Magnetic resonance elastography of slow and fast shear waves illuminates differences in shear and tensile moduli in anisotropic tissue

Abstract

Mechanical anisotropy is an important property of fibrous tissues; for example, the anisotropic mechanical properties of brain white matter may play a key role in the mechanics of traumatic brain injury (TBI). The simplest anisotropic material model for small deformations of soft tissue is a nearly incompressible, transversely isotropic (ITI) material characterized by three parameters: minimum shear modulus (µ), shear anisotropy (ϕ=µ1µ-1) and tensile anisotropy (ζ=E1E2-1). These parameters can be determined using magnetic resonance elastography (MRE) to visualize shear waves, if the angle between the shear-wave propagation direction and fiber direction is known. Most MRE studies assume isotropic material models with a single shear (µ) or tensile (E) modulus. In this study, two types of shear waves, "fast" and "slow", were analyzed for a given propagation direction to estimate anisotropic parameters µ, ϕ, and ζ in two fibrous soft materials: turkey breast ex vivo and aligned fibrin gels. As expected, the speed of slow shear waves depended on the angle between fiber direction and propagation direction. Fast shear waves were observed when the deformations due to wave motion induced stretch in the fiber direction. Finally, MRE estimates of anisotropic mechanical properties in turkey breast were compared to estimates from direct mechanical tests.

Keywords: Anisotropy; MR Elastography; Shear waves; Transversely isotropic material.

Conflict of interest statement

Conflict of Interest Statement

None of the authors have a conflict of interest that could influence the work described in this manuscript.

Copyright © 2016 Elsevier Ltd. All rights reserved.

Figures

Figure 1

The propagation direction (denoted by unit vector n) and polarization directions (unit vectors ms and mf) of slow and fast shear waves, respectively, in an incompressible, transversely isotropic, elastic material. The unit vector a denotes the normal to the plane of isotropy.

Figure 2

Schematic diagrams of: (a) cylindrical specimen with axial excitation; (b) cube specimen with tangential excitation in a plane parallel to the fiber direction to induce “fast” shear waves. (c) cube specimen with tangential excitation perpendicular to the dominant fiber direction to induce “slow” shear waves. (d) Photograph of cylindrical turkey breast specimen embedded in gelatin (corresponding to panel a). (e) Photograph of experimental setup for cube turkey breast (corresponding to panel b; actuator on left). (f) Photograph of a cylindrical sample placed in RF coil with actuator on right.

Figure 3

Fiber orientation estimated by DTI in (a) cylindrical and (b) cube specimens of turkey breast. Maximum principal diffusion direction vectors (cyan) are superimposed on fractional anisotropy maps (FA, grey) for each voxel.

Figure 4

Wave propagation in axially-excited, cylindrical specimens. (a–c) Representative images of elliptical waves exhibiting direction-dependent propagation with different wave speeds in different directions. (a) Representative sample #1 of turkey breast, 800 Hz; (b) Representative sample #2 of turkey breast, 800 Hz; (c) aligned fibrin gel, 200 Hz. (d) Circular waves in (isotropic) gelatin, 200 Hz. (e) Ellipses were fitted to the wave images (white and black lines in b–d) and the average ratios of their semi-axes are shown for the different materials.

Figure 5

Wave propagation in a cylindrically aligned fibrin gel (200 Hz actuation) specimen, illustrating analysis by directional filtering. (a) Elliptical waves exhibiting direction dependent propagation with different wave speeds in different directions. (b–c) Displacement field after directional filtering in each of two propagation directions specified by angle, θ, from the dominant fiber direction. (b) θ = 0° and (c) θ = 90°.

Figure 6

Wave propagation in a cube specimen of aligned fibrin with dominant fiber direction at 45° from horizontal (Figures 1(b,c)), illustrating analysis by directional filtering. (a) Excitation (600 Hz) in the mf direction (with a component along the fibers, as in Figure 1(b)) leads to predominantly downward-propagating fast shear waves. (b) Excitation (600 Hz) in the ms direction, perpendicular to the fibers, as in Figure 1(c), leads to predominantly downward-propagating slow shear waves. Panels (c,d): Directionally filtered waves in the [0 −1 0] direction corresponding to panels (a,b) respectively.

Figure 7

Average (± std. deviation) slow shear-wave speeds (blue *) plotted vs the angle between propagation direction and the horizontal axis of the cylinder, in cylindrical specimens. (a) Representative sample #1 of turkey breast (800 Hz). (b) Representative sample #2 of turkey breast (800 Hz). (c) Aligned fibrin gel (200 Hz). (d) Gelatin (200 Hz). Each plot is for a single sample; average values for each direction are computed over 5 slices. Theoretical curves (red lines) are obtained from Eq. 2 using values of μ and ϕ estimated by weighted, least-squares fitting for each sample.

Figure 8

Wave propagation visualized by MRE in cube samples with different directions of excitation relative to fiber orientation. Fibers are oriented approximately 45° from horizontal as in Figure 2(b,c). Top panels (a,b) show fast and slow wave propagation in turkey breast actuated at 800 Hz and bottom panels (c,d) show aligned fibrin actuated at 600 Hz. Left panels (a,c): Actuation in the mf direction with a component along the fibers (as in Figure 2(b)) leads to downward-propagating, fast shear waves. Right panels (b,d): Actuation in the ms direction, perpendicular to the fibers (as in Figure 2(c) leads to downward-propagating, slow shear waves.

Figure 9

Summary of shear-wave speeds in turkey breast (a, b) and aligned fibrin (c, d) at different angles θ of propagation direction relative to fiber direction. (a) Slow shear-wave speed in cylindrical turkey breast specimens (800 Hz, N = 4 samples). Estimated material parameters: μ = 33.1 ± 11.4 kPa, ϕ = 1.3 ± 0.7 (b) Fast and slow shear-wave speeds in cube specimens (800 Hz, N=5). Estimated parameters: μ = 33.2 ± 16.7, ζ = 9.2 ± 4.9. (c) Slow shear-wave speeds in a cylindrical fibrin specimen (200 Hz, N=3). Estimated material parameters: μ = 1.1 ± 0.5 kPa, ϕ = 1.1 ± 0.2 (d) Average fast and slow shear-wave speeds in a cube specimen of aligned fibrin (600 Hz, N=1). Estimated parameters: μ = 4.7 kPa, ζ = 2.7.

Figure 10

Storage (elastic) and loss (viscous) components of the complex shear modulus μ* = μ′ + iμ″ of turkey breast (N=33, 30–40 Hz) measured by direct mechanical testing (DST). The ratio of the storage moduli was μ‖′/μ⊥′=1.5±0.3, the ratio of the loss moduli was μ‖″/μ⊥″=2.0±0.3, and the ratio of the magnitudes was μ||/μ⊥ = 1.6 ± 0.3 (ϕ = 0.6 ± 0.3).