Biomedical Engineering Reference
In-Depth Information
Substituting equation (11.10) into (11.7) and (11.8), respectively, we obtain
∇•
φ
f
v
s
p
= 0
∂φ
f
∂t
−
κ
∇
+
(11.11)
(
v
s
∇•
−
κ
∇
p
) = 0
(11.12)
Equation (11.12) is a governing equation that links the cartilage matrix
deformation to the interstitial fluid motion. Equation (11.11) is an interme-
diate result that can simplify the (soon to be introduced) solute transport
governing equation.
In case of no mass sink, conservation of a solute phase (e.g., IGF) in a fluid
phase can be expressed as
∂
φ
f
c
I
∂t
J
I
= 0
+
∇•
(11.13)
where
c
I
+
φ
f
v
f
c
I
J
I
=
φ
f
D
I
∇
−
(11.14)
Here
J
I
is the mass flux of solute in the fluid phase and
D
I
is the effective
diffusion coecient of the solute in the cartilage including the tortuosity factor
for the cartilage matrix.
The cartilage solid matrix volumetric strain,
ε
v
, is given by the divergence
of the solid phase displacement
u
s
ε
v
=
∇•
(11.15)
For small strains,
φ
f
=
φ
f
0
+
ε
v
(11.16)
where
φ
f
0
represents the initial fluid volume faction.
Thus, the solute transport equation (11.13) can be simplified by using
equation (11.11)
φ
f
∂c
I
2
c
I
+
φ
f
v
s
−∇
φ
f
D
•∇
φ
f
D
I
∇
c
I
= 0
∂t
−
−
κ
∇
p
(11.17)
Equation (11.17) is the governing equation for the transport of solute in a
deformable porous medium (in this case cartilage).
11.2.2.2
Conservation of Linear Momentum
The incremental total stress tensor
σ
inside the tissue is the sum of the incre-
mental interstitial fluid pressure
p,
and the incremental elastic stress
σ
e
, result-
ing from deformation of the solid matrix [52]
p
I
+
σ
e
σ
=
−
(11.18)
where
I
is the identity tensor.
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