Urban, Matthew W.

Abstract The myocardium is known to be an anisotropic medium where the muscle fiber orientation changes through the thickness of the wall. Shear wave elastography methods use propagating waves which are measured by ultrasound or magnetic resonance imaging (MRI) techniques to characterize the mechanical properties of various tissues. Ultrasound- or MR-based methods have been used and the excitation frequency ranges for these various methods cover a large range from 24–500 Hz. Some of the ultrasound-based methods have been shown to be able to estimate the fiber direction. We constructed a model with layers of elastic, transversely isotropic materials that were oriented at different angles to simulate the heart wall in systole and diastole. We investigated the effect of frequency on the wave propagation and the estimation of fiber direction and wave speeds in the different layers of the assembled models. We found that waves propagating at low frequencies such as 30 or 50 Hz showed low sensitivity to the fiber direction but also had substantial bias in estimating the wave speeds in the layers. Using waves with higher frequency content (>200 Hz) allowed for more accurate fiber direction and wave speed estimation. These results have particular relevance for MR- and ultrasound-based elastography applications in the heart.

Evaluation of tissue engineering constructs is performed by a series of different tests. In many cases it is important to match the mechanical properties of these constructs to those of native tissues. However, many mechanical testing methods are destructive in nature which increases cost for evaluation because of the need for additional samples reserved for these assessments. A wave propagation method is proposed for characterizing the shear elasticity of thin layers bounded by a rigid substrate and fluid-loading, similar to the configuration for many tissue engineering applications. An analytic wave propagation model was derived for this configuration and compared against finite element model simulations and numerical solutions from the software package Disperse. The results from the different models found very good agreement. Experiments were performed in tissue-mimicking gelatin phantoms with thicknesses of 1 and 4 mm and found that the wave propagation method could resolve the shear modulus with very good accuracy, no more than 4.10% error. This method could be used in tissue engineering applications to monitor tissue engineering construct maturation with a nondestructive wave propagation method to evaluate the shear modulus of a material.

Shear wave elastography imaging (SWEI) with ultrasound uses an applied acoustic radiation force to obtain quantitative images of liver, breast, prostate, and other soft tissues. In SWEI, shear velocity waveforms are collected along a line, and then, the shear speed is estimated either with a k-space or a correlation-based method. Although these yield effective estimates in elastic media, both methods consistently overestimate the shear speed in viscoelastic soft tissue because neither considers the shear viscosity. To address this problem, we have created a new shear speed estimation approach that also accounts for the effect of shear viscosity. The new approach is evaluated in a computational model that calculates the three-dimensional (3D) intensity field generated by a linear array transducer in FOCUS (http://www.egr.msu.edu/~fultras-web) and then simulates the shear wave velocities in a viscoelastic medium with a 3D finite difference model. In the viscoelastic shear wave simulation model, the shear speed constant is 1.5 m/s, and the shear viscosity is 1 Pa·s. Correlation-based shear speed estimates are compared to the viscosity-compensated estimates, and the results show that the viscosity-compensated approach consistently achieves improved shear speed estimates relative to the correlation-based method. [Work supported in part by NIH Grant Nos. R01EB012079 and R01DK092255.]

Many methods for characterization of the mechanical properties of soft tissues using propagating shear waves have been developed over the past two decades. Most of these methods assume that the shear wave is traveling in an elastic, isotropic tissue. However, many soft tissues are viscoelastic and have material properties that are directionally dependent or anisotropic. We have been developing methods to measure waves propagating in soft tissues to estimate the anisotropic viscoelastic material properties. To refine our measurement methods, we have also developed techniques and models for simulating the wave propagation in these types of materials. We have developed specialized finite element and pseudo-spectral models that can simulate viscoelastic transversely isotropic materials and compared our results with measurements in ex vivo porcine muscle. Additionally, we have developed a finite element model that simulates wave propagation in the myocardium by creating a model of layered transverse isotropic media oriented at different angles. We examined the effect of frequency of the waves for estimating of the angle of propagation. We compared these simulation results with experimental results from an ex vivo porcine left ventricular wall. These simulation methods provide insight for optimizing our measurement methods for in vivo characterization of material properties.

Shear wave elastography with ultrasound applies an acoustic radiation force to generate shear waves in viscoelastic soft tissue. To address the need for more effective simulation tools that model shear waves generated by an applied acoustic radiation force, three-dimensional (3D) finite difference programs that simulate propagation of shear waves in an isotropic, homogeneous medium have been created. These programs simulate shear wave propagation in elastic and viscoelastic soft tissue models. The 3D finite difference implementation combines an explicit time-stepping approach and a staggered spatial grid with an absorbing boundary condition that reduces boundary reflections. The acoustic radiation force in these simulations is quickly and accurately simulated in FOCUS (http://www.egr.msu.edu/~fultras-web) for a focused linear ultrasound array with an f/# of 2. The compressional wave speed is 1500 m/s and the shear wave speed is 1.5m/s in the elastic and viscoelastic tissue models, and the shear viscosity is 1 Pa·s in the viscoelastic model. The simulation completes 90,000 time steps of a 10 ms simulation in an 80 mm × 80 mm × 80 mm region with spatial sampling equal to the wavelength of compressional component in three days on a 3.4 MHz Intel i7 processor. [Supported in part by NIH Grants R01EB012079 and R01DK092255.]

We implemented a three-dimensional solver for simulating the ultrasound radiation force induced shear wave motion in viscoelastic media using a mapped Chebyshev pseudospectral method. The solver used a body force field that is simulated by the Field II program as the excitation. Then, a velocity-stress formulation was used to calculate the shear wave motion. We used a Voigt model to describe tissue viscoelasticity, and the viscosity is introduced by memory variables and auxiliary differential equations. The top boundary of the domain had a fixed boundary condition to mimic the interaction between an ultrasound probe and the tissue surface. All other sides of the domain were surrounded by perfectly matched layers to minimize the reflections. Time stepping was implemented with the LSODE integrator with variable step sizes. Results show that the waveform of the simulated shear wave is close to experimental observations and equivalent FEM simulations. We used a Fourier-based method to estimate the phase velocities of the shear waves between 100–1000 Hz and a nonlinear regression to estimate the Voigt model parameters. The reconstructions matched within 10% error compared with the true values.

Green’s functions effectively simulate shear waves produced by an applied acoustic radiation force in elastic and viscoelastic soft tissue. In an effort to determine the optimal parameters for these simulations, the convergence of Green's function-based calculations is evaluated for realistic spatial distributions of the initial radiation force “push.” The input to these calculations is generated by FOCUS, the “Fast Object-oriented C + + Ultrasound Simulator,” which computes the approximate intensity fields generated by a Phillips L7-4 ultrasound transducer array for both focused and unfocused beams. The radiation force in the simulation model, which is proportional to the simulated intensity, is applied for 200 μs, and the resulting displacements are calculated with the Green's function model. Simulation results indicate that, for elastic media, convergence is achieved when the intensity field is sampled at roughly one-tenth of the wavelength of the compressional component that delivers the radiation force “push.” Aliasing and oscillation artifacts are observed in the model for an elastic medium at lower sampling rates. For viscoelastic media, spatial sampling rates as low as two samples per compressional wavelength are sufficient due to the low-pass filtering effects of the viscoelastic medium. [Supported in part by NIH Grants R01EB012079 and R01DK092255.]

In conventional shear wave elastography materials are assumed to be linear, elastic, homogeneous, and isotropic. These assumptions are important to account for in certain tissues because they are not always appropriate. Many tissues such as skeletal muscle, the kidney, and the myocardium are anisotropic. Shear waves can be used to investigate the directionally dependent mechanical properties of anisotropic media. To study these tissues in a systematic way and to account for the effects of the anisotropic architecture, laboratory-based phantoms are desirable. We will report on several phantom-based approaches for studying shear wave anisotropy, assuming that these materials are transversely isotropic. Phantoms with embedded fibers were used to mimic anisotropic tissues. Homogeneous phantoms were compressed to induce transverse isotropy according to the acoustoelastic phenomenon, which is related to nonlinear behavior of the materials. The fractional anisotropy of these phantoms was quantified to compare with measurements made in soft tissues. In addition, soft tissues are also viscoelastic, and we have developed a method to model viscoelastic transversely isotropic materials with the finite element method (FEM). The viscoelastic property estimation from phantom experiments and FEM simulations will also be discussed.

The Green's function approach describes propagating shear waves generated by an acoustic radiation force in elastic and viscoelastic soft tissue. Calculations with the Green's function approach are evaluated in elastic and viscoelastic soft tissue models for a line source and for a simulated focused beam. The results for the line source input are evaluated at 200 time samples in a 101 by 101 point grid that is perpendicular to the line source. For a shear wave speed of 1.4832 m/s and a compressional wave speed of 1500 m/s, shear wave simulations for a line source input in elastic and viscoelastic soft tissue models are completed in 431 and 2487 s with Matlab scripts, respectively, where the shear viscosity is 0.1 Pa.s in the viscoelastic model. Simulations are evaluated at a single point for an acoustic radiation force generated by a 128 element linear array operating at 4.09 MHz, and these simulations require 228 s and 1327 s for elastic and viscoelastic soft tissue models, respectively. The results show that these are effective models for simulating shear wave propagation in soft tissue, and plans to accelerate these simulations will also be discussed. [Supported in part by NIH Grants R01 EB012079 and R01 DK092255.]

Elasticity imaging has emerged as a viable clinical tool for assisting in diagnosis of certain diseases such as liver fibrosis and cancer. Many methods have been developed for generating and measuring shear waves in soft tissues. The advantage of using shear waves is that the shear wave velocity is proportional to the mechanical material properties of the tissues under investigation. Soft tissues are inherently viscoelastic and considerable effort has been made to quantify the dispersion, or variation of frequency, of the velocity and attenuation of shear waves. This is accomplished by measuring the shear wave motion and using Fourier-based techniques to extract the shear wave velocity and attenuation. Additional considerations are made in tissues where geometric dispersion is also present such as in the heart. Viscoelastic characterization of the shear viscoelasticity of soft tissues in vivo such as human liver, human kidney, and swine heart will be shown. We will also demonstrate parameterization of the results by using a model-free approach or by fitting the shear wave velocity dispersion to rheological models. The diagnostic value of the viscoelastic parameters will be discussed for each particular application. [This work was supported in part by NIH Grant Nos. DK092255, DK082408, and EB002167.]