Focused Shear Wave Beam Propagation in Tissue-Mimicking Phantoms.

OBJECTIVE: 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. METHODS: 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. RESULTS: 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. CONCLUSION: 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. SIGNIFICANCE: 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.

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.

Backscatter tensor imaging and 3D speckle tracking for simultaneous ex vivo structure and deformation measurement of myocardium.

OBJECTIVE: Biaxial mechanical testing is a common method for elucidation of mechanical properties of excised ventricular myocardium, especially in the context of structural remodeling that accompanies heart disease. Current imaging strategies in biaxial testing are based on optical camera imaging of the tissue surface, thus providing no information about the tissue microstructure and limiting strain measurements to two dimensions. Here, these limitations are overcome by replacing the camera with ultrasound imaging in order to measure both transmural fiber orientation and 3D tissue deformation during biaxial testing. METHODS: Quasi-static biaxial mechanical testing is applied to four samples of excised porcine ventricular myocardium (two left- and two right-ventricular tissues). During testing, a rotational scan of an ultrasound linear array provides data for both backscatter tensor imaging and 3D speckle tracking, from which transmural fiber orientation and tissue deformation are computed, respectively. Ultrasound-derived fiber orientation and tissue strain are validated against histology and camera surface imaging, respectively. DISCUSSION: Ultrasound-derived fiber angle and tissue strain exhibit good accuracy, with root-mean-square errors of 9.9° and 1.2% strain, respectively. Further investigation into the optimization of backscatter tensor imaging is warranted. Replacing the rotational scan of a linear array with volume imaging with a matrix array will improve the technique. CONCLUSION: Ultrasound imaging can replace the optical camera measurement during biaxial mechanical testing of ventricular myocardium in order to accurately provide measurements of transmural fiber orientation and tissue strain. In situ knowledge of transmural fiber structure and tissue deformation can enhance the inverse problem used to determine tissue mechanical properties from biaxial testing.

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.

Refraction-corrected backscatter tensor imaging of excised porcine ventricular myocardium.

Backscatter tensor imaging (BTI) is performed on excised porcine right- and left-ventricular myocardium to estimate the transmural myofiber orientation. Calculation of the backscatter spatial coherence employs measured sound speeds of the myocardium and the fluid that separates the tissue from the imaging array to account for effects of refraction during the delay-and-sum beamforming calculation. Compared to the assumption of a homogeneous sound speed in the imaging region, accounting for refraction yields significantly increased average spatial coherence as well as contrast of spatial coherence between the along- and across-fiber directions, thus improving sensitivity of BTI for myofiber orientation estimation.

Nonlinear piezoelectric surface acoustic waves.

The theory for nonlinear surface acoustic waves in crystals developed using Hamiltonian mechanics [Hamilton, Il'inskii, and Zabolotskaya, J. Acoust. Soc. Am. 105, 639 (1999)] is modified to account for piezoelectric material properties. The derived spectral evolution equations permit analysis of nonlinear surface wave propagation along a cut surface of any orientation with respect to the crystallographic axes and for piezoelectric crystals with any symmetry. Numerical simulations of waveform distortion in the particle velocity and electric field components are presented for surface wave propagation in Y-cut lithium niobate along the X- and Z-crystallographic axes. The influence of piezoelectricity is illustrated by comparing the nonlinear evolution of waveforms along a surface bounded by a vacuum (free space) and an ideal conductor (short circuit). Contributions to the nonlinearity from elasticity, piezoelectricity, electrostriction, and dielectricity are quantified separately for the two boundary conditions.

Evolution equation for nonlinear Lucassen waves, with application to a threshold phenomenon.

A nonlinear, fractional, surface wave equation with a spatial derivative of second order was developed by Kappler, Shrivastava, Schneider, and Netz [Phys. Rev. Fluids 2, 114804 (2017)] for propagation along an elastic interface coupled to a viscous incompressible liquid. Linear theory for the attenuation and dispersion was developed originally by Lucassen [Trans. Faraday Soc. 64, 2221 (1968)]. Kappler et al. introduced a fractional time derivative to account for the Lucassen wave attenuation and dispersion, and they included quadratic and cubic nonlinearity associated with compression of the elastic interface. Presented here is an integrated form of their time domain equation for progressive waves that is first order in the spatial derivative. Solutions of this evolution equation capture the main features of waveforms predicted by the full model equation of Kappler et al., especially the formation and propagation of shocks, while the evolution equation can be solved numerically with substantially less computational cost. Approximate analytical expressions obtained from the evolution equation for the nonlinear propagation speed and attenuation of a compression pulse reveal that a threshold phenomenon discussed by Kappler et al. is due to competition between quadratic and cubic nonlinearity associated with a lipid monolayer interface.

Plane nonlinear shear wave propagation in transversely isotropic soft solids.

Nonlinear wave equations are obtained for the two plane shear wave modes in a transversely isotropic soft solid. The material is modeled using a general expansion of the strain energy density up to fourth order in strain. Whereas, in an isotropic soft solid, leading order nonlinearity for plane wave propagation appears at cubic order in strain, elastic anisotropy in a transversely isotropic material introduces nonlinear effects at quadratic order, including interaction between the modes of a wave with two displacement components. Expressions for second harmonic generation in an elliptically polarized wave field illustrate the low efficiency of nonlinear interactions between the two displacement components, which results from the disparity between propagation speeds of the two shear wave modes. Coupled wave equations with up to cubic nonlinearity are presented and then approximated to describe linearly polarized waves by neglecting interaction between modes. Evolution equations are obtained for linearly polarized progressive waves, and explicit expressions are given in terms of elastic moduli and propagation direction for the coefficients of leading order nonlinearity. Expressions are presented for up to third harmonic generation from a time-harmonic source.

Elastic softening of sandstone due to a wideband acoustic pulse.

Elastic wave propagation experiments were performed on a thin bar sample composed of Texas "moss" sandstone in order to study nonlinear elastic effects in the time domain. The present experiments utilized a pendulous hammer to produce axially propagating transient signals with strain amplitude between 15 and 130 microstrain in the mid-audio band. Particle velocity along the bar axis was measured with a laser Doppler vibrometer, focused at various locations along the bar. Nonequilibrium dynamics and nonlinear elasticity effects were observed on the propagating pulse as it reflected between the ends of the bar. The same effects were also observed at a single location along the mid-length of the bar using a continuous-wave 1 MHz probe signal, propagating transverse to the bar axis. The results demonstrate larger strain amplitudes and greater reduction in the Young's modulus than in previously reported measurements that employed narrowband excitation. Nonlinear attenuation of the axial pulse is also observed, which increases with excitation amplitude. The present results indicate significant conditioning of the sandstone, particularly "softening" of the Young's modulus of up to 20%, primarily during the tensile phase of propagation, with a "slow dynamic" memory that is similar to that reported in previous investigations.

Transient solutions for the angular dependence of acoustical transmitters and receivers employing paraboloidal reflectors.

Time-domain solutions are presented for the angular dependence of waveforms in the far field of a point source at the focus of a rigid paraboloidal reflector, and also for waveforms at the focus as a function of the direction of a plane wave incident on the reflector. The main restriction is that the wavelength is small in relation to both the radius of the aperture and the minimum radius of curvature of the reflector, conditions which are satisfied for reflectors with appreciable gain. The solution in the far field due to a point source at the focus is related by the principle of reciprocity to the solution at the focus due to an incident plane wave. Both solutions are expressed as the convolution of an explicit expression for the unit step response with the time derivative of the pressure waveform incident on the reflector. Results are presented illustrating the angular dependence of the reflected pressure waveforms at the focus due to incident N waves and tone bursts.

Plane nonlinear shear waves in relaxing media.

Model equations with cubic nonlinearity are developed for a plane shear wave of finite amplitude in a relaxing medium. The evolution equation for progressive waves is solved analytically for a jump in stress that propagates into an undisturbed medium. Weak-shock theory is used to determine the amplitude and location of the shock when the solution predicts a multivalued waveform. The solution is similar to that obtained by Polyakova, Soluyan, and Khokhlov [Sov. Phys. Acoust. 8, 78-82 (1962)] for a compressional wave with quadratic nonlinearity in a relaxing fluid. Numerical simulations illustrate the effect of relaxation on shock formation in an initially sinusoidal shear wave. The minimum source amplitude required for an initially sinusoidal waveform to develop shocks in a relaxing medium is determined as a function of the dispersion and relaxation time. Limiting forms of the evolution equation are considered, and analytical solutions incorporating weak-shock theory are presented in the high-frequency limit. A Duffing-type model for a nonlinear shear-wave resonator is developed and investigated.