Biomedical Engineering Reference
In-Depth Information
either easily constructed regular meshes or irregular meshes that conform to
the geometry of the structure being registered and can be used to register a par-
ticular region of interest or the entire image space. Additionally, hyperelasticity
provides a physically realistic constraint for the registration of soft tissue defor-
mation. Hyperelasticity based constitutive relations have been used to describe
the behavior of a wide variety of soft tissues including the left ventricle [99-102],
arterial tissue [103, 104]. skin [105] and ligaments[106-109]. Hyperelastic Warp-
ing can be tailored to the type of soft tissue being registered through the appro-
priate choice of hyperelastic material model and material parameters.
Deformable image registration models based other material models have
been used extensively in the field of anatomical brain registration. As was de-
scribed above, an energy functional is minimized in order to achieve the regis-
tration solution. This functional consists of a measure of image similarity and
an internal energy term (Eq. 12.4). Measures of image similarity take the form
of differences in the square of the image intensities (Eq. 12.8) [15-17, 19, 110,
111] or are based on cross-correlation methods of the intensity or intensity
gradient values [112]. Since the internal energy term of the energy functional
is derived from the material model through the strain energy
W
, the registra-
tion process takes on the characteristics of the underlying material model. For
example, registration methods that use a viscous or inviscid fluid constitutive
model [15,17] have been shown to provide excellent registration results. How-
ever, these models have a tendency to underpenalize shear deformations, thus
producing physically unrealistic registration of solids. In other words, the defor-
mation of the deformable template resembles that of a fluid rather than that of a
solid.
Other continuum-based methods for deformable image registration use lin-
ear elasticity [12, 13, 15, 16] to regularize registration. The use of linear elasticity
is attractive due to the fact that it is relatively simple to implement. However,
for the large deformations involved in inter- or intra-subject registration, it has a
tendency to over-penalize large deformations. This is due to the fact that linear
elasticity is not rotationally invariant. For an isotropic linear elastic material,
the constitutive law is:
T
=
λ
tr(
e
)
+
µ
e
.
(12.37)
Here,
λ
and
µ
are the Lam e material coefficients, and
e
is the infinitesimal strain
“tensor” defined in terms of the displacement gradients. The infinitesimal strain
