Biomedical Engineering Reference
In-Depth Information
metabolism [
144
] to human walk [
50
]. Furthermore, the lungs are an optimal gas
exchanger by means of fractal structure of the peripheral airways, whereas diffusion
in the entire body (e.g. respiratory, metabolic, drug uptake, etc.) can be modeled by
a fractional derivative.
1
Based on similar concepts, the blood vascular network also
has a fractal design, and so do neural networks, branching trees, seiva networks in a
leaf, cellular growth and membrane porosity [
50
,
74
,
81
].
It is clear that a major contribution of the concept of FC has been and remains
still in the field of biology and medicine [
151
,
152
]. Is it perhaps because it is an
intrinsic property of natural systems and living organisms? This topic will try to
answer this question in a quite narrow perspective, namely (just) the human lungs.
Nevertheless, this example offers a vast playground for the modern engineer since
three major phenomena are interwoven into a complex, symbiotic system: fractal
structure, viscoelastic material properties, and diffusion.
1.2 Short History of Fractional Calculus and Its Application
to the Respiratory System
From the 1970s, FC has inspired an increasing awareness in the research commu-
nity. The first scientific meeting was organized as the First Conference on Frac-
tional Calculus and its Applications at the University of New Haven in June 1974
[
151
,
152
]. In the same year appeared the monograph of K.B. Oldham and J. Spanier
[
113
], which has become a textbook by now together with the later work of Pod-
lubny [
126
].
Signal processing, modeling, and control are the areas of intensive FC research
over the last decades [
146
,
147
]. The pioneering work of A. Oustaloup enabled
the application of fractional derivatives in the frequency domain [
118
], with many
applications of FC in control engineering [
20
,
117
].
Fractional calculus generously allows integrals and derivatives to have any order,
hence the generalization of the term
fractional order
to that of
general order
.Ofall
applications in biology, linear viscoelasticity is certainly the most popular field, for
their ability to model hereditary phenomena with long memory [
9
]. Viscoelasticity
has been shown to be the origin of the appearance of FO models in polymers (from
the Greek:
poly
, many, and
meros
, parts) [
2
] and resembling biological tissues [
30
,
68
,
143
].
Viscoelasticity of the lungs is characterized by compliance, expressed as the vol-
ume increase in the lungs for each unit increase in alveolar pressure or for each unit
decrease of pleural pressure. The most common representation of the compliance is
given by the pressure-volume (PV) loops. Changes in elastic recoil (more, or less:
stiffness) will affect these pressure-volume relationships. The initial steps under-
taken by Salazar to characterize the pressure-volume relationship in the lungs by
1
The reader is referred to the appendix for a brief introduction to FC.
