Image Processing Reference
In-Depth Information
14.5.1
E
M
QUATIONS
OF
OTION
The mechanical quantities we wish to characterize are those that relate strain to
stress, and because the displacements in MRE are very small (microns to tens of
microns), a linear relationship can be assumed between these. In the general case,
stress and strain are related by a rank 4 tensor, with up to 21 independent quantities
[40]. If one assumes that the material is isotropic, this reduces to two independent
quantities, the Lame constants
, related to longitudinal and shear defor-
mation, respectively. The isotropic relation between stress and strain is given by
λ
and
µ
σµ δ
ij
=
2
e
+
e
(14.2)
ij
ij
nn
where e
is the Kronecker delta, and
summation over repeated indices is assumed. The strain tensor e
is one component of the stress tensor,
δ
ij
ij
is defined in
ij
terms of the displacement tensor u
as
ij
euu
ij
=+
(
)/
2
(14.3)
i j
,
j i
,
where indices after a comma indicate differentiation. Substituting these into the
equation of motion, we obtain the general equation for harmonic motion in an
isotropic, linearly elastic medium [41]:
[
λ
u
]
+
[
µ
(
u
+
u
)]
= −
ρω
2
u
(14.4)
jj i
,
,
ij
,
ji
,
,
j
i
the angular frequency of the
mechanical oscillation. The Lame constants can be considered to be complex
quantities, with the imaginary parts representing attenuation for a viscoelastic
medium. Solving this equation requires knowledge of the full 3-D displacement
because the equations for the individual components are coupled. MRE phase
difference measurements in all three spatial orientations are thus required.
Additional assumptions can be made to further simplify the equation. If one
assumes local homogeneity,
with
ρ
being the density of the material and
ω
become single unknowns instead of functions
of position, and Equation 14.4 becomes an algebraic matrix equation that can be
solved locally by direct inversion, as described by the following equation (terms
in boldface are column vectors):
λ
and
µ
µ
∇++∇∇⋅ =−
2
u
(
λ µ
)
(
u
)
ρω
2
u
(14.5)
or more). This makes it difficult to
estimate both parameters simultaneously, and the longitudinal wavelength is so
long in tissues (tens of meters) that accurate estimation of
In soft tissues,
λ
>>
µ
(typically by 10
4
is very challenging
in any case. It is possible to partially filter out the effects of the longitudinal wave
because its contributions are at very low (near zero) spatial frequency. To remove
λ
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