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

Abstract

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.

Separable solutions to nonlinear anisotropic diffusion equation in elliptic coordinates

“…Anisotropic solutions to other classical partial differential equations might also be of value, e.g. extending the theory developed herein to wave propagation in anisotropic media will be useful in modelling or quantifying of the viscoelastic properties of biological tissues [17].…”

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

“…The transverse isotropic elasticity matrix using nomenclature from Guidetti et al 45 and taking the z-axis at the axis of isotropy is Adding only normal (no shear) static pre-stresses aligned with the x, y, and z directions, respectively, leads to the following governing equations of motion, 46,47 where u , v , and w refer to the displacement component in the x, y, and z direction, respectively, and subscripted and t after a comma refer to partial derivatives with respect to that spatial or time dimension: …”

“… (Color online) Slow shear waves in the y-z plane generated by a harmonic point torque at the origin polarized orthogonal to the axis of isotropy (z-axis) and direction of uniaxial stress , using material properties from our previous work (Refs. 45 and 51–53 ) that correspond approximately to passive muscle with fibers along the z-axis (Table II ). Simulations are for no pre-stress (b) and (c) or a uniaxial prestress from elongation along fiber direction that is equivalent to the stress from a 10% MVC contraction (d) and (e).…”

“…The FE prediction was validated against the exact integral solution for the unstressed point force and taking the radius of the cylinder to be large enough that waves attenuate before reaching the boundary. The case study parameters shown in Table II are typical of soft biological tissue and match those used previously by the last author, 45,51–53 but with the addition of cases with a nonzero uniaxial tensile stress (it is either 0 or equal to ).…”

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

“…Elastography measurements can be made using different imaging modalities, such as ultrasound [1,2,12,13,14], optics [15,16,17,18], or magnetic resonance imaging (MRI) [7,19]; in particular, the last approach is referred to as Magnetic Resonance Elastography (MRE), which involves motion-encoding MRI. MRE offers resolution up to hundreds of micrometers [20,21], great penetration depth [22] and added value in a multiparametric MRI approach [23]. MRE has high diagnostic accuracy for the staging of hepatic fibrosis [24] and has diagnostic potential for the detection of breast [25], thyroid [26] and prostate cancers [27,28].…”