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Research Papers

Mechanical Properties of Viscoelastic Media by Local Frequency Estimation of Divergence-Free Wave Fields

[+] Author and Article Information
Erik H. Clayton

e-mail: clayton@wustl.edu

Ruth J. Okamoto

Department of Mechanical Engineering and Materials Science,
Washington University in St. Louis,
One Brookings Drive,
Campus Box 1185,
Saint Louis, MO 63130

Philip V. Bayly

Department of Mechanical Engineering and Materials Science,
Washington University in St. Louis,
One Brookings Drive,
Campus Box 1185,
Saint Louis, MO 63130;
Department of Biomedical Engineering,
Washington University in St. Louis,
One Brookings Drive,
Campus Box 1185,
Saint Louis, MO 63130

1Corresponding author.

Contributed by the Bioengineering Division of ASME for publication in the JOURNAL OF BIOMECHANICAL ENGINEERING. Manuscript received October 1, 2012; final manuscript received January 14, 2013; accepted manuscript posted January 18, 2013; published online February 7, 2013. Editor: Victor H. Barocas.

J Biomech Eng 135(2), 021025 (Feb 07, 2013) (6 pages) Paper No: BIO-12-1447; doi: 10.1115/1.4023433 History: Received October 01, 2012; Revised January 14, 2013; Accepted January 18, 2013

Magnetic resonance elastography (MRE) is an imaging modality with which mechanical properties can be noninvasively measured in living tissue. Magnetic resonance elastography relies on the fact that the elastic shear modulus determines the phase velocity and, hence the wavelength, of shear waves which are visualized by motion-sensitive MR imaging. Local frequency estimation (LFE) has been used to extract the local wavenumber from displacement wave fields recorded by MRE. LFE -based inversion is attractive because it allows material parameters to be estimated without explicitly invoking the equations governing wave propagation, thus obviating the need to numerically compute the Laplacian. Nevertheless, studies using LFE have not explicitly addressed three important issues: (1) tissue viscoelasticity; (2) the effects of longitudinal waves and rigid body motion on estimates of shear modulus; and (3) mechanical anisotropy. In the current study we extend the LFE technique to (1) estimate the (complex) viscoelastic shear modulus in lossy media; (2) eliminate the effects of longitudinal waves and rigid body motion; and (3) determine two distinct shear moduli in anisotropic media. The extended LFE approach is demonstrated by analyzing experimental data from a previously-characterized, isotropic, viscoelastic, gelatin phantom and simulated data from a computer model of anisotropic (transversely isotropic) soft material.

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Figures

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Fig. 1

A time history of the axial displacement field produced by the closed-form analytical solution. The field is shown through a series of eight phase offsets (φ1,φ2,...,φ8) over one complete cycle. Outwardly propagating shear waves are excited by the axial vibration of a central filament. In addition to the Bessel function solution, a harmonic axial rigid body displacement was added. Parameters: G' = 1.2 kPa, G″ = 0.18 kPa, density ρ = 1100 kg/m3, frequency f = 400 Hz, and cylinder radius R = 22.5 mm.

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Fig. 2

(a) Transversely isotropic material model with the axis of symmetry (fiber axis) aligned with the x1-axis. (b)-(d) Application of shear stress along a face produces a corresponding shear strain in proportion to the shear modulus. In transversely isotropic media there are two shear moduli: μT describes shear in a plane normal to the symmetry axis, and μL in planes parallel to the symmetry axis. Biological media often exhibit μL>μT (γij = 2εij, for i≠j). Reproduced from Ref. [26].

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Fig. 3

Axial displacement fields from finite element simulations. (a) Fiber orientation aligned with the x1-axis. (b) Outwardly propagating shear waves excited by the axial vibration of a central filament. (c) Inwardly-propagating waves excited by the axial vibration of the outer boundary. Parameters: shear moduli μL = 2000 Pa, μT = 1000 Pa, density ρ = 1000 kg/m3, frequency f = 5 Hz, and cylinder radius R = 1 m. The loss factor (η = G″/G') of the material is 0.20.

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Fig. 4

Homogeneous isotropic viscoelastic inversion validation in the presence of rigid body (zero strain) motion. Storage G' and loss G" moduli are shown for inversion of the (a) distortion field Γ (G' = 1.32 ± 0.37 kPa; G″ = 0.21 ± 0.09 kPa), and (b) displacement field (G' = 7.48 ± 7.78 kPa; G″ = 5.55 ± 9.89 kPa). (Gtrue' = 1.2 kPa; Gtrue″ = 0.18 kPa.)

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Fig. 5

Comparison of the viscoelastic distortion-based LFE inversion (red dots) with total least-squares (TLS) direct inversion (black lines) performed by Okamoto [20]. Results in this study were not corrected for test-order temperature variations as they were in the Okamoto study (dashed black lines). Error bars represent ± 1 std. dev.

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Fig. 6

(a) Estimates of shear modulus assuming an isotropic material model. (b) Estimates of shear modulus μL (mean ± std. dev. = 1761 ± 111 Pa) obtained using the LFE-based approach for transversely isotropic materials. (c) Estimates of shear modulus μT (1045 ± 152 Pa). The true values are μL = 2000 and μT = 1000. In the anisotropic case, the estimation error increases in regions where the shear wave does not sufficiently activate a plane of shear governed by either μL or μT.

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