Biomedical Engineering Reference
In-Depth Information
d
0
.
5
dt
0
.
5
s
0
.
5
appears in several types of
problems [
10
]. The transmission lines, the heat flow, or the diffusion of neutrons in
a nuclear reactor are examples where the half-operator is the fundamental element.
Diffusion is in fact a part of transport phenomena, being one of the three essen-
tial partial differential equations of mathematical physics. Molecular diffusion is
generally superimposed on, and often masked by, other transport phenomena such
as convection, which tend to be much faster. However, the slowness of diffusion
can be the reason for its importance: diffusion is often encountered in chemistry,
physics, and biology as a step in a sequence of events, and the rate of the whole
chain of events is that of the slowest step. Transport due to diffusion is slower over
long length scales: the time it takes for diffusion to transport matter is proportional
to the square of the distance. In cell biology, diffusion is a main form of transport
for necessary materials such as amino acids within cells. Metabolism and respira-
tion rely in part upon diffusion in addition to bulk or active processes. For example,
in the alveoli of mammalian lungs, due to differences in partial pressures across
the alveolar-capillary membrane, oxygen diffuses into the blood and carbon diox-
ide diffuses out. Lungs contain a large surface area to facilitate this gas exchange
process. Hence, the spreading of any quantity that can be described by the diffusion
equation or a random walk model (e.g. concentration, heat, momentum, ideas, price)
can be called diffusion, and this is an ubiquitously present property of nature.
Finally, let us consider the fractal geometry; e.g. self-similarity and recurrence.
Much work has been done on the fundamental property of percolation using self-
similar fractal lattices such as the Sierpinski gasket and the Koch tree [
91
,
97
,
118
,
130
]. Examples from real life include the coastline, invasion-front curve, lightning,
broccoli and cauliflower, and several human organs such as lungs, vascular tree, and
brain surface [
9
]. Other studies involve the temporal dynamics of biological signals
and systems, which also pose recurrence [
37
,
144
].
It is generally acknowledged that dynamical systems (e.g. electrical circuits) in-
volving such geometrical structures would lead to the appearance of a fractional-
order transfer function [
118
]. Although this topic has been investigated for the res-
piratory tree, in this topic the relation to viscoelasticity will be made, to offer a
broader image of their interplay.
→
known that the fractional-order operator
1.3.3 Relation Between Fractal Structure and Fractal Dimension
A fractal is a set of points which at smaller scales resembles the entire set. Thus the
essential characteristic of the fractal is self-similarity. Its details at a certain scale
are similar to those at other scales, although not necessarily identical. The textbook
example of such a fractal is the Koch curve, depicted in Fig.
1.3
.
The concept of fractal dimension (
F
d
) originates from fractal geometry and it
emerges as a measure of how much space an object occupies between Euclidean di-
mensions, e.g. the fractal structure from Fig.
1.3
. In practice, the
Fd
of a waveform
