Analytical solution for converging elliptic shear wave in a bounded transverse isotropic viscoelastic material with nonhomogeneous outer boundary

Quantifying uniaxial prestress and waveguide effects on dynamic elastography estimates for a cylindrical rod

Dynamic elastography attempts to reconstruct quantitative maps of the viscoelastic properties of materials by noninvasively measuring mechanical wave motion in them. The target motion is typically transversely-polarized relative to the wave propagation direction, such as bulk shear wave motion. In addition to neglecting waveguide effects caused by small lengths in one dimension or more, many reconstruction strategies also ignore nonzero, non-isotropic static preloads. Significant anisotropic prestress is inherent to the functional role of some biological materials of interest, which also are small in size relative to shear wavelengths in one or more dimensions. A cylindrically shaped polymer structure with isotropic material properties is statically elongated along its axis while its response to circumferentially-, axially-, and radially-polarized vibratory excitation is measured using optical or magnetic resonance elastography. Computational finite element simulations augment and aid in the interpretation of experimental measurements. We examine the interplay between uniaxial prestress and waveguide effects. A coordinate transformation approach previously used to simplify the reconstruction of un-prestressed transversely isotropic material properties based on elastography measurements is adapted with partial success to estimate material viscoelastic properties and prestress conditions without requiring advanced knowledge of either.

Mechanical stiffness and anisotropy measured by MRE during brain development in the minipig

The relationship between brain development and mechanical properties of brain tissue is important, but remains incompletely understood, in part due to the challenges in measuring these properties longitudinally over time. In addition, white matter, which is composed of aligned, myelinated, axonal fibers, may be mechanically anisotropic. Here we use data from magnetic resonance elastography (MRE) and diffusion tensor imaging (DTI) to estimate anisotropic mechanical properties in six female Yucatan minipigs at ages from 3 to 6 months. Fiber direction was estimated from the principal axis of the diffusion tensor in each voxel. Harmonic shear waves in the brain were excited by three different configurations of a jaw actuator and measured using a motion-sensitive MR imaging sequence. Anisotropic mechanical properties are estimated from displacement field and fiber direction data with a finite element- based, transversely-isotropic nonlinear inversion (TI-NLI) algorithm. TI-NLI finds spatially resolved TI material properties that minimize the error between measured and simulated displacement fields. Maps of anisotropic mechanical properties in the minipig brain were generated for each animal at all four ages. These maps show that white matter is more dissipative and anisotropic than gray matter, and reveal significant effects of brain development on brain stiffness and structural anisotropy. Changes in brain mechanical properties may be a fundamental biophysical signature of brain development.

Decoupling Uniaxial Tensile Prestress and Waveguide Effects From Estimates of the Complex Shear Modulus in a Cylindrical Structure Using Transverse-Polarized Dynamic Elastography

Dynamic elastography, whether based on magnetic resonance, ultrasound, or optical modalities, attempts to reconstruct quantitative maps of the viscoelastic properties of biological tissue, properties altered by disease and injury, by noninvasively measuring mechanical wave motion in the tissue. Most reconstruction strategies that have been developed neglect boundary conditions, including quasi-static tensile or compressive loading resulting in a nonzero prestress. Significant prestress is inherent to the functional role of some biological tissues currently being studied using elastography, such as skeletal and cardiac muscle, arterial walls, and the cornea. In the present article a configuration, inspired by muscle elastography but generalizable to other applications, is analytically and experimentally studied. A hyperelastic polymer phantom cylinder is statically elongated in the axial direction while its response to transverse-polarized vibratory excitation is measured. We examine the interplay between uniaxial prestress and waveguide effects in this muscle-like tissue phantom using computational finite element simulations and magnetic resonance elastography measurements. Finite deformations caused by prestress coupled with waveguide effects lead to results that are predicted by a coordinate transformation approach that has been previously used to simplify reconstruction of anisotropic properties using elastography. Here, the approach estimates material viscoelastic properties that are independent of the nonhomogeneous prestress conditions without requiring advanced knowledge of those stress conditions.

Separable solutions to nonlinear anisotropic diffusion equation in elliptic coordinates

Two- and three-dimensional exact solutions of the nonlinear diffusion equation are proved to exist in elliptic coordinates subject to an arbitrary piecewise constant azimuthal anisotropy. Degrees of freedom traditionally used to satisfy boundary conditions are instead employed to ensure continuity and conservation of mass across contiguity surfaces between subdomains of distinct diffusivities. Not all degrees of freedom are exhausted thereby, and conditions are given for the inclusion of higher harmonics. Degrees of freedom associated with one isotropic subdomain are always available to satisfy boundary conditions. The second harmonic is pivotal in the solution construction as well as the identification of partial symmetries in the domain partition. The anisotropy gives rise to an unconventional mixed type critical point that combines saddle and node-like characteristics. This article is part of the theme issue 'New trends in pattern formation and nonlinear dynamics of extended systems'.

Biaxial Tensile Prestress and Waveguide Effects on Estimates of the Complex Shear Modulus Using Optical-Based Dynamic Elastography in Plate-Like Soft Tissue Phantoms

Dynamic elastography attempts to reconstruct quantitative maps of the viscoelastic properties of biological tissue, properties altered by disease and injury, by noninvasively measuring mechanical wave motion in the tissue. Most reconstruction strategies that have been developed neglect boundary conditions, including quasi-static tensile or compressive loading resulting in a nonzero prestress. Significant prestress is inherent to the functional role of some biological tissues, such as skeletal and cardiac muscle, arterial walls, and the cornea. In the present article a novel configuration, inspired by corneal elastography but generalizable to other applications, is studied. A polymer phantom layer is statically elongated via an in-plane biaxial normal stress while the phantom's response to transverse vibratory excitation is measured. We examine the interplay between biaxial prestress and waveguide effects in this plate-like tissue phantom. Finite static deformations caused by prestressing coupled with waveguide effects lead to results that are predicted by a novel coordinate transformation approach previously used to simplify reconstruction of anisotropic properties. Here, the approach estimates material viscoelastic properties independent of the nonzero prestress conditions without requiring advanced knowledge of those stress conditions.

Mapping heterogenous anisotropic tissue mechanical properties with transverse isotropic nonlinear inversion MR elastography

The white matter tracts of brain tissue consist of highly-aligned, myelinated fibers; white matter is structurally anisotropic and is expected to exhibit anisotropic mechanical behavior. In vivo mechanical properties of tissue can be imaged using magnetic resonance elastography (MRE). MRE can detect and monitor natural and disease processes that affect tissue structure; however, most MRE inversion algorithms assume locally homogenous properties and/or isotropic behavior, which can cause artifacts in white matter regions. A heterogeneous, model-based transverse isotropic implementation of a subzone-based nonlinear inversion (TI-NLI) is demonstrated. TI-NLI reconstructs accurate maps of the shear modulus, damping ratio, shear anisotropy, and tensile anisotropy of in vivo brain tissue using standard MRE motion measurements and fiber directions estimated from diffusion tensor imaging (DTI). TI-NLI accuracy was investigated with using synthetic data in both controlled and realistic settings: excellent quantitative and spatial accuracy was observed and cross-talk between estimated parameters was minimal. Ten repeated, in vivo, MRE scans acquired from a healthy subject were co-registered to demonstrate repeatability of the technique. Good resolution of anatomical structures and bilateral symmetry were evident in MRE images of all mechanical property types. Repeatability was similar to isotropic MRE methods and well within the limits required for clinical success. TI-NLI MRE is a promising new technique for clinical research into anisotropic tissues such as the brain and muscle.

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.

The combined importance of finite dimensions, anisotropy, and pre-stress in acoustoelastography

Dynamic elastography, whether based on magnetic resonance, ultrasound, or optical modalities, attempts to reconstruct quantitative maps of the viscoelastic properties of biological tissue, properties that are altered by disease and injury, by noninvasively measuring mechanical wave motion in the tissue. Most reconstruction strategies that have been developed neglect boundary conditions, including quasistatic tensile or compressive loading resulting in a nonzero prestress. Significant prestress is inherent to the functional role of some biological tissues currently being studied using elastography, such as skeletal and cardiac muscle, arterial walls, and the cornea. In the present article, we review how prestress alters both bulk mechanical wave motion and wave motion in one- and two-dimensional waveguides. Key findings are linked to studies on skeletal muscle and the human cornea, as one- and two-dimensional waveguide examples. This study highlights the underappreciated combined acoustoelastic and waveguide challenge to elastography. Can elastography truly determine viscoelastic properties of a material when what it is measuring is affected by both these material properties and unknown prestress and other boundary conditions?

Axially- and torsionally-polarized radially converging shear wave MRE in an anisotropic phantom made via Embedded Direct Ink Writing

Magnetic Resonance Elastography (MRE) is a non-invasive imaging method to quantitatively map the shear viscoelastic properties of soft tissues. In this study, Embedded Direct Ink Writing is used to fabricate a muscle mimicking anisotropic phantom that may serve as a standard for imaging studies of anisotropic materials. The technique allowed us to obtain a long shelf life silicone-based phantom expressing transverse isotropic mechanical properties. Another goal of the present investigation is to introduce a torsionally-polarized, radially-converging shear wave actuation method for MRE. The implemented design for this novel setup was first validated via its application to isotropic and homogeneous gelatin phantoms. Then, a comparison of the resulting complex wave images from axially- and torsionally-polarized MRE on the developed anisotropic phantom and on a skeletal muscle murine sample is presented, highlighting the value of using multiple actuation and motion encoding polarization directions when studying anisotropic materials.

Micro Air-Pulse Spatial Deformation Spreading Characterizes Degree of Anisotropy in Tissues

In optical coherence elastography (OCE), air-pulse stimulation has been widely used to produce propagation of mechanical waves for elastic characterization of tissues. In this paper, we propose the use of spatial deformation spreading (SDS) on the surface of samples produced by air-pulse stimulation for the OCE of transverse isotropic tissues. Experiments in isotropic tissue-mimicking phantoms and anisotropic chicken tibialis muscle were conducted using a spectral-domain optical coherence tomography system synchronized with a confocal air-pulse stimulation. SDS measurements were compared with wave speeds values calculated at different propagation angles. We found an approximately linear relationship between shear wave speed and SDS in isotropic phantoms, which was confirmed with predictions made by the numerical integration of a wave propagation model. Experimental measurements in chicken muscle show a good agreement between SDS and surface wave speed taken along and across the axis of symmetry of the tissues, also called degree of anisotropy. In summary, these results demonstrated the capabilities of SDS produced by the air-pulse technique in measuring the shear elastic anisotropy of transverse isotropic tissues.

Measuring viscoelastic parameters in Magnetic Resonance Elastography: a comparison at high and low magnetic field intensity

Magnetic Resonance Elastography (MRE) is a non-invasive imaging technique which involves motion-encoding MRI for the estimation of the shear viscoelastic properties of soft tissues through the study of shear wave propagation. The technique has been found informative for disease diagnosis, as well as for monitoring of the effects of therapies. The development of MRE and its validation have been supported by the use of tissue-mimicking phantoms. In this paper we present our new MRE protocol using a low magnetic field tabletop MRI device at 0.5 T and sinusoidal uniaxial excitation in a geometrical focusing condition. Results obtained for gelatin are compared to those previously obtained using high magnetic field MRE at 11.7 T. A multi-frequency investigation is also provided via a comparison of commonly used rheological models: Maxwell, Springpot, Voigt, Zener, Jeffrey, fractional Voigt and fractional Zener. Complex shear modulus values were comparable when processed from images acquired with the tabletop low field scanner and the high field scanner. This study serves as a validation of the presented tabletop MRE protocol and paves the way for MRE experiments on ex-vivo tissue samples in both normal and pathological conditions.

Analytical solution based on spatial distortion for a time-harmonic Green's function in a transverse isotropic viscoelastic solid

A strategy of spatial distortion to make an anisotropic problem become isotropic has been previously validated in two-dimensional transverse isotropic (TI) viscoelastic cases. Here, the approach is extended to the three-dimensional problem by considering the time-harmonic point force response (Green's function) in a TI viscoelastic material. The resulting wave field, exactly solvable using a Radon transform with numerical integration, is approximated via spatial distortion of the closed form analytical solution to the isotropic case. Different distortions are used, depending on whether the polarization of the wave motion is orthogonal to the axis of isotropy, with the approximation yielding differing levels of accuracy.

Shear wave speeds in nearly-incompressible fibrous materials with two fiber families

An analytical and numerical investigation of shear wave behavior in nearly-incompressible soft materials with two fiber families was performed, focusing on the effects of material parameters and imposed pre-deformations on wave speed. This theoretical study is motivated by the emerging ability to image shear waves in soft biological tissues by magnetic resonance elastography (MRE). In MRE, the relationships between wave behavior and mechanical properties can be used to characterize tissue properties non-invasively. We demonstrate these principles in two material models, each with two fiber families. One model is a nearly-incompressible linear elastic model that exhibits both shear and tensile anisotropy; the other is a two-fiber-family version of the widely-used Holzapfel-Gasser-Ogden (HGO) model, which is nonlinear. Shear waves can be used to probe nonlinear material behavior using infinitesimal dynamic deformations superimposed on larger, quasi-static "pre-deformations." In this study, closed-form expressions for shear wave speeds in the HGO model are obtained in terms of the model parameters and imposed pre-deformations. Analytical expressions for wave speeds are confirmed by finite element simulations of shear waves with various polarizations and propagation directions. These studies support the feasibility of estimating the parameters of an HGO material model noninvasively from measured shear wave speeds.

Converging super-elliptic torsional shear waves in a bounded transverse isotropic viscoelastic material with nonhomogeneous outer boundary

A theoretical approach was recently introduced [Guidetti and Royston, J. Acoust. Soc. Am. 144, 2312-2323 (2018)] for the radially converging slow shear wave pattern in transverse isotropic materials subjected to axisymmetric excitation normal to the axis of isotropy at the outer boundary of the material. This approach is enabled via transformation to an elliptic coordinate system with isotropic properties. The approach is extended to converging fast shear waves driven by axisymmetric torsional motion polarized in a plane containing the axis of isotropy. The approach involves transformation to a super-elliptic shape with isotropic properties and use of a numerically efficient boundary value approximation.

Converging super-elliptic torsional shear waves in a bounded transverse isotropic viscoelastic material with nonhomogeneous outer boundary

A theoretical approach was recently introduced [Guidetti and Royston, J. Acoust. Soc. Am. 144, 2312-2323 (2018)] for the radially converging slow shear wave pattern in transverse isotropic materials subjected to axisymmetric excitation normal to the axis of isotropy at the outer boundary of the material. This approach is enabled via transformation to an elliptic coordinate system with isotropic properties. The approach is extended to converging fast shear waves driven by axisymmetric torsional motion polarized in a plane containing the axis of isotropy. The approach involves transformation to a super-elliptic shape with isotropic properties and use of a numerically efficient boundary value approximation.

An investigation into the relationship between inhomogeneity and wave shapes in phantoms and ex vivo skeletal muscle using Magnetic Resonance Elastography and finite element analysis

Soft biological tissues such as skeletal muscle and brain white matter can be inhomogeneous and anisotropic due to the presence of fibers. Unlike biological tissue, phantoms with known microstructure and defined mechanical properties enable a quantitative assessment and systematic investigation of the influence of inhomogeneities on the nature of shear wave propagation. This study introduces a mathematical measure for the wave shape, which the authors call as the 1-Norm, to determine the conditions under which homogenization may be a valid approach. This is achieved through experimentation using the Magnetic Resonance Elastography technique on 3D printed inhomogeneous fiber phantoms as well as on ex-vivo porcine lumbus muscle. In addition, Finite Element Analysis is used as a tool to decouple the effects of directional anisotropy from those of inhomogeneity. A correlation is then established between the values of 1-Norm derived from the wave front geometry, and the spacing (d) between neighboring inhomogeneities (spherical inclusions or fibers and fiber intersections in phantoms and muscle). Smaller values of 1-Norm indicate less wave scattering at the locations of fiber intersections, which implies that the wave propagation may be approximated to that of a homogeneous medium; homogenization may not be a valid approximation when significant scattering occurs at the locations of inhomogeneities. In conclusion, the current study proposes 1-Norm as a quantitative measure of the magnitude of wave scattering in a medium, which can potentially be used as a homogeneity index of a biological tissue.

Dilatational and shear waves in poro-vioscoelastic media

Dynamic elastography methods are being developed for quantitatively and noninvasively mapping the viscoelastic properties of biological tissue that are often altered by disease and injury, as well as response to treatment. This involves inducing mechanical wave motion that also can be affected by the multiphase porous nature of the tissue, whether it be consideration of blood perfusion in the vascular network found in many regions of interest, or consideration of air movement in the complex bronchial tree within the lungs. Elastographic mapping requires reconstructing material properties based on interpretation of the measured wave motion. Reconstruction methods that explicitly incorporate poroelastic behavior are an active area of development. In the present article the equivalence of two theoretical approaches to modeling poroelastic behavior is demonstrated specifically in the frequency domain using parameter values that span the range expected in vivo for analysis of blood and air-infused regions. The two methods are known as (1) the mixture or biphasic formulation and (2) the poroelastic approach. The case of acoustic wave propagation in the lungs is specifically addressed by comparison of analytical predictions to recently reported experimental measurements. Establishing and validating this equivalence of theoretical approaches not only strengthens our fundamental understanding of the relevant physics, but also may lead to improved numerical methods for simulation and elastography reconstruction.

Analytical solution for diverging elliptic shear wave in bounded and unbounded transverse isotropic viscoelastic material with nonhomogeneous inner boundary

A theoretical approach was recently introduced by Guidetti and Royston [J. Acoust. Soc. Am. 144, 2312-2323 (2018)] for the radially converging elliptic shear wave pattern in transverse isotropic materials subjected to axisymmetric excitation normal to the fiber axis at the outer boundary of the material. This approach is enabled via a transformation to an elliptic coordinate system with isotropic properties. The approach is extended to the case of diverging shear waves radiating from a cylindrical rod that is axially oscillating perpendicular to the axis of isotropy and parallel to the plane of isotropy.

Analytical solution for converging elliptic shear wave in a bounded transverse isotropic viscoelastic material with nonhomogeneous outer boundary

Dynamic elastography methods-based on optical, ultrasonic, or magnetic resonance imaging-are being developed for quantitatively mapping the shear viscoelastic properties of biological tissues, which are often altered by disease and injury. These diagnostic imaging methods involve analysis of shear wave motion in order to estimate or reconstruct the tissue's shear viscoelastic properties. Most reconstruction methods to date have assumed isotropic tissue properties. However, application to tissues like skeletal muscle and brain white matter with aligned fibrous structure resulting in local transverse isotropic mechanical properties would benefit from analysis that takes into consideration anisotropy. A theoretical approach is developed for the elliptic shear wave pattern observed in transverse isotropic materials subjected to axisymmetric excitation creating radially converging shear waves normal to the fiber axis. This approach, utilizing Mathieu functions, is enabled via a transformation to an elliptic coordinate system with isotropic properties and a ratio of minor and major axes matching the ratio of shear wavelengths perpendicular and parallel to the plane of isotropy in the transverse isotropic material. The approach is validated via numerical finite element analysis case studies. This strategy of coordinate transformation to equivalent isotropic systems could aid in analysis of other anisotropic tissue structures.