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Research Papers

Measurement of the Dynamic Shear Modulus of Mouse Brain Tissue In Vivo by Magnetic Resonance Elastography

[+] Author and Article Information
Stefan M. Atay, Arash Sabet

Department of Mechanical and Aerospace Engineering, Washington University, 1 Brookings Drive, Box 1185, St. Louis, MO 63130

Christopher D. Kroenke

Advanced Imaging Research Center, Oregon Health and Science University, 3181 S. W. Sam Jackson Park Road, Portland, OR 97239-3098

Philip V. Bayly1

Department of Mechanical and Aerospace Engineering, Department of Biomedical Engineering, Washington University, 1 Brookings Drive, Box 1185, St. Louis, MO 63130pvb@me.wustl.edu

1

Corresponding author.

J Biomech Eng 130(2), 021013 (Mar 31, 2008) (11 pages) doi:10.1115/1.2899575 History: Received November 08, 2006; Revised June 20, 2007; Published March 31, 2008

In this study, the magnetic resonance (MR) elastography technique was used to estimate the dynamic shear modulus of mouse brain tissue in vivo. The technique allows visualization and measurement of mechanical shear waves excited by lateral vibration of the skull. Quantitative measurements of displacement in three dimensions during vibration at 1200Hz were obtained by applying oscillatory magnetic field gradients at the same frequency during a MR imaging sequence. Contrast in the resulting phase images of the mouse brain is proportional to displacement. To obtain estimates of shear modulus, measured displacement fields were fitted to the shear wave equation. Validation of the procedure was performed on gel characterized by independent rheometry tests and on data from finite element simulations. Brain tissue is, in reality, viscoelastic and nonlinear. The current estimates of dynamic shear modulus are strictly relevant only to small oscillations at a specific frequency, but these estimates may be obtained at high frequencies (and thus high deformation rates), noninvasively throughout the brain. These data complement measurements of nonlinear viscoelastic properties obtained by others at slower rates, either ex vivo or invasively.

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Copyright © 2008 by American Society of Mechanical Engineers
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Figures

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Figure 1

The MR elastography pulse sequence. A standard spin-echo MR imaging sequence was modified by the addition of sinusoidal motion-sensitizing gradients that oscillate at the frequency of vibration. The basic spin-echo sequence consists of rf excitation in conjunction with gradients in the slice-select (GSS), readout (GRO), and phase-encode (GPE) directions. This figure depicts harmonic motion-sensitizing gradients in the PE direction. The dashed lines indicate that motion-sensitizing gradients could also be applied in the RO and SS directions.

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Figure 2

Illustration of phase accumulation using MRE. The three rows of circles represent three individual “spin packets,” and the portion filled in represents the phase of a spin at a particular time. The five columns represent a complete cycle of vibration as well as gradient modulation of period T=2π∕ω, where the ω is the frequency of vibration measured in rad/s. The amount of phase that a spin accumulates at a given time is directly proportional to the magnetic field strength at that point. Thus, at t2, the upper and lower spins accrue more phase than the middle spin because they have been displaced by vibration into a higher magnetic field. At t4, the spins are displaced in the opposite direction; however, the gradient field has also switched direction and the upper and lower spins again accrue more phase than the middle spin. The net result is an image whose phase is proportional to displacement at a particular time during 1cycle, as seen on the right. An image of the displacements at a different point in the cycle can be obtained by shifting the motion-sensitizing gradients temporally. Time series of periodic displacements (and animations of wave propagation) can be obtained by incrementally varying this temporal delay between the mechanical excitation and the imaging gradients.

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Figure 3

(a) Top view of the wave-generating actuator. When a sinusoidal current i is sent through the coil in the longitudinal magnetic field Bo, an electromagnetic torque Tmag is developed, causing the actuator arm to vibrate back and forth. (b) A side view of the shaker apparatus showing the connection between the arm and a plastic machine screw nut glued to the skull. The coronal imaging plane is perpendicular to both views.

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Figure 4

MRE displacement images of a gel phantom (Gel 2) showing four time points in a complete cycle of wave motion at 400Hz. Waves can be most clearly seen in the PE (lateral) direction, which was the direction of excitation. The maximum amplitude in the PE direction was 33μm. Each frame is 18×17.5mm2. Directions are RO, inferior-superior; PE, lateral; and SS, anterior-posterior.

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Figure 5

Images of displacement from a FE simulation of shear wave propagation in a 3D viscoelastic solid. Parameters: shear modulus μ=1600N∕m2, loss factor η=0.1, and excitation frequency 400Hz. Image size is 25×25×6.25mm3; displacements were interpolated onto an array of 64×64×16 “voxels.”

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Figure 6

Shear modulus estimates and residual error for data from 400Hz excitation of gel phantom and FE simulation. Panel (a): Shear modulus estimate for Gel 2 (Fig. 4). The mean(±std.dev.) estimate was 1560±70N∕m2. Panel (b): Shear modulus estimate for the 400Hz FE simulation (Fig. 5). The mean(±std.dev.) estimate is 1760±90N∕m2. Panel (c): Residual error of wave equation fit for the gel phantom. (d) Residual error of wave equation fit for the FE simulation.

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Figure 7

(a) Waves in Gel 2 at 200Hz; ∼2.5 wavelengths/domain. (b) Map of shear modulus estimates, illustrating edge artifacts. (c) Map of residual error of fit to the wave equation, including edge artifacts. Edge effects are attributable to truncation error in Helmholtz decomposition and Laplacian estimation. With edge voxels masked out as in Fig. 6 above, the mean(±std.dev.) estimate was 1460±20N∕m2 with the Laplacian estimated in the frequency domain (shown); 1680±140N∕m2 with Laplacian estimated by finite differences in space.

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Figure 10

MRE images of displacement in a midcoronal mouse brain section showing four points in time during a cycle of wave propagation at 1200Hz with motion in all three directions. The maximum amplitude in the PE direction was ∼10μm. Each frame is 11.25×7.5mm2. Directions are RO, inferior-superior, PE, lateral; and SS, anterior-posterior.

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Figure 11

Dynamic shear modulus in anatomical sections. (a) Spin-echo “scout” images of anterior, middle, and posterior coronal sections of the mouse brain. (b) Representative images of dynamic shear modulus. Areas with residual error higher than 0.5 are masked out (dark blue). (c) Residual errors from the fit to the wave equation. Each frame is 11.25×7.5mm2.

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Figure 12

Dynamic shear modulus estimates (mean±std.dev.) in the cortical gray matter of the anterior, middle, and posterior mouse brain sections for six animals in vivo. Each bar shading represents a single mouse. The average estimates of the shear modulus of all six mice in the anterior, middle, and posterior sections were 14,800±2030N∕m2, 13,800±1490N∕m2, and 12,600±1990N∕m2, respectively.

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Figure 13

Dynamic shear modulus estimates (mean±std.dev.) in the subcortical gray matter in the anterior, middle, and posterior mouse brain sections for six animals in vivo. Each bar shading represents one mouse. The average estimates of the shear modulus of all six mice in the anterior, middle, and posterior sections were 18,700±2080N∕m2, 15,300±1480N∕m2, and 16,500±3060N∕m2, respectively.

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Figure 14

Dynamic shear modulus estimates (mean±std.dev.) in the anterior, middle, and posterior mouse brain sections for two animals postmortem. Each bar represents a single mouse. (a) Cortical gray matter. The average estimates of the shear modulus of both mice in the anterior, middle, and posterior sections were 14,600±50N∕m2, 14,100±1290N∕m2, and 13,900±70N∕m2, respectively. (b) Subcortical gray matter. The average estimates of the shear modulus of both mice in the anterior, middle, and posterior sections were 15,400±2180N∕m2, 14,200±800N∕m2, and 15,400±410N∕m2, respectively. No estimate of shear modulus was significantly different from the corresponding value observed in the living tissue (Student’s t-test).

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Figure 9

MRE images of displacement in the PE (lateral) direction in anterior, middle, and posterior mouse brain sections at four points in time during a cycle of wave propagation at 1200Hz. Excitation was in the PE direction. The maximum amplitude in the PE direction was approximately 10μm. Each frame is 11.25×7.5mm2.

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Figure 8

Estimates of dynamic shear modulus for (a) gel phantoms and (b) FE simulations as a function of frequency. (a) Filled-in markers represent the shear moduli determined by shear plate rheometry at 80Hz for three gel materials. Open markers represent the shear modulus estimates determined using MRE at 80Hz, 200Hz, 400Hz, and 800Hz. The gels exhibit frequency-dependent viscoelastic behavior; the dynamic shear modulus determined using elastography increases with increasing frequency. (b) Elastography estimates of shear modulus from FE simulations. Parameters: μ0=1600N∕m2 and loss factor η=0.1. Elastography yielded similar results for all frequencies of the FE model. Estimates are within 10% of each other, and are consistent with approximate values obtained from an estimate of wavelength in the RO direction (see Eq. 11, Table 2).

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