Biomedical Engineering Reference
In-Depth Information
A similar computation for the mechanical strain energy term
W
leads to the
weak form of the momentum equations (see, e.g., [24]):
(
T
−
S
)
∂
S
∂
d
J
=
0
.
G
(
ϕ,
η
):
=
DE
(
ϕ
)
·
η
=
β
σ
:
∇
η
d
ν
−
(12.13)
β
λ
ϕ
·
η
Here,
σ
is the 2nd order symmetric Cauchy stress tensor,
1
J
F
∂
W
∂
C
F
T
(12.14)
σ
=
.
Thus, the forces applied to the physical model of the deforming template due to
the differences in the image data are opposed by internal forces that arise from
the deformation of the material through the constitutive model. The particular
form of
W
depends on the material and its symmetry (i.e., isotropic, transversely
isotropic, etc.) [26, 30-33].
The linearization of Eq. (12.13) yields:
(
T
−
S
)
∂
S
∂
d
J
L
ϕ
∗
G
(
ϕ,
η
)
=
β
σ
:
∇
η
d
ν
−
β
λ
ϕ
·
η
s
s
(
u
)
d
ν
(12.15)
+
β
∇
η
:
σ
:
∇
(
u
)
d
ν
+
β
∇
η
:
c
:
∇
·
u
d
J
+
β
η
·
k
Here,
c
is the 4th order spatial elasticity tensor [1]:
2
W
∂
C
IJ
∂
C
KL
,
4
J
F
iI
F
jJ
F
kK
F
lL
∂
(12.16)
c
ijkl
=
s
[
·
]
is the symmetric gradient operator:
and
∇
T
∂
[
·
]
∂
1
2
∂
[
·
]
∂
s
[
·
]
=
∇
ϕ
+
.
(12.17)
ϕ
In the field of computational mechanics, the first two terms in the second line
of Eq. (12.15) are referred to as the
geometric
and
material stiffnesses
, respec-
tively [1]. The 2nd order tensor representing the image stiffness for Hyperelastic
Warping is:
∂
S
∂
∂
S
∂
−
(
T
−
S
)
∂
2
U
2
S
=
∂
(12.18)
k
ϕ
=
λ
⊗
.
∂
ϕ∂
ϕ
ϕ
∂
ϕ∂
ϕ
These three terms form the basis for evaluating the relative influence of the
image-derived forces and the forces due to internal stresses on the converged
solution to the deformable image registration problem, as illustrated in the fol-
lowing two sections.
