Biomedical Engineering Reference
In-Depth Information
is not a true tensor since it does not obey the transformation laws for 2
nd
order
tensors. In detail:
∂
u
∂
X
T
1
2
+
∂
u
∂
X
e
=
·
(12.38)
∂
X
=
∂
(
x
−
X
)
∂
u
But
,
∂
X
=
F
−
1
.
(12.39)
For any deformation gradient
F
, we can use the polar decomposition to write
F
as
F
=
RU
, where
R
is a proper orthogonal rotation and
U
is the positive
definite symmetric right stretch tensor. With this substitution,
1
2
(
RU
−
1
)
T
+
(
RU
−
1
)
.
e
=
(12.40)
As indicated in Eq. (12.40), the strain
e
depends directly on
R
, which describes
the local rigid body rotation. As a result, even the smallest rotation of mate-
rial axes induces stress in a linear elastic solid, making the constitutive model
nonobjective.
This work has demonstrated that Hyperelastic Warping may be used to ana-
lyze a wide variety of image registration problems, using standard medical image
modalities such as ultrasound, MRI, and CT. The types of analyses demonstrated
range from anatomical matching typical of nonphysical image registration, to the
large physical deformations present in the deformation of the left ventricle over
the cardiac cycle. As demonstrated in the presented work, the method allows for
the estimation of the stress distribution within the structure(s) being registered,
an attribute that has not been demonstrated by other registration methods.
Acknowledgments
Financial support from NSF Grant # BES-0134503 ( JAW, AIV, NP ), NIH grant
# R01-EB000121 ( JAW, AIV ) and NIH Grant # PO1-DC01837 (AIV) is gratefully
acknowledged. An allocation of computer time was provided by the Center for
High Performance Computing (CHPC) at the University of Utah. The authors
thank the following individuals for their contributions to this work and their
continued collaboration: Grant T. Gullberg, Richard D. Rabbitt, Willem F. De-
craemer, Anton E. Bowden, Bradley N. Maker, Steve A. Maas, Geoffrey D. Vince,
Robert J. Gillies, Edward V. R. DiBella and Jean-Philippe Galons.
