A finite element model for analyzing shear wave propagation observed in magnetic resonance elastography

Numerical simulation of wave propagation through interfaces using the extended finite element method for magnetic resonance elastography

Magnetic resonance elastography (MRE) is an elasticity imaging technique for quantitatively assessing the stiffness of human tissues. In MRE, finite element method (FEM) is widely used for modeling wave propagation and stiffness reconstruction. However, in front of inclusions with complex interfaces, FEM can become burdensome in terms of the model partition and computationally expensive. In this work, we implement a formulation of FEM, known as the eXtended finite element method (XFEM), which is a method used for modeling discontinuity like crack and heterogeneity. Using a level-set method, it makes the interface independent of the mesh, thus relieving the meshing efforts. We investigate this method in two studies: wave propagation across an oblique linear interface and stiffness reconstruction of a random-shape inclusion. In the first study, numerical results by XFEM and FEM models revealing the wave conversion rules at linear interface are presented and successfully compared to the theoretical predictions. The second study, investigated in a pseudo-practical application, demonstrates further the applicability of XFEM in MRE and the convenience, accuracy, and speed of XFEM with respect to FEM. XFEM can be regarded as a promising alternative to FEM for inclusion modeling in MRE.

A 2D finite element model for shear wave propagation in biological soft tissues: Application to magnetic resonance elastography

Dynamic elastography is a virtual palpation tool that aims at investigating the mechanical response of biological soft tissues in vivo. The objective of this study is to develop a finite element model (FEM) with low computational cost for reproducing realistically wave propagation for magnetic resonance elastography in heterogeneous soft tissues. Based on the first-order shear deformation theory for moderately thick structures, this model is developed and validated through comparison with analytical formulations of wave propagating in heterogeneous, viscoelastic infinite medium. This 2D-FEM is then compared to experimental data and a 3D-FEM using a commercial software. Our FEM is a powerful promising tool for investigations of magnetic resonance elastography.

ELASTICITY MEASUREMENTS BY SHEAR WAVE ELASTOGRAPHY: COMPARISON AND SELECTION OF SHEAR WAVE, RAYLEIGH WAVE AND LAMB WAVE THEORY

Shear wave elastography (SWE) is a powerful method for the diagnosis of tissue disorders or degeneration based on tissue elasticity. In SWE application, it was recognized that the wave speed depends not only on the tissue elasticity but also on the structural shape, leading to different theoretical models. For liver, skin and myocardium, the appropriate theoretical model is known to be shear wave, Rayleigh wave and Lamb wave theory, respectively. Therefore, appropriate theoretical model should be adopted for the proper application of SWE. In this study, we verify these theoretical models in gelatin samples of different thicknesses, using experimental and numerical SWE tests. The results indicate that the wave speed was influenced by the ratio of the wavelength and sample thickness and the measurement region. Based on these results, the selection of theoretical model could be divided into three cases, and the appropriate theoretical model can be selected accordingly.

Numerical simulations of magnetic resonance elastography using finite element analysis with a linear heterogeneous viscoelastic model

Abstract

Magnetic resonance elastography (MRE) is a technique to identify the viscoelastic moduli of biological tissues by solving the inverse problem from the displacement field of viscoelastic wave propagation in a tissue measured by MRI. Because finite element analysis (FEA) of MRE evaluates not only the viscoelastic model for a tissue but also the efficiency of the inversion algorithm, we developed FEA for MRE using commercial software called ANSYS, the Zener model for displacement field of a wave inside tissue, and an inversion algorithm called the modified integral method. The profile of the simulated displacement field by FEA agrees well with the experimental data measured by MRE for gel phantoms. Similarly, the value of storage modulus (i.e., stiffness) recovered using the modified integral method with the simulation data is consistent with the value given in FEA. Furthermore, applying the suggested FEA to a human liver demonstrates the effectiveness of the present simulation scheme.

Graphical abstract

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.

Finite element analysis to investigate variability of MR elastography in the human thigh

PURPOSE

To develop finite element analysis (FEA) of magnetic resonance elastography (MRE) in the human thigh and investigate inter-individual variability of measurement of muscle mechanical properties.

METHODS

Segmentation was performed on MRI datasets of the human thigh from 5 individuals and FEA models consisting of 12 muscles and surrounding tissue created. The same material properties were applied to each tissue type and a previously developed transient FEA method of simulating MRE using Abaqus was performed at 4 frequencies. Synthetic noise was applied to the simulated data at various levels before inversion was performed using the Elastography Software Pipeline. Maps of material properties were created and visually assessed to determine key features. The coefficient of variation (CoV) was used to assess the variability of measurements in each individual muscle and in the groups of muscles across the subjects. Mean measurements for the set of muscles were ranked in size order and compared with the expected ranking.

RESULTS

At noise levels of 2% the CoV in measurements of |G^*| ranged from 5.3 to 21.9% and from 7.1 to 36.1% for measurements of ϕ in the individual muscles. A positive correlation (R² value 0.80) was attained when the expected and measured |G^*| ranking were compared, whilst a negative correlation (R² value 0.43) was found for ϕ.

CONCLUSIONS

Created elastograms demonstrated good definition of muscle structure and were robust to noise. Variability of measurements across the 5 subjects was dramatically lower for |G^*| than it was for ϕ. This large variability in ϕ measurements was attributed to artefacts.

The simulation of magnetic resonance elastography through atherosclerosis

The clinical diagnosis of atherosclerosis via the measurement of stenosis size is widely acknowledged as an imperfect criterion. The vulnerability of an atherosclerotic plaque to rupture is associated with its mechanical properties. The potential to image these mechanical properties using magnetic resonance elastography (MRE) was investigated through synthetic datasets. An image of the steady state wave propagation, equivalent to the first harmonic, can be extracted directly from finite element analysis. Inversion of this displacement data yields a map of the shear modulus, known as an elastogram. The variation of plaque composition, stenosis size, Gaussian noise, filter thresholds and excitation frequency were explored. A decreasing mean shear modulus with an increasing lipid composition was identified through all stenosis sizes. However the inversion algorithm showed sensitivity to parameter variation leading to artefacts which disrupted both the elastograms and quantitative trends. As noise was increased up to a realistic level, the contrast was maintained between the fully fibrous and lipid plaques but lost between the interim compositions. Although incorporating a Butterworth filter improved the performance of the algorithm, restrictive filter thresholds resulted in a reduction of the sensitivity of the algorithm to composition and noise variation. Increasing the excitation frequency improved the techniques ability to image the magnitude of the shear modulus and identify a contrast between compositions. In conclusion, whilst the technique has the potential to image the shear modulus of atherosclerotic plaques, future research will require the integration of a heterogeneous inversion algorithm.

Rescaled Local Interaction Simulation Approach for Shear Wave Propagation Modelling in Magnetic Resonance Elastography

Properties of soft biological tissues are increasingly used in medical diagnosis to detect various abnormalities, for example, in liver fibrosis or breast tumors. It is well known that mechanical stiffness of human organs can be obtained from organ responses to shear stress waves through Magnetic Resonance Elastography. The Local Interaction Simulation Approach is proposed for effective modelling of shear wave propagation in soft tissues. The results are validated using experimental data from Magnetic Resonance Elastography. These results show the potential of the method for shear wave propagation modelling in soft tissues. The major advantage of the proposed approach is a significant reduction of computational effort.

Magnetic resonance elastography of abdomen

Many diseases cause substantial changes in the mechanical properties of tissue, and this provides motivation for developing methods to noninvasively assess the stiffness of tissue using imaging technology. Magnetic resonance elastography (MRE) has emerged as a versatile MRI-based technique, based on direct visualization of propagating shear waves in the tissues. The most established clinical application of MRE in the abdomen is in chronic liver disease. MRE is currently regarded as the most accurate noninvasive technique for detection and staging of liver fibrosis. Increasing experience and ongoing research is leading to exploration of applications in other abdominal organs. In this review article, the current use of MRE in liver disease and the potential future applications of this technology in other parts of the abdomen are surveyed.

Stiffness of resting lumbar myofascia in healthy young subjects quantified using a handheld myotonometer and concurrently with surface electromyography monitoring

This study aimed to non-invasively quantify passive stiffness of superficial myofascia at a lower lumbar (L3-L4) anatomical level in young healthy male and female subjects and investigate its possible morphological variation. Resting prone lumbar myofascial measurements were quantified using MyotonPro(®) and statistically analyzed in 20 young healthy individuals over 3-weekly intervals, concurrently with surface electromyography (sEMG). Averaged mean ± SE stiffness (Newton/meter) over three weeks was significantly (p < 0.001) greater in males (247.8 ± 11.3) than females (208.4 ± 11.3), on the right (237.7 ± 12.8) than left sides (218.5 ± 12.3), at 10-min (231.4 ± 9.1) than initial baseline (224.8 ± 9.1) values. A polymorphism of stiffness values in 10 male and 10 female subjects was suggested by box plot analyses of the 3 weekly measurements and greater inter-individual than intra-individual variances. Greater knowledge of lumbar myofascial stiffness can improve understanding of their contributions in health and chronic low back disorders.

Analysis of shear wave propagation derived from MR elastography in 3D thigh skeletal muscle using subject specific finite element model

The purpose of this study was to develop a subject specific finite element model derived from MRI images to numerically analyze the MRE (magnetic resonance elastography) shear wave propagation within skeletal thigh muscles. A sagittal T2 CUBE MRI sequence was performed on the 20-cm thigh segment of a healthy male subject. Skin, adipose tissue, femoral bone and 11 muscles were manually segmented in order to have 3D smoothed solid and meshed models. These tissues were modeled with different constitutive laws. A transient modal dynamics analysis was applied to simulate the shear wave propagation within the thigh tissues. The effects of MRE experimental parameters (frequency, force) and the muscle material properties (shear modulus: C10) were analyzed through the simulated shear wave displacement within the vastus medialis muscle. The results showed a plausible range of frequencies (from 90Hz to 120 Hz), which could be used for MRE muscle protocol. The wave amplitude increased with the level of the force, revealing the importance of the boundary condition. Moreover, different shear displacement patterns were obtained as a function of the muscle mechanical properties. The present study is the first to analyze the shear wave propagation in skeletal muscles using a 3D subject specific finite element model. This study could be of great value to assist the experimenters in the set-up of MRE protocols.

Maximum likelihood estimation of shear wave speed in transient elastography

Ultrasonic transient elastography (TE), enables to assess, under active mechanical constraints, the elasticity of the liver, which correlates with hepatic fibrosis stages. This technique is routinely used in clinical practice to assess noninvasively liver stiffness. The Fibroscan system used in this work generates a shear wave via an impulse stress applied on the surface of the skin and records a temporal series of radio-frequency (RF) lines using a single-element ultrasound probe. A shear wave propagation map (SWPM) is generated as a 2-D map of the displacements along depth and time, derived from the correlations of the sequential 1-D RF lines, assuming that the direction of propagation (DOP) of the shear wave coincides with the ultrasound beam axis (UBA). Under the assumption of pure elastic tissue, elasticity is proportional to the shear wave speed. This paper introduces a novel approach to the processing of the SWPM, deriving the maximum likelihood estimate of the shear wave speed when comparing the observed displacements and the estimates provided by the Green's functions. A simple parametric model is used to interface Green's theoretical values of noisy measures provided by the SWPM, taking into account depth-varying attenuation and time-delay. The proposed method was evaluated on numerical simulations using a finite element method simulator and on physical phantoms. Evaluation on this test database reported very high agreements of shear wave speed measures when DOP and UBA coincide.

Reconstructing 3-D maps of the local viscoelastic properties using a finite-amplitude modulated radiation force

A modulated acoustic radiation force, produced by two confocal tone-burst ultrasound beams of slightly different frequencies (i.e. 2.0 MHz ± Δf/2, where Δf is the difference frequency), can be used to remotely generate modulated low-frequency (Δf ≤ 500 Hz) shear waves in attenuating media. By appropriately selecting the duration of the two beams, the energy of the generated shear waves can be concentrated around the difference frequency (i.e., Δf ± Δf/2). In this manner, neither their amplitude nor their phase information is distorted by frequency-dependent effects, thereby, enabling a more accurate reconstruction of the viscoelastic properties. Assuming a Voigt viscoelastic model, this paper describes the use of a finite-element-method model to simulate three-dimensional (3-D) shear-wave propagation in viscoelastic media containing a spherical inclusion. Nonlinear propagation is assumed for the two ultrasound beams, so that higher harmonics are developed in the force and shear spectrum. Finally, an inverse reconstruction algorithm is used to extract 3-D maps of the local shear modulus and viscosity from the simulated shear-displacement fields based on the fundamental and second-harmonic component. The quality of the reconstructed maps is evaluated using the contrast between the inclusion and the background and the contrast-to-noise ratio (CNR). It is shown that the shear modulus can be accurately reconstructed based on the fundamental component, such that the observed contrast deviates from the true contrast by a root-mean-square-error (RMSE) of only 0.38 and the CNR is greater than 30 dB. If the second-harmonic component is used, the RMSE becomes 1.54 and the corresponding CNR decreases by approximately 10-15 dB. The reconstructed shear viscosity maps based on the second harmonic are shown to be of higher quality than those based on the fundamental. The effects of noise are also investigated and a fusion operation between the two spectral components is applied to enhance the reconstruction quality. Finally, a modified shear-wave spectroscopy technique, shown to be more robust to noise, is described for the estimation of the viscoelastic properties inside and outside the spherical inclusion under conditions of increased noise.

Theoretical prediction of ultrasound elastography for detection of early osteoarthritis

Ultrasound elastography could be used as a new noninvasive technique for detecting early osteoarthritis. As the first critical step, this study theoretically predicted the excitation power and the measurement errors in detecting cartilage detect. A finite element model was used to simulate wave propagation of elastography in the cartilage. The wave was produced by a force F, and the wave speed C was calculated. The normal cartilage model was used to define the relationship between the wave speed and elastic modulus. Various stiffness values were simulated. F = 10 N with a duration of 0.5 ms was required for having measurable deformation (10 μm) at the distal site. The deformation had a significant rise when the wave crossed the defect. The relationship between the wave speed and elastic parameters was found as C = 1.57 × (E)/(2 × ρ(1+μ)))^1/2, where E was the elastic modulus, μ was Poisson's ratio, and ρ was the density. For the simulated defect with an elastic modulus of 7 MPa which was slightly stiffer than the normal cartilage, the measurement error was 0.1 MPa. The results suggested that, given the simulated conditions, this new technique could be used to detect the defect in early osteoarthritis.

Identification process based on shear wave propagation within a phantom using finite element modelling and magnetic resonance elastography

Magnetic resonance elastography (MRE), based on shear wave propagation generated by a specific driver, is a non-invasive exam performed in clinical practice to improve the liver diagnosis. The purpose was to develop a finite element (FE) identification method for the mechanical characterisation of phantom mimicking soft tissues investigated with MRE technique. Thus, a 3D FE phantom model, composed of the realistic MRE liver boundary conditions, was developed to simulate the shear wave propagation with the software ABAQUS. The assumptions of homogeneity and elasticity were applied to the FE phantom model. Different ranges of mesh size, density and Poisson's ratio were tested in order to develop the most representative FE phantom model. The simulated wave displacement was visualised with a dynamic implicit analysis. Subsequently, an identification process was performed with a cost function and an optimisation loop provided the optimal elastic properties of the phantom. The present identification process was validated on a phantom model, and the perspective will be to apply this method on abdominal tissues for the set-up of new clinical MRE protocols that could be applied for the follow-up of the effects of treatments.

Abdominal magnetic resonance elastography

Magnetic resonance elastography (MRE) is a magnetic resonance imaging-based technique for quantitatively assessing the mechanical properties of tissues based on the propagation of shear waves. Multiple studies have described many potential applications of MRE, from characterizing tumors to detecting diffuse disease processes. Studies have shown that MRE can be successfully implemented to assess abdominal organs. The first clinical application of MRE to be well documented is the detection and characterization of hepatic fibrosis, which systematically increases the stiffness of liver tissue. In this diagnostic role, it offers a safer, less expensive, and potentially more accurate alternative to invasive liver biopsy. Emerging results suggest that measurements of liver and spleen stiffness may provide an indirect way to assess portal hypertension. Preliminary studies have demonstrated that it is possible to use MRE to evaluate the mechanical properties of other abdominal structures, such as the pancreas and kidneys. Steady technical progress in developing practical protocols for applying MRE in the abdomen and the pelvis provides opportunities to explore many other potential applications of this emerging technology.

Synthetic materials in the study of cell response to substrate rigidity

While it has long been understood that cells can sense and respond to a variety of stimuli, including soluble and insoluble factors, light, and externally applied mechanical stresses, the extent to which cells can sense and respond to the mechanical properties of their environment has only recently begun to be studied. Cell response to substrate stiffness has been suggested to play an important role in processes ranging from developmental morphogenesis to the pathogenesis of disease states and may have profound implications for cell and tissue culture and tissue engineering. Given the importance of this phenomenon, there is a clear need for systems for cell study in which substrate mechanics can be carefully defined and varied independently of biochemical and other signals. This review will highlight past work in the field of cell response to substrate rigidity as well as areas for future study.

Ability of magnetic resonance elastography to assess taut bands

BACKGROUND

Myofascial taut bands are central to diagnosis of myofascial pain. Despite their importance, we still lack either a laboratory test or imaging technique capable of objectively confirming either their nature or location. This study explores the ability of magnetic resonance elastography to localize and investigate the mechanical properties of myofascial taut bands on the basis of their effects on shear wave propagation.

METHODS

This study was conducted in three phases. The first involved the imaging of taut bands in gel phantoms, the second a finite element modeling of the phantom experiment, and the third a preliminary evaluation involving eight human subjects-four of whom had, and four of whom did not have myofascial pain. Experiments were performed with a 1.5 T magnetic resonance imaging scanner. Shear wave propagation was imaged and shear stiffness was reconstructed using matched filtering stiffness inversion algorithms.

FINDINGS

The gel phantom imaging and finite element calculation experiments supported our hypothesis that taut bands can be imaged based on its outstanding shear stiffness. The preliminary human study showed a statistically significant 50-100% (P=0.01) increase of shear stiffness in the taut band regions of the involved subjects relative to that of the controls or in nearby uninvolved muscle.

INTERPRETATION

This study suggests that magnetic resonance elastography may have a potential for objectively characterizing myofascial taut bands that have been up to now detectable only by the clinician's fingers.

Identification and quantification of myofascial taut bands with magnetic resonance elastography

OBJECTIVE

To explore the feasibility of using a new magnetic resonance imaging (MRI) technique--magnetic resonance elastography (MRE)--to identify and quantitate the nature of myofascial taut bands.

DESIGN

This investigation consisted of 3 steps. The first involved proof of concept on gel phantoms, the second involved numeric modeling, and the third involved a pilot trial on 2 subjects. Imaging was performed with a 1.5 T MRI machine. Shear waves were produced with a custom-developed acoustically driven pneumatic transducer with gradient-echo image collection gated to the transducer's motion. Shear wave propagation were imaged by MRE.

SETTING

An MRI research laboratory.

PARTICIPANTS

Two women, one with a 3-year history of myofascial pain and the other serving as the control.

INTERVENTIONS

Not applicable.

MAIN OUTCOME MEASURES

MRE images, finite element analysis calculations, and tissue and phantom stiffness determinations.

RESULTS

Results of the phantom measurements, finite element calculations, and study patients were all consistent with the concept that taut bands are detectable and quantifiable with MRE imaging. The findings in the subjects suggest that the stiffness of the taut bands (9.0+/-0.9 KPa) in patients with myofascial pain may be 50% greater than that of the surrounding muscle tissue.

CONCLUSIONS

Our findings suggest that MRE can quantitate asymmetries in muscle tone that could previously only be identified subjectively by examination.

Differential effects of pre-tension on shear wave propagation in elastic media with different boundary conditions as measured by magnetic resonance elastography and finite element modeling

Magnetic resonance elastography (MRE) can non-invasively determine material stiffness based on the propagating shear wavelength. Shear wave propagation in a finite homogenous isotropic material can be affected by multiple factors. In this study we examined the effects of pre-tension and frequency on MRE shear measurements of gel phantoms with different boundary conditions, frequencies, and geometries. Results from MRE measurements were compared to wave motion theory in elastic solids and qualitatively to a finite element (FE) model. Results indicated that boundary conditions, geometry and pre-tension are important factors to be considered when performing MRE tests on a finite material, and that FE modeling can help explore how the shear wave propagation is affected under various boundary conditions and axial stresses, among other potential factors.