Biomedical Engineering Reference
In-Depth Information
1.3.2 Fractional-Order Dynamical Systems
Let us consider the rheological properties of soft biological tissue, i.e. viscoelastic-
ity. Typical cases are the arterial wall [
23
] and lung parenchyma [
8
], which clearly
show viscoelastic behavior. In these recent reports, the authors acknowledge that
integer-order models to capture these properties can reach high orders and that frac-
tional derivative models with fewer parameters have proven to be more efficient in
describing rheological properties. Both of these authors define the complex modulus
of elasticity as being determined by a real part, i.e. the storage modulus, capturing
the elastic properties, and, respectively, by an imaginary part, i.e. the dissipation
modulus, capturing the viscous properties:
σ(ω)
ε(ω)
=
E
∗
(j ω)
=
E
S
(ω)
+
jE
D
(ω)
(1.14)
with
σ
the stress and
ε
the strain,
E
S
and
E
D
the real and imaginary parts of the
complex modulus. This complex modulus
E
∗
(jω)
shows partial frequency depen-
dence within the physiologic range in both soft tissue examples. A typical example
of an integer-order lumped rheological model is the Kelvin-Voigt body, consist-
ing of a perfectly elastic element (spring) in parallel with a purely viscous element
(dashpot):
η
dε(t)
dt
σ(t)
=
Eε(t)
+
(1.15)
with
E
the elastic constant of the spring and
η
the viscous coefficient of the dash-
pot. One of the limitations of this model is that it shows creep but does not show
relaxation, the latter being a key feature of viscoelastic tissues [
2
,
68
]. The classical
definition of fractional-order derivative (i.e. the Riemann-Liouville definition) of an
arbitrary function
f(t)
is given by [
113
,
126
]
d
n
f
dt
n
t
1
d
dt
f(τ)
=
τ)
n
dτ
(1.16)
(
1
−
n)
(t
−
0
where
is the Euler gamma function. Hence, the FO derivative can be seen in the
context of (
1.15
) as the convolution of
ε(t)
with a
t
−
n
function, anticipating some
kind of memory capability and power-law response. It follows that the
spring-pot
element
can be defined based on (
1.16
)as
η
d
n
ε
σ
=
dt
n
,
1
≥
n
≥
0
(1.17)
in which the value for
n
can be adjusted to incorporate either a purely elastic com-
ponent (
n
1). Both Bates and Craiem acknowl-
edged the fact that the soft biological tissue follows both elastic and viscous be-
havior under baseline and stimulated case. Therefore, if one needs to derive a gen-
eral model for characterizing soft tissue rheological properties, two instead of one
spring-pot elements may be necessary.
Now, let us consider the diffusive properties; e.g. heat transfer [
91
], gas exchange
[
66
] and water transfer through porous materials [
12
,
91
]. Diffusion is of funda-
mental importance in many disciplines of physics, chemistry, and biology. It is well
=
0), either a pure viscous one (
n
=