Introduction - The Human Respiratory System - page 6

order equations is diffusion and some papers discuss this aspect [ 91 ], but current
state-of-art lacks a mathematical basis for modeling diffusion in the lungs. The sub-
ject in itself is challenging due to its complexity and requires an in-depth study of
alveolar dynamics. This is not treated in this topic, but the reader is encouraged to
check the provided literature for latest advances in this topic.
The study of the interplay between fractal structure, viscoelasticity, and breathing
pattern did not capture the attention of both medical and engineering research com-
munities. This is surprising, since interplay clearly exists and insight into its mecha-
nisms may assist diagnosis and treatment. This topic will address this issue and will
establish several relations between recurrent geometry (symmetric and asymmetric
tree) and the appearance of the fractional-order models, viscoelasticity, and effects
of pulmonary disease on these properties.

1.3 Emerging Tools to Analyze and Characterize
the Respiratory System

1.3.1 Basic Concepts of Fractional Calculus

The FC is a generalization of integration and derivation to non-integer (fractional)
order operators. At first, we generalize the differential and integral operators into
one fundamental operator D t ( n the order of the operation) which is known as
fractional calculus . Several definitions of this operator have been proposed (see,
e.g. [ 126 ]). All of them generalize the standard differential-integral operator in two
main groups: (a) they become the standard differential-integral operator of any order
when n is an integer; (b) the Laplace transform of the operator D t is s n (provided
zero initial conditions), and hence the frequency characteristic of this operator is
(jω) n . The latter is very appealing for the design of control systems by using spec-
ifications in the frequency domain [ 117 ].
A fundamental D t operator, a generalization of integral and differential operators
( differintegration operator), is introduced as follows:

⎧
⎨

⎫
⎬

d n

dt n ,

n > 0

D t = (1.7)

1 , n = 0

⎩ t ⎭

0 (dτ ) − n ,n< 0

where n is the fractional order and dτ is a derivative function. Since the entire topic
will focus on the frequency-domain approach for fractional-order derivatives and
integrals, we shall not introduce the complex mathematics for time-domain analysis.
The Laplace transform for integral and derivative order n are, respectively:
L D − n
t

f(t) =

s − n F(s) (1.8)

L D t f(t) =

s n F(s) (1.9)