Biomedical Engineering Reference
In-Depth Information
(spring in parallel with a spring-dashpot), whose deformation response during a creep/relaxation
test is represented by:
1
σ
=
E
R
1
ε
+
τ
ε
d
dt
d
dt
+
τ
σ
where
E
R
is the relaxed modulus,
τ
ε
is the relaxation time for constant strain, and
τ
σ
is the relaxation
time for constant stress. The spring elements in the model describe the stiffness of the tissue while
the dashpot helps describe the time-dependent deformation. When fit to experimental data, these
can be used to calculate the instantaneous modulus and apparent viscosity of the material:
E
0
=
E
R
1
τ
σ
−
τ
ε
τ
ε
+
τ
ε
) .
While viscoelastic models are useful for providing a basic description of biological tissue deforma-
tion, they are not particularly representative of the actual mechanical characteristics associated with
articular cartilage. As discussed previously, cartilage can be described as having two phases: one
solid, one fluid. More complex mathematical models of the tissue, such as the biphasic/poroelastic
model [
58
,
655
], take into account this composition, providing parameters that describe the stiff-
ness, fluid flow, and deformation characteristics of the tissue. One example of this is the following
equation that describes confined creep compression of cartilage using the biphasic solution [
58
]:
μ
=
E
R
(τ
σ
−
1
exp
2
α
0
) h
2
n
−
2
π
2
n
2
H
A
kt/
(
1
∞
F
0
H
A
2
π
2
1
2
1
2
ε
zz
(t)
=
−
+
−
+
+
n
=
0
where
ε
zz
is the observed strain,
F
0
is the applied constant load,
H
A
is the aggregate modulus,
k
is the
permeability,
h
is the sample thickness, and
α
0
is the solid content ratio. When fit to experimental
data, values for
H
A
and
k
can be extracted. Other equations exist in the literature for the various
geometries and device configurations that are possible for cartilage testing.
More complex models of articular cartilage exist and are useful for identifying particular
parameters that might be of interest. For example, an alternative to the biphasic model of cartilage
is the poroviscoelastic model, which accounts for the different phases of the tissue and well as their
short and long time responses to loading [
46
]. Articular cartilage can be modeled in even more
complexity than as just a two-phased tissue. A third phase, the ionic phase, can significantly affect
the motion of fluid through the solid matrix, and hence, the deformation characteristics of the
tissue. The triphasic model accounts for contributions from the solid, fluid, and ionic phases of the
tissue but results in the same parameters as the biphasic solution, with the addition of fixed charge
density [
656
].
Other characteristics that might be of interest include the frictional and torsional properties
of the tissue. The coefficient of friction,
μ
, can be determined using a simple relationship between
the normal,
N
, and friction,
F
f
, forces measured during friction tests:
F
f
N
μ
=
