Magnetic resonance elastography in nonlinear viscoelastic materials under load

Abstract

Characterisation of soft tissue mechanical properties is a topic of increasing interest in translational and clinical research. Magnetic resonance elastography (MRE) has been used in this context to assess the mechanical properties of tissues in vivo noninvasively. Typically, these analyses rely on linear viscoelastic wave equations to assess material properties from measured wave dynamics. However, deformations that occur in some tissues (e.g. liver during respiration, heart during the cardiac cycle, or external compression during a breast exam) can yield loading bias, complicating the interpretation of tissue stiffness from MRE measurements. In this paper, it is shown how combined knowledge of a material's rheology and loading state can be used to eliminate loading bias and enable interpretation of intrinsic (unloaded) stiffness properties. Equations are derived utilising perturbation theory and Cauchy's equations of motion to demonstrate the impact of loading state on periodic steady-state wave behaviour in nonlinear viscoelastic materials. These equations demonstrate how loading bias yields apparent material stiffening, softening and anisotropy. MRE sensitivity to deformation is demonstrated in an experimental phantom, showing a loading bias of up to twofold. From an unbiased stiffness of [Formula: see text] Pa in unloaded state, the biased stiffness increases to 9767.5 [Formula: see text]1949.9 Pa under a load of [Formula: see text] 34% uniaxial compression. Integrating knowledge of phantom loading and rheology into a novel MRE reconstruction, it is shown that it is possible to characterise intrinsic material characteristics, eliminating the loading bias from MRE data. The framework introduced and demonstrated in phantoms illustrates a pathway that can be translated and applied to MRE in complex deforming tissues. This would contribute to a better assessment of material properties in soft tissues employing elastography.

Keywords: Biorheology; Elastic waves; Magnetic resonance elastography; Nonlinear mechanics; Tissue mechanics.

Fig. 1

Planar waves through a loaded phantom; (Top left) Phantom in undeformed state. (Top right) Phantom in compressed state, compared against the undeformed state. (Bottom left) Planar wave created by moving the front face of the phantom (in blue) along the ${\mathit{e}}_1$ direction. (Bottom right) Planar wave created by moving the top face of the phantom (in blue) along the ${\mathit{e}}_2$ direction

Fig. 2

Illustration of experimental setup, protocol and data in rheological tests. (Left) Phantom in the rheological instrument. The moving platen is compressing the phantom and oscillates vertically, while the loading cell records the force. (Right, top) The traction measurements in one phantom for the six tests: four micro-oscillations and one macro-oscillation (sweeping over frequencies) and a relaxation test. (Right, bottom) Platen displacements, corresponding to phantom compression levels. Zoomed panel: exemplification of a micro-oscillatory test displacements, showing the frequency sweep; illustration of a cycle in the lowest frequency regime—one period, starting from the lowest point in the compression

Fig. 3

Illustration of the MRE test setup. (Top) 3D view of the setup. A coil vibrating to the frequency established by the MRE sequence is connected to a flexible lamella, which causes an attached rod to vibrate longitudinally. A piston fit at the end of the rod is indenting the phantom, thus generating compressional waves. The phantom is resting on a smooth support and is in contact with the back support plate, which prevents the phantom from slipping and thus helps converting the compressional waves into shear waves. An upper plate compressing the phantom is kept in place by bolts fixed to the side plates. Reception coils are placed around the phantom, for signal enhancement. (Bottom) 2D top view of the setup

Fig. 4

Images corresponding to the uncompressed (top row) and compressed (middle and bottom rows) phantoms (here illustrated in phantom 12). (First column) Phantom depicted at the different deformation states. The uncompressed state is kept for reference. The piston (grey bar) is indenting the phantom perpendicularly during the MRE scan. Slices are acquired in the coronal plane (depicted in blue). (Second column) Cross-sectional (coronal) view from the T2 weighted images. The piston indentation can be seen, as well as the expansion of the phantom in the ${\mathit{e}}_1-{\mathit{e}}_2$ directions under compression. Wave displacements from the MRE imaging protocol can be seen in the ${\mathit{e}}_1$ direction (third column), in the ${\mathit{e}}_2$ direction (fourth column) and in the ${\mathit{e}}_3$ direction (fifth column). It can be observed in the wave displacements, between the three rows, that the wavelength increases under compression. The wave images in the third, fourth and fifth columns have been cropped around the phantom area, to exclude the surrounding noise

Fig. 5

The minimum error (per Eq. 20) obtained by fitting the model to the data for each phantom, sweeping over fixed values of $\alpha$ between 0.01 and 1. (Left) The error for the 5-parameter model. (Right) The error for the 3-parameter model. The minimal error for each phantom, obtained by employing the parameters in Table 1 (5-parameter model) and Table 3 (3-parameter model), is enhanced

Fig. 6

Illustration of the model fit considering the 5-parameter model or 3-parameter model in phantom 3. (Left) The model fit to the data using all parameters: $C_1$, $C_2$, ${\delta }_1$, ${\delta }_2$ and $\alpha$ (with an error of 1.50%). (Right) The model fit to the data using a reduced number of parameters: $C_2$, ${\delta }_1$, and $\alpha$ (with an error of 1.78%). The parameter values can be seen in Tables 1 and 3

Fig. 7

Contribution of each parameter to the model (black) compared to the data (red) (here, in phantom 4). Each of the four linear parameters ($C_1$ (top left), $C_2$ (top right), ${\delta }_1$ (bottom left), ${\delta }_2$ (bottom right)) was fit to the data, while setting the other 3 to be 0 ($\alpha$ was kept constant at 0.03). In each quadrant, the traction is depicted along the y-axis and the time along the x-axis (waiting times between tests not plotted for convenience). The parameters and errors can be seen in Table 2

Fig. 8

Stiffness maps $G^{\prime }$ and corrected maps M corresponding to the uncompressed (top row) and compressed (middle and bottom rows) phantoms (here illustrated in phantom 12). (First column) Phantom depicted at the different deformation states. The uncompressed state is kept for reference. The piston (grey bar) is indenting the phantom perpendicularly during the MRE scan. Slices are acquired in the coronal plane (depicted in blue). (Second column) Stiffness estimates using the UR. The region of interest (ROI) of the apparent stiffness map $G^{\prime }$ was obtained by eroding the phantoms’ margins. (Third column) Estimates of map M using the CR. The ROI of maps M was obtained by eroding the phantoms’ margins

Fig. 9

Average of stiffness maps $G^{\prime }$ and corrected maps M over all pixels in a phantom (excluding boundaries), in all deformation states. (Left) Average of stiffness map $G^{\prime }$ obtained using UR (Eq. 9) on the MRE data. In undeformed state, when $\lambda =1$, the average stiffness of the phantoms lies within 4.5-5.3 kPa. With increased compression, the phantoms appear to be stiffer, following a nonlinear trend. (Right) Average of corrected maps M, obtained by employing CR (Eq. 22) on the MRE data. The average value of the corrected maps M spans between 0.96 and 1.11. The standard deviation for all phantoms in all compression states cross the line at the ideal value $M=1$