Biomedical Engineering Reference
In-Depth Information
knowledge are insufficient to comprehend the laws governing the
development of its complex structures and the operation of its
functions therein.
The essence of somewhat simpler systems, known as “com-
plex”, still cannot be revealed from the behavior of their isolated
components because these components do have interactions with
one another, and the overall behavior of this complex system is in
fact determined by the very ways its elements are coupled. Even
a few tightly coupled elements can produce very complex dynam-
ics known as “chaos” on purely deterministic ground
(3)
.The
degree of freedom in these chaotic systems is low, suggesting that
a few coupled differential equations govern the complex behavior
(4)
. When a complex system is assembled from a large number
of components, these cannot possibly be tightly coupled. Hence
the degree of freedom for the system is rendered to be high. The
complexity of this system is manifested in a peculiar pattern of
spatial and/or temporal correlation known as self-similar or frac-
tal behavior.
Here, we overview the application of the fractal approach on
one such complex system, the cerebral hemodynamics. We briefly
introduce the fractal properties, and show how they relate to com-
plex systems. Because many aspects of complexity are present in
hemodynamics, we will demonstrate that to extract these features
from such signals requires an array of measuring and analytical
tools. We give an overview of fractal descriptors for the 1 dimen-
sional signal up to the 4 dimensions of time and space.
2. Fractal
Characterization
No straightforward axiom exists to determine if an observed
object is fractal or not, hence this decision can only be made based
on identifying the presence of its fractal properties
(5)
.These
are fundamental, interrelated features like the self-similarity (or
self-affinity), the power-law scaling relationship of features, scale
invariance, scaling range and fractal (non-integer) dimension of
the object. (For a more detailed explanation see ref.
(6)
).
2.1. Self-similarity
Pieces of a fractal object when enlarged are similar to larger pieces
or to that of the whole. We can classify the self-similar properties
from two points of view:
1. Similarity
. If the pieces are identical,
rescaled replica of each other, the fractal is exact. When the simi-
larity is present only in between statistical populations of observed
data of a given feature assessed at different scales, the fractal is
referred to as statistical;
2. Scaling
. If the scaling is uniform in all
directions (isotropic), then the fractal is self-similar. If the scal-
ing is anisotropic, i.e. in one direction, the proportions between
