Image Processing Reference
In-Depth Information
from a small quasi-static longitudinal stress applied once each time the pulse
sequence is repeated [31]. Often in these methods, the strain alone is used as a
surrogate for stiffness, that is, low strain means high stiffness and high strain results
in softer or low-stiffness regions. The distribution of strain can also be related to
the predicted distribution of stress and the material parameters deduced through
elasticity equations [32]. Such calculations are usually global in nature (and thus
computationally intensive), and they require knowledge of boundary conditions.
Dynamic methods, as described in the following text, rely on the wave equation,
which in its differential form is local in character. Therefore, the distribution of
dynamic displacement (a 3-D vector) and its second-order partial derivatives in
time and space, due to a propagating shear wave in a small region of the tissue,
are enough to completely characterize the shear moduli of the tissue in that region.
Strictly speaking, quasi-static methods, mentioned in the preceding text, as well as
the dynamic or harmonic methods, described in the following text, can be termed
MRE; nevertheless this chapter will henceforth concentrate only on the latter.
14.4
MRE
Dynamic MRE uses propagating mechanical waves as a probe for the elastic prop-
erties of tissues. Shear waves of frequencies in the 50 to 1000 Hz range are suitable
as probes because they are much less attenuated than those of higher frequencies,
their wavelength in tissue-like materials is in the useful range of millimeters to tens
of millimeters, and the shear modulus varies so widely in body tissues. High-
frequency longitudinal acoustic waves (ultrasound) are not directly suitable for use
as probes because their propagation is governed by the bulk modulus, which varies
little in soft tissue. Longitudinal acoustic waves at lower frequencies are also not
suitable because they have long wavelengths (on the order of meters for waves of
frequency below 1 kHz) [10].
In MRE, a phase-contrast MRI technique is used to spatially map and measure
displacement patterns corresponding to harmonic shear waves with amplitudes in the
range of microns or less. A conventional MRI system is used with an additional
motion-sensitizing gradient imposed along a specific direction and switched in polar-
ity at an adjustable frequency (
Figure 14.1
). Trigger pulses synchronize an oscillator
and amplifier unit that drives an electromechanical actuator coupled to the surface of
the object to be imaged. The actuator induces shear waves in the object at the same
frequency as the motion-sensitizing gradient. The harmonic motion of the spins at
the frequency of the motion-sensitizing gradients causes a measurable phase shift in
the received MR signal. From the measured phase shift, it is possible to calculate the
displacement at each voxel and directly image the acoustic waves within the object.
The phase shift caused by a propagating mechanical wave with a wave
vector within a medium at a given frequency (1/ T) in the presence of a cyclic
motion-encoding gradient at the same frequency is given by [8,9]
k
(
)
γ
NT G
•
ξ
0
0
φθ
(, )
r
=
cos(
k
•+
r
θ
)
(14.1)
2
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