Characterizing dispersion in bovine liver using ARFI-based Shear Wave Rheometry

Abstract

Background:

Dispersion presents both a challenge and a diagnostic opportunity in shear wave elastography (SWE). Shear Wave Rheometry (SWR) is an inversion technique for processing SWE data acquired using an acoustic radiation force impulse (ARFI) excitation. The main advantage of SWR is that it can characterize the shear properties of homogeneous soft media over a wide frequency range. Assumptions associated with SWR include tissue homogeneity, tissue isotropy, and axisymmetry of the ARFI excitation).

Objective:

Evaluate the validity of the SWR assumptions in ex vivo bovine liver.

Approach:

SWR was used to measure the shear properties of bovine liver tissue as function of frequency over a large frequency range. Assumptions associated with SWR (tissue homogeneity, tissue isotropy, and axisymmetry of the ARFI excitation) wore evaluated through measurements performed at multiple locations and probe orientations. Measurements focused on quantities that would reveal violations of the assumptions.

Main results:

Measurements of shear properties were obtained over the 25–250 Hz range, and showed a 4-fold increase in shear storage modulus (from 1 to 4 kPa) and over a 10-fold increase in the loss modulus (from 0.2 to 3 kPa) over that decade-wide frequency range. Measurements under different conditions were highly repeatable, and model error was low in all cases.

Significance and Conclusion:

SWR depends on modeling the ARFI-induced shear wave as a full vector viscoelastic shear wave resulting from an axisymmetric source; it is agnostic to any specific rheological model. Despite this generality, the model makes three main simplifying assumptions. These results show that the modeling assumptions used in SWR are valid in bovine liver, over a wide frequency band.

1. Introduction

Methods for the non-invasive quantification of shear wave properties, collectively referred to as shear wave elastography (SWE), continue to show promise in the stratification and clinical management of chronic liver disease [1, 2, 3, 4, 5, 6]. Several SWE implementations, predominantly using either ultrasound or magnetic resonance imaging (MRI), are now available to physicians. These tools typically report one of two related parameters, the shear wave speed (SI unit: m/s), or the shear modulus (SI unit: kPa) of liver tissue. Estimates of these parameters obtained from different SWE implementations exhibit statistically significant variation in both phantom studies [7, 8, 9] and clinical applications [10]. For example, when ultrasound-based acoustic radiation force impulse (ARFI)-SWE measurements were compared to those from magnetic resonance elastography (MRE) for 50 liver patients [11], good correlation between the computed shear modulus values from both methods was found, however, the shear modulus estimates from US measurements were approximately twice larger than those obtained from MRE. Such variance confounds direct comparisons between techniques/systems, raises questions about their accuracy, and thus hinders their clinical adoption [10], [12].

The dispersive nature of liver tissue coupled with the fact that different techniques operate in different frequency ranges may explain the observed discrepancies. For example, ARFI-SWE typically uses shear wave pulses with energy in the 100–400 Hz range [13] while MRE typically operates at individual shear wave frequencies in the 50–60 Hz range. Moreover, the dispersion characteristics of liver tissue, i.e. the nature of the frequency-dependence, may have diagnostic value [14, 15, 16, 17, 18, 5]. For these reasons, there is interest in measuring frequency dependence of shear wave properties, such as the shear wave phase velocity [19, 20], shear wave attenuation [21], and shear wave dispersion slope [22]. As a material property, frequency dependence of the shear wave speed is intrinsically linked to shear viscosity [23].

Extracting shear viscosity from measurements of shear wave propagation is complicated by the need to separate different sources of wave attenuation, including geometrical spreading (or focusing), medium inhomogeneity, and viscosity [24]. Early attempts to measure shear viscosity made idealized assumptions about geometrical spreading of the shear wavefront [21, 25]. This can lead to significant biases, as high as 40% [26]. The so-called “frequency-shift” method is widely believed to be insensitive to geometrical spreading. It depends, however, on the assumption that the field at any point is that due to a single ray [27] passing through that point. Even when that assumption is satisfied, the $\mathrm{\Delta }x$ that appears in the formulation must be aligned with the ray direction to yield an accurate estimate of attenuation [28, 29]. Finally, it implicitly assumes a specific frequency dependence of wave attenuation [27], and is thus not “model-free.” Most authors neglect the impact of medium inhomogeneity on wave attenuation estimates. Many related approaches to ultrasound viscoelastography have been recently reviewed [30].

A model-based inversion technique for ARFI-SWE called Shear Wave Rheometry (SWR) was presented in [31] that addresses many of the shortcomings of other approaches. In the characterization of methods reviewed in [30], SWR would be classified as a rheological model-free inverse problem approach based on axisymmetric full-vector shear wave propagation. Axisymmetric waves include cone shaped wavefronts, cylindrical wavefronts, and spherical wavefronts as special cases. Earlier work on measures of ARFI excited wavefields [31] has shown that inversion based on axisymmetric vector shear waves more accurately measures viscous loss compared to the use of more restrictive plane, spherical, or cylindrical wave models. Being rheological model-free also allows SWR to avoid potential bias from fitting a prescribed frequency dependence to the viscoelastic parameters[30]. SWR was used to characterize dispersion in homogeneous soft gels [32] over a frequency range of more than three octaves.

The model used in SWR rests on three key assumptions. The first is medium homogeneity, i.e. elastic properties are the same everywhere. The second is medium isotropy, i.e. elastic properties are the same regardless of the direction/orientation chosen to measure them. These assumptions allows for spatial averaging which can improve SNR at individual frequencies. The third modeling assumption in SWR is axisymmetric propagation of the shear wavefront generated by the ARFI-push. The validity of these assumptions in gels was demonstrated previously [31, 32], The purpose of the present contribution is to evaluate these assumptions in liver tissue.

To evaluate feasibility of the SWR approach in liver, complex frequency-dependent shear modulus estimates were obtained in ex vivo bovine liver tissue using SWR over a wide frequency bandwidth. Measurements were acquired at multiple locations and over multiple probe orientations and compared. Homogeneity and anistropy are tested by assessing repeatability from location to location and orientation to orientation, respectively. The axisymmetry of the shear wave propagation was evaluated using the mismatch between measurement and the axisymmetric model, and by assessing the left-vs-right asymmetry of the shear wavefield. The results obtained from SWR were also compared to shear modulus estimates obtained from group speed measurements and our implementation of shear wave spectroscopy (SWS) [13].

2. Methods

2.1. Liver Tissue Preparation

A whole liver from a small adult bull, weighing about 4.5 kg, was obtained from a slaughterhouse (Adam’s Farm, Athol, MA) 3–4 hours after slaughter. The liver was immersed in an isotonic saline solution (deionized water with sea salt, 0.9% concentration by weight) and degassed (vacuum chamber with pump DAA-V715A-EB, Gast Mfg. Co., MI, USA) in a cold room at 4°C for two hours.

The small size of the vacuum chamber required the liver to be cut to size. Two roughly-square pieces, approximately 15 to 20 cm in size and 5 to 10 cm in thickness, were placed in plastic containers with an ultrasound absorptive rubber pad at the bottom. An ice bath maintained the tissues at about 4°C throughout the data acquisition.

2.2. Shear Wave Elastography

An ATL L7–4 linear array transducer (128 elements, 38.4mm aperture, 5 MHz) was connected to a Verasonics research ultrasound system (Model V1, Verasonics, Inc. Kirkland, WA, USA). Ultrasound imaging and ARFI-SWE were performed through the liver capsule with ultrasound gel used as the coupling medium between the transducer and tissues.

A shear disturbance was excited using three consecutive ARFI pushes along the center axis of the transducer aperture, at depths of 21.6, 20.2, and 18.8 mm, in this order, with an F number of approximately 0.5. Consecutive 100 μs pushes were separated by a 100 μs wait time, so that three pushes were completed over a 500 μs total period. The distance between the push locations was small compared to the shear wavelength, and as such the three pushes approximated the effect of a single strong push. The resulting shear wave was tracked for 15 ms starting 1 ms after the last push pulse, with an imaging pulse repetition frequency (PRF) of 10 kHz. Compounding was performed with 3 angles (−8°, 0°, 8°). An algorithm based on 2D-autocorrelation [33] was used to compute the axial shear wave particle velocity from the high-speed image data. Visualization of a typical shear wave disturbance generated by the ARFI-pushes is shown in Fig. 1(a–c).

Figure 1.

(a-c) Time-domain snapshots of the shear wave disturbance initiated at $t=0$ (location 1, 0°orientation). The full dataset has 0.1 ms temporal resolution over the 1 to 15 ms interval, and examples are shown here at 1, 3, and 5 ms. (d-f) Corresponding shear wavefields were obtained, with 25 Hz frequency resolution over the 25–250 Hz interval, with maximum amplitude around 100 Hz. Examples are shown here at 25, 150, and 250 Hz.

The shear wave pulse propagation measured in the time domain was separated into its temporal frequency components via a Fourier transform to obtain shear wavefields at discrete frequencies. These are visualized in Fig. 1(d–f). The shear wavefield encodes the complex wave number of the shear wave, which is related to the complex shear modulus $\mu ={\mu }^{\prime }-i{\mu }^{\prime \prime }$, which in turn relates the shear stress $\tau$ to the shear strain $\gamma$, at each discrete frequency $\omega$, as:

$$\tau =\mu \left(i\omega \right)\gamma ,$$ (1)

where we assume a time dependence of $e^{-i\omega t}$. The density of the medium was assumed to be 1000 kg/m³ and the shear modulus was estimated using the aforementioned SWR model-based inversion technique [31].

2.3. Study Design

The L7–4 transducer was mounted in a 3 degrees-of-freedom positioning stage (Velmex Inc., New York, NY) while the tissue rested on a rotation stage, for a total of four degrees of freedom (three translational and one rotational). The axis of rotation of the rotation stage was aligned with the transducer axis; the alignment was performed based on B-mode imaging, performed in a water bath that contained a metallic hexagonal nut located at the center of rotation.

Two locations in the largest piece of liver and a third location in the second piece were identified for subsequent ARFI-SWE measurements. These locations were chosen based on B-mode imaging, to identify relatively uniform tissue regions devoid of major blood vessels. Several separate ARFI-SWE measurements were performed at each one of these locations, by using the turntable to rotate the liver tissue relative to the ultrasound probe’s long axis: thirteen measurements were performed with 30 deg rotation increments (including measurements at 0 and 360° which should in principle be identical). Some additional measurements were made to test repeatability and variability within the liver. For example, at locations 1 and 2, measurements were made at ±5 deg incremental rotations about the 0° orientation. At all three locations, two additional measurements with a 3 mm and 6 mm translation of the transducer in the elevation direction were also made. Furthermore, at location 3, one additional measurement after pressing the transducer 1 mm further down along its axis was made to see the effect, if any, of increased contact pressure. Thus, overall there were 5 measurements at 0° orientation, 2 additional measurements at 0+/−5° and a total of 20, 18, and 17 measurements, respectively, at locations 1, 2 and 3. This measurement scheme is explained in detail in Table 1.

Table 1.

The transducer position and orientation used for the ARFI-SWE measurements are listed, for each one of the three locations sampled within the bovine liver tissues. As seen from this list, a total of 20, 18, and 17 measurements were performed at locations 1, 2 and 3, respectively.

Orientation & Position	Loc-1	Loc-2	Loc-3
0°	1,2	1	1
30–330° in 30° steps	3–13	2–12	2–12
360°	14,15	13	13
365°	16	14	X
360°	17	15	14
355°	18	16	X
360° +3mm	19	17	15
360° +6mm	20	18	16
360° +6mm; 1mm down	x	x	17

Tissue temperature was monitored at the beginning and end of each set of measurements (non-contact infrared thermometer, Fluke 62 Max+). For measurements at locations 1 and 3, the rotation axis was aligned with the median long axis of the L7–4 transducer, i.e., along the imaging-depth direction. In contrast, to achieve a larger sampled volume, measurements at location 2 were made by rotating the liver sample around an axis parallel to the transducer long axis, but offset by 40 mm in the elevation direction. As such, rotations further entailed a translation of the liver tissue with respect to the transducer, and thus a larger, potentially more inhomogeneous, and arguably more representative volume of tissue was sampled around location 2 than at locations 1 and 3.

2.4. Measuring Dispersion

The shear wave disturbance created at $t=0$ was captured with 0.1 ms temporal resolution over the 1 to 15 ms interval, allowing shear wavefields to be obtained for specific frequencies from 25–250 Hz with 25 Hz resolution. These measured wavefields were used to estimate the complex shear wavenumber in the medium using the SWR model-based inversion scheme [31]. In short, the inversion scheme fitted the measured complex wavefield with a forward shear wave propagation model in the domain outside of the ARFI-push region. The best fit was obtained by minimizing an error functional defined as the “energy mismatch” between the model and the measurement. The “energy mismatch” is defined as the sum over all pixels of the squared amplitude of the difference in the complex particle velocity at that frequency.

The ARFI-push region is assumed to be a thin cylinder of radius $r_0$ and is excluded from the model-fit. The parameter $r_0$ is also estimated as part of this minimization scheme, and in general, its value depends on the medium, the push parameters, and the frequency. In our measurements, the imaging field of view (FOV) of the L7–4 transducer was 38.4 mm wide by 35 mm deep, and the estimated $r_0$ varied between 2 – 3 mm. The model was therefore fit over two regions that were each roughly 16 × 35 mm in size on either side of the push region. The measured volume of the liver tissue at locations 1 and 3 was a cylindrical region of diameter approximately 39 mm and height 35 mm, while that at location 2 was a somewhat larger cylindrical region due to the offset in the center of rotation, as explained earlier. The shear wavelength obtained at 250 Hz was approximately 8 mm, while the wavelength of ultrasound waves used for imaging and the ARFI-push was 0.3 mm at 5 MHz.

3. Results

3.1. Dispersion Measurements

An example of the measured shear wave propagation is visualized in Fig. 1. Figure 1a–c shows three different time points, and one can see the propagation of the shear wave from a moment shortly after it was created (Fig. 1a) to later time points (Fig. 1b, then 1c). Any temporal variation can be Fourier transformed to get its temporal frequency content, and three different such frequencies are shown in Fig. d-f, from a lower frequency in Fig 1d to higher ones in Fig. 1e and 1f. The SWR model-based inversion was performed for all shear wave measurements, at all three locations. An example is shown in Fig. 2 for data from location 1 at 150 Hz and a orientation of 90°. From these data, the estimated complex shear modulus is plotted in Fig. 3 for all measured frequencies, for location 1. Figure 3(a) was limited to similar measurements performed near orientations 0°or 360° and provided an estimate of measurement repeatability. The contact pressure applied in measurement 17 at location 3 also yielded a further repeatability test. The larger standard deviations in Fig. 3(b) compared to Fig. 3(a) may be caused by inhomogeneity and/or anisotropy in the studied bovine liver tissues. The same data as in Fig. 3 were also used to obtain an energy spectrum of the shear waves generated here, shown in Fig. 4, which peaks in the vicinity of 100Hz. In Fig. 5, SWR results were compared to those obtained from our implementation of shear wave spectroscopy (SWS) with a cylindrical spreading assumption and from a group speed estimation. In the regime where SWS can be expected to work well, to evaluate the shear storage modulus at relatively low frequencies, SWS and SWR results were found to be in reasonable agreement. Otherwise, for higher frequencies and/or for evaluating the shear loss modulus, SWS results were affected by large standard deviations, and the smaller standard deviations of SWR results were consistent with advantageous SNR properties for SWR, a main rationale for the present work. In contrast, estimations based on group speed (magenta line in Fig. 5(a)) do not resolve the imaginary component, nor the frequency dependence of the shear modulus.

Figure 2.

Estimation of unknown material parameters was done using the model-based inversion procedure developed in [31]. The real part of the measured wavefield (a) is shown alongside its model-fitted counterpart (b). The fit error shown in (c) had an ‘energy’ below 10% of that of the fitted data. Note that even though the measured wavefield had artifacts in the top-left quadrant (a), the model-based fit suppressed these artifacts (b). The plots (d), (e) and (f) compare the measured wavefield (red) and its fitted counterpart (blue) in terms of shear wave phase (d), shear wave amplitude (e) and shear wave displacement (f), at the depth of the push. Black vertical lines show the demarcation between the ARFI-excitation region ($r<r_0$) and the source-free region, where the model-data fit is performed. These data are from location 1, with 90° transducer orientation, and the shear modulus estimate for this measurement was $\mu =\left(2.8-i1.5\right)\mathrm{kPa}$

Figure 3.

The shear storage and loss modulus estimates obtained at location 1 are plotted as functions of frequency. In (a), the 7 measurements near O°orientation were combined: measurements 1–2 & 14–18 with transducer orientation of 0° and 355–365°. In contrast, all 20 measurements were included in (b). The fact that the standard deviation is noticeably higher in (b) vs. (a) indicates that the estimated shear modulus has some orientation dependence, presumably caused by tissue inhomogeneity and/or anisotropy.

Figure 4.

The energy spectrum of the generated shear wave is shown here in the blue curve. Energy in the wavefields at individual frequencies from 25–250 Hz is quantified and normalized by the sum of wavefield energies in the same frequency range. It is notable that the shear wave energy peaks near 75 Hz and drops rapidly on both sides. The left-right asymmetric energy in the wavefield is shown in the red curve. The plot shows 7 blue and red curves corresponding to 7 measurements at location 1 with transducer orientation set to 0 (or 360)°

Figure 5.

SWR results from the 20 measurements at location 1 are compared with the corresponding results from our implementation of shear wave spectroscopy (SWS). The mean and standard deviation over 20 measurements of the storage modulus are plotted in (a) and of the loss modulus in (b). While SWS and SWR measurements are in rough agreement, the smaller standard deviation associated with SWR results demonstrate the key advantage of the approach, which is to perform better than SWS in low-SNR scenarios [31]. Shear modulus estimates obtained using group speed estimation are shown in magenta with one standard deviation shown by the grey region. Note that group speed estimates only yield the real part of the complex shear modulus (i.e. they contain no information about the the shear loss modulus) and also do not resolve the frequency dependence.

The data from Fig. 3(b), for location 1, is combined in Fig. 6 with that from locations 2 and 3. These show agreement within the error bars of the measurements. The larger standard deviation for data from location 2 (shown in red in Fig. 6) is attributed to the offset that was introduced between the axis of the rotation stage and the transducer long axis, as described earlier, which further diversified the tissue regions being sampled. Note the important dispersion effect captured in Fig. 6, as the shear storage and loss modulus increase by about 4-fold and 10-fold, respectively, over the 25–250 Hz interval.

Figure 6.

The storage and loss modulus estimates at locations 1, 2 and 3 are shown. The mean and standard deviation values were obtained over all measurements at each location. Artificial ±3 Hz horizontal shifts were introduced to avoid data points from locations 1–3 occluding one another. Note that bovine liver exhibits significant dispersion over this 1 decade of frequency from 25–250 Hz: a factor of 4 increase in shear storage modulus and a factor of 10 increase in the loss modulus. We note that while the loss modulus is not linear in frequency, it scales approximately linearly over the range examined.

3.2. Orientation Dependence

Ideally, in homogeneous isotropic tissues, the orientation of the probe should have no impact on the measured shear properties. As such, the presence/absence of anisotropy was assessed here by analyzing results obtained with different probe orientations. Results from measurements 1 through 12, corresponding to rotation angles from 0 to 330°, were Fourier transformed along the ‘rotation angle’ dimension. The resulting onesided rotation-frequency components are shown in Fig. 7 for the shear storage modulus. The components ranging from n = 1 to n = 6 in Fig. 7 represent undesirable orientation-dependent variations that should be absent in an ideally homogeneous isotropic medium. These have been normalized by the n = 0 component which represents the mean value of the modulus over all measured orientations. Of particular interest is the n = 2 cyclic frequency component, which provides a measure of anisotropy, while values in the range n = 3 to n = 6 reflect the variance of the measurements. Similar results were obtained for the shear loss modulus.

Figure 7.

The 12 measurements of the shear storage modulus over the 0–330° orientation range at location-1 are analyzed using a circular Fourier transform. The (n = 1 to 6) components are normalized by the (n = 0) principal component which represents the mean over the measured orientations. The (n = 1 to 6) components are the orientation dependent variations, of which 1 and 2 are of most interest because they may indicate transducer asymmetry, material inhomogeneity, and/or material anisotropy. It is seen that at 150 Hz and higher, where the imaging FOV can accommodate at least one full shear wavelength on each side of the ARFI-push, all these circular frequency components are small, justifying the modeling assumptions of homogeneity and isotropy in SWR.

3.3. Quality Metrics

The model-based inversion procedure leads to two measurement metrics that may serve as a quality-check on the results obtained from SWR. First, the model-measurement mismatch can be quantified using an L2-energy norm of the difference in the measured and model-predicted wavefields (see Fig. 2(c)). This quantity was employed as the cost-function in the minimization procedure leading to fitted wavefields (see Fig. 2(b)) [31]. A low mismatch, typically less than 10%, would mean that the model agrees fairly well with the measurement and thus that the estimated complex shear wave number is a good representation of the shear properties of the medium under investigation.

Second, even before the model-based inversion procedure is tried, the left-right asymmetry in the wavefield can be quantified as an “asymmetric energy norm.” Given that the excitation is left-right symmetric, we expect to measure a symmetric wavefield in a medium that is both homogeneous and isotropic. Asymmetric Energy (or Asymmetry) is defined as the sum-squared magnitudes of the difference between left and right halves of the wavefield (see red line in Fig. 4). This can be normalized by the total ‘energy’ in the wavefield at each frequency (see blue line in Fig. 4), leading to the normalized asymmetry measure shown in magenta in Fig. 8, for 7 measurements performed at location 1. The other quality metric, the “mismatch”, is shown in black in the same Fig. 8, for the same measurements. The asymmetry ratio and the model mismatch both remained under 15% in the 25–250 Hz frequency range employed here, consistent with the frequency range of significant signal in the shear wave pulse.

Figure 8.

The ratio of the red to blue curves in Fig. 4, a normalized measure of asymmetry, is shown here in magenta, for seven measurements made at location 1 at 0° orientation. The measurement-model mismatch is shown in black, for the same measurements. Note that the model mismatch is correlated with the measurement asymmetry.

4. Discussion

The present feasibility study demonstrates that SWR can measure the frequency-dependent complex shear modulus of ex vivo bovine liver tissues. Moreover, these measurements are possible over a wider frequency band and with greater precision than shear wave spectroscopy (SWS). Considering the four major sources of measurement variability, i.e. (1) heterogeneity (2) anisotropy (3) lack of axisymmetry and (4) dispersion, results from Fig. 6 suggest that dispersion is by far the largest. More specifically, standard deviations in Fig. 6 capture variation caused by the first three of these factors and measurement noise, yet the variation with frequency of the shear storage and loss moduli over the 25–250 Hz range was far greater in size than the measured standard deviations. For example, a measurement at 200 Hz at one location is significantly different from a measurement at 50 Hz at the same location, while all measurements at 200 Hz in all locations, and at all orientations, are substantially the same. Furthermore, analysis of the orientation dependence of shear parameters, e.g. Fig. 7, shows that the circular frequency components are small, especially at 150 Hz and higher, where the imaging FOV can accommodate at least one full shear wavelength on each side of the ARFI-push. This is an indication that the modeling assumptions of homogeneity and isotropy of bovine liver tissue were justified here.

The 25–250 Hz frequency range was selected here based on the shear wave energy spectrum shown in Fig. 4, and the two quality metrics, wavefield asymmetry and model mismatch, shown in Fig. 8. We and others have found that the usable frequency range depends not only on the ARFI-SWE parameter settings but also on the properties of the medium [34, 35]. For example, in this study, SWR obtained useful measurements in the frequency range 25–250 Hz in bovine liver, however, in our gelatin measurements, 100–1000 Hz and 200–1800 Hz were found to be suitable for soft and hard gelatin samples, respectively [31], even though the same diagnostic ultrasound transducer (L7–4), and identical ARFI-SWE parameters were used.

By fitting the entire measured wavefield at individual frequencies, SWR resolves the frequency dependence of shear parameters and gains a SNR advantage over other techniques that use a smaller portion of the measured domain. SWR thus allows material parameter estimation over a wider frequency range. Moreover, SWR provides viscoelastic characterization of the shear properties of soft tissue that is independent of any particular rheological material model. At the same time, however, the medium homogeneity and isotropy assumptions rule out its application to anisotropic tissues such as muscle, or to inhomogeneous tissues that contain focal lesions.

The main limitations of this study were the imaging of excised tissues rather than in vivo tissues, and the small number of tissue samples measured here. Further limitations include: (1) The fact that SWR modeling assumptions were validated only under the conditions described in this study, i.e. when fresh liver tissue was degassed while immersed in a saline solution and kept refrigerated at 4°C. (2) Major blood vessels were excluded from the SWE FOV guided by ultrasound B-mode imaging, and (3) shear wave propagation was investigated primarily in the 25–250 Hz frequency range in a shallow depth range of 5–35 mm below the liver outer surface.

5. Conclusion

Shear Wave Rheometry (SWR) enabled frequency-dependent measurements of the complex shear modulus of bovine liver tissues over a decade-wide frequency range. Bovine liver exhibited significant dispersion over 25–250 Hz, a factor of 4 increase in shear storage modulus and over a factor of 10 increase in the loss modulus. Errors caused by anisotropy and/or inhomogeneity were smaller and on the same order as the measurement variance. Moreover, of all the factors considered (lack of axisymmetry, anisotropy, inhomogeneity, dispersion, and measurement noise), dispersion was by far the biggest source of measurement variability. The variation found in the storage modulus suggests that simple rheological models like Kelvin-Voigt may not apply to liver tissues over a decade-wide frequency range.

Two quality metrics were proposed in this study, i.e. wavefield asymmetry and model-to-measurement energy mismatch, to check the quality of the measurements obtained and the validity of the modeling assumptions in given tissue samples. Since the model used in SWR does not require detailed apriori knowledge of the source distribution, it is hoped to prove suitable for clinical applications, where patient-to-patient variability can lead to differences in the temporal and spatial characteristics of the generated shear waves. It is hoped that the resulting frequency-dependent characterization of the shear properties will enable quantitative comparisons between disparate elastography techniques and that the nature of this frequency dependence will have diagnostic value.