that bifurcate towards smaller and smaller daughter vessels. In fact, the
vessels supplying the liver, lungs, kidney, and heart have been found to exhibit
scaling relationships for branch diameter, branch length, pressure, and radius-to-
length ratios [ 5 , 21 , 22 ]. Javanaud estimated the fractal dimension of the liver to be
d f ≈

2[ 29 ].
Due to differences in the global and regional nature of blood flow to organs,
Macheras developed a homogeneous-heterogeneous distribution model [ 42 ]. The
homogeneous conditions are considered “well-stirred,” and the heterogeneous
conditions near tissues are considered “under-stirred.” Figure 4 illustrates this
composite system.
While the homogeneous portions of the circulatory system can be described using
conventional kinetics, regional areas such as those feeding the liver are fractal and
thus should display fractal kinetics. Essentially, the geometry of a surface where a
chemical process is taking place affects the rate at which the process can occur. This
reaction rate can be expressed as follows [ 25 ]:

t − ( 1 − d S / 2 ) . k ( t ) ∝ (48)

The theoretical model [ 16 ] combining well-stirred Euclidean and fractal com-
partments led to analytical solutions for the time evolution of the drug concentration
which were obtained using perturbation analysis. The equations were then tested