Shear Waves In Soft Tissues Research Articles

Nonlinear least-squares estimation of shear modulus and shear viscosity in viscoelastic media

Time-domain estimates of the shear wave speed are obtained by performing cross-correlations or by evaluating the time-to-peak for two laterally separated shear wave particle velocity waveforms, where the distance divided by the propagation time yields an approximate value for the shear wave speed. Challenges associated with these time-domain estimation approaches include increasing errors as the shear viscosity increases and the absence of an estimated value for the shear viscosity parameter which is responsible for the rapid attenuation of shear waves in soft tissue. To obtain a time-domain estimate of the shear viscosity that also provides an estimate for the shear wave speed, a nonlinear least squares estimation approach is applied to three-dimensional (3D) shear wave particle velocities computed for a shear elasticity of 1.5 kPa, shear viscosities of 1, 2, 3, and 4 Pa.s, and a realistic simulated 3D acoustic radiation force excitation. The results show cross-correlations tend to over-estimate the value of the shear wave speed, the nonlinear least squares approach tends to under-estimate the value of the shear wave speed, and the nonlinear least square approach produces more accurate estimates as the shear viscosity increases. Two-dimensional maps of each estimated parameter are also shown.

Limitations of the Caputo time-fractional wave equation when applied to shear waves in pig liver

When applied to compressional waves, the Caputo time-fractional wave equation describes power law attenuation in soft tissue. However, when applied to shear waves, the Caputo wave equation produces some unexpected behavior. To demonstrate examples of the time-domain waveforms generated by this fractional calculus model, the Caputo wave equation is numerically evaluated for multiple orders of the time-fractional derivative between 0 and 1, for an extended range of values for the relaxation time, and for a fixed value of the shear wave speed extracted from ex vivo pig liver. The computed waveforms are then compared to the measured shear wave particle velocities obtained from ex vivo pig liver. These comparisons reveal that the full-width at half maximum (FWHM) of the positive component of the main shear wave particle velocity waveform measured in pig liver is at least an order of magnitude greater than the FWHM obtained from the Caputo wave equation for all parameter combinations evaluated. These preliminary results indicate that the Caputo wave equation produces time-domain waveforms that are inconsistent with measured shear wave data in pig liver and that further efforts are required to establish more effective fractional calculus models for shear waves in soft tissue.

Surface excitation of focused shear wave beams in soft elastic media: Theory

Shear wave propagation is employed in medical ultrasound imaging, because it reveals variation in the viscoelastic properties of tissue. Frequencies below 1 kHz are required for imaging with shear waves in soft tissue due to their high attenuation and low propagation speeds, compared to compressional waves with frequencies above 1 MHz used for ultrasound imaging. Shear waves exhibiting particle motion in the direction of propagation, referred to as longitudinally polarized shear waves, can be generated by applying longitudinal motion of a circular disk to the surface of a soft elastic medium. This approach is used in practice because it permits imaging of the longitudinal shear wave with a conventional ultrasound transducer that is coaxial with the source of the shear wave. Presented here are the theoretical framework and numerical simulations that illustrate effects of focusing on longitudinally polarized shear waves. Longitudinal, transverse, radial, and torsional source polarizations are considered. The present investigation was motivated initially by an experimental study in optics due to Dorn et al. [Phys. Rev. Lett. 91, 233901 (2003)]. Our predictions for shear wave beams support their measurements of light beams revealing that the longitudinal electric field component produces a smaller focal spot than the transverse field component.

Finding consensus among rheological models for shear waves in soft tissues

Shear wave viscoelastic properties of tissue are now measured by a wide variety of methods including elastography, imaging scanners, rheological shear viscometers, and a variety of calibrated stress-strain analyzers. Because absorption and sound speed can be strong functions of frequency, fitting the data to an viscoelastic model which best describes observed behavior is a common step in understanding and comparing data sets among tissues and disease states. Because the same data can be represented by different model constants, there is a need to reach consistency and consensus on the most effective models among different sub-fields in acoustics, biomechanics, and elastography. To this end, we examined many established rheological models as well as data sets. We argue that the long history of biomechanics, including the concept of the extended relaxation spectrum, and the theoretical framework of multiple relaxation models which model the multi-scale nature of physical tissues, consistent with power law data extending over several decades of frequency and time, all lead to the conclusion that fractional derivative models represent the most succinct and meaningful models of soft tissue viscoelastic behavior.

Group versus Phase Velocity of Shear Waves in Soft Tissues.

Across the varieties of waves that have been studied in physics, it is well established that group velocities can be significantly greater than or less than phase velocities measured within comparable frequency bands, depending on the particular mechanisms involved. The distinction between group and phase velocities is important in elastography, because diagnoses are made based on shear wave speed estimations from a variety of techniques. We review the general definitions of group and phase velocity and examine their specific relations within an important general class of rheological models. For the class of tissues and materials exhibiting power law dispersion, group velocity is significantly greater than phase velocity, and simple expressions are shown to interrelate the commonly measured parameters. Examples are given from phantoms and tissues.

Passive elastography in soft-tissues: Phase velocity measurement

Elastography is an imaging technique used on medical ultrasound devices. It consists in measuring shear waves in soft tissues in order to give a tomography reconstruction of the shear elasticity. A method of measurement, usually referred as passive elastography, is to use noise correlation techniques on diffuse shear wave fields present in the medium. In human body, these fields can be naturally created by activities like heart beating, arteries pulsatility. Passive elastography allows to locally estimate the group velocity in the medium and this method has already been used for some organs as the liver, the thyroid or the brain. The present study is therefore devoted to improve this method with the calculation of the phase velocity. For example, this information, obtained for each frequency, could allow to measure the attenuation with Kramers-Kronig relations.

Contactless remote induction of shear waves in soft tissues using a transcranial magnetic stimulation device

This study presents the first observation of shear waves induced remotely within soft tissues. It was performed through the combination of a transcranial magnetic stimulation device and a permanent magnet. A physical model based on Maxwell and Navier equations was developed. Experiments were performed on a cryogel phantom and a chicken breast sample. Using an ultrafast ultrasound scanner, shear waves of respective amplitudes of 5 and 0.5 μm were observed. Experimental and numerical results were in good agreement. This study constitutes the framework of an alternative shear wave elastography method.

Development of a phantom mimicking the functional and structural behaviors of the thigh muscles characterized with magnetic resonance elastography technique.

Magnetic Resonance Elastography (MRE) is a non invasive technique based on the propagation of shear waves in soft tissues providing the quantification of the mechanical properties [1]. MRE was successfully applied to healthy and pathological muscles. However, the MRE muscle methods must be further improved to characterize the deep muscles. A way will be to develop phantom mimicking the muscle behavior in order to set up new MRE protocol. Thus, the purpose of this study is to create a phantom composed of a similar skeletal muscle architecture (fiber, aponorosis) and equivalent elastic properties as a function of the muscle state (passive or active). Two homogeneous phantoms were manufactured with different concentrations of plastisol to simulate the elastic properties in relaxed (50% of plastisol) and contracted (70% of plastisol) muscle conditions. Moreover, teflon tubing pipes (D = 0.9 mm) were thread in the upper part of the phantom (50%) to represent the muscle fibers and a plastic sheet (8 × 15 cm) was also included in the middle of the phantom to mimic the aponeurosis structure. Subsequently, MRE tests were performed with two different pneumatic drivers, tube and round, (f = 90Hz) to analyze the effect of the type of driver on the wave propagation. Then, the wavelength was measured from the phase images to obtain the elastic properties (shear modulus). Both phantoms revealed elastic properties which were in the same range as in vivo muscle in passive (μ(50%) = 2.40 ± 0.18 kPa ) and active (6.24 ± 0.21 kPa) states. The impact of the type of driver showed higher values (about 1.2kPa) with the tube. The analysis of the wave behavior revealed a sliding along the plastic sheet as it was observed for in vivo muscle study. The wave was also sensitive to the presence of the fibers where gaps were identified. The present study demonstrates the ability of the phantom to mimic the structural and functional properties of the muscle.

Measuring viscoelasticity of soft tissues using shear and guided waves

Elasticity imaging has emerged as a viable clinical tool for assisting in diagnosis of certain diseases such as liver fibrosis and cancer. Many methods have been developed for generating and measuring shear waves in soft tissues. The advantage of using shear waves is that the shear wave velocity is proportional to the mechanical material properties of the tissues under investigation. Soft tissues are inherently viscoelastic and considerable effort has been made to quantify the dispersion, or variation of frequency, of the velocity and attenuation of shear waves. This is accomplished by measuring the shear wave motion and using Fourier-based techniques to extract the shear wave velocity and attenuation. Additional considerations are made in tissues where geometric dispersion is also present such as in the heart. Viscoelastic characterization of the shear viscoelasticity of soft tissues in vivo such as human liver, human kidney, and swine heart will be shown. We will also demonstrate parameterization of the results by using a model-free approach or by fitting the shear wave velocity dispersion to rheological models. The diagnostic value of the viscoelastic parameters will be discussed for each particular application. [This work was supported in part by NIH Grant Nos. DK092255, DK082408, and EB002167.]

Optimized Shear Wave Generation Using Hybrid Beamforming Methods

Elasticity imaging is a medical imaging modality that measures tissue elasticity as an aid in the diagnosis of certain diseases. Shear wave-based methods have been developed to perform elasticity measurements in soft tissue. These methods often use the radiation force mechanism of focused ultrasound to induce shear waves in soft tissue such as liver, kidney, breast, thyroid and skeletal muscle. The efficiency of the ultrasound beam in producing broadband extended shear waves in soft tissue is very important to the widespread use of this modality. Hybrid beamforming combines two types of focusing, conventional spherical focusing and axicon focusing, to produce a beam for generating a shear wave that has increased depth-of–field (DOF) so that measurements can be made with a shear wave with a consistent wave front. Spherical focusing is used in many applications to achieve high lateral resolution, but has low DOF. Axicon focusing, with a cone-shaped transducer, can provide good lateral resolution with large DOF. We describe our linear aperture design and beam optimization performed using angular spectrum simulations. We performed a large parametric simulation study in which we varied the focal depth for the spherical focusing portion of the aperture, the numbers of elements devoted to the spherical and axicon focusing portions of the aperture and the opening angle used for axicon focusing. The hybrid beamforming method was experimentally tested in two phantoms, and shear wave speed measurement accuracy and DOF for each hybrid beam were evaluated. We compared our results with those for shear waves generated using only spherical focusing. The results of this study indicate that hybrid beamforming is capable of producing a beam with increased DOF over which accurate shear wave speed measurements can be made for different-size apertures and at different focal depths.

Shear Wave Dispersion Ultrasound Vibrometry Based on a Different Mechanical Model for Soft Tissue Characterization

Ultrasound vibrometry can measure the propagation velocity of shear waves in soft tissue noninvasively, and the shear moduli of tissue can be estimated inversely from the velocities at multiple frequencies. It is possible to choose the appropriate model for tissue characterization from mathematical methods and analysis of model behaviors. The three classic models, Voigt, Maxwell, and Zener, were applied to fit the velocity measurements and estimate shear moduli of porcine livers with different thermal damage levels and different storage times. The Zener model always provided the best estimation of the moduli with the minimum errors in our experiments. Unlike the Voigt and Maxwell models, the moduli of the Zener model cannot be used to indicate damage levels in porcine livers directly, but the creep and relaxation behaviors of the Zener model are effective.

Improving the Robustness of Time-of-Flight Based Shear Wave Speed Reconstruction Methods Using RANSAC in Human Liver in vivo

The stiffness of tissue can be quantified by measuring the shear wave speed (SWS) within the medium. Ultrasound is a real-time imaging modality capable of tracking the propagation of shear waves in soft tissue. Time-of-flight (TOF) methods have previously been shown to be effective for quantifying SWS from ultrasonically tracked displacements. However, the application of these methods to in vivo data is challenging due to the presence of additional sources of error, such as physiologic motion or spatial inhomogeneities in tissue. This article introduces the use of random sample consensus (RANSAC), a model fitting paradigm robust to the presence of gross outlier data, for estimating the SWS from ultrasonically tracked tissue displacements in vivo. SWS reconstruction is posed as a parameter estimation problem and the RANSAC solution to this problem is described. Simulations using synthetic TOF data show that RANSAC is capable of good stiffness reconstruction accuracy (mean error 0.5 kPa) and precision (standard deviation 0.6 kPa) over a range of shear stiffness (0.6–10 kPa) and proportion of inlier data (50%–95%). As with all TOF SWS estimation methods, the accuracy and precision of the RANSAC reconstructed shear modulus decreases with increasing tissue stiffness. The RANSAC SWS estimator was applied to radiation force induced shear wave data from 123 human patient livers acquired with a modified SONOLINE Antares ultrasound system (Siemens Healthcare, Ultrasound Business Unit, Mountain View, CA, USA) in a clinical setting before liver biopsy was performed. Stiffness measurements were not possible in 19 patients due to the absence of shear wave propagation inside the liver. The mean liver stiffness for the remaining 104 patients ranged from 1.3 to 24.2 kPa and the proportion of inliers for the successful reconstructions ranged between 42% to 99%. Using RANSAC for SWS estimation improved the diagnostic accuracy of liver stiffness for delineating fibrosis stage compared with ordinary least squares (OLS) without outlier removal (AUROC = 0.94 for F ≥ 3 and AUROC = 0.98 for F = 4). These results show that RANSAC is a suitable method for estimating the SWS from noisy in vivo shear wave displacements tracked by ultrasound. (E-mail: michael.h.wang@duke.edu)

Shear wave elastography of cetacean tissues.

An ultrasound‐based elastography system is being developed to noninvasively measure the complex shear speed in cetacean head tissues, including brain, jaw fat, and melon. The system instrumentation features a pair of dual‐element confocal ultrasound transducers, one of which is used to remotely generate low frequency shear waves in soft tissues via radiation force, while the other is used as a vibrometer to measure the resulting shear wave displacements. Results are presented for in situ measurements on formalin fixed tissues from a bottlenose dolphin (tursiops truncatus). [Work supported by ONR.]

Shear Wave Spectroscopy for In Vivo Quantification of Human Soft Tissues Visco-Elasticity

In vivo assessment of dispersion affecting the propagation of visco-elastic waves in soft tissues is key to understand the rheology of human tissues. In this paper, the ability of the supersonic shear imaging (SSI) technique to generate planar shear waves propagating in tissues is fully exploited. First, by strongly limiting shear wave diffraction in the imaging plane, this imaging technique enables to discriminate between the usually concomitant influences of both medium rheological properties and diffraction affecting the shear wave dispersion. Second, transient propagation of these plane shear waves in soft tissues can be measured using echographic images acquired at very high frame. In vitro and in vivo experiments demonstrate that dispersion curves, which characterize the rheological behavior of tissues by measuring the frequency dependence of shear wave speed and attenuation, can be recovered in the 75-600 Hz frequency range. Based on a phase difference algorithm, the dispersion curves are computed in 1 cm2 regions of interest from the acquired propagation movie. In vivo measurements in Biceps Brachii muscle and liver of three healthy volunteers show important differences in the rheological behavior of these different tissues. Liver tissue appears to be much more dispersive with a phase velocity ranging from approximately 1.5 m/s at 75 Hz to approximately 3 m/s at 500 Hz whereas muscle tissue shows an important anisotropy, shear waves propagating longitudinally to the muscular fibers are almost nondispersive while those propagating transversally are very dispersive with a shear wave speed ranging from 0.5 to 2 m/s between 75 and 500 Hz. The estimation of dispersion curves is local and can be performed separately in different regions of the organ. This signal processing approach based on the SSI modality introduces the new concept of in vivo shear wave spectroscopy (SWS) that could become an additional tool for tissue characterization. This paper demonstrates the in vivo ability of this SWS to quantify both local shear elasticity and dispersion in real time.

Dual confocal ultrasound system for shear wave elastography

A dual confocal transducer system for ultrasound‐based elastography is presented. The system is intended to noninvasively measure the complex shear speed in cetacean head tissues, including brain, jaw fat, and melon. The system instrumentation features a pair of dual‐element confocal ultrasound transducers, one of which is used to remotely generate low frequency (100‐1000 Hz) shear waves in soft tissues via radiation force, and the other is used to measure the resulting shear wave displacements using Doppler techniques. One transducer is configured as an open ring into which the other transducer can be placed and translated. The relative positions of the transducers are mechanically manipulated in order to measure short‐path propagation and estimate shear wave speed and loss. Work supported by ONR.

Measurement of elastic nonlinearity of soft solid with transient elastography.

Transient elastography is a powerful tool to measure the speed of low-frequency shear waves in soft tissues and thus to determine the second-order elastic modulus mu (or the Young's modulus E). In this paper, it is shown how transient elastography can also achieve the measurement of the nonlinear third-order elastic moduli of an Agar-gelatin-based phantom. This method requires speed measurements of polarized elastic waves measured in a statically stressed isotropic medium. A static uniaxial stress induces a hexagonal anisotropy (transverse isotropy) in solids. In the special case of uniaxially stressed isotropic media, the anisotropy is not caused by linear elastic coefficients but by the third-order nonlinear elastic constants, and the medium recovers its isotropic properties as soon as the uniaxial stress disappears. It has already been shown how transient elastography can measure the elastic (second-order) moduli in a media with transverse isotropy such as muscles. Consequently this method, based on the measurement of the speed variations of a low-frequency (50-Hz) polarized shear strain waves as a function of the applied stress, allows one to measure the Landau moduli A, B, C that completely describe the third-order nonlinearity. The several orders of magnitude found among these three constants can be justified from the theoretical expression of the internal energy.