Biomedical Engineering Reference
In-Depth Information
Due to its charged nature, cartilage also exhibits complex electrochemical
phenomena (e.g., streaming, diffusion potential, and Donnan osmotic pres-
sure) [5]. For example, the inhomogeneous distribution of proteoglycans may
lead to depth-dependent fixed charge density distribution [53]. In turn, inho-
mogeneities in the fixed charge density can potentially affect the mobile ion
concentration, internal electrical potential and osmotic pressure [5]. Because
of the composite structure of cartilage, the response of cartilage can be sig-
nificantly different under tension, compression, and shear or throughout the
cartilage thickness [53]. To account for different cartilage properties in tension
and compression, Soltz and Ateshian [54] proposed that the mechanical behav-
ior of cartilage's solid phase can be described by employing the “orthotropic
octant-wise linear elasticity” model of Curnier et al. [55]. The model can be
simplified to the more specialized case of cubic symmetry to reduce the num-
ber of material constants. The elastic stress
σ
e
resulting from deformation of
the solid matrix can then be defined as
⎨
⎬
3
3
σ
e
=
λ
1
[
A
a
:
E
]tr(
A
a
E
)
A
a
+
λ
2
tr (
A
a
E
)
A
b
+2
µ
E
(11.19)
⎩
⎭
a
=1
b
=1
b
=
a
where
E
is the infinitesimal strain tensor related to the solid-phase displace-
ment
u
defined as
E
=
2
u
)
T
,
λ
1
and
µ
are the Lame constants,
∇
u
+(
∇
A
a
is a texture tensor (
A
a
=
a
a
⊗
a
a
) corresponding to each of the three pre-
ferred material directions defined by unit vector
a
a
(
a
a
•
a
a
= 1, no sum over
a
) with
a
1
parallel to the split line direction;
a
2
perpendicular to the split line
direction; and
a
3
normal to the articular cartilage surface (Figure 11.2). The
term
λ
1
[
A
a
:
E
] denotes that
λ
1
is a function of
A
a
:
E
λ
1
[
A
a
:
E
]=
λ
−
1
,
A
a
:
E
<
0
(11.20)
λ
+1
,
A
a
:
E
>
0
which suggests that the material properties,
λ
1
, differ whether the normal
strain component along the
a
a
is compressive or tensile, and so can incorpo-
rate the bimodulus response of the cartilage into the solute transport model.
Equation (11.19) is adopted throughout this study.
The momentum equation for the mixture, under quasistatic conditions (in
the absence of body forces) is given by
∇•
σ
= 0
(11.21)
Equations (11.12), (11.17), and (11.21) form the basic set of governing
equations describing the fully coupled mechanical and solute transport behav-
ior in a three-dimensional porous media system with solute diffusion only in
the fluid phase. This basic model can be further extended to explore, for exam-
ple, the influence of solute binding, IGF-I, and mechanical stimuli mediated
biosynthesis and transport of matrix molecules (e.g., aggrecan).
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