Analysis of Dynamic Magnetic Resonance Elastography Data - Advance Image Processing in Magnetic Resonance Imaging - page 450

where in the direction of propagation k.

The attenuation can also be expressed as the attenuation per wavelength, which
is the acoustic quality factor Q

α denotes attenuation by the factor e

k

[36].
The “true” shear modulus is the real part

= c/ ωη

or c, which describes the behavior
of a static object in equilibrium. However, some processing techniques described
in the following text calculate only the local wavelength and do not consider
attenuation. These techniques essentially estimate the wave speed, and we can
speak of an “effective” shear modulus or “shear stiffness” that is defined as the
square of the wave speed by analogy to the lossless case. The results are usually
presented in terms of this shear stiffness at a given frequency. Other techniques
calculate both

µ

r

) directly, which can be converted to wave
speed and attenuation using Equation 14.8 and Equation 14.9. This determination
of

µ and µ (or c and η

r I

µ or α tends to be very sensitive to noise. A more stable way to determine the

i

parameters is to calculate the shear stiffness at several different frequencies
and fit the result to expressions derived from Equation 14.8 and Equation 14.9.
Spatial wavelength and attenuation decrease and increase, respectively, as the
mechanical frequency increases. This has two competing effects on stiffness
determination: (1) higher resolution because the wavelength is smaller, and (2)
lower displacement and hence lower signal. The best frequency for a particular
application depends on trade-off between these two effects.

c and η

14.5.3 P G

HASE RADIENT

After extracting the harmonic component at the driving frequency, the amplitude
and phase (relative to an arbitrary zero point) that characterize the harmonic
oscillation at each pixel in the image are obtained. If the motion is a simple

propagating shear wave, the gradient of this phase directly yields the change in
phase per pixel, easily convertible to a local frequency and thus to shear stiffness.
This analysis can have very high resolution, but is very sensitive to noise, and
data smoothing is usually necessary. This technique yields inaccurate results when
two or more waves are superimposed (e.g., reflected waves) or when the motion
is complex because the phase values then do not represent a single propagating
wave [44]. However, it is useful in specialized situations in which simple plane-
wave propagation is a good approximation. The other approaches, described in
the following text, do not suffer from this drawback; they correctly handle reflec-
tions and other complex interactions because they are based on the underlying
equations of motion.

14.5.4 L F E (LFE)

OCAL REQUENCY STIMATION

The local spatial frequency of the shear-wave propagation pattern can be calcu-
lated using an algorithm that combines local estimates of instantaneous frequency
over several scales [45]. These estimates are derived from filters that are a product
of radial and directional components and can be considered to be oriented log-
normal quadrature wavelets. The shear stiffness is then given by

µ = f mech

2

/f spatial

2

,