Introduction

Full 3D wavefield magnetic resonance elastography (MRE) data are usually acquired with multi-slice spin-echo planar sequences. Bulk motion during the motion encoding causes spurious phase jitter from slice to slice. The curl calculation that is often used to remove the effects of longitudinal waves amplifies untreated jitter. Currently, phase jitter is reduced using in-plane filtering techniques that either subtract away slowly varying in-plane phase,¹ or subtract a constant phase from each slice to minimize second-order differences in phase across slices.² Our experience is with the former method, referred to here as low-pass subtraction (LPS). We observe that artifact is incompletely mitigated in some cases, causing errors in stiffness images generated by 3D inversion processing. We hypothesized that higher performance would be achieved by a neural network trained to recognize and remove slice-to-slice phase discontinuities, while simultaneously denoising the data.

Method

Algorithm training
An Inception-like convolutional network was trained to learn a phasor corresponding only to the shear displacements. Training data were complex-valued image patches generated according to a prescribed forward model. For each training example, the magnitude was taken from in vivo brain MRE data while the phase was prescribed by 6 components:

1. Simulated shear wave field (u) computed using an analytical wave equation (maximum stiffness of 5 kPa, maximum damping ratio of 0.6)
2. Simulated background phase field (3rd order 3D polynomial) modeling the composite effects of B0 inhomogeneity, eddy currents, and concomitant fields
3. Slice-to-slice perturbation (max of 25%) of the background phase
4. Linear phase ramp (max range of ±70 radians) in the slice direction
5. Slice-to-slice perturbation of the linear phase ramp (corresponding to motion up to 3 mm)
6. Random constant (0-2π) to shift the location of phase wrapping.

Normally distributed i.i.d. noise was then added to the complex image patch. A brain mask was applied to isolate the desired measurements. The network inputs included a simulated positive- and negative-motion encoding direction (sign of u was flipped and random processes newly sampled for negative encoding) in a 15×15×15 voxel patch. The targets of the neural network were the real and imaginary parts of exp(i*u).

Simulation experiment
A simulated data set was generated to test the algorithm against known displacements. Displacements were computed using a finite difference model (FDM) of the governing equation for linear, elastic, isotropic materials with brain geometry derived from the T1-weighted image of a 55-year-old healthy male volunteer.³ Brain voxels were assigned a stiffness of 2 kPa and a damping ratio of 0.25; cerebrospinal fluid (CSF) voxels were assigned 0.15 kPa stiffness and 0.7 damping ratio. Using Dirichlet boundary conditions, a 60-Hz harmonic bulk displacement in the anterior-posterior direction was applied to CSF voxels, excluding the ventricular system, and the FDM was solved using 20,000 iterations of the conjugate gradient method. The forward model described above was applied to create a simulated MRE data set with known displacements. The SNR (10-50) and background phase field (3rd polynomial, max of 0-2pi) were random for each motion direction and time offset. A second simulated data set without i.i.d. noise was generated in the same manner to separate the effects of artifact removal and denoising. Displacement images were computed in three ways:

1. Phase subtraction of positive and negative encoding directions → phase unwrapping⁴ → temporal Fourier transform
2. LPS → phase subtraction of encoding directions → phase unwrapping → temporal Fourier transform
3. Net denoising → phase unwrapping → temporal Fourier transform

Displacement images (true and estimated) were mean-centered for each direction. The estimated displacements were then compared to the true values via linear regression.

In vivo experiment
This experiment used previously acquired data from a brain MRE repeatability study in which 10 volunteers were scanned 3 times each.⁵ First, displacement images were computed in three ways, as in the simulation experiment. The curl was then computed with a neural network trained to perform numerical differentiation, and stiffness maps were computed by neural network inversion.⁶ For each filtering method, repeatability of global brain stiffness was summarized as the coefficient of variation (CV) in each volunteer. Differences in CV between filtering methods were tested with a linear mixed-effects model (P<0.05 significant).

Results

Correlation coefficients of the first harmonics of the displacements (Fig 1&2) were calculated and plotted for each of the processing methods: no filter, LPS filter, and neural net denoising. This was done for both the noisy and noise-free simulated data. Example in vivo curl images are shown in Fig 3. Both filtering methods significantly reduced CVs relative to no filtering, but the two filters did not have significantly different CVs (Fig 4).

Conclusion

The neural net-based phase correction best reproduced the true displacement values in simulation (Fig 1&2). The improvement in performance for both noisy and noise-free data suggests the higher correlation given by the net is due to improved slice to slice dejittering. In vivo, the net further reduces jitter artifact relative to LPS (Fig 3). Moreover, an examination of the CV in the repeatability study shows that the net reduces CV relative to LPS, though not to a statistically significant degree in this sample.

Acknowledgements

This work was supported by NIH grants EB001981 and EB027064.