Saturable Fractal Pharmacokinetics and Its Applications - Mathematical Methods and Models in Biomedicine - page 328

( )

generates a concentration-time curve, C , that first rises as absorption of the drug

dominates (invasion) and then decreases after a maximum concentration value,
C max , is reached (elimination). This decline may be relatively short, taking place
over minutes or hours, or may last for several days, and it is mainly governed by
the rate of elimination of the drug from the body and drug distribution including
such processes as tissue binding. The goal of pharmacokinetic modeling is to use
these curves to describe, compare, and predict a drug's course in the body, as well
as to determine optimum dosing regimes, potential toxicity, and possible drug-
drug interactions. One of the main difficulties is to combine organ-level modeling
with molecular-level modeling that provides insights into drug distribution and
binding with proteins and lipids. This is still a key challenge in pharmacokinetic
modeling and misconceptions have occurred in the past due to the difficulties
in combining these two vastly different temporal and spatial scales [ 37 ]. Drug
distribution processes cover complicated molecular and cellular details of the
hepatic elimination processes of drugs. Sensitivity analysis has shown that complex
models of hepatic elimination cannot be identified on the basis of whole body drug
disposition data (plasma concentration-time curves) making this aspect in need of
appropriate model development.
While classical compartmental models are the most common type of pharma-
cokinetic mathematical models and they can provide adequate agreement with
clinical pharmacokinetic data sets, they often fail to provide a good fit to the tail
regions, where non-exponential time-dependence can occur that is better fit by
power laws or gamma functions [ 52 , 64 ]. Since all data sets are finite in size,
they can always be fit with a sufficiently large number of compartments and an
associated large number of adjustable parameters associated with a chosen basis set
of functions (e.g. exponentials). However, this does not address the fundamental
origin of the frequently encountered non-exponential behavior in pharmacokinetics.
A link has been made previously [ 9 ] between concentration-time curves with power-
law tails and fractal kinetics. We believe that nonlinearities in pharmacokinetic
models are of crucial importance. Therefore, to provide a clear contrast, we first
introduce standard ideas in pharmacokinetics which are based on linearity.
In general, a system is considered to be linear when its output is directly
proportional to its input. The concept of linearity in the body's handling of a drug
is very important; it implies that the concentration of the drug as well as any
derived parameters scale easily with both dose and time. The associated kinetic
processes can be described by a set of linear differential equations that follow
the superposition principle (which states that the whole is equal to the sum of its
individual components).
According to traditional pharmacokinetic models, the body can be divided
into compartments, and a drug's journey between two connected compartments is
described by a rate coefficient [ 39 ]. In a linear model, these rate coefficients k are
assumed to be constant. The concentration in each compartment can be described
by the following differential equation:

t

d C )

d t =

( t

kC ( t ) . (1)