Cemre Ariyurek^{1,2}, Bilal Tasdelen^{1,2}, Alireza Sadeghi-Tarakameh^{1,2}, Yusuf Ziya Ider^{1}, and Ergin Atalar^{1,2}

^{1}Department of Electrical and Electronics Engineering, Bilkent University, Ankara, Turkey, ^{2}National Magnetic Resonance Research Center (UMRAM), Ankara, Turkey

### Synopsis

Reconstructed elastography
maps suffer from noise due to derivative operations in the inversion algorithms. Considering
the noise on the complex MRI signal, SNR of the reconstructed elastogram is
derived analytically for k-MDEV inversion. Verified SNR derivations by Monte
Carlo Simulations, an SNR-weighted k-MDEV inversion was proposed to maximize
SNR of the elastogram. Quality of the proposed reconstruction was assessed by
estimation error, noise performance and edge response analyses. SNR of the
elasticity improved twice as the conventional method. Results on phantom and human liver data are also provided. As future work, this analysis will be performed
for other inversion methods.

### Introduction

Although SNR of the elasticity map is critical
for diagnostic purposes, there are only a few studies on the SNR of MR
elastography (MRE)^{1,2}. Moreover, the SNRs of the elasticity
reconstruction algorithms have not been investigated analytically. Considering the
noise on the complex MRI signal, SNR of the reconstructed elasticity map can be
formulated. This analysis can be beneficial for deciding weights for combining
multi-directional, multi-frequency MRE data in order to obtain high quality
elasticity maps. ### Theory

The primary noise source on complex MRI signal is
thermal and modelled as additive white Gaussian noise (WGN). With the
assumption of high SNR, the noise characteristics of the wave speed obtained
using k-MDEV inversion^{3} technique can be linearized and analytical
expression for SNR of the wave-speed (SNR_{c}), shown in Figure 1, can be
obtained. It can be seen that SNR_{c} is directly proportional with SNR
of the MRI (SNR_{MRI}) signal, voxel size, number of directional filters, and inversely
proportional with the number of phase offsets and size of the Gaussian kernel.### Methods

To verify analytical derivations, SNR_{c}
was computed by Monte Carlo Simulations (MCS) conducted on a simulated plane
wave data by adding WGN with SNR swept from 1 to 28 and compared
with analytically derived SNR of the wave speed map. The minimum SNR value (SNR_{min})
that high SNR assumption holds for the analytical derivations was determined. Using
this information, an SNR-weighted elasticity reconstruction was implemented by
modifying k-MDEV inversion, as depicted in Figure 1. In the proposed
reconstruction, computed wave numbers at each frequency, motion encoding direction
and directional filter are weighted averaged with the square of SNR of wave
numbers (SNR_{k}). SNR values of the filtered displacements at the
output of directional filtering are computed analytically and corresponding SNR_{k}s
are set to zero if they are under the threshold, determined by SNR_{min}.
To test the SNR-weighted reconstruction; estimation error, noise performance
and edge response analyses were conducted. Estimation error and noise performance analyses
were conducted on a 2D homogeneous phantom. For estimation error, the
mean-squared-error between the reconstructed wave speed and the ground truth
was calculated. Moreover, noise performance of the proposed SNR-weighted
inversion was assessed by adding WGN with SNR swept and calculating standard
deviation of the wave speed map over repetitions. For edge response analysis, simulations
on a 2D step phantom with two mediums having different elasticities were
conducted and response of the reconstruction was investigated at the edge between
mediums. For validation on experimental data, multifrequency MRE phantom data^{4},
reported in previous studies^{3,5}, and multifrequency MRE liver data^{6} were used. For comparison, all the analyses were additionally performed on amplitude-weighted
(conventional) and Octahedral Shear Strain (OSS)^{1}-weighted^{7} k-MDEV. ### Results

Comparison of computed and analytical SNR_{c}
is shown in Figure 2(a,b). Estimation errors for all reconstructions were
computed to be less than 0.2%, indicating the accuracy of the methods. Results
of MCS to assess the noise performance and edge response of the reconstructions
are depicted in Figure 3(a,b). Furthermore, reconstructed wave speed maps for
MRE phantom and human liver are depicted in Figure 4 and 5, respectively. ### Discussion

Analytically derived SNR is verified by
computed SNR using MCS, which holds for SNR_{MRI}>6 (Figure 2). This
implies that the high SNR assumption used in the derivations fails below this value.
According to MCS results, depicted in Figure 3(a), SNR of the elastogram was improved twice
and four times when SNR-weights were used instead of amplitude-weights
and OSS-weights, respectively. As seen in Figure 3(b), there are no differences
for the edge response of the reconstructions; hence resolutions are measured to
be the same for the parameters used. In Figure 4, some wave artifacts are
observed at the background of the amplitude-weighted inversion which are
suppressed in other reconstructions. Mean wave speed of liver, stomach, intervertebral
disk and spinal cord were found to be consistent with the values reported in
the literature^{3}. In addition, contrast of the SNR-weighted wave
speed map of the liver is slightly better than other reconstructions. Despite the absence
of a ground truth for human liver results, based on significant improvements in
the SNR of the elastography maps without any loss in resolution and accuracy, one
may argue that this weighting can be useful to achieve SNR maximization for MRE
inversion.
### Conclusion

SNR of the elasticity map for the k-MDEV algorithm
has been derived and used as weights to combine multiple data to obtain
possible maximal SNR for the elastogram. In addition, this analysis could be
useful for tuning parameters of the inversion to increase SNR of the elastograms.
Moreover, SNR performances of inversion algorithms could be compared. As a future work, this analysis will be performed to other inversion techniques. ### Acknowledgements

No acknowledgement found.### References

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