Biomedical Engineering Reference
In-Depth Information
That is, under this loading condition, the governing equation (11.27)
depends only on the tensile aggregate modulus
H
+
A
, which is largely deter-
mined by the collagen matrix.
11.2.2.4
Boundary Conditions
At the center of the cartilage (i.e., at
r
= 0),
u
r
(0
,t
)=0
,
∂p
∂r
=0
,
∂c
I
∂r
= 0
(11.28)
r
=0
r
=0
At the outer edge of the cartilage (i.e., at
r
=
r
0
), the following boundary
conditions apply:
I
(
r
0
,t
)=
c
I
0
p
(
r
0
,t
)=0
,
(11.29)
Furthermore, at this cartilage-solute bath interface, the quantities
φ
f
and
φ
s
each exhibit a discontinuity. The traction condition across the interface
implies that
∂u
r
∂r
u
r
|
r
=
r
0
r
0
+
ε
z
υ
−
=
(11.30)
1
−
υ
r
=
r
0
λ
2
H
+
A
+
λ
2
where
υ
is Poisson's ratio defined as
υ
=
.
11.2.2.5
Initial Conditions
For simplicity, the following initial conditions can be assumed
u
r
(
r,
0)=0
,
(
r,
0)=0
,
I
(
r,
0) = 0
(11.31)
That is, both, the radial displacement of the solid matrix and the fluid
pressure are initially set to zero throughout the tissue and we have assumed
that there is no growth factor within the tissue.
11.2.2.6
Numerical Method
Numerical results obtained throughout this chapter were obtained by solving
the derived governing equations using the commercial finite element software
COMSOL MULTIPHYSICS [59]. The applied strain protocol was discretized
into time steps. At each time step, the solute concentration,
c
I
, solid phase
displacement,
u
r
, solid phase velocity,
v
r
, and the interstitial fluid pressure,
p,
were determined. A one-dimensional domain using 300 quadratic Galerkin
elements was used for all calculations, and a relative tolerance 10
−
10
and an
absolute tolerance 10
−
11
were adopted. The Finite Element (FEM) discretiza-
tion of the time-dependent partial differential equation (PDE) was solved
using an implicit solver of COMSOL MULTIPHYSICS so that oscillations in
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