Acoustoelasticity in soft solids: assessment of the nonlinear shear modulus with the acoustic radiation force

Mechanical characterization of functionally graded soft materials with ultrasound elastography

Functionally graded soft materials (FGSMs) with microstructures and mechanical properties exhibiting gradients across a spatial volume to satisfy specific functions have received interests in recent years. How to characterize the mechanical properties of these FGSMs in vivo/in situ and/or in a non-destructive manner is a great challenge. This paper investigates the use of ultrasound elastography in the mechanical characterization of FGSMs. An efficient finite-element model was built to calculate the dispersion relation for surface waves in FGSMs. For FGSMs with large elastic gradients, the measured dispersion relation can be used to identify mechanical parameters. In the case where the elastic gradient is smaller than a certain critical value calculated here, our analysis on transient wave motion in FGSMs shows that the group velocities measured at different depths can infer the local mechanical properties. Experiments have been performed on polyvinyl alcohol (PVA) cryogel to demonstrate the usefulness of the method. Our analysis and the results may not only find broad applications in mechanical characterization of FGSMs but also facilitate the use of shear wave elastography in clinics because many diseases change the local elastic properties of soft tissues and lead to different material gradients. This article is part of the theme issue 'Rivlin's legacy in continuum mechanics and applied mathematics'.

Bidirectional Ultrasound Elastographic Imaging Framework for Non-invasive Assessment of the Non-linear Behavior of a Physiologically Pressurized Artery

Studies of non-destructive bidirectional ultrasound assessment of non-linear mechanical behavior of the artery are scarce in the literature. We hereby propose derivation of a strain-shear modulus relationship as a new graphical diagnostic index using an ultrasound elastographic imaging framework, which encompasses our in-house bidirectional vascular guided wave imaging (VGWI) and ultrasound strain imaging (USI). This framework is used to assess arterial non-linearity in two orthogonal (i.e., longitudinal and circumferential) directions in the absence of non-invasive pressure measurement. Bidirectional VGWI estimates longitudinal (μ_L) and transverse (μ_T) shear moduli, whereas USI estimates radial strain (ɛ_r). Vessel-mimicking phantoms (with and without longitudinal pre-stretch) and in vitro porcine aortas under static and/or dynamic physiologic intraluminal pressure loads were examined. ɛ_r was found to be a suitable alternative to intraluminal pressure for representation of cyclic loading on the artery wall. Results revealed that μ_T values of all samples examined increased non-linearly with ε_r magnitude and more drastically than μ_L, whereas μ_L values of only the pre-stretched phantoms and aortas increased with ɛ_r magnitude. As a new graphical representation of arterial non-linearity and function, strain-shear modulus loops derived by the proposed framework over two consecutive dynamic loading cycles differentiated sample pre-conditions and corroborated direction-dependent non-linear mechanical behaviors of the aorta with high estimation repeatability.

A comparison of hyperelastic constitutive models applicable to shear wave elastography (SWE) data in tissue-mimicking materials

Shear wave elastography (SWE) techniques have received substantial attention in recent years. Strong experimental data in SWE suggest that shear wave speed changes significantly due to the known acoustoelastic effect (AE). This presents both challenges and opportunities toward in vivo characterization of biological soft tissues. In this work, under the framework of continuum mechanics, we model a tissue-mimicking material as a homogeneous, isotropic, incompressible, hyperelastic material. Our primary objective is to quantitatively and qualitatively compare experimentally measured acoustoelastic data with model-predicted outcomes using multiple strain energy functions. Our analysis indicated that the classic Neo-Hookean and Mooney-Rivlin models are inadequate for modeling the AE in tissue-mimicking materials. However, a subclass of strain energy functions containing both high-order/exponential term(s) and second-order invariant dependence showed good agreement with experimental data. Based on data investigated, we also found that discrepancies may exist between parameters inversely estimated from uniaxial compression and SWE data. Overall, our findings may improve our understanding of clinical SWE results.

Acoustoelasticity Analysis of Transient Waves for Non-Invasive In Vivo Assessment of Urinary Bladder

A non-invasive method for measurement of the bladder wall nonlinear elastic behavior is presented. The method is based on acoustoelasticity modeling of the elasticity changes in bladder tissue modulus at different volumetric strain levels. At each volume, tissue strain is obtained from the real-time ultrasound images. Using acoustic radiation force, a transient Lamb wave is excited on the bladder wall and instantaneous modulus of shear elasticity is obtained from the 2-D Fourier analysis of the spatial-temporal dispersion maps. Measured elasticity and strain data are then used in an acoustoelasticity formulation to obtain the third order elastic coefficient, referred to as nonlinearity parameter A, and initial resting elasticity μ₀. The method was tested in ex vivo porcine bladder samples (N = 9) before and after treatment with formalin. The estimated nonlinearity parameter, A, was significantly higher in the treated samples compared to intact (p < 0.00062). The proposed method was also applied on 16 patients with neurogenic bladders (10 compliant and 6 non-compliant subjects). The estimated nonlinearity parameter A was significantly higher in the non-compliant cases compared to the compliant (p < 0.0293). These preliminary results promise a new method for non-invasive evaluation of the bladder tissue nonlinearity which may serve as a new diagnostic and prognostic biomarker for management of the patients with neurogenic bladders.

Shear Wave Elastography in the Evaluation of the Kidneys in Pediatric Patients with Unilateral Vesicoureteral Reflux

OBJECTIVES

To evaluate the ability of shear wave elastography (SWE) to detect renal parenchymal scar formation in patients with vesicoureteral reflux.

METHODS

We prospectively evaluated 49 patients with unilateral grade 2 or higher-degree VUR. All patients underwent dimercaptosuccinic acid (DMSA) scintigraphy for evaluation of the renal parenchymal scar. After the DMSA scan, 2 radiologists, who were blinded to clinical data and each other's measurements, evaluated the kidneys of the patients using SWE. The kidneys were divided into 3 parts: upper pole, middle region, and lower pole, and 3 regions of interest were placed to each part. Shear wave velocity (SWV) values were calculated using meters per second as a unit and recorded for each region. Afterward, SWV values were compared to DMSA results.

RESULTS

There was no significant difference between the observers' mean SWV values of kidneys with VUR without scar formation (mean ± SD, 2.11 ± 0.06 and 2.09 ± 0.05 m/s) and the contralateral normal kidney SVW values (2.11 ± 0.06 and 2.10 ± 0.05 m/s; P = .936 and .724, respectively). We observed a significant difference between the mean SWV values of the kidneys with VUR accompanied by scar formation (2.28 ± 0.10 and 2.27 ± 0.11 m/s) and the mean SWV values of the contralateral normal kidneys (2.09 ± 0.05 and 2.10 ± 0.04 m/s; P < .001 for both observers).

CONCLUSIONS

Shear wave elastography could detect scar tissue in kidneys; however, the variability of the stiffness due to the kidney's complex structure, and variations in blood perfusion and the glomerular filtration rate of the kidney might limit the use of SWE in current clinical diagnostic algorithms for VUR.

Magnetic resonance elastography in nonlinear viscoelastic materials under load

Characterisation of soft tissue mechanical properties is a topic of increasing interest in translational and clinical research. Magnetic resonance elastography (MRE) has been used in this context to assess the mechanical properties of tissues in vivo noninvasively. Typically, these analyses rely on linear viscoelastic wave equations to assess material properties from measured wave dynamics. However, deformations that occur in some tissues (e.g. liver during respiration, heart during the cardiac cycle, or external compression during a breast exam) can yield loading bias, complicating the interpretation of tissue stiffness from MRE measurements. In this paper, it is shown how combined knowledge of a material's rheology and loading state can be used to eliminate loading bias and enable interpretation of intrinsic (unloaded) stiffness properties. Equations are derived utilising perturbation theory and Cauchy's equations of motion to demonstrate the impact of loading state on periodic steady-state wave behaviour in nonlinear viscoelastic materials. These equations demonstrate how loading bias yields apparent material stiffening, softening and anisotropy. MRE sensitivity to deformation is demonstrated in an experimental phantom, showing a loading bias of up to twofold. From an unbiased stiffness of [Formula: see text] Pa in unloaded state, the biased stiffness increases to 9767.5 [Formula: see text]1949.9 Pa under a load of [Formula: see text] 34% uniaxial compression. Integrating knowledge of phantom loading and rheology into a novel MRE reconstruction, it is shown that it is possible to characterise intrinsic material characteristics, eliminating the loading bias from MRE data. The framework introduced and demonstrated in phantoms illustrates a pathway that can be translated and applied to MRE in complex deforming tissues. This would contribute to a better assessment of material properties in soft tissues employing elastography.

Ultrasonic waves in uniaxially stressed multilayered and one-dimensional phononic structures: Guided and Floquet wave analysis

This paper shows that acoustoelasticity in one-dimensional (1D) multilayered isotropic hyperelastic materials can be understood through the analysis of elastic wave velocities as a function of applied stress. This theoretical framework is used for eigenvalue analyses in stressed elastic structures through a reformulation of the stiffness matrix method, obtaining modal solutions, as well as reflection and transmission coefficients for different multilayered configurations. Floquet wave analysis for the stressed 1D structures is supported using numerical results.

Fourier-Domain Shift Matching: A Robust Time-of-Flight Approach for Shear Wave Speed Estimation

Our primary objective of this work was to design and test a new time-of-flight (TOF) method that allows measurements of shear wave speed (SWS) following impulsive excitation in soft tissues. Particularly, under the assumption of the local plane shear wave, this work named the Fourier-domain shift matching (FDSM) method, estimates SWS by aligning a series of shear waveforms either temporally or spatially using a solution space deduced by characteristic curves of the well-known 1-D wave equation. The proposed SWS estimation method was tested using computer-simulated data, and tissue-mimicking phantom and ex vivo tissue experiments. Its performance was then compared with three other known TOF methods: lateral time-to-peak (TTP) method with robust random sampling consensus (RANSAC) fitting method, Radon sum transformation method, and a modified cross correlation method. Hereafter, these three TOF methods are referred to as the TTP-RANSAC, Radon sum, and X-corr methods, respectively. In addition to an adapted form of the 2-D Fourier transform (2-D FT)-based method in which the (group) SWS was approximated by averaging phase SWS values was considered for comparison. Based on data evaluated, we found that the overall performance of the above-mentioned temporal implementation of the proposed FDSM method was most similar to the established Radon sum method (correlation = 0.99, scale factor = 1.03, and mean difference = 0.07 m/s), and the 2-D FT (correlation = 0.98, scale factor = 1.00, and mean difference = 0.10 m/s) at high signal quality. However, results obtained from the 2-D FT method diverged (correlation = 0.201) from these of the proposed temporal implementation in the presence of diminished signal quality, whereas the agreement between the Radon sum approach and the proposed temporal implementation largely remained the same (correlation = 0.98).

Gauging force by tapping tendons

Muscles are the actuators that drive human movement. However, despite many decades of work, we still cannot readily assess the forces that muscles transmit during human movement. Direct measurements of muscle-tendon loads are invasive and modeling approaches require many assumptions. Here, we introduce a non-invasive approach to assess tendon loads by tracking vibrational behavior. We first show that the speed of shear wave propagation in tendon increases with the square root of axial stress. We then introduce a remarkably simple shear wave tensiometer that uses micron-scale taps and skin-mounted accelerometers to track tendon wave speeds in vivo. Tendon wave speeds are shown to modulate in phase with active joint torques during isometric exertions, walking, and running. The capacity to non-invasively assess muscle-tendon loading can provide new insights into the motor control and biomechanics underlying movement, and could lead to enhanced clinical treatment of musculoskeletal injuries and diseases.

Application of Acoustoelasticity to Evaluate Nonlinear Modulus in Ex Vivo Kidneys

Currently, dynamic elastography techniques estimate the linear elastic shear modulus of different body tissues. New methods that investigate other properties of soft tissues such as anisotropy, viscosity, and shear nonlinearity would provide more information about the structure and function of the tissue and might provide a better contrast than tissue stiffness and hence provide more effective diagnostic tools for some diseases. It has previously been shown that shear wave velocity in a medium changes due to an applied stress, a phenomenon called acoustoelasticity (AE). Applying a stress to compress a medium while measuring the shear wave velocity versus strain provides data with which the third-order nonlinear shear modulus can be estimated. To evaluate the feasibility of estimating , we evaluated ten ex vivo porcine kidneys embedded in 10% porcine gelatin to mimic the case of a transplanted kidney. Under assumptions of an elastic incompressible medium for AE measurements, the shear modulus was quantified at each compression level and the applied strain was assessed by measuring the change in the thickness of the kidney cortex. Finally, was calculated by applying the AE theory. Our results demonstrated that it is possible to estimate a nonlinear shear modulus by monitoring the changes in strain and due to kidney deformation. The magnitudes of are higher when the compression is performed progressively and when using a plate attached to the transducer. Nevertheless, the values obtained for are similar to those previously reported in the literature for breast tissue.

A model for acoustic vaporization dynamics of a bubble/droplet system encapsulated within a hyperelastic shell

Nanodroplets have great, promising medical applications such as contrast imaging, embolotherapy, or targeted drug delivery. Their functions can be mechanically activated by means of focused ultrasound inducing a phase change of the inner liquid known as the acoustic droplet vaporization (ADV) process. In this context, a four-phases (vapor + liquid + shell + surrounding environment) model of ADV is proposed. Attention is especially devoted to the mechanical properties of the encapsulating shell, incorporating the well-known strain-softening behavior of Mooney-Rivlin material adapted to very large deformations of soft, nearly incompressible materials. Various responses to ultrasound excitation are illustrated, depending on linear and nonlinear mechanical shell properties and acoustical excitation parameters. Different classes of ADV outcomes are exhibited, and a relevant threshold ensuring complete vaporization of the inner liquid layer is defined. The dependence of this threshold with acoustical, geometrical, and mechanical parameters is also provided.

Simulation of nonlinear transient elastography: finite element model for the propagation of shear waves in homogeneous soft tissues

In this study, visco-hyperelastic Landau's model, which is widely used in acoustical physic field, is introduced into a finite element formulation. It is designed to model the nonlinear behaviour of finite amplitude shear waves in soft solids, typically, in biological tissues. This law is used in finite element models based on elastography, experiments reported in Jacob et al, the simulations results show a good agreement with the experimental study: It is observed in both that a plane shear wave generates only odd harmonics and a nonplane wave generates both odd and even harmonics in the spectral domain. In the second part, a parametric study is performed to analyse the influence of different factors on the generation of odd harmonics of plane wave. A quantitative relation is fitted between the odd harmonic amplitudes and the non-linear elastic parameter of Landau's model, which provides a practical guideline to identify the non-linearity of homogeneous tissues using elastography experiment.

Effect of renal perfusion and structural heterogeneity on shear wave elastography of the kidney: an in vivo and ex vivo study

Background

To evaluate the effect of perfusion status on elasticity measurements of different compartments in the kidney using shear wave elastography (SWE) both in vivo and ex vivo.

Methods

Thirty-two rabbit kidneys were used to observe the elasticity variation caused by renal artery stenosis and vein ligation in vivo, and six beagle kidneys were studied ex vivo to explore the effect of renal perfusion on elasticity. Supersonic SWE was applied to quantify the elasticity values of different renal compartments (cortex, medulla and sinus). Additionally, histopathological examination was performed to explore the possible mechanisms.

Results

The elasticity of the cortex was higher than that of the medulla, and the elasticity of the sinus was lowest among the compartments in native kidneys. The Young’s modulus (YM) of the cortex, medulla and sinus increased gradually as the duration of renal vein ligation increased, from 16.34 ± 1.01 kPa to 55.06 ± 5.61 kPa, 13.71 ± 1.16 kPa to 39.63 ± 2.91 kPa, and 12.61 ± 0.84 kPa to 29.30 ± 2.04 kPa, respectively. In contrast, the YM of the three compartments respectively decreased with progressive artery stenosis, from 16.34 ± 1.83 kPa to 11.21 ± 1.79 kPa, 13.31 ± 1.67 kPa to 8.07 ± 1.37 kPa, and 12.78 ± 2.66 kPa to 6.72 ± 0.95 kPa. Artery perfusion was the main factor influencing elasticity in ex vivo. The cortical elasticity was more prone to change with renal perfusion both in vivo and ex vivo. Histopathological examination showed progressive changes in the structure and content of the three compartments, consistent with the elasticity variation.

Conclusions

Both the complex structure/anisotropy and the perfusion of the kidney obviously influence the evaluation of renal elasticity. The measurement of SWE should be performed at a specific location along a certain angle or direction, and renal perfusion status should also be taken into account to ensure reproducible detection.

Coded excitation waveform engineering for high frame rate synthetic aperture ultrasound imaging

Coded excitation was initially introduced to ultrasound imaging as a method for enhancing the signal-to-noise ratio (SNR). However, this method was also shown to be helpful in conjunction with synthetic aperture transmission for high frame rate imaging. Recently, we introduced two families of mismatched coded excitations based on frequency modulation chirp and combined frequency modulation and Golay code. Here "mismatched" indicates that the coded excitations generate very small cross-correlations among themselves while each has a very strong autocorrelation. Employing weakly correlated coded excitations enables performing simultaneous insonifications from several elements of the ultrasonic transducer and receiving distinguishable responses to each code. In this work, we propose and experimentally demonstrate another set of mismatched correlated coded excitations based on Golay codes. The generated phase codes share identical duration and center frequency which results in similar SNR and image resolution.

Mechanics of ultrasound elastography

Ultrasound elastography enables in vivo measurement of the mechanical properties of living soft tissues in a non-destructive and non-invasive manner and has attracted considerable interest for clinical use in recent years. Continuum mechanics plays an essential role in understanding and improving ultrasound-based elastography methods and is the main focus of this review. In particular, the mechanics theories involved in both static and dynamic elastography methods are surveyed. They may help understand the challenges in and opportunities for the practical applications of various ultrasound elastography methods to characterize the linear elastic, viscoelastic, anisotropic elastic and hyperelastic properties of both bulk and thin-walled soft materials, especially the in vivo characterization of biological soft tissues.

Numerical Simulation of Focused Shock Shear Waves in Soft Solids and a Two-Dimensional Nonlinear Homogeneous Model of the Brain

Shear waves that propagate in soft solids, such as the brain, are strongly nonlinear and can develop into shock waves in less than one wavelength. We hypothesize that these shear shock waves could be responsible for certain types of traumatic brain injuries (TBI) and that the spherical geometry of the skull bone could focus shear waves deep in the brain, generating diffuse axonal injuries. Theoretical models and numerical methods that describe nonlinear polarized shear waves in soft solids such as the brain are presented. They include the cubic nonlinearities that are characteristic of soft solids and the specific types of nonclassical attenuation and dispersion observed in soft tissues and the brain. The numerical methods are validated with analytical solutions, where possible, and with self-similar scaling laws where no known solutions exist. Initial conditions based on a human head X-ray microtomography (CT) were used to simulate focused shear shock waves in the brain. Three regimes are investigated with shock wave formation distances of 2.54 m, 0.018 m, and 0.0064 m. We demonstrate that under realistic loading scenarios, with nonlinear properties consistent with measurements in the brain, and when the shock wave propagation distance and focal distance coincide, nonlinear propagation can easily overcome attenuation to generate shear shocks deep inside the brain. Due to these effects, the accelerations in the focal are larger by a factor of 15 compared to acceleration at the skull surface. These results suggest that shock wave focusing could be responsible for diffuse axonal injuries.

Ultrasound-based imaging methods of the kidney-recent developments

In recent years, several novel ultrasound (US)-based techniques have emerged for kidney diagnostic imaging, including tissue stiffness assessment with elastography, Ultrasensitive Doppler techniques, and contrast-enhanced ultrasonography to assess renal microvascularization. Renal elastography has become available with the development of noninvasive quantitative techniques, following the rapidly growing field of liver fibrosis diagnosis. With the increased incidence of chronic kidney disease, noninvasive diagnosis of renal fibrosis can be of critical value. However, it is difficult to simply extend the application of US elastography from one organ to the other due to anatomic and technical issues. Today, renal elastography appears to be a promising application that, however, still requires optimization and validation. New ultrasensitive Doppler techniques improve the detection of slow blood flow and can be used alone or after administration of US contrast agents. These microbubble-based agents are extremely well tolerated and can be administered even in cases of impaired renal function. Despite the lack of approval, they improve the characterization of atypical renal masses, complex cystic renal masses, and peripheral vascular disorders. Dynamic contrast-enhanced US is based on quantification of the signal intensity from region of interest and mathematical fits of the time-intensity curves. Perfusion-related parameters can be extracted for the monitoring of vascular changes in the renal parenchyma and in tumors in order to evaluate drug response. This estimation of renal perfusion depends on many parameters that should be kept constant for follow-up studies, and, when possible, an internal reference should be used to normalize the measurements.

Shear-Wave Elastography Assessments of Quadriceps Stiffness Changes prior to, during and after Prolonged Exercise: A Longitudinal Study during an Extreme Mountain Ultra-Marathon

In sports medicine, there is increasing interest in quantifying the elastic properties of skeletal muscle, especially during extreme muscular stimulation, to improve our understanding of the impact of alterations in skeletal muscle stiffness on resulting pain or injuries, as well as the mechanisms underlying the relationships between these parameters. Our main objective was to determine whether real-time shear-wave elastography (SWE) can monitor changes in quadriceps muscle elasticity during an extreme mountain ultra-marathon, a powerful mechanical stress model. Our study involved 50 volunteers participating in an extreme mountain marathon (distance: 330 km, elevation: +24,000 m). Quantitative SWE velocity and shear modulus measurements were performed in most superficial quadriceps muscle heads at the following 4 time points: before the race, halfway through the race, upon finishing the race and after recovery (+48 h). Blood biomarker levels were also measured. A significant decrease in the quadriceps shear modulus was observed upon finishing the race (3.31±0.61 kPa) (p<0.001) compared to baseline (3.56±0.63 kPa), followed by a partial recovery +48 h after the race (3.45±0.6 kPa) (p = 0.002) across all muscle heads, as well as for each of the following three muscle heads: the rectus femoris (p = 0.003), the vastus medialis (p = 0.033) and the vastus lateralis (p = 0.001). Our study is the first to assess changes in muscle stiffness during prolonged extreme physical endurance exercises based on shear modulus measurements using non-invasive SWE. We concluded that decreases in stiffness, which may have resulted from quadriceps overuse in the setting of supra-physiological stress caused by the extreme distance and unique elevation of the race, may have been responsible for the development of inflammation and muscle swelling. SWE may hence represent a promising tool for monitoring physiologic or pathological variations in muscle stiffness and may be useful for diagnosing and monitoring muscle changes.

In Vivo Quantification of the Nonlinear Shear Modulus in Breast Lesions: Feasibility Study

Breast cancer detection in the early stages is of great importance since the prognosis, and the treatment depends more on this. Multiple techniques relying on the mechanical properties of soft tissues have been developed to help in early detection. In this study, we implemented a technique that measures the nonlinear shear modulus (NLSM) (μ(NL)) in vivo and showed its utility to detect breast lesions from healthy tissue. The technique relies on the acoustoelasticity theory in quasi-incompressible media. In order to recover μ(NL), static elastography and supersonic shear imaging are combined to subsequently register strain maps and shear modulus maps while the medium is compressed. Then, μ(NL) can be recovered from the relationship between the stress, deduced from strain maps, and the shear modulus. For this study, a series of five nonlinear phantoms were built using biological tissue (pork liver) inclusions immersed in an agar-gelatin gel. Furthermore, 11 in vivo acquisitions were performed to characterize the NLSM of breast tissue. The phantom results showed a very good differentiation of the liver inclusions when measuring μ(NL) with a mean value of -114.1 kPa compared to -34.7 kPa for the gelatin. Meanwhile, values for the shear modulus for the liver and the gelatin were very similar, 3.7 and 3.4 kPa, respectively. In vivo NLSM mean value for the healthy breast tissue was of -95 kPa, while mean values of the benign and the malignant lesions were -619 and -806 kPa with a strong v ariability, respectively. This study shows the potential of the acoustoelasticity theory in quasi-incompressible medium to bring a new parameter for breast cancer diagnosis.

Assessment of the Cervix in Pregnant Women Using Shear Wave Elastography: A Feasibility Study

The quantitative assessment of the cervix is crucial for the estimation of pre-term delivery risk and the prediction of the success of labor induction. We conducted a cross-sectional study using shear wave elastography based on the supersonic shear imaging technique. The shear wave speed (SWS) of the lower anterior part of the cervix was quantified over an 8-mm region of interest in 157 pregnant women. Cervical SWS is slightly but significantly reduced in patients diagnosed with pre-term labor and in patients who actually delivered pre-term.

Coexisting and mixing phenomena of thermoacoustic and magnetoacoustic waves in water

Concurrent generation and mixing phenomenon of thermoacoustic (TA) and magnetoacoustic (MA) waves in water are predicted and observed. A theory unifying TA and MA is further put forward to analyze it. By scaling down the radio frequency in thermoacoustics to the low mega Hertz range and by incorporating appropriately a static magnetic field, TA and MA waves are simultaneously generated in the conductive matter. The two waves propagate concurrently in water and produce dynamic acoustic radiation force due to water absorption. Such dynamic radiation force vibrates the absorbing water and consequently yields acoustic emissions at the inter-modulation frequencies of TA and MA waves, creating mixing effect similar to that of vibro-acoustography. The mixing effect can be potentially utilized to mimic vibro-acoustography imaging without firing external ultrasound towards intrinsic dual-contrast (elasticity and conductivity) imaging.

Characterization of transverse isotropy in compressed tissue-mimicking phantoms

Tissues such as skeletal muscle and kidneys have well-defined structure that affects the measurements of mechanical properties. As an approach to characterize the material properties of these tissues, different groups have assumed that they are transversely isotropic (TI) and measure the shear wave velocity as it varies with angle with respect to the structural architecture of the organ. To refine measurements in these organs, it is desirable to have tissue-mimicking phantoms that exhibit similar anisotropic characteristics. Some approaches involve embedding fibers into a material matrix. However, if a homogeneous solid is under compression due to a static stress, an acoustoelastic effect can manifest that makes the measured wave velocities change with the compression stress. We propose to exploit this characteristic to demonstrate that stressed tissue mimicking phantoms can be characterized as a TI material. We tested six phantoms made with different concentrations of gelatin and agar. Stress was applied by the weight of a water container centered on top of a plate on top of the phantom. A linear array transducer and a V-1 Verasonics system were used to induce and measure shear waves in the phantoms. The shear wave motion was measured using a compound plane wave imaging technique. Autocorrelation was applied to the received in-phase/quadrature data. The shear wave velocity, c, was estimated using a Radon transform method. The transducer was mounted on a rotating stage so measurements were made every 10° over a range of 0° to 360°, where the stress is applied along 0° to 180° direction. The shear moduli were estimated. A TI model was fit to the data and the fractional anisotropy was evaluated. This approach can be used to explore many configurations of transverse isotropy with the same phantom, simply by applying stress to the tissue-mimicking phantom.

Characterization of transverse isotropy in compressed tissue-mimicking phantoms

Tissues such as skeletal muscle and kidneys have well-defined structure that affects the measurements of mechanical properties. As an approach to characterize the material properties of these tissues, different groups have assumed that they are transversely isotropic (TI) and measure the shear wave velocity as it varies with angle with respect to the structural architecture of the organ. To refine measurements in these organs, it is desirable to have tissue-mimicking phantoms that exhibit similar anisotropic characteristics. Some approaches involve embedding fibers into a material matrix. However, if a homogeneous solid is under compression due to a static stress, an acoustoelastic effect can manifest that makes the measured wave velocities change with the compression stress. We propose to exploit this characteristic to demonstrate that stressed tissue mimicking phantoms can be characterized as a TI material. We tested six phantoms made with different concentrations of gelatin and agar. Stress was applied by the weight of a water container centered on top of a plate on top of the phantom. A linear array transducer and a V-1 Verasonics system were used to induce and measure shear waves in the phantoms. The shear wave motion was measured using a compound plane wave imaging technique. Autocorrelation was applied to the received in-phase/quadrature data. The shear wave velocity, c, was estimated using a Radon transform method. The transducer was mounted on a rotating stage so measurements were made every 10° over a range of 0° to 360°, where the stress is applied along 0° to 180° direction. The shear moduli were estimated. A TI model was fit to the data and the fractional anisotropy was evaluated. This approach can be used to explore many configurations of transverse isotropy with the same phantom, simply by applying stress to the tissue-mimicking phantom.

The 'sixth sense' of ultrasound: probing nonlinear elasticity with acoustic radiation force

Prompted by a recent finding that the magnitude of the acoustic radiation force (ARF) in isotropic tissue-like solids depends linearly on a particular third-order modulus of elasticity-hereon denoted by C, this study investigates the possibility of estimating C from the amplitude of the ARF-generated shear waves. The featured coefficient of nonlinear elasticity, which captures the incipient nonlinear interaction between the volumetric and deviatoric modes of deformation, has so far received only a limited attention in the context of soft tissues due to the fact that the latter are often approximated as (i) fluid-like when considering ultrasound waves, and (ii) incompressible under static deformations. On establishing the analytical and computational platform for the proposed sensing methodology, the study proceeds with applying the prototype technique toward estimating via ARF the third-order modulus C in a series of tissue-mimicking phantoms. To help validate the concept and its implementation, the germane third-order modulus is independently estimated in each phantom via an established technique known as acoustoelasticity. The C-estimates obtained respectively via acoustoelasticity and the new theory of ARF show a significant degree of consistency. The key features of the new sensing methodology are that: (a) it requires no external deformation of a material other than that produced by the ARF, and (b) it estimates the nonlinear C-modulus locally, over the focal region of an ultrasound beam-where the shear waves are being generated.

Comb-push ultrasound shear elastography of breast masses: initial results show promise

Purpose or Objective

To evaluate the performance of Comb-push Ultrasound Shear Elastography (CUSE) for classification of breast masses.

Materials and Methods

CUSE is an ultrasound-based quantitative two-dimensional shear wave elasticity imaging technique, which utilizes multiple laterally distributed acoustic radiation force (ARF) beams to simultaneously excite the tissue and induce shear waves. Female patients who were categorized as having suspicious breast masses underwent CUSE evaluations prior to biopsy. An elasticity estimate within the breast mass was obtained from the CUSE shear wave speed map. Elasticity estimates of various types of benign and malignant masses were compared with biopsy results.

Results

Fifty-four female patients with suspicious breast masses from our ongoing study are presented. Our cohort included 31 malignant and 23 benign breast masses. Our results indicate that the mean shear wave speed was significantly higher in malignant masses (6 ± 1.58 m/s) in comparison to benign masses (3.65 ± 1.36 m/s). Therefore, the stiffness of the mass quantified by the Young’s modulus is significantly higher in malignant masses. According to the receiver operating characteristic curve (ROC), the optimal cut-off value of 83 kPa yields 87.10% sensitivity, 82.61% specificity, and 0.88 for the area under the curve (AUC).

Conclusion

CUSE has the potential for clinical utility as a quantitative diagnostic imaging tool adjunct to B-mode ultrasound for differentiation of malignant and benign breast masses.

Measuring the linear and nonlinear elastic properties of brain tissue with shear waves and inverse analysis

We use supersonic shear wave imaging (SSI) technique to measure not only the linear but also the nonlinear elastic properties of brain matter. Here, we tested six porcine brains ex vivo and measured the velocities of the plane shear waves induced by acoustic radiation force at different states of pre-deformation when the ultrasonic probe is pushed into the soft tissue. We relied on an inverse method based on the theory governing the propagation of small-amplitude acoustic waves in deformed solids to interpret the experimental data. We found that, depending on the subjects, the resulting initial shear modulus [Formula: see text] varies from 1.8 to 3.2 kPa, the stiffening parameter [Formula: see text] of the hyperelastic Demiray-Fung model from 0.13 to 0.73, and the third- [Formula: see text] and fourth-order [Formula: see text] constants of weakly nonlinear elasticity from [Formula: see text]1.3 to [Formula: see text]20.6 kPa and from 3.1 to 8.7 kPa, respectively. Paired [Formula: see text] test performed on the experimental results of the left and right lobes of the brain shows no significant difference. These values are in line with those reported in the literature on brain tissue, indicating that the SSI method, combined to the inverse analysis, is an efficient and powerful tool for the mechanical characterization of brain tissue, which is of great importance for computer simulation of traumatic brain injury and virtual neurosurgery.

Quantification of elasticity changes in the myometrium during labor using Supersonic Shear Imaging: a feasibility study

Very little is known about the myometrium's physiology in terms of its elasticity but shear wave elastography could be an efficient tool to better understand it. This could considerably help the prevention of difficult births, the consequences of which are tremendous for neonate morbidity and pathologies. The purpose of this paper is to show the feasibility of the in vivo monitoring of myometrial stiffness changes in contraction and relaxation during pregnancy. In this study, Supersonic Shear Wave Imaging, a real-time and quantitative imaging technique that has been proven efficient for the investigation of tissue elasticity, was used to quantify the uterus shear-wave speed and stiffness in 6 patients, through the abdomen, using an 8-MHz linear ultrasound probe. Changes in shear wave speed were tracked in real time during the uterine contraction and were well correlated with the uterine pressure, which is currently considered to be a gold standard. These results open a new way to better understand the myometrium contraction during labour.

Characterization of the nonlinear elastic properties of soft tissues using the supersonic shear imaging (SSI) technique: inverse method, ex vivo and in vivo experiments

Dynamic elastography has become a new clinical tool in recent years to characterize the elastic properties of soft tissues in vivo, which are important for the disease diagnosis, e.g., the detection of breast and thyroid cancer and liver fibrosis. This paper investigates the supersonic shear imaging (SSI) method commercialized in recent years with the purpose to determine the nonlinear elastic properties based on this promising technique. Particularly, we explore the propagation of the shear wave induced by the acoustic radiation force in a stressed hyperelastic soft tissue described via the Demiray-Fung model. Based on the elastodynamics theory, an analytical solution correlating the wave speed with the hyperelastic parameters of soft tissues is first derived. Then an inverse approach is established to determine the hyperelastic parameters of biological soft tissues based on the measured wave speeds at different stretch ratios. The property of the inverse method, e.g., the existence, uniqueness and stability of the solution, has been investigated. Numerical experiments based on finite element simulations and the experiments conducted on the phantom and pig livers have been employed to validate the new method. Experiments performed on the human breast tissue and human heel fat pads have demonstrated the capability of the proposed method for measuring the in vivo nonlinear elastic properties of soft tissues. Generalization of the inverse analysis to other material models and the implication of the results reported here for clinical diagnosis have been discussed.

Imaging of shear waves induced by Lorentz force in soft tissues

This study presents the first observation of elastic shear waves generated in soft solids using a dynamic electromagnetic field. The first and second experiments of this study showed that Lorentz force can induce a displacement in a soft phantom and that this displacement was detectable by an ultrasound scanner using speckle-tracking algorithms. For a 100 mT magnetic field and a 10 ms, 100 mA peak-to-peak electrical burst, the displacement reached a magnitude of 1 μm. In the third experiment, we showed that Lorentz force can induce shear waves in a phantom. A physical model using electromagnetic and elasticity equations was proposed. Computer simulations were in good agreement with experimental results. The shear waves induced by Lorentz force were used in the last experiment to estimate the elasticity of a swine liver sample.

Quantifying the passive stretching response of human tibialis anterior muscle using shear wave elastography

BACKGROUND

Quantifying passive stretching responses of individual muscles helps the diagnosis of muscle disorders and aids the evaluation of surgical/rehabilitation treatments. Utilizing an animal model, we demonstrated that shear elastic modulus measured by supersonic shear wave elastography increases linearly with passive muscle force. This study aimed to use this state-of-the-art technology to study the relationship between shear elastic modulus and ankle dorsi-plantarflexion angle of resting tibialis anterior muscles and extract physiologically meaningful parameters from the elasticity-angle curve to better quantify passive stretching responses.

METHODS

Elasticity measurements were made at resting tibialis anterior of 20 healthy subjects with the ankle positioned from 50° plantarflexion to up to 15° dorsiflexion at every 5° for two cycles. Elasticity-angle data was curve-fitted by optimizing slack angle, slack elasticity, and rate of increase in elasticity within a piecewise exponential model.

FINDINGS

Elasticity-angle data of all subjects were well fitted by the piecewise exponential model with coefficients of determination ranging between 0.973 and 0.995. Mean (SD) of slack angle, slack elasticity, and rate of increase in elasticity were 10.9° (6.3°), 5.8 (1.9) kPa, and 0.0347 (0.0082) respectively. Intraclass correlation coefficients of each parameter were 0.852, 0.942, and 0.936 respectively, indicating excellent test-retest reliability.

INTERPRETATION

This study demonstrated the feasibility of using supersonic shear wave elastography to quantify passive stretching characteristics of individual muscle and provided preliminary normative values of slack angle, slack elasticity, and rate of increase in elasticity for human tibialis anterior muscles. Future studies will investigate diagnostic values of these parameters in clinical applications.

Biomechanical properties of the calcaneal tendon in vivo assessed by transient shear wave elastography

OBJECTIVE

The purpose of this study is to assess the elastic and anisotropic properties of normal calcaneal tendon in vivo by transient shear wave elastography (SWE).

MATERIALS AND METHODS

This study was approved by our institutional ethics committee. Eighty healthy subjects over 18 years of age were prospectively included. Data on the patients' height, weight, sporting activities, and take-off foot were assessed. The thickness, width, and cross-sectional area of the calcaneal tendons were measured. The shear wave propagation velocity (Vmean) was measured by three radiologists on axial and sagittal SWE images at four different degrees of ankle flexion, enabling to calculate elasticity modulus (Emean), and relative anisotropy coefficient (A) values.

RESULTS

In complete plantar flexion, Vmean was 6.8 ± 1.4 m.s(-1) and 5.1 ± 0.8 m.s(-1), respectively, on the sagittal and axial SWE image, resulting in an elastographic anisotropy A of 0.24 ± 0.16. The best interobserver correlation coefficient of Emean and Vmean was 0.43 and 0.46, respectively, in the sagittal SWE for complete plantar flexion. Vmean and Emean significantly increase when the tendon is stretched by ankle dorsiflexion. The maximal values in sagittal SWE were Vmean = 16.1 ± 0.7 m.s(-1), Emean = 779.5 ± 57.1kPa and A = 0.63 ± 0.07.

CONCLUSIONS

SWE allows the elastic properties of the calcaneal tendon to be evaluated quantitatively in vivo, but interobserver reproducibility is questionable. It confirms the tendinous elastographic anisotropy and stiffness augmentation of stretched tendon.

Acoustic waves in medical imaging and diagnostics

Up until about two decades ago acoustic imaging and ultrasound imaging were synonymous. The term ultrasonography, or its abbreviated version sonography, meant an imaging modality based on the use of ultrasonic compressional bulk waves. Beginning in the 1990s, there started to emerge numerous acoustic imaging modalities based on the use of a different mode of acoustic wave: shear waves. Imaging with these waves was shown to provide very useful and very different information about the biological tissue being examined. We discuss the physical basis for the differences between these two basic modes of acoustic waves used in medical imaging and analyze the advantages associated with shear acoustic imaging. A comprehensive analysis of the range of acoustic wavelengths, velocities and frequencies that have been used in different imaging applications is presented. We discuss the potential for future shear wave imaging applications.

Ultrasonic characterization of the nonlinear properties of canine livers by measuring shear wave speed and axial strain with increasing portal venous pressure

Elevated hepatic venous pressure is the primary source of complications in advancing liver disease. Ultrasound imaging is ideal for potential noninvasive hepatic pressure measurements as it is widely used for liver imaging. Specifically, ultrasound based stiffness measures may be useful for clinically monitoring pressure, but the mechanism by which liver stiffness increases with hepatic pressure has not been well characterized. This study is designed to elucidate the nonlinear properties of the liver during pressurization by measuring both hepatic shear wave speed (SWS) and strain with increasing pressure. Tissue deformation during hepatic pressurization was tracked in 8 canine livers using successively acquired 3-D B-mode volumes and compared with concurrently measured SWS. When portal venous pressure was increased from clinically normal (0-5mmHg) to pressures representing highly diseased states at 20mmHg, the liver was observed to expand with axial strain measures up to 10%. At the same time, SWS estimates were observed to increase from 1.5-2m/s at 0-5mmHg (baseline) to 3.25-3.5m/s at 20mmHg.

Supersonic shear wave elastography of in vivo pig kidney: influence of blood pressure, urinary pressure and tissue anisotropy

The in vivo influence of renal anisotropy and of urinary and vascular pressure on elasticity values using ultrasonic supersonic shear wave elastography was studied in pigs. Experiments were conducted in agreement with the European Commission guidelines and directives of the French Research Ministry. Six kidneys in three pigs were studied in vivo. Elasticity of renal cortex and medulla was quantified through the shear modulus (μ) by using the supersonic shear imaging technique with an 8 MHz linear ultrasound probe. All measurements were done peroperatively both in the axis and perpendicular to the main axis of pyramids, in normal condition, after progressive increase of urinary pressure, and after renal artery and renal vein ligation. In normal conditions, cortical (C) and medullary (M) elasticity values were always higher when acquisitions were realized with the ultrasound main axis perpendicular to main pyramid axis (C(//): 7.7 ± 2.3 kPa; M(//): 8.7 ± 2.5 kPa) than parallel (C(⊥): 6.9 ± 1.4 kPa; M(⊥): 6.6 ± 2.3 kPa), demonstrating an effect of renal anisotropy. In renal cortex, two bands were separated, inner cortex showing higher elasticity values (IC(⊥): 8.1 ± 1.9 kPa) than outer cortex (OC(⊥): 6.9 ± 1.4 kPa). Renal artery and renal vein ligation induced a decrease and an increase of elasticity respectively. Parenchymal elasticity increased linearly with elevation of urinary pressure. Intrarenal elasticity values vary with tissue anisotropy and, with vascular and urinary pressure levels. These parameters have to be taken into account for interpretation of tissue changes. Separation of outer and inner cortex could be attributable to perfusion differences.

Shear wave elastography may add a new dimension to ultrasound evaluation of thyroid nodules: case series with comparative evaluation

Although elastography can enhance the differential diagnosis of thyroid nodules, its diagnostic performance is not ideal at present. Further improvements in the technique and creation of robust diagnostic criteria are necessary. The purpose of this study was to compare the usefulness of strain elastography and a new generation of elasticity imaging called supersonic shear wave elastography (SSWE) in differential evaluation of thyroid nodules. Six thyroid nodules in 4 patients were studied. SSWE yielded 1 true-positive and 5 true-negative results. Strain elastography yielded 5 false-positive results and 1 false-negative result. A novel finding appreciated with SSWE, were punctate foci of increased stiffness corresponding to microcalcifications in 4 nodules, some not visible on B-mode ultrasound, as opposed to soft, colloid-inspissated areas visible on B-mode ultrasound in 2 nodules. This preliminary paper indicates that SSWE may outperform strain elastography in differentiation of thyroid nodules with regard to their stiffness. SSWE showed the possibility of differentiation of high echogenic foci into microcalcifications and inspissated colloid, adding a new dimension to thyroid elastography. Further multicenter large-scale studies of thyroid nodules evaluating different elastographic methods are warranted.

Strain-induced damage reduces echo intensity changes in tendon during loading

Tendon functionality is related to its mechanical properties. Tendon damage leads to a reduction in mechanical strength and altered biomechanical behavior, and therefore leads to compromised ability to carry out normal functions such as joint movement and stabilization. Damage can also accumulate in the tissue and lead to failure. A noninvasive method with which to measure such damage potentially could quantify structural compromise from tendon injury and track improvement over time. In this study, tendon mechanics are measured before and after damage is induced by "overstretch" (strain exceeding the elastic limit of the tissue) using a traditional mechanical test system while ultrasonic echo intensity (average gray scale brightness in a B-mode image) is recorded using clinical ultrasound. The diffuse damage caused by overstretch lowered the stress at a given strain in the tissue and decreased viscoelastic response. Overstretch also lowered echo intensity changes during stress relaxation and cyclic testing. As the input strain during overstretch increased, stress levels and echo intensity changes decreased. Also, viscoelastic parameters and time-dependent echo intensity changes were reduced.

Quantitative imaging of nonlinear shear modulus by combining static elastography and shear wave elastography

The study of new tissue mechanical properties such as shear nonlinearity could lead to better tissue characterization and clinical diagnosis. This work proposes a method combining static elastography and shear wave elastography to derive the nonlinear shear modulus by applying the acoustoelasticity theory in quasi-incompressible soft solids. Results demonstrate that by applying a moderate static stress at the surface of the investigated medium, and by following the quantitative evolution of its shear modulus, it is possible to accurately and quantitatively recover the local Landau (A) coefficient characterizing the shear nonlinearity of soft tissues.

The impact of hepatic pressurization on liver shear wave speed estimates in constrained versus unconstrained conditions

Increased hepatic venous pressure can be observed in patients with advanced liver disease and congestive heart failure. This elevated portal pressure also leads to variation in acoustic radiation-force-derived shear wave-based liver stiffness estimates. These changes in stiffness metrics with hepatic interstitial pressure may confound stiffness-based predictions of liver fibrosis stage. The underlying mechanism for this observed stiffening behavior with pressurization is not well understood and is not explained with commonly used linear elastic mechanical models. An experiment was designed to determine whether the stiffness increase exhibited with hepatic pressurization results from a strain-dependent hyperelastic behavior. Six excised canine livers were subjected to variations in interstitial pressure through cannulation of the portal vein and closure of the hepatic artery and hepatic vein under constrained conditions (in which the liver was not free to expand) and unconstrained conditions. Radiation-force-derived shear wave speed estimates were obtained and correlated with pressure. Estimates of hepatic shear stiffness increased with changes in interstitial pressure over a physiologically relevant range of pressures (0-35 mmHg) from 1.5 to 3.5 m s(-1). These increases were observed only under conditions in which the liver was free to expand while pressurized. This behavior is consistent with hyperelastic nonlinear material models that could be used in the future to explore methods for estimating hepatic interstitial pressure noninvasively.

On the third- and fourth-order constants of incompressible isotropic elasticity

Consider the constitutive law for an isotropic elastic solid with the strain-energy function expanded up to the fourth order in the strain and the stress up to the third order in the strain. The stress-strain relation can then be inverted to give the strain in terms of the stress with a view to considering the incompressible limit. For this purpose, use of the logarithmic strain tensor is of particular value. It enables the limiting values of all nine fourth-order elastic constants in the incompressible limit to be evaluated precisely and rigorously. In particular, it is explained why the three constants of fourth-order incompressible elasticity μ, Ā, and D are of the same order of magnitude. Several examples of application of the results follow, including determination of the acoustoelastic coefficients in incompressible solids and the limiting values of the coefficients of nonlinearity for elastic wave propagation.

Nonlinear elastic behavior of phantom materials for elastography

The development of phantom materials for elasticity imaging is reported in this paper. These materials were specifically designed to provide nonlinear stress/strain relationship that can be controlled independently of the small strain shear modulus of the material. The materials are mixtures of agar and gelatin gels. Oil droplet dispersions in these materials provide further control of the small strain shear modulus and the nonlinear parameter of the material. Since these materials are mostly water, they are assumed to be incompressible under typical experimental conditions in elasticity imaging. The Veronda-Westman model for strain energy density provided a good fit to all materials used in this study. Materials with a constant gelatin concentration (3.0% dry weight) but varying agar concentration (0.6-2.8% dry weight) demonstrated the same power law relationship between elastic modulus and agar concentration found for pure agar (1.89 +/- 0.02), consistent with percolation theory, and provided a consistent nonlinearity parameter of 4.5 +/- 0.3. The insights provided by this study will form the basis for stable elastography phantoms with stiffness and nonlinear stress/strain relationships in the background that differ from those in the target.

Third- and fourth-order constants of incompressible soft solids and the acousto-elastic effect

Acousto-elasticity is concerned with the propagation of small-amplitude waves in deformed solids. Results previously established for the incremental elastodynamics of exact non-linear elasticity are useful for the determination of third- and fourth-order elastic constants, especially in the case of incompressible isotropic soft solids, where the expressions are particularly simple. Specifically, it is simply a matter of expanding the expression for ρv(2), where ρ is the mass density and v the wave speed, in terms of the elongation e of a block subject to a uniaxial tension. The analysis shows that in the resulting expression: ρv(2) = a+be+ce(2), say, a depends linearly on μ; b on μ and A; and c on μ, A, and D, the respective second-, third, and fourth-order constants of incompressible elasticity, for bulk shear waves and for surface waves.

Third- and fourth-order elasticities of biological soft tissues

In the theory of weakly nonlinear elasticity, Hamilton et al. [J. Acoust. Soc. Am. 116, 41-44 (2004)] identified W=muI(2)+(A/3)I(3)+DI(2) (2) as the fourth-order expansion of the strain-energy density for incompressible isotropic solids. Subsequently, much effort focused on theoretical and experimental developments linked to this expression in order to inform the modeling of gels and soft biological tissues. However, while many soft tissues can be treated as incompressible, they are not in general isotropic, and their anisotropy is associated with the presence of oriented collagen fiber bundles. Here the expansion of W is carried up to fourth order in the case where there exists one family of parallel fibers in the tissue. The results are then applied to acoustoelasticity, with a view to determining the second- and third-order nonlinear constants by employing small-amplitude transverse waves propagating in a deformed soft tissue.

Nonlinear reflection of shock shear waves in soft elastic media

For fluids, the theoretical investigation of shock wave reflection has a good agreement with experiments when the incident shock Mach number is large. But when it is small, theory predicts that Mach reflections are physically unrealistic, which contradicts experimental evidence. This von Neumann paradox is investigated for shear shock waves in soft elastic solids with theory and simulations. The nonlinear elastic wave equation is approximated by a paraxial wave equation with a cubic nonlinear term. This equation is solved numerically with finite differences and the Godunov scheme. Three reflection regimes are observed. Theory is developed for shock propagation by applying the Rankine-Hugoniot relations and entropic constraints. A characteristic parameter relating diffraction and non-linearity is introduced and its theoretical values are shown to match numerical observations. The numerical solution is then applied to von Neumann reflection, where curved reflected and Mach shocks are observed. Finally, the case of weak von Neumann reflection, where there is no reflected shock, is examined. The smooth but non-monotonic transition between these three reflection regimes, from linear Snell-Descartes to perfect grazing case, provides a solution to the acoustical von Neumann paradox for the shear wave equation. This transition is similar to the quadratic non-linearity in fluids.

Theoretical limitations of the elastic wave equation inversion for tissue elastography

This article examines the theoretical limitations of the local inversion techniques for the measurement of the tissue elasticity. Most of these techniques are based on the estimation of the phase speed or the algebraic inversion of a one-dimensional wave equation. To analyze these techniques, the wave equation in an elastic continuum is revisited. It is proven that in an infinite medium, harmonic shear waves can travel at any phase speed greater than the classically known shear wave speed, mu/rho, by demonstrating this for a special case with cylindrical symmetry. Hence in addition to the mechanical properties of the tissue, the phase speed depends on the geometry of the wave as well. The elastic waves in an infinite cylindrical rod are studied. It is proven that multiple phase speeds can coexist for a harmonic wave at a single frequency. This shows that the phase speed depends not only on the mechanical properties of the tissue but also on its shape. The final conclusion is that the only way to avoid theoretical artifacts in the elastograms obtained by the local inversion techniques is to use the shear wave equation as expressed in the curl of the displacements, i.e., the rotations, for the inversion.