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
(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
=
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