Biomedical Engineering Reference
In-Depth Information
the strain. Such a material also allows the use of a zero Poisson's ratio, which
prevents coupling of side strain to the axial strain and therefore makes it ideal for
approximating the contact in the ventricles. The first-order strain energy density
function for this material is given by [96]
J
β
h
α
h
−
1
.
W
hyperfoam
=
2
µ
h
α
h
β
h
(
λ
α
h
1
+
λ
α
h
2
+
λ
α
h
3
−
3) +
(5)
Here, the stiffness of this material is chosen to be a fraction of that used for the
brain tissue,
α
h
=
−
4
.
7 (the same as the value of
α
used by Miller et al. [92] for
the brain tissue with a similar energy function), and
β
h
=0(for fully compressible
behavior, i.e.,
υ
h
=0). Initial experiments have shown that using this material
model for the ventricles, displacements propagated to the ventricle wall opposite
to that which proximal to the simulated tumor were minimal.
4.3.5. boundary value problem statement
If body forces, such as gravity, are ignored, the desired deformation mapping
ϕ
:
κ
r
→
κ
t
can be found by solving the static equilibrium equation:
∇
o
·
P
=
0
,
(6)
where
is the first Piola-Kirchhoff tensor, which is related to strain via the ma-
terial constitutive law [96]. For brain tissues, simulations are performed with the
constitutive law described in Section 4.3.3. The following boundary conditions
(BCs) complete the statement of the BVP (refer to Figure 13):
P
ϕ
(
X
)
·
N
(
X
)=0
,
X
∈
∂B
01
,
and
ϕ
(
X
)=0
,
X
∈
∂B
02
,
(7)
e
th
e
th
(
X
)=
e,
X
∈
D
r
and
(
X
)=0
,
X
∈
∂
B
,
(8)
F
−T
SNX
=
PJ
N
(
X
)
,
X
∈
∂B
T
,
(9)
SN
(
X
)=
0
,
X
∈
∂B
V
,
(10)
where
in the relaxed configuration.
Equation (7) implies a sliding BC over the brain surface, except for locations
where the falx meets the inner surface of the skull which are pinned. Equation (8)
implies that the expansive strain due to edema is restricted to
D
r
. Equation (9)
is the traction BC implied by the tumor pressure, expressed in terms on normals
in
κ
r
. In simulations where no contact in the ventricles is anticipated, Eq. (10)
implies that the ventricles are assumed void with zero intraventricular pressure. If
hyperfoam material is used for approximating ventricle contact, Eq. (10) should
be replaced with the constitutive model for this material (Eq. (4)).
N
(
X
) is the outward surface normal at
X
