Saturable Fractal Pharmacokinetics and Its Applications - Mathematical Methods and Models in Biomedicine - page 337

where C i is the concentration of reactant i and k is the kinetic rate coefficient in
units of inverse time. The law of mass action “mass-balance” assures the equality
between the amounts of material that leave one compartment and the amounts that
enter into another. The reaction order n i is the number of concentration terms that
must be multiplied together to get the rate of the reaction [ 57 ]. For a single step, n i is
typically equal to the molecularity, which is the number of molecules that are altered
during the reaction. When only one molecule is modified, the reaction is given by

v = kC . (19)

The movement of the drug molecules between compartments is proportional
to the volume rate of fluid flow between the connected compartments and the
concentration of the compound in the originating compartment, and inversely
proportional to the volume of the original compartment. Given an initial distribution
of the compounds in the various compartments, the model is integrated forwards in
time, using a 4th order Runge-Kutta algorithm, with variable time-steps.

2.4 Enzyme Kinetics and the Michaelis-Menten Equation

The rate of enzyme-catalyzed reactions can deviate from those predicted by classical
kinetics. At high concentrations, saturation of the enzymes limits the maximum
reaction rate that can be achieved, while at low concentrations, the rate of formation
of the enzyme-substrate complex becomes significant and the reaction becomes
dependent on the substrate concentration [ 13 ].
Consider the reaction:

k 1
GGGGGB

k 2
−→

+ + ,

E S ES E P (20)

F GGGGG
k − 1

where E , S , ES ,and P represent the enzyme, substrate, enzyme-substrate complex,
and product, respectively. If we denote the concentration of the substrate as C ,the
concentration of the enzyme-substrate as x , and the total concentration of enzymes
as e 0 , the system is described by the following ordinary differential equations:

d x
d t =

k 1 ( e 0 − x ) C − ( k − 1 + k 2 ) x , (21)

d p
d t =

k 2 x . (22)

Using the Briggs-Haldane treatment [ 57 ] to simplify the problem, a quasi-
steady-state assumption is made where the concentration of the substrate-enzyme
complex is taken to be constant, i.e. d x / d t = 0. Therefore,