Magnetic Resonance Measurement of Transient Shear Wave Propagation in a Viscoelastic Gel Cylinder

Abstract

A magnetic resonance measurement technique was developed to characterize the transient mechanical response of a gel cylinder subjected to angular acceleration. The technique employs tagged magnetic resonance imaging (MRI) synchronized to periodic impact excitation of a bulk specimen. The tagged MRI sequence provides, non-invasively, an array of distributed displacement and strain measurements with high spatial (here, 5 mm) and temporal (6 ms) resolution. The technique was validated on a cylindrical gelatin sample. Measured dynamic strain fields were compared to strain fields predicted using (1) a closed-form solution and (2) finite element simulation of shear waves in a three-parameter "standard" linear viscoelastic cylinder subjected to similar initial and boundary conditions. Material parameters used in the analyses were estimated from measurements made on the gelatin in a standard rheometer. The experimental results support the utility of tagged MRI for dynamic, non-invasive assays such as measurement of shear waves in brain tissue during angular acceleration of the skull. When applied in the inverse sense, the technique has potential for characterization of the mechanical behavior of gel biomaterials.

Figure 1

(A) The cylindrical container of gelatin was secured in a freely rotating assembly via two collars. An off-axis weight provided a gravitational torque. (B,C) On release of a latch, the container of gel cylinder rotated until it was abruptly stopped by a round plastic bar, which was cushioned by a rubber sleeve.

Figure 2

Tagged MR images of the central plane of the gel cylinder. Tag lines move with the deforming material. (A) Reference image. (B) Deformed gel at time of maximal positive radial-circumferential shear. (C) Deformed gel at time of maximal negative shear strain.

Figure 3

Tagged MR images superimposed with isocontours of phase of the dominant harmonic components of the tagged image. The intersections of these contours are tracked to measure displacement and strain.

Figure 4

Angular acceleration, α, angular velocity, ω, and angular displacement, θ, of the outer boundary of the gel cylinder.

Figure 5

Components of the complex shear modulus G* estimated by shear plate rheometry. The storage modulus G′= Re[G*] and the loss modulus G″= Im[G*]. Dots (●) represent experimental data; solid lines represent the frequency response of the linear viscoelastic model (Eq. 4) with parameters given in Table 1.

Figure 6

Components of the Lagrangian strain tensor from experiment (A) and nonlinear simulation (B). Strain components are shown in Cartesian coordinates (top) and polar coordinates (bottom).

Figure 7

Radial-circumferential Lagrangian shear strain, ε_rθ, fields for the nonlinear simulation (top) and experiment (bottom).

Figure 8

Radial-circumferential shear strain, ε _rθ, as a function of time and radial position, shown as image, surface, and time series. (a-c) Analytical solution (linearized strain). (d-f) FE simulation (linear). (g-i) FE simulation (with geometric nonlinearity). (j-l) Experiment (Lagrangian strain).

Figure 9

Comparison of key statistics (maximum, minimum, and root mean square values) of radial-circumferential Lagrangian shear strain in the gel cylinder. Strain statistics from analysis of tagged MR images (dots) and from the nonlinear FE model (circles).