Biomedical Engineering Reference
In-Depth Information
the concomitant determination of stresses within the deforming body. The ap-
proach may be applied to physical deformations that arise in solid and fluid
mechanics as well as to non-physical deformations such as the inter- and intra-
subject registration of image data. For the physical deformation case, the goal
is to quantify the kinematics and the kinetics of the deformations. In the non-
physical case, only the kinematics of the deformations are sought.
12.2
Hyperelastic Warping
The standard notation and symbols of modern continuum mechanics are em-
ployed in the following presentation [23-25]. In particular, direct notation is
used, with boldface italics for vector and tensor fields. The outer product is de-
noted with “
⊗
”, a matrix inner product is denoted with “:”, and a matrix-vector
product is denoted with “
·
”. Index notation is incorporated for quantities that
cannot be readily written in with direct notation. The condensed Voigt notation
typically employed in finite element (FE) analysis is utilized as needed [1].
12.2.1
Finite Deformation Theory
A Lagrangian reference frame is assumed in the following presentation, and
thus the kinematics of material points corresponding to the template image are
tracked with respect to their original positions. However, it should be noted that
the approach could be adapted readily to an Eulerian framework. The template
and target images are assumed to have spatially varying scalar intensity fields
defined with respect to the reference configuration and denoted by
T
and
S
,
respectively. The deformation map is denoted
ϕ
(
X
)
=
x
=
X
+
u
(
X
) where
x
are the current (deformed) coordinates corresponding to
X
and
u
(
X
)isthe
displacement field.
F
is the deformation gradient [26]:
F
(
X
)
=
∂
ϕ
(
X
)
∂
X
.
(12.1)
The local change in density is directly related to the deformation gradient
through the Jacobian,
J
:
=
det(
F
)
=
ρ
0
/ρ
, where det (
F
) is the determinant
of the deformation gradient,
ρ
0
is the density in the reference configuration and
ρ
is the density in the deformed configuration. At this point, it is assumed that
T
and
S
have a general dependence on position in the reference configuration
X
and the deformation map
ϕ
(
X
).
