Low-frequency Shear Research Articles

Using level set based inversion of arrival times to recover shear wave speed in transient elastography and supersonic imaging

Transient elastography and supersonic imaging are promising new techniques for characterizing the elasticity of soft tissues. Using this method, an ‘ultrafast imaging’ system (up to 10 000 frames s−1) follows in real time the propagation of a low-frequency shear wave. The displacement of the propagating shear wave is measured as a function of time and space. Here we develop a fast level set based algorithm for finding the shear wave speed from the interior positions of the propagating front. We compare the performance of level curve methods developed here and our previously developed (McLaughlin J and Renzi D 2006 Shear wave speed recovery in transient elastography and supersonic imaging using propagating fronts Inverse Problems 22 681–706) distance methods. We give reconstruction examples from synthetic data and from data obtained from a phantom experiment accomplished by Mathias Fink's group (the Laboratoire Ondes et Acoustique, ESPCI, Université Paris VII).

Fibroscan ® : un outil non invasif de mesure de la dureté du foie par élastométrie

Fibroscan ® is a new medical device that measures liver stiffness noninvasively. The method is based on one-dimensional transient elastography. Ultrasound signals (3,5 MHz) are used to follow the propagation of a low-frequency shear elastic wave (50 Hz) that propagates at a velocity that depends on liver stiffness. Measurement is performed on the right lobe of the liver using a probe that is placed on the skin surface. Result is obtained at the end of examination that takes about 5 minutes. Pilot and multicentre studies have shown that liver stiffness is strongly correlated to fibrosis stage obtained from liver biopsy on patients with chronic hepatitis C. Fibroscan ® could be used to improve the evaluation of patients with liver chronic diseases by reducing the number of liver biopsies and by allowing the following of treatments.

Effect of Added Poly(oxyethylene)dodecyl Ether on the Phase and Rheological Behavior of Wormlike Micelles in Aqueous SDS Solutions

Upon the addition of a short EO chain nonionic surfactant, poly(oxyethylene) dodecyl ether (C12EOn), to dilute micellar solution of sodium dodecyl sulfate (SDS) above a particular concentration, a sharp increase in viscosity occurs and a highly viscoelastic micellar solution is formed. The oscillatory-shear rheological behavior of the viscoselastic solutions can be described by the Maxwell model at low shear frequency and combined Maxwell-Rouse model at high shear frequency. This property is typical of wormlike micelles entangled to form a transient network. It is found that when C12EO4 in the mixed system is replaced by C12EO3 the micellar growth occurs more effectively. However, with the further decrease in EO chain length, phase separation occurs before a viscoelastic solution is formed. As a result, the maximum zero-shear viscosity is observed at an appropriate mixing fraction of surfactant in the SDS-C12EO3 system. We also investigated the micellar growth in the mixed surfactant systems by means of small-angle X-ray scattering (SAXS). It was found from the SAXS data that the one-dimensional growth of micelles was obtained in all the SDS-C12EOn (n=0-4) aqueous solutions. In a short EO chain C12EOn system, the micelles grow faster at a low mixing fraction of nonionic surfactant.

Microscopic magnetic resonance elastography (μMRE)

Magnetic resonance elastography (MRE) was extended to the microscopic scale to image low-frequency acoustic shear waves (typically less than 1 kHz) in soft gels and soft biological tissues with high spatial resolution (34 micromx34 micromx500 microm). Microscopic MRE (microMRE) was applied to agarose gel phantoms, frog oocytes, and tissue-engineered adipogenic and osteogenic constructs. Analysis of the low-amplitude shear wave pattern in the samples allowed the material stiffness and viscous loss properties (complex shear stiffness) to be identified with high spatial resolution. microMRE experiments were conducted at 11.74 T in a 56-mm vertical bore magnet with a 10 mm diameterx75 mm length cylindrical space available for the elastography imaging system. The acoustic signals were generated at 550-585 Hz using a piezoelectric transducer and high capacitive loading amplifier. Shear wave motion was applied in synchrony with the MR pulse sequence. The field of view (FOV) ranged from 4 to 14 mm for a typical slice thickness of 0.5 mm. Increasing the agarose gel concentration resulted in an increase in shear elasticity and shear viscosity. Shear wave motion propagated through the frog oocyte nucleus, enabling the measurement of its shear stiffness, and in vitro shear wave images displayed contrast between adipogenic and osteogenic tissue-engineered constructs. Further development of microMRE should enable its use in characterizing stiffer materials (e.g., polymers, composites, articular cartilage) and assessing with high resolution the mechanical properties of developing tissues.

Longitudinal and lateral low frequency head wave analysis in soft media

This article studies the influence of the head wave in the lateral and longitudinal components of the displacements generated by the radiation of low-frequency elastic waves in an isotropic and homogeneous soft solid. Low-frequency shear waves are used to characterize elastic properties of soft tissues. In this context, it is useful to have a detailed study of the low-frequency wave field in this kind of material. A soft medium is characterized by the fact that the head wave is found in the source's axis. Even though its amplitude is small compared with the shear wave, it is possible to be observed experimentally by recording consecutive ultrasonic A-lines while the low-frequency wave propagates inside the medium. A standard one-dimensional speckle tracking technique is employed to measure the displacements. Experimental results were interpreted through the exact Green's function solution to the half-space problem. According to the theoretical and experimental analysis, the head wave and surface related terms in general contribute to the displacements in the low-frequency range. This article thoroughly analyzes and experimentally shows the contribution of the head wave for the lateral component, which is not fully addressed by the literature.

Rheological Behavior and Nonisothermal Crystallization of Polypropylene/Epoxidized Natural Rubber Blends

Blends of polypropylene (PP) with epoxidized natural rubber (ENR) were prepared using melt mixing. Dynamic mechanical analysis (DMA) was employed to investigate the miscibility of the two polymers. The morphology evolution with changes of composition was studied using scanning electron microscopy (SEM). Dynamic shear tests were carried out within a frequency range of 0.01 to 100 rad/s at different temperatures. It was found that the dynamic viscosity of the blend increases significantly with increasing ENR content. The shear thickening can take place at low shear frequencies, depending on temperature and composition. Isothermal plots of log G′ (storage modulus) against log G″ (loss modulus) were obtained. According to the temperature dependence of the plots, the miscibility of the blends is discussed. Differential scanning calorimetry (DSC) was used to investigate the nonisothermal crystallization of the blends. The crystallization process can be described by the modified Ozawa kinetic equation. The values of the kinetic parameter, F(T), were calculated. It was found that the crystallization capability of PP in the blend strongly depends on the composition.

Formation and Disruption of Viscoelastic Wormlike Micellar Networks in the Mixed Surfactant Systems of Sucrose Alkanoate and Polyoxyethylene Alkyl Ether

Addition of lipophilic nonionic amphiphiles, such as short EO-chain polyoxyethylene dodecyl ether (C12EOn, n = 1−4) and monolaurin (ML), to the dilute micellar solution of sucrose hexadecanoate (P1695) induces a sharp increase in viscosity that is attributed to the one-dimensional micellar growth. With successive addition of the nonionics, the viscosity gradually decreases and, eventually, a phase separation occurs. In the maximum-viscosity region of the mixed systems, a viscoelastic solution with a rheological behavior typical of the interwoven network of wormlike micelles is obtained. The oscillatory shear behavior of the systems can be described by the Maxwellian model in the region of low shear frequency. The ability of the lipophilic nonionics to induce micellar growth in the mixed system decreases in the following order: ML ≈ C12EO1 ≈ C12EO2 < C12EO3 < C12EO4. A similar effect of the hydropilic group is observed in the sucrose dodecanoate (SM1200)/C16EOn aqueous system, although the micellar growth...

Broad frequency range characterization of molten polymers

We report a technique for obtaining the continuous retardation spectrum or relaxation spectrum over a very broad range of relaxation times. The major challenge is to obtain reliable data at the long-time end of the spectrum. It is often not practical to use low-frequency oscillatory shear to obtain the necessary data because of the very long time required for the measurements. Polyolefins pose special problems in long-time characterization, because time-temperature superposition is not very useful. Information about the long-time behavior can be obtained by means of creep experiments, but now another problem arises. If we use a stress sufficiently large to yield precise data, the strain may go beyond the limit of linear viscoelastic behavior before steady state is achieved. This problem can be addressed by eliminating the stress well before nonlinearity becomes a problem and tracking the resulting recovery. Using a method proposed by Meissner, the combined creep and recovery data can be used to construct the entire creep curve. The final challenge is to combine the data from the oscillatory shear and creep experiments to obtain a composite linear spectrum. We propose a new method for accomplishing this. Retardation spectra are inferred from both sets of data and compared to reveal the range of times over which both techniques provide reliable data and the range in which each technique is uniquely reliable. The two spectra can then be combined to form a composite spectrum, which can be used to calculate any linear material function. This procedure is demonstrated for two branched polypropylenes whose terminal zones could not be accessed by means of oscillatory shear. For each of these materials, we found that the spectra obtained by the two experimental techniques superposed perfectly within the region of overlapping experimental windows, making it possible to construct a reliable composite spectrum valid over a wide range of times. The composite continuous spectrum is more directly related to molecular structure than rheological properties, such as the complex modulus, which involve integrals of the spectrum.

3D shear wave generation in soft tissues using the time reversal kaleidoscopes

The time reversal kaleidoscope provides elegant possibilities for both temporal and spatial concentration of acoustic energy in a wide frequency band. Using such ultrasound focused beams, it is possible to generate mechanical sources radiating low frequency shear waves inside tissues. Shear waves are of great interest to investigate tissue viscoelastic properties. In most techniques relying on this concept, generation and imaging of shear waves are realized with a single ultrasonic probe, limiting the capabilities concerning the generation of the shear source. Here we propose a system able to generate and image shear waves in a full 3D volume with any kind of polarization. The focused beam is generated by the time reversal kaleidoscope. By combining the concepts of time reversal mirrors and chaotic reverberating cavities, the time reversal kaleidoscope is able to focus ultrasound beams in a large 3D volume with only some tens of transducers and presents equivalent performances than conventional 2D matrices made of thousands of transducers. The kaleidoscope is triggered by an ultrafast imaging system able to image the resulting shear waves. In vitro results in tissue mimicking phantoms are presented. Chaotic cavities of different geometries are tested and compared. [Work supported in part by NIH grant.]

Shear-induced grain boundary motion for lamellar phases in the weakly nonlinear regime.

We study the effect of an externally imposed oscillatory shear on the motion of a grain boundary that separates differently oriented domains of the lamellar phase of a diblock copolymer. A direct numerical solution of the Swift-Hohenberg equation in shear flow is used for the case of a transverse/parallel grain boundary in the limits of weak nonlinearity and low shear frequency. We focus on the region of parameters in which both transverse and parallel lamellae are linearly stable. Shearing leads to excess free energy in the transverse region relative to the parallel region, which is in turn dissipated by net motion of the boundary toward the transverse region. The observed boundary motion is a combination of rigid advection by the flow and order parameter diffusion. The latter includes break up and reconnection of lamellae, as well as a weak Eckhaus instability in the boundary region for sufficiently large strain amplitude that leads to slow wave number readjustment. The net average velocity is seen to increase with frequency and strain amplitude, and can be obtained by a multiple scale expansion of the governing equations.

Prospects for elasticity reconstruction in the heart

The elastic moduli in anisotropic media can be estimated using either direct mechanical or sound speed measurements. Here we compare moduli in the passive heart estimated with different methods and demonstrate that high-frequency (i.e., ultrasonic) sound speed measurements are inconsistent with static deformations and low-frequency shear wave results. Both tissue fixation and the high-operating frequency of ultrasonic measurements contribute to these discrepancies. Moreover, the precision of ultrasonic sound speed measurements required to estimate elastic moduli describing static deformations of a nearly incompressible anisotropic medium such as the heart appears to be beyond the scope of current methods. We conclude that an incompressible anisotropic elastic model is appropriate for elasticity reconstruction in the heart, in which three independent constants characterize small strain behavior, but four are needed for a fully nonlinear description of finite deformations.

Prospects for Elasticity Reconstruction in the Heart

The elastic moduli in anisotropic media can be estimated using either direct mechanical or sound speed measurements. Here we compare moduli in the passive heart estimated with different methods and demonstrate that high-frequency (i.e., ultrasonic) sound speed measurements are inconsistent with static deformations and low-frequency shear wave results. Both tissue fixation and the high-operating frequency of ultrasonic measurements contribute to these discrepancies. Moreover, the precision of ultrasonic sound speed measurements required to estimate elastic moduli describing static deformations of a nearly incompressible anisotropic medium such as the heart appears to be beyond the scope of current methods. We conclude that an incompressible anisotropic elastic model is appropriate for elasticity reconstruction in the heart, in which three independent constants characterize small strain behavior, but four are needed for a fully nonlinear description of finite deformations.

Low-Frequency Shear Electromagnetic Waves in a Nonuniform Pair Plasma

The hydrodynamic equations for a pair plasma along with Poisson's equation and Ampère's law are used to derive nonlinear equations for low-frequency (in comparison with the electron gyrofrequency) shear electromagnetic waves in a nonuniform hot magnetoplasma. In the linear approximation, the frequencies of the dispersive magnetic drift-shear electromagnetic waves are obtained. The results should help to understand the properties of low-frequency electromagnetic waves in strongly magnetized nonuniform astrophysical and laboratory electron-positron plasmas.

Measurement of elastic nonlinearity of soft solid with transient elastography.

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

Motion of a Transverse/Parallel Grain Boundary in a Block Copolymer under Oscillatory Shear Flow

A mesoscopic model of a diblock copolymer is used to study the motion of a grain boundary separating two regions of perfectly ordered lamellar structures under an oscillatory but uniform shear flow. The case considered is a grain boundary separating lamellae along the so-called parallel orientation (with wavevector parallel to the velocity gradient direction) and along the transverse orientation (wavevector parallel to the shear direction). In the model considered, lamellae in the parallel orientation are marginal with respect to the shear, whereas transverse lamellae are uniformly compressed instead. A multiple scale expansion valid in the weak segregation regime and for low shear frequencies leads to a pair of envelope equations for the grain boundary. These equations show that the grain boundary moves by the action of the shear, with a net average velocity toward the transverse region. Three different dynamical effects are at play: a rigid deformation of the transverse region by the shear which increases its free energy, diffusive relaxation of the order parameter in the grain boundary region leading to relative phase motion between the two domains during a shear cycle, and wavenumber adjustment in the transverse region. We show that the average velocity of the boundary is an increasing function of shear frequency and that, except at very low frequencies, it can be expressed as the product of a mobility coefficient and a driving force given by the excess energy stored in the transverse phase being sheared.

In vivo breast tumor detection using transient elastography

This paper presents first in vivo experiments for breast tumor detection using transient elastography. This technique has been developed for detection and quantitative mapping of hard lesions in soft tissues. It consists in following the propagation inside soft tissues of very low-frequency shear waves (∼ 60 Hz) generated by a vibrating system located at the body surface. Because transient shear waves propagate through the medium in less than 0.1 s, the shear propagation imaging is performed with an ultrafast echographic scanner able to reach frame rates up to 6000 Hz. The local shear wave speed is directly linked to the local shear Young's modulus of the medium. The shear elasticity map of the medium can then be computed using an inversion algorithm. In vivo experiments were conducted on 15 women who had palpable breast lesions. For clinical adaptability reasons, shear waves were generated by the surface of the ultrasound (US) imaging transducer itself, which was linked to a mechanical vibrator. Our preliminary in vivo results demonstrate the clinical applicability of the transient elastography technique for breast lesion detection. (E-mail: michael.tanter@espci.fr)

Phase and Rheological Behavior of Salt-Free Alkyltrimethylammonium Bromide/Alkanoyl-N-methylethanolamide/Water Systems

Addition of N-hydroxyethyl-N-methyldodecanamide (NMEA-12) and N-hydroxyethyl-N-methylhexadecanamide (NMEA-16) to dilute solutions of hexadecyltrimethylammonium bromide (CTAB) and dodecyltrimethylammonium bromide (DTAB) in the micellar (Wm) phase results in an increase in viscosity. It is found that the CTAB−NMEA-12 or CTAB−NMEA-16 and DTAB−NMEA-16 surfactant systems show viscoelastic behavior typical of systems containing wormlike micelles. The dynamic viscoelastic behavior of the viscoelastic micellar phase follows the Maxwell model at low shear frequency at the composition of maximum viscosity. The mixing fraction of NMEA in total amphiphile for the maximum viscosity increases with decreasing the total amphiphile concentration. Most probably, in a dilute region, as, the effective cross-sectional area per surfactant at the hydrophobic interface in the micelle, increases and more NMEA is needed to decrease the average as. Assuming that one-dimensional growth of micelles takes place upon addition of NMEA and as for the surfactant and NMEA are constant, the rod-micellar length was calculated as a function of the mixing fraction of NMEA in total amphiphile. As a result, the rodlike micellar length is not largely increased up to certain amount of added NMEA, above which the enormous increase in micellar length takes place. The micellar growth can be simply explained by decreasing the effective cross-sectional area per amphiphile upon addition of NMEA.

Transient elastography in anisotropic medium: application to the measurement of slow and fast shear wave speeds in muscles.

From the measurement of a low frequency (50-150 Hz) shear wave speed, transient elastography evaluates the Young's modulus in isotropic soft tissues. In this paper, it is shown that a rod source can generate a low frequency polarized shear strain waves. Consequently this technique allows to study anisotropic medium such as muscle. The evidence of the polarization of low frequency shear strain waves is supported by both numeric simulations and experiments. The numeric simulations are based on theoretical Green's functions in isotropic and anisotropic media (hexagonal system). The experiments in vitro led on beef muscle proves the pertinent of this simple anisotropic pattern. Results in vivo on man biceps shows the existence of slow and fast shear waves as predicted by theory.

Shear viscoelastic properties of liquids and their boundary layers

An acoustical resonance method with piezoquartz vibrator was used in the experimental determination of shear elasticity modulus and a tangent of mechanical loss angle of studied liquids and their boundary layers. It has been shown that liquid has an earlier unknown low frequency (approx. 100 kHz) viscoelastic relaxation process. The experimental results of investigation of low frequency shear elasticity of different class of liquids and their solutions have been presented. An experimental research of shear properties in dependence on shear deformation rate has been carried out. The possibility of the discovery of anomalous high viscosity of liquids has also been considered.

Surface characterization of cross-linked elastomers by shear modulation force microscopy

Abstract Brominated poly(isobutylene- co -4-methylstyrene) (BIMS) is a synthetic terpolymer which can be stoichiometrically cross-linked by N , N ′-dicinnamylidene-1,6-hexanediamine (DIAK). The degree of cross-links on bulk samples was determined by measuring the low frequency shear modulus with parallel plate method. The surface modulus of the same samples was measured using a new method, shear modulation force microscopy (SMFM). The moduli of the bulk sample and the surface were found to have the same scaling with the cross-link density, and good agreement with rubber elasticity theory was obtained in both cases. The SMFM was then used to monitor the cross-linking process, as a function of DIAK concentration and curing time, in thin films where standard rheological methods cannot be applied. The results show that in all cases the cross-linking reaction at the surface reaches saturation in less than 10 min as compared to at least 1 h in the bulk samples.