Introduction
Elastographic methods based on shear wave propagation and described in details by ^{Dewall (2013)} and ^{Urban et al. (2012)} have been applied, in the last two decades, to investigate the mechanical properties of biological tissues, such as liver (^{Chen et al., 2013}; ^{Zhang et al., 2008a}; ^{Zhu et al., 2014}), whole blood and/or blood clot (^{Gennisson et al., 2006}; ^{Huang et al., 2013}), breast (^{Garra et al., 1997}; ^{Meng et al., 2011}) and prostate (^{Sumura et al., 2007}; ^{Zhang et al., 2008b}). Some of these investigations were designed to determine the correspondence between the mechanical properties of normal and diseased organs and consequently to establish elastographic methods as a potential diagnostic tool.
The propagation of the shear wave produces microvibrations in the medium and the vibration waveform can be determined employing a probing ultrasonic pulse-echo system and processing a sequence of radiofrequency (RF) echo signals (^{Catheline et al., 2003}; ^{Chen et al., 2013}; ^{Mitri et al., 2011}; ^{Urban et al., 2009}; ^{Zheng et al., 2007}). In this context, the processing techniques commonly used with the sequence of RF echo signals are: cross-spectral analysis (^{Hasegawa and Kanai, 2006}), cross-correlation (^{Céspedes et al., 1995}; ^{Ophir et al., 1991}) and quadrature demodulation (^{Zheng and Greenleaf, 1999}). Once the vibration waveform is detected, then the shear wave properties such as phase velocity and attenuation coefficient can be measured and used in calculating the medium viscoelastic parameters such as shear modulus and viscosity.
A previous study by ^{Costa-Júnior and Machado (2011)} presented the development of an ultrasonic method, named UDmV, to detect microvibrations of a medium and the current investigation applies the UDmV method in determining the vibrations caused by a shear wave propagating in a 7% gelatin tissue mimicking phantom. A method proposed by ^{Zheng and Greenleaf (1999)} was used to process the in-phase and quadrature components of a sequence of RF echo signals from the phantom and to yield the vibration waveform due to the propagation of a shear wave induced in the phantom. The values for the shear phase velocity and attenuation coefficient, obtained from the detected vibration waveform, were used following a procedure similar to the transient elastography (TE) technique (^{Catheline et al., 2004}), in order to estimate the phantom shear modulus and viscosity, at 24.4 ± 0.2 °C and considering the propagation of shear waves with different amplitudes. Furthermore, the current work added another two processing techniques to determine the initial phase of the reference sine and cosine signals used for phase-quadrature demodulation and compared their results with those obtained by the previous technique described previously in ^{Costa-Júnior and Machado (2011)}.
Theoretical foundations
Considering a continuous shear wave propagating in a homogeneous, isotropic and viscoelastic medium represented by the Voigt model, then the shear wave phase velocity,
where
Solving (1) and (2) for
Therefore, according to (3) and (4) then the medium viscoelastic parameters
According to ^{Chen et al. (2004)}, the shear wave generated as described above has a cylindrical wave front and at large distances (over one-tenth of the shear wavelength) from the focus of
where
Additionally, the attenuation coefficient of a cylindrical wave,
where
The estimation of
where
Additionally,
Equations 7 and 8 are valid for
As demonstrated by ^{Costa-Júnior and Machado (2011)}, the UDmV method allows determining the in phase and quadrature components of the complex envelopes of RF echo signals. Furthermore, applying the Kalman filter to the velocity vibration waveform given in (7) enables to estimate the vibration phases,
Methods
Modification in UDmV method
^{Costa-Júnior and Machado (2011)} estimated the frequency of a sinusoidal signal employing the Kalman filter repeatedly over the signal with the input frequency value of the filter spanning over a frequency bandwidth containing the frequency of the signal. In this case, for each input frequency value, the amplitude of the signal estimated by Kalman filter was stored and the frequency corresponding to the highest amplitude value of the estimated signal was considered to be the signal frequency. This approach was employed in the current study to estimate the frequency of the signal that excited the transducer, so two frequency spannings were implemented, the first over a bandwidth corresponding to ± 10% of the nominal frequency, with steps of 1 kHz, and the second over a bandwidth of ± 5% of the frequency estimated in the previous stage, but in steps of 1 Hz. Knowing the frequency of the signal used to excite the transducer, then its initial phase can become estimated using the Kalman filter again (^{Zheng et al., 2007}). A flow chart, in Figure 1, summarizes the process, named here KF_{AP}, used to estimate the frequency and the initial phase of the transducer excitation signal based on the Kalman filtering.
Another technique proposed in this current investigation to estimate the frequency of the transducer excitation signal, and named KF_{MS}, was also implemented. It consists in performing the two spannings in frequency, as done with the KF_{AP} technique. However, the estimated frequency is the one that resulted in the minimum mean square error between the input signal and the sinusoidal signal estimated by the Kalman filter. Once that frequency of the excitation signal was determined, then the signal initial phase was estimated employing the Kalman filter again, as done for the KF_{AP} technique.
The approaches KF_{AP} and KF_{MS} implemented in current work and incorporated into the UDmV method, presented by ^{Costa-Júnior and Machado (2011),} provided the modified versions of the UDmV method, named UDmV_{AP} and UDmV_{MS}, respectively. In order to determine the performances of KF_{AP}, KF_{MS} and the previous approach used UDmV in estimating the initial phase of the transducer excitation signal, two metrics commonly employed in the literature, namely the bias and the jitter (^{Urban et al., 2008}; ^{Urban and Greenleaf, 2008}) were considered. The bias,
where
Computational simulations were accomplished using the software Matlab® (R2011a; Mathworks, Natick, MA) to compare the performances of the three approaches based on the estimated phase bias and jitter. Then, 200 sinusoidal signals were computer simulated with center frequency of 5 MHz, sampling frequency of 50 MHz and 1 µs duration, for different signal-to-noise ratio (SNR), keeping the initial phase of 120°, and for different initial phases (keeping an SNR of 30 dB). The initial phase for each configuration of the simulated signals was determined based on KF_{AP}, KF_{MS} and the previous approach used UDmV and then
Experimental procedure
The experimental setup illustrated in Figure 2 was used to generate a shear wave in the phantom and also to detect the micro-vibrations caused in the phantom due to the propagation of the shear wave.
The longitudinal wave emitted by
The phantom used in the experiment was made with pork skin gelatin (300 Sigma Bloom; Sigma-Aldrich, St. Louis, MO), in a concentration of 7% of the volume of deionized water, and plastic particles, to act as scatterers, with 1.2 mm in diameter and with a concentration of 5% of the volume. Additionally, another plastic sphere, with a diameter of 3.92 ± 0.07 mm and density of 2575.93 ± 32.23 kg∙m^{-3}, was embedded in the central region of the phantom, at a region free of particles (volumetric central portion of the phantom without the plastic particles with the diameter of 1.2 mm), to enable the focal alignment of
After collecting the RF ultrasonic echo-signal with
Two positioning systems, each one consisting of three manual linear stages (M-443, Newport Corporation, Irvine, Ca) and an angle bracket (360-90, Newport Corporation, Irvine, Ca), were used to adjust the position of the transducers.
After acquiring all the RF echo-signals with a sampling frequency of 50 MHz, these signals were resampled at 500 MHz employing the ‘resample.m’ function of Matlab. Additionally, the RF ultrasonic signals were time-gated with a Hanning window having a full width half maximum duration of approximately 1 μs, which corresponds to a range gate of 0.75 mm. The low-pass filter cutoff frequency used in phase-quadrature demodulation was 1 MHz.
The acquired RF echo-signals were phase-quadrature demodulated, based on the the UDmV method as presented by ^{Costa-Júnior and Machado (2011)} or modified versions UDmV_{AP} and UDmV_{MS}, and corresponding in phase and quadrature components applied in (7) to extract the motion waveform. Thereafter the Kalman filter was employed to calculate the phase and amplitude of all motion waveforms, which were used in (5) and (6) to estimate the shear wave phase velocity and the attenuation coefficient, respectively.
Measurements of phase and amplitude of the vibration waveform at the four positions along the propagation path of the shear wave were repeated seven times. For each experimental run, then a linear fit to the phase values as a function of distance was implemented and the angular coefficient of the straight line was used to calculate the phase velocity in accordance to Equation 5. Similarly, a linear fit to
Statistical analysis
The statistical analyses were performed with SPSS software (version 20.0; SPSS Inc., Chicago, USA). The results for all statistical tests were considered significant for p-value <0.05. The purpose of the statistical study was to detect significant difference between the phase velocity attenuation, shear modulus or shear viscosity values obtained using UDmV_{AP}, UDmV_{MS} and UDmV techniques, for each EA of
Shapiro-Wilk test was used to verify the normality of the distribution and the Levene's test was applied to assess the equality of variances for
When the distributions were normal and the groups presented equality of variances, one-way analysis of variance (ANOVA) was employed. Multiple comparisons (post hoc) tests were performed applying Tukey test when the ANOVA results were statistically significant. On the other hand, when one of these condition has been not satisfied, the nonparametric Kruskal-Wallis test followed by Dunn's multiple comparison test were utilized to determine which measurements are statistically different.
In addition, these statistical tests were also employed to assess the measured values of μ and η of the gelatin phantom, obtained based on the waveform vibration extracted applying the modified UDmV method to the RF ultrasonic echo-signals acquired under three different excitation conditions of the transducer
Results
The phase bias and jitter obtained from the phase values estimated with UDmV (Costa and Machado, 2011), UDmV_{AP} and UDmV_{MS} approaches, as a function of SNR and the initial phase, are depicted in Figure 3a and 3b, respectively.
Figure 4 presents the vibration velocity waveform signals of the medium and obtained based on the UDmV_{MS} method applied to the RF signals collected at four positions equally spaced, considering an amplitude of 170 V_{pp} (peak-to-peak) for the excitation of
Figure 5a illustrates the mean and standard deviation of the shear wave phase increment obtained at four positions along the shear wave propagation path and 5b the function
The shear wave phase velocity and attenuation coefficient values were obtained and substituted in (3) and (4) to estimate the gelatin phantom shear modulus and viscosity. Table 1 contains the average (std) values of the phase velocity, attenuation coefficient, shear modulus and viscosity obtained based on the propagation of the 97.644 Hz cylindrical shear wave on the phantom with transducer
Method |
EA (VPP) |
(m·s-1) |
(Np·m-1) |
(kPa) |
(Pa·s) |
---|---|---|---|---|---|
UDmV_{MS} | 170 | 1.306 ± 0.003 | 43.026 ± 1.448 | 1.663 ± 0.008 | 0.501 ± 0.017 |
140 | 1.290 ± 0.009 | 39.439 ± 2.582 | 1.630 ± 0.022 | 0.444 ± 0.029 | |
110 | 1.273 ± 0.014 | 34.809 ± 5.060 | 1.594 ± 0.034 | 0.377 ± 0.056 | |
UDmV_{AP} | 170 | 1.306 ± 0.003 | 43.089 ± 1.421 | 1.662 ± 0.007 | 0.501 ± 0.017 |
140 | 1.290 ± 0.009 | 39.445 ± 2.581 | 1.630 ± 0.022 | 0.444 ± 0.029 | |
110 | 1.272 ± 0.014 | 34.801 ± 5.070 | 1.594 ± 0.034 | 0.377 ± 0.056 | |
UDmV | 170 | 1.316 ± 0.003 | 44.145 ± 2.189 | 1.685 ± 0.006 | 0.525 ± 0.027 |
140 | 1.301 ± 0.007 | 38.904 ± 2.389 | 1.659 ± 0.018 | 0.449 ± 0.028 | |
110 | 1.294 ± 0.016 | 38.079 ± 4.049 | 1.642 ± 0.041 | 0.432 ± 0.043 |
The Shapiro-Wilk test indicated that phase velocity, attenuation, shear modulus and viscosity values presented normal distribution, regardless the excitation conditions of transducer
EA (VPP) |
Compared pairs | Mean difference | p-value | |
---|---|---|---|---|
(m·s^{-1}) |
170 | UDmV_{AP}-UDmV_{MS} | 0.000 | 1.00 |
UDmV_{MS}-UDmV | -0.010 | p<0.05 | ||
UDmV_{AP}-UDmV | -0.010 | p<0.05 | ||
140 | UDmV_{AP}-UDmV_{MS} | 0.000 | 1.00 | |
UDmV_{MS}-UDmV | -0.011 | p<0.05 | ||
UDmV_{AP}-UDmV | -0.011 | p<0.05 | ||
110 | UDmV_{AP}-UDmV_{MS} | -0.001 | 1.00 | |
UDmV_{MS}-UDmV | -0.021 | p<0.05 | ||
UDmV_{AP}-UDmV | -0.022 | p<0.05 | ||
(kPa) |
170 | UDmV_{AP}-UDmV_{MS} | -0.001 | 1.00 |
UDmV_{MS}-UDmV | -0.022 | p<0.05 | ||
UDmV_{AP}-UDmV | -0.023 | p<0.05 | ||
140 | UDmV_{AP}-UDmV_{MS} | 0.000 | 1.00 | |
UDmV_{MS}-UDmV | -0.029 | p<0.05 | ||
UDmV_{AP}-UDmV | -0.029 | p<0.05 | ||
110 | UDmV_{AP}-UDmV_{MS} | 0.000 | 1.00 | |
UDmV_{MS}-UDmV | -0.048 | p<0.05 | ||
UDmV_{AP}-UDmV | -0.048 | p<0.05 |
Table 3 presents the difference between the average μ or
Discussion
The methods UDmV_{AP}, UDmV_{MS} and UDmV used to estimate the shear wave phase presented good accuracy (values below 1.03°) and precision (values below 2.23°) with SNR values above 30 dB. However, the method implemented in this investigation (UDmV_{MS}) to estimate the phase presented the lowest values of phase bias and jitter, regardless the SNR. In addition, the highest values of phase bias magnitude and jitter were 0.53 and 12.14°, respectively, with a SNR of 0 dB. On the other hand, the UDmV_{AP} and UDmV_{MS} techniques yielded similar precision when the SNR is above 10 dB, even though UDmV_{AP} technique presented a phase bias of approximately -2°, independent of SNR.
When the computational simulation was employed to evaluate the accuracy and precision of methods used to estimate the phase, corresponding to the nominal initial phase of the simulated signal, the UDmV_{AP} and UDmV_{MS} techniques presented phase jitter values less than 2.23°, with the largest value obtained for a nominal phase of 120°. However, disregarding this values then the phase jitter becomes lower than 0.6° for both techniques. The overall phase bias of UDmV_{AP} and UDmV_{MS} methods is less than 4.5°. Nevertheless, the highest value of phase bias estimated with UDmV_{MS} was 1.03°. Thus, the modification made to the UDmV method allowed the estimation of the initial phase of
As in the previous study by ^{Costa-Júnior and Machado (2011)}, the UDmV method was employed to detect microvibrations, but this time the microvibrations corresponded to the propagation of the shear wave in a gelatin phantom. In the worst case, when the amplitude of the excitation signal of transducer
Shear wave phase velocity, shear modulus and viscosity of pig skin gelatin phantom, at a concentration of 7%, measured based on shear wave propagation are 1.32 m·s^{-1}, 1.61 kPa and 0.85 Pa∙s (^{Amador et al., 2011}), respectively, at a temperature undisclosed by the authors. Additionally, the viscoelasticity of the same material, at a temperature between 24.5 and 25.5 °C, estimated by ^{Huang et al. (2013)} comprehends shear modulus values of 1.59 ± 0.03 and 1.63 ± 0.07 kPa, and viscosity values of 0.86 ± 0.05 and 0.32 ± 0.02 Pa∙s, determined by means of shear-wave dispersion ultrasound vibrometry (SDUV) and embedded-sphere method (ESM), respectively. Furthermore, the shear wave velocity was about 1.29 m∙s^{-1} for shear wave frequency of 100 Hz. The results obtained in the present research approach for phase velocity and shear modulus of a 7% gelatin phantom at 24.4 ± 0.2 °C, using the EA of 170 V_{PP} for
The gelatin phantom employed in the present work was prepared with ultrasound scatterers having a diameter (1.2 mm) larger than those (approximately 50 μm) of the particles generally used for ultrasound tissue-mimicking phantoms. Consequently, the larger scatterers resulted in RF-echo signals with amplitudes larger than those that would exist if smaller scatterers were considered in the phantom preparation and this effect improved the SNR of the UDS echo signal. Although using larger scatterers could present a limitation to the experimental results of the present work, the results for the shear modulus and viscosity of the gelatin phantom are closely related to the results reported in the literature.
The theoretical foundations presented in this paper are based on the propagation of shear wave in an infinite medium, which would eliminate the existence of standing waves. On the other hand, the propagation medium (phantom) used is finite (diameter = 12.82 cm and height = 4 cm), with the plastic sphere positioned in the central region of the phantom. Considering the location distant
The results from the computational simulation indicated that the UDmV technique should present the worst results in estimating the shear wave velocity, when compared to UDmV_{AP} and UDmV_{MS} methods, once the UDmV technique had the larger values for phase bias and jitter. This fact was confirmed with multiple comparison of the Tukey test applied to the phase velocity values determined using the phase values estimated with the UDmV method for all EA conditions of transducer
Kruskal-Wallis test results indicated that μ and η values presented significant difference when the excitation configuration of
According to the data presented in Table 2 and considering the three techniques to estimate the initial phase of the reference sinusoidal signals used for phase-quadrature demodulation, there is a statistical difference between the means of
In view of the results obtained so far, there is a great motivation to carry out a new experiment, which could be performed to characterize the rheology of biological tissue in vitro. For this purpose, the experimental setup would be modified in order to generate the harmonic shear waves and viscoelastic coefficients would be assessed only through the dispersion of phase velocity as a function of the frequency of the shear wave.
The computational simulation demonstrated that the modification in UmVD method resulted in more accurate and precise estimates of the phase of the sinusoidal signal, which is used to generate the sine and cosine necessary to calculate the in phase and quadrature components of the RF echo-signals. The experimental results revealed that when the push and probing systems are synchronized, the UDmV method is able to estimate the in phase and quadrature components, which were employed to form the signal that represents the vibration waveform of the medium, due to the propagation of the shear wave. In addition, it was possible to use the method to estimate the phase velocity, the attenuation coefficient, shear modulus and viscosity of the 7% gelatin phantom, based on the propagation of shear waves in the phantom.