Image Processing Reference
In-Depth Information
λ
from consideration, the assumption can be made that displacements due to the
longitudinal wave vary slowly and are thus negligible (this corresponds to assum-
ing that
0). The large difference between longitudinal and shear waves
in tissue make this a reasonable assumption. The equation then simplifies to a
single vector equation in
λ
(
∇⋅
u
)
=
µ
, but all three components of motion are still required:
[(
∇∇⋅ ∇
u
)
2
u
]
µω
=−
2
[]
u
(14.6)
Alternatively, one can assume incompressibility (
∇⋅
u
=
0), and the equation
then simplifies to the Helmholtz equation:
µ
∇=−
2
u
ρω
2
u
(14.7)
The terms involving components in the different orthogonal directions are
now decoupled, and each component satisfies the equation separately. Thus,
measurements in only one sensitization direction (and an estimate of the Laplacian
of that component) suffice to determine
. Experiments with tissue-simulating
phantom data sets have shown little difference among inversion results that use
Equation 14.5, Equation 14.6, and Equation 14.7, suggesting that the incompress-
ibility assumption is valid in practice [42,43].
Filtering approaches can also be designed based on the fact that the displacement
field corresponding to the longitudinal wave is curl free, whereas that corresponding
to the shear wave is divergence free [43]. Taking the curl of Equation 14.5 leads
directly to the Helmholtz equation (with the curl of the displacement replacing the
displacement itself) but with no need for the incompressibility assumption. This
technique can remove artifacts present in the standard inversion in certain situations,
but it is also more susceptible to noise because it involves additional derivative
operations [43].
µ
14.5.2
S
M
M
F
HEAR
ODULUS
AND
ECHANICAL
REQUENCY
In the earlier treatment, the Lame constants were complex quantities, with the
imaginary parts representing attenuation for a viscoelastic medium. Because the
damping term involves the time derivative of the strain, for harmonic motion this
can be denoted as
[40]. The simplest case is an isotropic,
homogeneous, and incompressible medium (Equation 14.7). With no attenuation,
a simple shear wave propagates with a specific wavelength or spatial frequency f
µ
=
µ
+
i
µ
=
c
+
i
ωη
r
i
.
sp
The shear modulus is
µρ
=
f
2
/f
s
2
=
ρ
v
s
2
, where
f
is the mechanical driving
mech
mech
frequency and v
is the wave speed (or phase velocity). We will henceforth assume
s
~1.0 for all soft tissues [11]. If there is attenuation, the wave speed and
attenuation are functions of frequency and are given by
that
ρ
c
cc
+
++
2
(
2
ωη
ωη
2
2
)
ω
2
(
c
2
+
ω η
ωη
2
2
)
/
1 2
−
c
v
2
=
and
α
2
=
(14.8, 14.9)
s
(
)
/
2
c
+
2
2
2
1 2
2
2
2
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