Biomedical Engineering Reference
In-Depth Information
Chapter 1
Introduction
1.1 The Concept of Fractional Signals and Systems
in Biomedical Engineering
The seminal concepts risen from two mathematicians, the bourgeois L'Hopital and
the philosopher Leibnitz, have proven yet again that old ideas have long shadows.
Three hundred years after this cross-fertilization, modern sciences are plucking its
fruits at a logarithmically ever-increasing speed. About half a century ago, frac-
tional calculus has emerged from the shadows of its abstract form into the light of
a very broad application field, varying from ecology, economics, physics, biology,
and medicine. Of course, it all became possible with a little aid from the revolu-
tion in computer science and microchip technology, allowing to perform complex
numerical calculations in a fraction of a millisecond. Nowadays, it turns out that
Mother Nature has a very simple, yet extremely effective design tool: the fractal.
For those not yet aware of this notion, the concise definition coined by Mandel-
brot is that a fractal structure is a structure where its scale is invariant under a(ny)
number of transformations and that it has no characteristic length [
97
]. Fractals and
their relative dimensions have been shown to be natural models to characterize var-
ious natural phenomena, e.g. diffusion, material properties, e.g. viscoelasticity, and
repetitive structures with (pseudo)recurrent scales, e.g. biological systems.
The emerging concepts of fractional calculus (FC) in biology and medicine have
shown a great deal of success, explaining complex phenomena with a startling sim-
plicity [
95
,
167
]. For some, such simplicity may even be cause for uneasiness, for
what would the world be without scepticism? It is the quest to prove, to show, to sus-
tain one's ideas by practice that allows progress into science and for this, one must
acknowledge the great amount of results published in the last decades and nicely
summarized in [
149
-
152
].
To name a few examples, one cannot start without mentioning the work of Man-
delbrot, who, in his quest to decipher the Geometry of Life, showed that fractals are
ubiquitous features [
97
]. An emerging conclusion from his investigations was that
in Nature there exists the so called “magic number”, which allows to generically
describe all living organisms. Research has shown fractal properties from cellular
