Biomedical Engineering Reference
In-Depth Information
model. Since the subchondral plate is difficult to identify on MR images, a uniform thickness of 1.5
mm was assumed, in accordance with previous research (Milz and Putz 1994).
6.2.2 f
inite
e
lement
d
iScretization
The FE model of the human knee joint was developed based on a three-dimensional geometrical
model using the finite element software ABAQUS (Simulia, Inc., USA). Discretizations were imple-
mented in the cortical bone, cancellous bone, cartilage, and meniscus with three-dimensional four-
node tetrahedral elements. Four ligaments (ACL, PCL, MCL, and LCL) were modeled as bundles of
two-node truss elements. The nodes of the truss elements were uniformly fixed within the ligament
insertion sites on the bone. Four meniscal horn attachments were also modeled as truss elements,
with one side fixed to the meniscal horn and the other side to bone.
The ligament was assumed to be a nonlinear hyperelastic material with the force-strain
relationship:
0,
ε≤
0,
1
4
k
εε
2
,
f
=
0
≤ε≤ε
2
,
(6.1)
1
1
(
)
2
,
k
ε−ε
,
ε≤ε
1
1
where
f
is the in situ force of the ligament, ε is the strain on the ligament, ε
l
is the reference strain,
assumed to be 0.03, and
k
is the stiffness coefficient. The
k
of ACL, PCL, MCL, and LCL are
10,000, 18,000, 6,000, and 8,250, respectively (Blankevoort et al. 1991; Li et al. 2002; Netravali
et al. 2011). The meniscal attachments were assumed to be a noncompressive elastic material with
an elastics modulus of 600 MPa. The medial and lateral menisci were assumed to be transversely
isotropic, linear elastic materials. The moduli in circumferential (E
θ
), radial (E
R
), and axial (E
Z
)
directions were 125, 27.5, and 27.5 MPa; the Poisson ratios (ν
θR
, ν
θZ
, and ν
RZ
) were 0.1, 0.1, and 0.33,
and the shear moduli (G
θR
and G
θZ
) were 2.0 MPa (Yao et al. 2006). Cortical bone, cancellous bone,
subchondral bone, and cartilage were assumed to be homogeneous linear elastic materials. The
elastic moduli and Poisson ratios are shown in Table 6.1 (Taylor et al. 1998; Au et al. 2005; Yao et al.
2006). Since the friction on the articular surface is extremely small, a frictionless and finite sliding
algorithm was assigned on cartilage-cartilage and cartilage-meniscus contact pairs.
6.2.3 m
odel
V
alidation
For an effective FE simulation, validation is critical to ensure that the model can accurately predict
the realistic deformation of tissue. In the present study, validation was implemented by comparing
realistic meniscal deformation with the deformation predicted by the FE model. The realistic three-
dimensional shape of the meniscus (including information on meniscal movement and deformation)
table 6.1
material Properties of bone and Cartilage
tissue
young's modulus
Poisson ratio
Cortical bone
17 GPa
0.33
Subchondral bone
1.15 GPa
0.25
Cancellous bone
0.4 GPa
0.33
Cartilage
5 MPa
0.35
Source:
Taylor et al.,
J Biomech
, 31, 4, 303-310, 1998; Au et al.,
J Biomech
38, 4, 827-832, 2005;
Yao et al.,
J Biomech Eng
, 128, 1, 135-141, 2006.
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