Biomedical Engineering Reference
In-Depth Information
distribution function of their time constants, a linear model has been able to cap-
ture the hysteresis loop of the lungs, capturing the relaxation function decreasing
linearly with the logarithm of time [
49
]. This is a class of systems which may be
successful in acknowledging the origin of the constant-phase behavior, but there
is no clearly defined micro-structural basis. Some attempts to establish this origin
have been made [
9
].
•
Class 4
:
complex dynamic systems exhibiting self-similar properties (fractals
).
This class is based on the fact that the scale-invariant behavior is ubiquitous in
nature and the stress relaxation is the result of the rich dynamic interactions of
tissue strips independent of their individual properties [
8
,
91
]. Although interest-
ing, this theory does not give an explanation for the appearance of constant-phase
behavior.
•
Class 5
:
systems with input-output relationships including fractional-order
equations
; borrowed from fractional calculus theory, several tools were used
to describe viscoelasticity by means of fractional-order differential equations
[
8
,
23
,
143
].
Referring to the specific application of respiratory mechanics, Classes 3-5 are most
likely to characterize the properties of lung parenchyma. The work presented in this
topic deals primarily with concepts from Class 4, but addresses also several items
from Class 5.
Following the direction pointed out hitherto, several studies have been performed
to provide insight on fiber viscoelasticity at macro- and microscopic levels, using tis-
sue strips from animals [
162
]. For instance, Maksym attempted to provide a model
based on Hookean springs (elastin) in parallel with a nonlinear string element (col-
lagen) to fit measurements of stress-strain in tissue strips in dogs [
96
]. Their theory
is based on the seminal work of Salazar and Hildebrandt and the results suggest that
the dominant parameter in (
1.1
)is
n
. This parameter has been found to increase in
emphysema and decrease in fibrosing alveolitis. They interpret the changes in this
variable as related to alterations in collagen and elastin networks.
About a decade later, Bates provided another mechanistic interpretation of the
quasi-linear viscoelasticity of the lung, suggesting a model consisting of series
spring-dashpot elements (Maxwell bodies) [
8
]. He also suggests the genesis of
power-law behavior arising from:
•
the intrinsic complexity of dynamic systems in nature, ubiquitously present;
•
the property of being self-organized critically, posing an avalanche behavior (e.g.
sandpile);
•
the rich-get-richer mechanism (e.g. internet links).
whereas the common thread which sews all them together is
sequentiality
. By allow-
ing two FO powers in the model of Maxwell bodies arranged in parallel (a spring
in parallel with a dashpot), he discussed viscoelasticity in simulation studies. Sim-
ilar attempts have been done by Craiem and Armentano in models of the arterial
wall [
23
].
Hitherto, the research community focused on the aspect of viscoelasticity in soft
biological tissues. The other property of the lungs which can be related to fractional-
