Biomedical Engineering Reference
In-Depth Information
By taking the log of both sides, we obtain:
log
g
(
x
)=
log
A
+
α
log
x
.
(14)
Thus when plotted on a log-log scale, this type of function produces a straight
line with slope
. Power laws are attractive because they are scale-invariant. Multi-
plying the
x
-variable by a factor of
a
merely changes the constant of proportionality:
α
ax
α
)=(
x
α
.
g
(
x
)=
A
(
aA
)
(15)
The shape of such a system is the same irrespective of the scale. Because the
characteristics of the system remain the same as we zoom in or out of the physical
space occupied by the system, this allows us to extrapolate from shorter time or
space intervals to longer ones and vice versa.
Fractals are objects that have non-integer power exponents. They can be
described by the following equation:
d
1
−
D
L
(
d
)
∝
,
(16)
where
D
is the fractal dimension of the system and can be thought of as a measure
of the plane-filling properties of a structure [
43
,
44
].
Fractals describe systems behaving under constraints. For example, the circula-
tory system consists of a series of bifurcating vessels, and the time it takes for a
circulating drug to reach its target will depend on the confines of its path. Similarly,
most organs in the body are complex structures and the diffusion of a drug across
them will be limited by the available surfaces. Because their structure persists down
to smaller and smaller scales, fractals have the unique ability to fill the available
space as efficiently as possible.
Power laws and fractals can be generated in many different ways and can exist
in both space and time. That is, a system characteristic can scale either with length
or with time. In pharmacokinetics, for example, the concentration of a drug can
depend on the path it must follow through the body or on characteristic times such
as diffusion or residence rates. It should be noted that both types of fractals can exist
simultaneously in a system. In the sections that follow, we will consider the calcium
antagonist drug mibefradil, whose elimination is governed by the fractal geometry
of its site of metabolism, as well as the chemotherapeutic drug paclitaxel, whose
elimination is proposed to be governed by its residence time within the cells.
2.2
Power Laws in Pharmacokinetics
Power laws have been applied in the analysis of both dose- and time-dependences
in pharmacokinetic systems. When a drug is injected into the body via a single
bolus intravenous dose, the resulting plasma concentration time curve is called a
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