Analysis of wave patterns in MR elastography of skeletal muscle using coupled harmonic oscillator simulations

The ability to study muscle elasticity in vivo would be of great clinical interest. Magnetic resonance elastography (MRE) has the potential to quantify noninvasively the distribution of the shear modulus in muscle tissue. Elasticity information may be derived by extracting frequencies from the wave patterns of phase-contrast MRE images. In a new approach, MRE wave patterns were reconstructed using 3D coupled harmonic oscillator calculations (CHO). To analyze in vivo MRE measurements of the biceps brachii of healthy volunteers, different anisotropic fibrous structures for the couplings between the muscle elements have to be assumed. V-shaped wave patterns as observed when excitation was applied on the tendon were reproduced by a model, where in a central band of stiff fascicles wave propagation was about twice as fast as that in surrounding tissue. Planar waves were observed for excitation near the muscle surface. They could be reconstructed by assuming a simultaneous wave excitation of all muscle fibers, where fibers along the main muscle axis were coupled more strongly than those perpendicular to the axis. The results show that CHO calculations provide a fast and reliable method for incorporating anatomical information of the investigated tissue in the reconstruction of complex wave patterns.     

Effect of off-frequency sampling in magnetic resonance elastography

In magnetic resonance elastography (MRE), shear waves at a certain frequency are encoded through bipolar gradients that switch polarity at a controlled encoding frequency and are offset in time to capture wave propagation using a controlled sampling frequency. In brain MRE, there is a possibility that the mechanical actuation frequency is different from the vibration frequency, leading to a mismatch with encoding and sampling frequencies. This mismatch can occur in brain MRE from causes both extrinsic and intrinsic to the brain, such as scanner bed vibrations or active damping in the head. The purpose of this work was to investigate how frequency mismatch can affect MRE shear stiffness measurements. Experiments were performed on a dual-medium agarose gel phantom, and the results were compared with numerical simulations to quantify these effects. It is known that off-frequency encoding alone results in a scaling of wave amplitude, and it is shown here that off-frequency sampling can result in two main effects: (1) errors in the overall shear stiffness estimate of the material on the global scale and (2) local variations appearing as stiffer and softer structures in the material. For small differences in frequency, it was found that measured global stiffness of the brain could theoretically vary by up to 12.5% relative to actual stiffness with local variations of up to 3.7% of the mean stiffness. It was demonstrated that performing MRE experiments at a frequency other than that of tissue vibration can lead to artifacts in the MRE stiffness images, and this mismatch could explain some of the large-scale scatter of stiffness data or lack of repeatability reported in the brain MRE literature.     

Magnetic resonance elastography and diffusion-weighted imaging of the sol/gel phase transition in agarose

The dynamics of the sol/gel phase transition in agarose was analyzed with magnetic resonance elastography (MRE) and diffusion-weighted imaging, providing complementary information on a microstructural as well as on a macroscopic spatial scale. In thermal equilibrium, the diffusion coefficient of agarose is linearly correlated with temperature, independent of the sol/gel phase transition. In larger agarose samples, the transition from the sol to the gel state was characterized by a complex position and temperature dependency of both MRE shear wave patterns and apparent diffusion coefficients (ADC). The position dependency of the temperature was experimentally found to be qualitatively similar to the behavior of the ADC maps. The dynamics of the temperature could be described with a simplified model that described the heat exchange between sol and gel compartments. The experiments supported the approach to derive temperature maps from the ADC maps by a linear relationship. The spatially resolved dynamics of the temperature maps were therefore employed to determine the elasticities. For this reason, experimental MRE data were simulated using a model of coupled harmonic oscillators. The calculated images agreed well with the experimentally observed MRE wave patterns.     

Diffraction-biased shear wave fields generated with longitudinal magnetic resonance elastography drivers

Magnetic resonance elastography (MRE) is a technique for quantifying the acoustic response of biological tissues to propagating waves applied at low frequencies in order to evaluate mechanical properties. Application-specific MRE drivers are typically required to effectively deliver shear waves within the tissue of interest. Surface MRE drivers with transversely oriented vibrations have often been used to directly generate shear waves. These drivers may have disadvantages in certain applications, such as poor penetration depth and inflexible orientation. Therefore, surface MRE drivers with longitudinally oriented vibrations are used in some situations. The purpose of this work was to investigate and optimize a longitudinal driver system for MRE applications. A cone-like hemispherical distribution of shear waves being generated by these drivers and the wave propagation being governed by diffraction in the near field are shown. Using MRE visualization of the vector displacement field, we studied the properties of the shear wave field created by longitudinal MRE drivers of various sizes to identify optimum shear wave imaging planes. The results offer insights and improvements in both experimental design and imaging plane selection for 2-D MRE data acquisition.     

基于双带宽高斯滤波器的磁共振弹性图局域频率估算算法研究与实现

本文对生物体软组织的生物力学模型进行了分析推导和简化,得到了磁共振相位图与生物组织弹性之间的力学关系,根据该关系研究了局域频率估算算法,应用Matlab对算法进行了编程实现,体模弹性图的实现验证了算法有效性,为后续开展临床肝纤维化分级研究奠定基础. 关键词： 磁共振弹性成像 弹性生物力学模型 局域频率估算     

On the feasibility of elastic wave visualization within polymeric solids using magnetic resonance elastography

In this paper, the feasibility of extending previously described magnetic resonance elastography (MRE) dynamic displacement (and associated elasticity) measurement techniques, currently used successfully in tissue, to solid materials which have much higher shear rigidity and much lower nuclear spin densities, is considered. Based on these considerations, the MRE technique is modified in a straightforward manner and used to directly visualize shear wave displacements within two polymeric materials, one of which is relatively stiff.     

Theoretical Analysis of Shear Wave Interference Patterns by Means of Dynamic Acoustic Radiation Forces

Acoustic radiation forces associated with high intensity focused ultrasound stimulate shear wave propagation allowing shear wave speed and shear viscosity estimation of tissue structures. As wave speeds are meters per second, real time displacement tracking over an extend field-of-view using ultrasound is problematic due to very high frame rate requirements. However, two spatially separated dynamic external sources can stimulate shear wave motion leading to shear wave interference patterns. Advantages are shear waves can be imaged at lower frame rates and local interference pattern spatial properties reflect tissue's viscoelastic properties. Here a theoretical analysis of shear wave interference patterns by means of dynamic acoustic radiation forces is detailed. Using a viscoelastic Green's function analysis, tissue motion due to a pair of focused ultrasound beams and associated radiation forces are presented. Overall, this paper theoretically demonstrates shear wave interference patterns can be stimulated using dynamic acoustic radiation forces and tracked using conventional ultrasound imaging.     

Lateral field excitation properties of langasite single crystal

In this work, bulk acoustic wave propagation properties of langasite single crystal excited by lateral electric field have been investigated. Three important crystal cuts have been identified for different operational modes of lateral field excitation (LFE) on langasite substrate, namely the (yxl)65o (pure-LFE mode), (yxl)45o (quasi-LFE mode), and (yxl)0o (pseudo-LFE mode). Devices on langasite substrate with the above cuts were fabricated and tested, and the experimental results agree well with the theoretical analysis. It is found that a pure thickness shear mode exists in the (yxl)65o langasite LFE device with the bare side facing liquid, and no spurious mode is found due to its moderately large piezoelectric coupling factor. In addition, (yxl)0o langasite LFE device is also found suitable for liquid phase sensing applications.     

Stiffness change of the supraspinatus muscle can be detected by magnetic resonance elastography

Magnetic resonance elastography (MRE) and ultrasound shear wave elastography (SWE) are imaging techniques to measure stiffness of the soft tissue using magnetic resonance imaging (MRI) and ultrasound images, respectively. The purpose of this study was to explore the feasibility of the MRE measurement to evaluate the change in supraspinatus (SSP) muscle stiffness before and after rotator cuff tear, and to compare the result with those of SWE. Six swine shoulders were used. The skin and subcutaneous fat were removed, and the stiffness value of the SSP muscle was measured by MRE and SWE. The MRE measurement was performed with 0.3 T open MRI and the vibration from a pneumatic driver system with active driver to a passive driver to create the shear wave in the tissue. The passive driver was placed on the center of the SSP muscle. The stiffness was estimated from the wave images using local frequency estimation methods. In the SWE measurement, the probe of the ultrasound was placed on the center of the SSP muscle. The shear wave propagation speed was measured at a depth of 1 cm from the surface, and the stiffness was calculated. After those measurements, the rotator cuff tendon was detached from the greater tuberosity, and MRE and SWE measurements were then performed in the same manner again. The differences in the stiffness values were compared between before and after the rotator cuff tendon tear on both the MRE and SWE measurements. The results indicated that stiffness values on MRE and SWE were 9.3 ± 1.8 and 10.0 ± 1.2 kPa respectively before the rotator cuff tear, and 7.3 ± 1.3 and 8.0 ± 0.8 kPa respectively after the tendon detachment. Stiffness values were significantly lower after the tendon detachment on both the MRE and SWE measurements (p < 0.05). Our results demonstrated that stiffness values of the SSP muscle on MRE and SWE were lower after rotator cuff detachment. From this result, MRE may be a feasible method for quantification of the change in rotator cuff muscle stiffness.     

Influence of composition of carbonyl iron particles on dynamic mechanical properties of magnetorheological elastomers

Magnetorheological elastomers (MRE) are known as smart materials. However, the magnetorheological (MR) effect of MRE is not high enough at present, which limits its engineering applications. Prior studies have shown that magneto-induced shear storage modulus and MR effect were mainly determined by the performance of the ferromagnetic particles. In this paper, MRE samples were prepared by carbonyl iron particles (CIP) of different compositions based on silicon rubber under external magnetic field. Their microstructures were observed using an optical digital microscope and a scanning electron microscope. The dynamic mechanical properties of MRE samples were measured using a modified dynamic mechanical analyzer under varying magnetic field strength and frequency. The results show that the carbon content of CIP have a greater impact on the dynamic mechanical properties of MRE. The magneto-induced shear storage modulus and MR effect can be increased by selecting CIP of low carbon content. In addition, the damping property is also significantly influenced by the carbon content of the CIP. This study is expected to provide guidance for fabrication of high performance MRE.     

Lateral-field-excitation properties of LiNbO3 single crystal

LiNbO 3 has been found attractive for lateral field excitation (LFE) applications due to its high piezoelectric coupling. In this paper, bulk acoustic wave propagation properties for LiNbO 3 single crystal excited by a lateral electric field have been investigated using the extended Christoffel-Bechmann method. It is found that the LFE piezoelectric coupling factor for c mode reaches its maximum value of 95.46% when ψ = 0 for both (yxl)-58 and (yxwl)±60 /58 LiNbO 3 . The acoustic wave phase velocity of c mode TSM (thickness shear mode) changes from 3456 m/s to 3983 m/s as a function of ψ. Here ψ represents the angle between the lateral electric field and the crystallographic X-axis in the substrate major surface. A 5 MHz LFE device of (yxl)-58 LiNbO 3 with ψ = 0 was designed and tested in air. A major resonance peak was observed with the motional resistance as low as 17 and the Q-factor value up to 10353. The test result is well in agreement with the theoretical analysis, and suggests that the LFE LiNbO 3 device can be a good platform for high performance resonator or sensor applications.     

Magnetic resonance elastography of the brain in a mouse model of Alzheimer's disease: initial results

The increasing prevalence of Alzheimer's disease (AD) has provided motivation for developing novel methods for assessing the disease and the effects of potential treatments. Magnetic resonance elastography (MRE) is an MRI-based method for quantitatively imaging the shear tissue stiffness in vivo. The objective of this research was to determine whether this new imaging biomarker has potential for characterizing neurodegenerative disease. Methods were developed and tested for applying MRE to evaluate the mouse brain, using a conventional large bore 3.0T MRI system. The technique was then applied to study APP-PS1 mice, a well-characterized model of AD. Five APP-PS1 mice and 8 age-matched wild-type mice were imaged immediately following sacrifice. Brain shear stiffness measurements in APP-PS1 mice averaged 22.5% lower than those for wild-type mice (P = .0031). The results indicate that mouse brain MRE is feasible at 3.0T, and brain shear stiffness has merit for further investigation as a potential new biomarker for Alzheimer's disease.     

Modeling shear modulus distribution in magnetic resonance elastography with piecewise constant level sets

Magnetic resonance elastography (MRE) is designed for imaging the mechanical properties of soft tissues. However, the interpretation of shear modulus distribution is often confusing and cumbersome. For reliable evaluation, a common practice is to specify the regions of interest and consider regional elasticity. Such an experience-dependent protocol is susceptible to intrapersonal and interpersonal variability. In this study we propose to remodel shear modulus distribution with piecewise constant level sets by referring to the corresponding magnitude image. Optimal segmentation and registration are achieved by a new hybrid level set model comprised of alternating global and local region competitions. Experimental results on the simulated MRE data sets show that the mean error of elasticity reconstruction is 11.33% for local frequency estimation and 18.87% for algebraic inversion of differential equation. Piecewise constant level set modeling is effective to improve the quality of shear modulus distribution, and facilitates MRE analysis and interpretation.     

Feasibility of MR elastography of the intervertebral disc

Low back pain (LBP) is a costly and widely prevalent health disorder in the U.S. One of the most common causes of LBP is degenerative disc disease (DDD). There are many imaging techniques to characterize disc degeneration; however, there is no way to directly assess the material properties of the intervertebral disc (IVD) within the intact spine. Magnetic resonance elastography (MRE) is an MRI-based technique for non-invasively mapping the mechanical properties of tissues in vivo. The purpose of this study was to investigate the feasibility of using MRE to detect shear wave propagation in and determine the shear stiffness of an axial cross-section of an ex vivo baboon IVD, and compare with shear displacements from a finite element model of an IVD motion segment in response to harmonic shear vibration. MRE was performed on two baboon lumbar spine motion segments (L3–L4) with the posterior elements removed at a range of frequencies (1000–1500 Hz) using a standard clinical 1.5 T MR scanner. Propagating waves were visualized in an axial cross-section of the baboon IVDs in all three motion-encoding directions, which resembled wave patterns predicted using finite element modeling. The baboon nucleus pulposus showed an average shear stiffness of 79 ± 15 kPa at 1000 Hz. These results suggest that MRE is capable of visualizing shear wave propagation in the IVD, assessing the stiffness of the nucleus of the IVD, and can differentiate the nucleus and annulus regions.     

Comparison of the surface wave method and the indentation method for measuring the elasticity of gelatin phantoms of different concentrations

The speed of the surface Rayleigh wave, which is related to the viscoelastic properties of the medium, can be measured by noninvasive and noncontact methods. This technique has been applied in biomedical applications such as detecting skin diseases. Static spherical indentation, which quantifies material elasticity through the relationship between loading force and displacement, has been applied in various areas including a number of biomedical applications. This paper compares the results obtained from these two methods on five gelatin phantoms of different concentrations (5%, 7.5%, 10%, 12.5% and 15%). The concentrations are chosen because the elasticity of such gelatin phantoms is close to that of tissue types such as skin. The results show that both the surface wave method and the static spherical indentation method produce the same values for shear elasticity. For example, the shear elasticities measured by the surface wave method are 1.51, 2.75, 5.34, 6.90 and 8.40 kPa on the five phantoms, respectively. In addition, by studying the dispersion curve of the surface wave speed, shear viscosity can be extracted. The measured shear viscosities are 0.00, 0.00, 0.13, 0.39 and 1.22 Pa.s on the five phantoms, respectively. The results also show that the shear elasticity of the gelatin phantoms increases linearly with their prepared concentrations. The linear regressions between concentration and shear elasticity have R² values larger than 0.98 for both methods.     

Rayleigh-Lamb wave propagation on a fractional order viscoelastic plate

A previous study of the authors published in this journal focused on mechanical wave motion in a viscoelastic material representative of biological tissue [Meral et al., J. Acoust. Soc. Am. 126, 3278-3285 (2009)]. Compression, shear and surface wave motion in and on a viscoelastic halfspace excited by surface and sub-surface sources were considered. It was shown that a fractional order Voigt model, where the rate-dependent damping component that is dependent on the first derivative of time is replaced with a component that is dependent on a fractional derivative of time, resulted in closer agreement with experiment as compared with conventional (integer order) models, such as those of Voigt and Zener. In the present study, this analysis is extended to another configuration and wave type: out-of-plane response of a viscoelastic plate to harmonic anti-symmetric Lamb wave excitation. Theoretical solutions are compared with experimental measurements for a polymeric tissue mimicking phantom material. As in the previous configurations the fractional order modeling assumption improves the match between theory and experiment over a wider frequency range. Experimental complexities in the present study and the reliability of the different approaches for quantifying the shear viscoelastic properties of the material are discussed.     

Coupled flexural-longitudinal wave motion in a periodic beam

Periodic structure theory is used to study the interactions between flexural and longitudinal wave motion in a beam (representing a plate) to which offset spring-mounted masses (representing stiffeners) are attached at regular intervals. An equation for the propagation constants of the coupled waves is derived. The response of a semi-infinite periodic beam to a harmonic force or moment at the finite end is analyzed in terms of the characteristic free waves corresponding to these propagation constants. Computer results are presented which show how the propagation constants are affected by the coupling, and how the forced response varies with distance from the excitation point. The spring-mounted masses can provide very high attenuation of both longitudinal and flexural waves when no coupling is present, but when coupling is introduced the two waves combine to give very low (or zero) attenuation of the longitudinal wave. The influence of different damping levels on spatial attenuation is also studied.     

Investigation of quasi lateral-field-excitation on (yxl)-17° LiNbO3 single crystal

Quasi lateral-field-excitation (LFE) on LiNbO₃ crystal is investigated both theoretically and experimentally. It is found that when the driving electric field direction is parallel to the crystallographic X-axis of the piezoelectric substrate, (yxl)-17° LiNbO₃ LFE bulk acoustic wave devices work on quasi-LFE mode. The experimental results agreed with the theoretical prediction well. The results provide the cut of LiNbO₃ crystal for quasi-LFE bulk acoustic wave devices, which is important for designing high performance LFE sensors on LiNbO₃ substrates.     

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.