Recently several Magnetic Resonance Elastography (MRE) studies have shown elastograms with increased fine feature detail, posing a challenge to stiffness estimation illustrated in Fig. 1: fine features and interfaces add detail within a broader wavelength, differentiating fine-scale elasticity estimates from coarse.

Here we sought to investigate the following questions: 1) Do MRE stiffness estimates scale with level of detail (i.e. smoothing parameter value) or can a stable stiffness estimate be found? 2) Does such an estimate contain fine feature detail? 3) If not, what is the optimal reconciliation of the two sources of radiological information: fine features and stable property estimates?

Metrics: Two multifrequency MRE acquisitions were subjected to progressive smoothing and we evaluated the relationship between two variables: complex shear modulus magnitude |G*|, as determined by Helmholtz inversion, and image detail level or sharpness, as determined by the Reduced Energy Ratio (RER), a technique common to computational photography [1]. Here the RER measurement was adapted to three dimensions, applied to an 8x8x8 block of retained frequencies.

Acquisitions: Full-field displacements of a commercial QA elasticity phantom (CIRS, Norfolk, VA), were acquired at 50, 62.5, and 75 Hz using an EPI protocol in [2]. The phantom's prescribed background shear modulus is 8.3Kpa +/- 0.4 Kpa (Young’s modulus of 25 Kpa +/- 5%). In vivo axial acquisitions of brain data were also studied at the three frequencies of 30, 35, and 40 Hz, with the protocol in [3]. Images were phase unwrapped using the LBE technique in [2], and inverted using MDEV Inversion [4].

Progressive Smoothing: It was shown in previous work [5] that progressive Gaussian smoothing causes increasing artefact at borders and interfaces in MRE results. Consequently, in the present study Gaussian smoothing was paired with soft thresholding in a complex dualtree wavelet (CDTW) basis, an approach expected to better handle waves with interfaces and discontinuities [6], applied to MRE in recent work [7] . Protocol 1: images were smoothed with an isotropic 3D Gaussian kernel, with standard deviation (SD) stepped from 0.5 to 3px in intervals of 0.1 (support of 15px). Protocol 2: images were progressively de-noised in an CDTW basis using soft thresholding with threshold (T) stepped from 0.05 to 3 in intervals of 0.05.

Plots of stiffness against sharpness measurements are seen in Figure 2.

Stiffness: Gaussian stiffness estimates ranged, for phantom, 6.6-28.4KPa, and for brain, 816- 4,398 Pa, without a stable plateau. Wavelet estimates stabilised at mean values of 8248 Pa (phantom) and 1396 Pa (brain).

Sharpness: Peak RER values occurred early in all smoothing protocols. Phantom sharpness was highest in the Gaussian protocol at SD = 0.6 and highest in the CDTW protocol at T = 0.25 . Brain sharpness was highest in the Gaussian protocol at SD = 0.6 and highest in the CDTW protocol at T = 0.2.

Figure 3 illustrates the CDWT protocol results by contrasting three elastograms: an image that is noisy and of low sharpness (RER) : the image of maximum RER, but lower stiffness estimate than the stable estimate; and a later image with stable |G*| estimate, but low RER.

As anticipated, the Gaussian approach did not yield stable stiffness estimates for the phantom or brain acquisitions. The instability of the Gaussian smoothing confirms the findings in [5]. The wavelet estimate however remained stable with increasing thresholding, supporting the CDTW as a stable way to handle noise and feature scale, and deliver a spatially averaged estimate for the brain.

In the CDTW case, the image of highest sharpness came in the first 10% of the elasticity estimate, while values were lower than the stable value. This suggests that fine-featured elastography, which will contain additional fine features and interfaces, will produce lower elasticity estimates for the brain than more coarsely resolved elastography, in which the wavelengths will not contain these small interfaces. Thus more coarsely resolved feature estimates are not a lower-resolution average of fine-featured estimates, but will be in a different value range corresponding to the coarser feature scale.

This preliminary investigation suggests that optimal sharpness of detail, and stable stiffness estimate, may occur at different threshold values for a denoising protocol. This may explain value discrepancies between high-resolution and low-resolution elastograms in some cases. As an approach to deriving a stable, global stiffness estimate, the CDTW soft thresholding protocol seems robust and warrants further development. Sharpness estimates may also warrant future use as measurements of feature scale fineness in elastograms, aiding the reconciliation of elasticity estimates at different scales, and for this the RER seems effective.