Saturable Fractal Pharmacokinetics and Its Applications - Mathematical Methods and Models in Biomedicine - page 338

( − ) − ( + ) = .

k 1 e 0 x C k − 1 k 2 x 0 (23)

Collecting the terms in x and rearranging gives:

k 1 e 0 C
k − 1 +

x =

k 1 C .

(24)

k 2 +

Using the fact that the rate of the reaction is v = k 2 x gives

k 2 e 0 C
k − 1 +

v =

C .

(25)

k 2
k 1 +

= =( + ) /

Finally, denoting v max k 2 e 0 and K M k − 1 k 2 k 1 gives the Michaelis-

Menten equation:

v max C
k M +

v =

C .

(26)

The parameter v max is the maximum velocity of the reaction, and the Michaelis-
Menten constant K M is the substrate concentration at half the maximum velocity.
In the low-concentration case, where C

K M ,Eq.( 26 ) reduces to

v max
K M

v = C , (27)

which describes first-order kinetics with k = v max / K M . In the high-concentration

case, where C K M ,Eq.( 26 ) becomes

v = v max , (28)

which is steady-state kinetics with a constant reaction rate.

2.5 Asymptotics of the Concentration-Time Curve

The solution to a compartmental model with constant coefficients takes the form
of a linear superposition of exponential terms, and the resulting concentration-
time curve exhibits an exponentially decaying tail. However, there is evidence that
the concentration-time curves of many drugs exhibit long-time power law tails of
the form

t − γ C ( t ) ∼ for t > T , (29)

where T marks the time of the onset of the tail. Negative power laws were
first applied, empirically, to describe the washout of bone-seeking radioisotopes
[ 3 , 6 , 51 ]. Subsequently, other types of clearance curves have been fit by a single