Data in this study show that shear wave speeds produced with radiation force of a focused ultrasound beam and time-of-flight can be transducer dependent, depth dependent, and lateral range dependent. It was found that shear wave speed measured by the C4-2 transducer increased when measurements were made closer to the transducer surface. This finding is consistent with results reported by D'Onofrio, et al
. (D'Onofrio et al. 2010
), where ARFI with a different curvilinear transducer was used to measure liver shear wave speed in 20 healthy volunteers. They reported that the mean shear wave speed was 1.56 m/s in deep portion versus 1.90 m/s in shallow portion of the right lobe. Another study in healthy volunteers reported similar findings (Kaminuma et al. 2011
). However, this difference was thought to be due to inherent liver “stiffness” variation or compression from transducer or cardiovascular system. Results from this study suggest that this difference might instead be due to depth dependent measurement bias. Referring to Eq. (1)
, a small bias in shear wave speed can result in a bigger bias in elasticity because the shear modulus is proportional to square of shear wave speed. For example, the shear wave speed measured by the C4-2 in phantom 2 was about 1.7 m/s at 30 mm depth and 1.5 m/s at 70 mm depth, giving values of μ
of 2.3 kPa and 2.9 kPa from Eq. (1)
. Using the cutoff values from a MRE study with 133 liver biopsy patients (Huwart et al. 2008
), this can lead to a difference between F0 (no fibrosis) and F ≥ 2 (stage 2 fibrosis or higher). Therefore, bias occurring in clinically relevant measurement conditions can be large enough to change diagnosis results and clinical decision making.
One possible cause of depth dependent bias in C4-2 measurements is the undesired intensity field of the push beam. The force field F
, which is proportional to the intensity field where the force per unit volume, can be written as F
are the ultrasound attenuation and sound speed of the medium. According to , the push beam generating the radiation force has two peaks with equal distance from the mid-elevational plane. is a simplified illustration of the force field in the transducer focal plane. The broken line rectangular box represents the transducer with its ultrasound beam directing into plane of this figure. The two peaks are represented by Source 1 and Source 2 which are confined within this figure but have longer axial extension out of the figure plane. Shear wave speed is estimated from the arrival time along the mid-elevational line represented by the solid horizontal line in . At time t1
, the shear wave travels distance a
to intersect with the horizontal line where shear wave detections are made. If the real shear wave speed of the medium is c0
, then a
(the shear waves from source 2 are not shown in for clarity of presentation). However, the apparent shear wave speed by measuring the arrival time along the mid-elevational line is:
Therefore, the measured shear wave speed is always greater than the true wave speed. When the C4-2 is focused deeper, Sources 1 and 2 will be closer to the mid-elevational plane and the bias will be smaller. shows a single peak in elevational direction when the transducer is focused at 50 mm in depth. However, a relatively wide beam width in the elevational direction can still generate significant out-of-plane shear waves that introduce bias (though smaller) to speed measurements.
Schematic plot of bias caused by shear wave propagating from the split peaks in elevational direction.
The bias in shear wave speed measurement also depends on the lateral range used for speed measurement: bias is smaller when the measurement range d
in is farther away from the sources (smaller θ
and therefore smaller
). This is consistent with the results in .
also shows non-zero intensity before and beyond the transducer focus, which has an “X” shape in the x-z plane. Therefore, additional shear waves generated from these non-focal regions can propagate to the focal plane and interfere with the shear wave coming from the focal zone. The apparent speed of these additional shear waves will be different from the true shear wave speed if measured along the lateral direction at the focal plane, because the lateral measurement direction is not parallel with the propagation direction of these additional shear waves. Different aperture sizes (32, 64, and 96 elements) for push beam transmission used in this study will change the angle of the “X” shape. indicates that different aperture size does not change shear wave speed measured by the C4-2 significantly at any given depth. This seems to suggest that for the C4-2, bias in shear wave speed measurement may be influenced more by the push beam shape in the y-z plane than that in the x-z plane.
Focusing of the L7-4 is good in both lateral and elevational direction at a focal depth as small as 20 mm, and the shear wave speed measured with the L7-4 shows little dependence with focal depth. Compared to the C4-2, the L7-4 shows more obvious change with different aperture size. This is not surprising because wider aperture size should not change the X beam shape too much for the C4-2 (outer elements contribute little to the focused beam because they are oriented away from the focus due to the curvature of the transducer). Similar to the C4-2, the L7-4 consistently shows higher shear wave speed at lateral range close to the push beam.
The potential sources of error discussed above are associated with time-of-flight based estimators relying on a priori
information about the shear wave propagation direction from a focused beam. More advanced inversion algorithms that do not make such assumptions will not be affected by this source of error. However, it is not uncommon for ultrasound based direct inversion methods (Bercoff et al. 2004a
) to assume negligible out-of-plane propagations, the validity of which may require further investigations.
Other than the push beam shape, dispersion is another possible reason for the measurement biases observed in this study. Equation (1)
assumes a pure elastic medium. For viscoelastic materials, shear wave speed is dispersive and frequency dependent (Chen et al. 2009
). Therefore, if the frequency band of the shear wave changes with focus depth, aperture size, or lateral range, the measured shear wave speed can also change due to dispersion caused by material viscosity. shows the power spectra of shear waves (recorded 5 mm away from the push beam) generated by the C4-2 with different focal depths in phantom 1. The spectra are similar to each other. These results seem to suggest that viscous dispersion is not the main cause of the depth dependent bias for C4-2. Note that these experiments were performed in phantoms made with Zerdine®, which has low viscosity (Havre et al. 2008
). Tissue has higher viscosity and therefore the impact of dispersion effects requires further investigations.
Power spectra of shear waves (recorded 5 mm away from the push beam) generated by the C4-2 with different focal depths in phantom 1.
There are a couple of possible options to correct for the shear wave speed measurement bias. One can use homogenous phantoms with known viscoelasticity to calibrate measurements made at different depth and steered angle. The transducer, its aperture size used for the push beam, and the lateral range used in shear wave speed measurement are usually fixed for a given clinical application. The calibration would only need to cover the viscoelastic range and measurement locations expected in a particular clinical application. For a curvilinear array, the active aperture can be translated along the curved surface of the transducer to produce push beams at different angles without real “steering”. Therefore, the correction factor probably will be insensitive to “steering angle” in this case because the push beam shape and frequency content of the shear wave should not change significantly for different push beam angles. The shear wave speed was measured in phantom 2 using 5 different push beam angles (0°, 4.7°, 9.5°, 14.2°, and 18.9°) when focusing at 30 mm with the C4-2. Variation of shear wave speed measurements among these 5 angles were less than 0.06 m/s (data not shown). A calibration table can be generated and used to correct for measurements in real tissues. Interpolation may be needed to increase the “resolution” of such a calibration table.
Another correction option is through simulation. The intensity field of the push beam can be calculated and used to compute the force field of the push beam. Shear waves generated by the push beam can be simulated with Finite Element Method (Palmeri et al. 2005
) or Green's function (Bercoff et al. 2004b
). Measurement bias along the lateral direction at the focal depth can then be estimated from these simulations and used to correct real tissue measurements. A correction table similar to the phantom calibration approach can be generated by the simulation method. The 3D beam shape for pulse-echo detection can also introduce averaging to shear wave motion detection (Palmeri et al. 2006
) and therefore should also be accounted for in the simulation correction approach.
This study has some limitations. First, the possible reasons for these observed biases in shear wave speed measurement are hypothetical and not confirmed. This study is intended as a “case report” to alert investigators in this field of a potential problem in these measurements. Further investigations are required to understand and confirm the mechanism causing this phenomenon. The simulation approach outlined above may provide valuable insights into this investigation. Second, the MRE and 1D TE utilized different excitation sources, shear wave frequencies, reconstruction algorithms, and regions of interest that introduce additional unknowns for the comparison with shear wave speed measured by time-of-flight with radiation force, making them not ideal control experiments. Third, shear wave detection in this study used flash imaging, which is different from the typical focused “beam” detection in other methods such as ARFI. Therefore, results in this study may be different from other methods where beam detection is used. SNR of echoes from flash imaging may not be as high as that from the focused beam detection schemes. Displacement tracking errors associated with scatterer shearing within the tracking point spread function (Palmeri et al. 2006
) is also different for flash imaging. However, shear waves detected by flash imaging matched simulation by Green's function very well in viscoelastic phantom studies, suggesting that flash imaging can provide reliable tracking of shear waves (Bercoff et al. 2004b