Shear Wave Beams Research Articles

Predicting second-harmonic generation in shear wave beams in tissue-mimicking phantoms

Planar nonlinear shear waves in isotropic media are subject to only cubic nonlinear effects and generate only odd harmonics during propagation. However, wavefront curvature in shear wave beams breaks the symmetry and yields quadratic nonlinear effects, and therefore, a second harmonic may be present in shear wave beams depending on the polarization of the wave. Focused radially polarized shear wave beams have been generated in tissue-mimicking phantoms [Cormack et al., IEEE TBME (2024)], and it is postulated that the second harmonic may be used to estimate the quadratic nonlinearity in such a medium as an additional biomarker for diseased tissue. Here, an analytical solution obtained in the paraxial approximation for nonlinear propagation of a shear wave beam in an absorbing medium [Spratt et al., AIP Conf. Proc. 1685, 080007 (2015)] is used to characterize the strength of second-harmonic generation. Feasibility of measuring the second harmonic experimentally in a weakly nonlinear, radially polarized focused shear wave beam propagating in a tissue-like medium is explored. Second-harmonic generation in focused shear wave beams with other polarizations is discussed. [P.G.K. was supported by the ARL:UT Chester M. McKinney Graduate Fellowship in Acoustics.]

Focused shear wave beam propagation through a 3D printed human rib cage

Transient elastography (TE) is used in clinical screenings for liver fibrosis. TE generates shear wave motion in the liver by vibration of a piston at the skin. The small, flat piston in current devices radiates shear waves that diverge from the liver, resulting in low SNR and high failure rates, especially in obese patients. Our group recently introduced focused shear wave beams for TE [Cormack et al., IEEE TBME (2024)], in which vibration of a concave piston generates shear waves that converge towards the focal region, thereby increasing SNR for liver stiffness estimation. However, because piston sizes needed for efficient shear wave focusing are larger than the typical intercostal space, the rib cage that lies between the skin and the liver may cause shear wave aberration and influence stiffness measurement. Here, we present measurements of broadband focused shear wave propagation in tissue-mimicking gel in which are embedded 3D printed human ribs. Both straight elliptical rods and anatomically realistic 3D printed ribs are used as aberrators. Measurements are compared to 3D simulations of shear wave beam propagation [Archer et al., JASA (2023)]. Effects of shear wave aberration by the rib cage depend on shear wavelength, piston-to-rib distance, and piston radius and curvature.

Reconstructing the source condition for focused shear wave beams in soft elastic media

Shear waves are employed in medical ultrasound imaging because they reveal variations in viscoelastic properties of soft tissue. Focused shear waves produced by a longitudinally vibrating piston at several hundred hertz are investigated here experimentally in soft tissue phantoms and using an analytical model for shear wave beam generation and propagation. Experiments employed a spherically concave piston shaped to focus the shear wave beam at a depth of 4 cm in a soft tissue phantom. Analytical modeling of this data provided a good fit, but required treating the focal length and piston diameter as adjustable parameters. The largest source of error in the modeling is suspected to be that the source condition is assumed to exhibit zero displacement surrounding the piston, whereas the actual condition in the source plane is a traction-free surface surrounding the piston. Here, in order to elucidate the actual source condition, numerical back propagation of field measurements is employed. Simulations show that the shear source condition can be accurately recovered with information related only to the shear wave, i.e., without compressional wave information. Back propagation of measured beams is discussed in the context of more accurately modeling the source condition for future device optimization.

Focused Shear Wave Beam Propagation in Tissue-Mimicking Phantoms.

Ultrasound transient elastography (TE) technologies for liver stiffness measurement (LSM) utilize vibration of small, flat pistons, which generate shear waves that lack directivity. The most common cause for LSM failure in practice is insufficient shear wave signal at the needed depths. We propose to increase shear wave amplitude by focusing the waves into a directional beam. Here, we demonstrate the generation and propagation of focused shear wave beams (fSWBs) in gelatin. Directional fSWBs are generated by vibration at 200-400 Hz of a concave piston embedded near the surface of gelatin phantoms and measured with high-frame-rate ultrasound imaging. Five phantoms with a range of stiffnesses are employed. Shear wave speeds assessed by fSWBs are compared with those by radiation-force-based methods (2D SWE). fSWB amplitudes are compared to predictions using an analytical model. fSWB-derived shear wave speeds are in good agreement with 2D SWE. The amplitudes of fSWBs are localized to the LSM region and are significantly greater than unfocused shear waves. Overall agreement with theory is observed, with some discrepancies in the theoretical source condition. Focusing shear waves can increase the signal in the LSM region for TE. Challenges for translation include coupling piston vibration with the patient skin and increased attenuation in vivo compared to the phantoms employed here. Fibrosis is the most predictive measure of patient outcome in non-alcoholic fatty liver disease. Increased shear wave amplitude in the LSM region can reduce fibrosis assessment failure rates by TE, thus reducing the need for invasive methods like biopsy.

Experimental study on the nonlinear mixing of ultrasonic waves in concrete using an array technique

This research develops a procedure that combines ultrasonic array technology with non-collinear ultrasonic wave mixing to detect and scan internal microscale damage in concrete. Exploiting the underlying mechanics of nonlinear wave mixing, two incident shear wave beams generated by two ultrasonic transducer arrays (and with frequencies low enough to propagate without being overly scattered by the aggregate) are used to create a longitudinal mixed wave, the amplitude of which is a direct measure for the amount of localized, internal damage. The arrays enable beam steering, making it possible to scan for damage along an arc. The effectiveness of the proposed technique is demonstrated by characterizing different types of microscale damage embedded at known locations in a concrete prism specimen. These results establish the feasibility of using a nonlinear wave mixing array technique for the efficient inspection of internal microscale damage in a large concrete structure.

Evaluation of the Fresnel approximation for the longitudinal particle displacement in shear wave beams

The longitudinal particle displacement in shear wave beams generated with disks that vibrate longitudinally at several hundred hertz is used to assess stiffness of soft tissue in vivo. Nonlinear elasticity of tissue, although less commonly used than linear stiffness, may be a more sensitive or specific biomarker of disease. A mathematical model for linear propagation of both transverse and longitudinal motion in a shear wave beam in an isotropic elastic half-space, and comparisons with measurements in tissue-mimicking phantoms, was published earlier this year [Archer et al., JASA 153, 1591 (2023)]. When including elastic nonlinearity in a propagation model, the complexity is reduced substantially by deriving an evolution equation based on the Fresnel approximation [Zabolotskaya, Sov. Phys. Acoust. 32, 296 (1986)]. For application to elastography, the parameter space in which the Fresnel approximation provides an accurate description of the longitudinal particle displacement must be established. In this presentation, the accuracy of the Fresnel approximation is assessed for linear shear wave beams, both unfocused and focused, by comparison with the existing mathematical model. Attention is devoted primarily to the longitudinal particle displacement in nearly incompressible media. [PGK is supported by the ARL:UT Chester M. McKinney Graduate Fellowship in Acoustics.]

Nonlinear propagation of quasiplanar shear wave beams in soft elastic media with transverse isotropy.

Model equations are developed for shear wave propagation in a soft elastic material that include effects of nonlinearity, diffraction, and transverse isotropy. A theory for plane wave propagation by Cormack [J. Acoust. Soc. Am. 150, 2566 (2021)] is extended to include leading order effects of wavefront curvature by assuming that the motion is quasiplanar, which is consistent with other paraxial model equations in nonlinear acoustics. The material is modeled using a general expansion of the strain energy density to fourth order in strain that comprises thirteen terms defining the elastic moduli. Equations of motion for the transverse displacement components are obtained using Hamilton's principle. The coupled equations of motion describe diffraction, anisotropy of the wave speeds, quadratic and cubic plane wave nonlinearity, and quadratic nonlinearity associated with wavefront curvature. Two illustrative special cases are investigated. Spatially varying shear vertical wave motion in the fiber direction excites a quadratic nonlinear interaction unique to transversely isotropic soft solids that results in axial second harmonic motion with longitudinal polarization. Shear horizontal wave motion in the fiber plane reveals effects of anisotropy on third harmonic generation, such as beam steering and dependence of harmonic generation efficiency on the propagation and fiber directions.

Transient elastography with focused shear wave beams

Transient elastography (TE) is a clinically available ultrasound elastography technique in which mechanical vibration of a small piston at the tissue surface generates a short shear wave pulse that propagates into the body. The piston doubles as an ultrasonic transducer; thus, the shear wave propagation is measured with pulse-echo ultrasound along the beam axis, yielding the wave speed and tissue stiffness. The long shear wavelength compared to piston diameter used in current devices results in low shear wave signal that decays rapidly on axis as the shear wave spreads in all directions, thus limiting the signal-to-noise ratio and imaging depth in practice. We present an overview of our new technology of focused shear wave beams, which aims to overcome these limitations by concentrating shear wave energy in TE near to the beam axis with vibration of a concave circular piston. Analytical modeling of focused shear wave generation and propagation is validated against measurements in gelatin phantoms. Results indicate that focused shear wave beams have relatively larger amplitudes and penetration depths compared to unfocused shear wave beams. Ongoing modeling and experimental efforts towards the measurement of elastic anisotropy, such as of skeletal muscle, using focused shear wave beams will be discussed.

Longitudinal motion of focused shear wave beams in soft elastic media.

Shear waves are employed in medical ultrasound imaging because they reveal variations in viscoelastic properties of soft tissue. Frequencies below 1 kHz are required due to the substantially higher attenuation and lower propagation speeds than for compressional waves. Shear waves exhibiting particle motion in the direction of propagation, referred to as longitudinal shear waves, can be generated with longitudinal motion of a circular disk on the surface of a soft elastic medium. This approach permits imaging of the longitudinal shear wave with a conventional ultrasound transducer that is coaxial with the source of the shear wave. Presented here is a mathematical model describing the complete wave field generated by displacement at the surface of an isotropic elastic half-space. Numerical simulations are shown for longitudinal, transverse, torsional, and radial source polarizations, with emphasis on focused longitudinal shear waves. Predictions are consistent with measurements of light beams revealing that the longitudinal electric field component produces a smaller focal spot than the transverse field component [Dorn, Quabis, and Leuchs, Phys. Rev. Lett. 91, 233901 (2003)]. Simulations are compared with preliminary measurements of a focused longitudinal shear wave beam generated in a soft tissue phantom by longitudinal motion of a spherically concave piston.

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

Shear waves are employed in medical imaging to reveal variations in the viscoelastic properties of soft tissues, which are useful biomarkers for pathologies such as breast lesions and liver disease. Shear wave excitation methods that employ acoustic radiation force or surface vibration with a small piston have limitations associated with imaging depth and shear wave amplitude. We introduce a new method for surface excitation of shear waves that employs longitudinal motion of a concave-shaped piston source to generate a focused shear wave beam, thereby increasing shear wave amplitude and penetration depth. Focused shear waves are excited in a gel phantom in the frequency range of 200 Hz to 400 Hz using a concave piston with 40 mm radius of curvature and 50 mm diameter. The wave field comprises both transverse and longitudinal displacement components, which are polarized perpendicular and parallel to the propagation direction, respectively. Transversely and longitudinally polarized wave fields are measured using ultrasound speckle tracking. The longitudinally polarized shear wave is of interest because it is measured in elastography applications. Preliminary comparisons exhibit good agreement between measured beam patterns and those predicted by an analytical theory for shear wave beam propagation due to surface excitation with a piston source.

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.

Subwavelength twin ultrasound focused (STUF) beam generated by shear-to-longitudinal mode conversion in a triangular prism

Focusing ultrasound fields in regions below the incident wavelength using elementary objects (such as spherical and cylindrical) as acoustic lenses have been successfully demonstrated over the last five years. This unique way to tightly concentrate the energy of acoustic fields is interesting for the development of new high resolution ultrasound systems. Usually, an incident longitudinal beam interacts with these lenses to produce a subwavelength beams at its shadow region. However, no shear-wave beam has been reported to produce subwavelength beams using objects as acoustic lenses. In this work, we numerically and experimentally report the generation of a subwavelength twin ultrasound focusing (STUF) beam using a 1 MHz shear transducer coupled to a Rexolite triangular prism. Numerical simulations were performed in order to study the generated field as a function of the apical angle of the prism and to find the mechanical configuration for the experimental validation. The results show that by changing the apical angle of the triangular prism, the main features of the STUF beams can be changed. To validate the numerical model, a prism with apical angle of 90o was built and the acoustic field distribution was measured by a needle hydrophone. A STUF beam with width of 0.8λ and depth of focus (DOF) 3λ was generated. A good agreement between numerical and experimental results was reported. The proposed system open new possibilities to design and built a simple and low cost acoustic system for microparticle trapping applications.

Harmonic generation in a shear wave beam with spatially varying polarization in transversely isotropic soft tissue

Shear wave beams with spatially varying polarization are characterized here by stretching the medium in the plane perpendicular to the propagation direction. Such a beam is produced in a soft tissue by applying motion perpendicular to the source plane, as with a piston source, due to the quasi-incompressible nature of the medium. In the transversely isotropic (TI) soft tissue, such as muscle, this polarization causes extension and contraction in the fiber direction, resulting in nonlinear propagation effects that are unique to the TI tissue compared to propagation in an isotropic tissue [Cormack, J. Acoust. Soc. Am. 150, 2566 (2021)]. A KZ-type equation is presented that describes a special case of propagation of a shear wave beam with spatially varying polarization in the TI soft tissue, accounting for material anisotropy along with leading-order effects of nonlinearity and diffraction. Solutions for harmonic generation from a time-harmonic source are presented for focused and unfocused Gaussian-derivative distributions of transverse displacement in the source plane. Of practical interest is the longitudinal displacement in the shear wave field, which possesses a second-harmonic component on the axis that results from stretching in the fiber direction. This second harmonic, which is absent in isotropic tissues, may be measured using existing elastography techniques.

Second-harmonic generation in shear wave beams with different polarizations

A parabolic equation describing the propagation of collimated shear wave beams in isotropic elastic solids was derived by Zabolotskaya [Sov. Phys. Acoust. 32, 296–299 (1986)], and was seen to contain both cubic and quadratic nonlinear terms at leading order. While second-order nonlinear effects vanish for the quasi-planar case of linearly-polarized shear wave beams, the importance of quadratic nonlinearity for more complicated polarizations is not yet well understood. The current work investigates the significance of quadratic nonlinearity by considering second-harmonic generation in shear wave beams generated by a certain class of source polarizations that includes such cases as radial and torsional polarization, among others. Corresponding to such beams with Gaussian amplitude shading, analytic solutions are derived for the propagated beam at the source frequency and the second harmonic. Diffraction characteristics are discussed, and special attention is paid to the relationship between the source polarization of the beam and the polarization of the subsequently generated second harmonic. Finally, suggestions are made for possible experiments that could be performed in tissue phantoms, exploiting the theoretical results of this work. [Work supported by the ARL:UT McKinney Fellowship in Acoustics.]

Focused, radially polarized shear wave beams in tissue-like media

In the past decade there has been a surge in the optics literature regarding the unique characteristics of focused, radially-polarized light beams. Of particular interest is the existence of a longitudinal component to the electric field in the focal region of the beam, of comparable amplitude to the radial component and yet with a smaller beamwidth [cf. Q. Zhan, Adv. Opt. Photon. 1, 1-57 (2009)]. In the linear approximation there exists a direct analogy between these light beams and radially-polarized shear wave beams in incompressible elastic media, and hence we may interpret the results found in the optics literature as applying to low-frequency shear waves propagating through tissue-like media. When considering nonlinear effects, however, the fact that the gradient of the field experiences a larger gain due to focusing than the field itself implies that the shear wave case is more susceptible to nonlinear behavior than its optical analog. Second-harmonic generation in the focal region of a focused, radially-polarized shear wave beam in a soft solid is investigated. [Work supported by the ARL:UT McKinney Fellowship in Acoustics and by NIH DK070618.]

Propagation of radially polarized shear wave beams with cubic and quadratic nonlinearities.

This presentation describes an extension of the work by Wochner et al. [J. Acoust. Soc. Am. 123, 2488–2495 (2008)], wherein a coupled pair of nonlinear parabolic equations was derived for the two components of the particle motion perpendicular to the axis of a shear wave beam in an isotropic hyperelastic medium. Although the equations derived in that work contain both cubic and quadratic nonlinearities, the latter were discarded based on the fact that they are not present in plane shear waves, and hence assumed negligible within the quasiplane paraxial region of linearly polarized shear wave beams. However, the experimental observation of prominent second harmonic generation by nonplanar shear waves in soft solids [Jacob et al., J. Acoust. Soc. Am. 122, 1917–1926 (2007)] has motivated a more thorough understanding of these quadratic nonlinear terms. In the present work, the quadratic nonlinearity is explored, and, in particular, it is found that the form of this nonlinearity simplifies for axisymmetric radially polarized beams. Numerical results are presented comparing the effects of the cubic and quadratic nonlinearities on the generation of harmonics along the beam axis and the directivity patterns of those harmonics. [Work supported by the ARL:UT McKinney Fellowship in Acoustics.]

Variations of the nonlinear equation for diffracting beams in fluids to study different modes of propagation in elastic media.

Four variations of the equation for nonlinear acoustic beam propagation derived originally for a fluid will be reviewed. These variations describe finite amplitude beam propagation in isotropic elastic solids and in crystals, influence of diffraction on nonlinear Rayleigh waves, and nonlinear shear wave beams. The distinguishing features of each case will be discussed. The equation for nonlinear longitudinal wave beams in isotropic solids is based on nonlinear theory of elasticity and is close to the original equation for fluids. The equation for sound beams in crystals takes into account that the direction of energy propagation does not coincide with the direction normal to wavefronts. Nevertheless, the general form of the equation is similar to that of the original equation for fluids. Distinguishing features of Rayleigh waves are that they are two-dimensional and their nonlinearity is nonlocal. Because of the nonlocal nonlinearity, the evolution equation for this case was derived in the frequency domain using Hamiltonian formalism. The equation for nonlinear shear wave beams was derived in the cubic approximation in terms of particle displacement and it accounts for any polarization. Solutions of these evolution equations will be presented to illustrate phenomena specific to each case.

Cubic nonlinearity in shear wave beams with different polarizations

A coupled pair of nonlinear parabolic equations is derived for the two components of the particle motion perpendicular to the axis of a shear wave beam in an isotropic elastic medium. The equations account for both quadratic and cubic nonlinearity. The present paper investigates, analytically and numerically, effects of cubic nonlinearity in shear wave beams for several polarizations: linear, elliptical, circular, and azimuthal. Comparisons are made with effects of quadratic nonlinearity in compressional wave beams.

Time Reversed Elastic Waves within Soft Solids

When a scalar far-field wave is time reversed from the surface of a cavity, it starts to converge toward its initial point source location, then collapses and finally diverges. Without near field evanescent waves, the symmetric focus spot is limited by the Rayleigh criterion. We present an experimental observation of a time-reversal vectorial wave in the volume of a soft solid. The elastic field is measured using the transient elastography technique. It is observed that the time reversed far field wave collapses and gives birth to near field effects. Numerical computation based on elastodynamic Green's functions in a time-reversal cavity confirms and completes the experimental conclusions: the time symmetry with respect to collapse time is broken and the Rayleigh criterion is direction dependant; the spatial collapse is larger in the direction of the point body force than in the perpendicular direction. The authors anticipate that the method can be used for shear wave beam forming in soft tissues as well as to tissue elasticity assessment.

Relationship Between Closure Stress of Small Fatigue Crack and Ultrasonic Response

This paper describes a new methodology for analyzing the closure stress of small fatigue cracks using ultrasonics. Surface connected back-wall cracks in steel specimens are considered. The crack-corner echo of ultrasonic shear wave beam of 50° incidence in the material is used. In order to account for the exact effect of closure stress on ultrasonic response, the echo amplitude is analyzed based on artificially applying additional closure stress. The initial closure stress at no load condition is determined by the standard compliance technique. A relationship between the closure stress and the crack-echo response is established which provides the facility of getting a true model of crack closure stress. The relationship is found to be independent of crack depth. The present approach can be used to overcome the difficulties encountered by crack closure in recent nondestructive ultrasonic techniques and may lead the way in quantitative evaluation of crack depth as well as exact closure stress.