Saturable Fractal Pharmacokinetics and Its Applications - Mathematical Methods and Models in Biomedicine - page 341

A form of concentration-dependent fractal kinetics was developed by L opez-
Quintela and Casado [ 41 ], who proposed the following scaling relationship:

k eff
i

A i C 1 − d f

= 0 ≤ d f ≤ 1 , (36)

where d f is the fractal dimension of the space. By applying this equation to v max ,the
formula was obtained:

v ef max C 2 − d f
K ef M +

v =

C ,

(37)

where v ef max and K ef M are new constants. For d f =

1, the classical Michaelis-Menten

equation is recovered, and as d f → 0, the complexity of the reaction becomes more

and more important. Heidel and Maloney [ 26 ] performed an analytical exploration
of this equation, and Ogihara [ 53 ] applied it to model carrier-mediated transport
under heterogeneous conditions.
A seemingly different approach to concentration-dependent fractal kinetics is
the “power-law formalism” developed by Savageau [ 57 ], expressed through the
generalized mass-action representation:

r
k = 1 α ik

n
j = 1 C g ijk

r
k = 1 β ik

n
j = 1 C h ijk

d C i
d t =

− , (38)

j j

where are the kinetic rate coefficients and g and h are the kinetic rate

orders associated with each reactant. The equations for the power-law formalism
are complicated and Savageau admits that this model works best for large series
of reactions rather than one or more reactions catalyzed by only one enzyme [ 57 ].
Savageau justifies this formalism by showing that for homodimeric reactions, its
equations are equivalent to the fractal kinetics equations. However, this equivalence
has yet to be proven for any other reactions due to the complexity of the equations.
In principle, it is possible that Eq. ( 38 ) can be obtained by summing over several
Michaelis-Menten reactions.
To summarize, any reaction for which h

α and β

n is referred to as following
fractal-like kinetics. In this chapter, an alternative formulation of dose-dependent
fractal kinetics is proposed based on fractal reaction orders under steady state
conditions.

> 0or X >

2.8 Model Solutions

In a strict sense, a steady state regime means that the concentration of the reactant
is constant in time. One way in which this can be achieved is if the concentration of
drug molecules is much greater than the concentration of enzymes, even if the local
concentration values vary considerably. Even in the presence of drug elimination,