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directional-bandpass-filtered data (Z=-1.067, p=0.285). However, there were no significant differences between the no-filtered and bandpass-filtered data (Z=-3.724, p=-0.00019), or between the no-filtered and directional-filtered data (Z=-3.68, p=0.00023), or between the no-filtered and directional-bandpass-filtered data (Z=-3.724, p=0.00019), or between the bandpass-filtered and directional-filtered data (Z=-3.636, p=0.00027), or between the directional-filtered and directional-bandpass-filtered data (Z=-3.680, p=0.00023).

In case of elasticity value calculated by LFE and AIDE for both MD and LBE unwrapped left and right psoas major and MD-unwrapped right psoas major, the stiffness value calculated by using directional-bandpass filtered data was same as bandpass-filtered data.

6.3.3.5. Comparison of stiffness values between MD-unwrapped and LBE-unwrapped data

Wilcoxon signed-rank test showed statistically significant difference between the LFE and AIDE stiffness values, obtained from both MD-unwrapped and LBE-unwrapped data for no-filtered, bandpass-filtered, directional-filtered and directional-bandpass-filtered data pairs. The calculated p-value was less than 0.05. The stiffness values calculated by using LFE were significantly higher than the stiffness values calculated by using AIDE for both MD and LBE unwrapped data.

6.4. Discussion

127 6.4.1. MREWave Phantom Data

The stiffness value by applying AIDE was underestimated for large inclusion in no-filtered data due to the presence of noise. However, LFE stiffness value was not underestimated significantly for large inclusion in no-filtered data. This is the evidence of usage of lognormal filters in LFE^[54]. AIDE does not incorporate any filter in its algorithm and is simply a direct inversion of the equations of motion^[54,116]. The spatial resolution of the elastogram produced by LFE is lower^[188] and is determined by the width of the filters incorporated in it. The elastograms obtained by using bandpass-filtered and directional-bandpass filtered data appeared smoother in both LFE and AIDE maps. Yet, the stiffness map obtained by using directional-filtered data were not smooth because the directional-filtered data still constituted low and high frequency noise components.

Directional filter could only remove wave interference components^[106].

6.4.2. Fruit Jelly Phantom Data

The stiffness values obtained by using LFE were higher than the stiffness values calculated by using AIDE in all data groups. This may be because AIDE involves the calculation of Laplacian^[9]. The Laplacian could enhance noise and the presence of noise could decrease MRE signal and wave amplitude. This subsequently underestimate the elasticity values. However, LFE involves log-normal quadrature filters in its algorithm before it calculates the stiffness value^[9]. Hence, LFE is immune to noise. LFE could overestimate the elasticity value at low amplitude and noisy areas, as the local wave number calculated by LFE might not fall within the range of filters used in its algorithm^[116]. Particularly, the no-filtered and directional-filtered data showed higher values because the noise components were present in those data. Hence, the stability of

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the elastogram may not be achieved. Again, the stiffness map obtained by using AIDE from no-filtered and directional-filtered data showed inhomogeneous illuminations due to standing wave nodes. Standing wave are distinguished by nodes with zero displacement amplitude^[116]. The elastograms obtained from bandpass-filtered and directional-bandpass-filtered data were numerically stable^[116]. However, the window width of the bandpass was manually selected, and we recommend automatic filters for more accurate denoising of the data. There were two direction of wave propagation in fruit jelly phantom, R→L for right jelly and RL for left jelly. A uni-directional filter was applied twice for each direction of wave propagation. Nonetheless, a two-directional filter along the directions of propagation of wave could be applied^[106].

6.4.3. Psoas Major Muscle Data

The stiffness values of MD-unwrapped data were significantly higher than LBE-unwrapped data and the LFE values were higher than the AIDE values. LFE can overestimated the stiffness values for low amplitude areas in MD-unwrapped data. LFE might output incorrect local wave number if the local frequency fall outside the filter range incorporated in it, that might subsequently lead to incorrectly elasticity value^[116]. In case of AIDE, LBE-unwrapped data contained higher noise than MD-unwrapped data because of noise amplification by Laplacian operator in LBE^[105]. Higher amount of noise in LBE-unwrapped data could decrease MRE signal, that might underestimate the stiffness value as compared to MD-unwrapped data in AIDE. MD unwraps the data without using Laplacian operator, just by adding integral multiple of 2π to each phase wrapped pixel^[109]. Hence, MD-unwrapped data contained less noise compared to

LBE-6.4. Discussion

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unwrapped data. This suggested the elasticity values is dependent upon the type of usage of phase unwrapping algorithm and the wave inversion method.

The LFE stiffness values of directional-bandpass filtered and bandpass-filtered data for both MD and LBE unwrapped data were significantly lower than no-filtered and directional-filtered data. This implied that the no-filtered and directional filtered data contained noise components. The presence of noise and low amplitude motion might output incorrect local wave number, thus incorrect LFE elasticity value^[116]. The AIDE stiffness values of no-filtered and directional-filtered data for both MD and LBE unwrapped data were lower than bandpass-filtered and directional-bandpass filtered data.

The no-filtered and directional-filtered data contained noise. Furthermore, AIDE itself involves the calculation of Laplacian, that amplifies noise^[9].

The stiffness values of directional-bandpass-filtered data removed high frequency noise, low frequency noise and wave reflections. The bandpass filter provided numerical stability during wave inversion^[116] and directional filter removed wave reflections^[54]. In case of the stiffness values of bandpass-filtered and directional-bandpass-filtered data were not significantly different, the directional filter did not affect the stiffness output.

Bandpass filter was selected by prior windowing and the bandwidth of the filter can be changed manually. However, automatic windowing of filter bandwidth is recommended for future work for more precise elasticity reconstruction. Again, there were two main wave propagation directions, waves travelling in the form ripple from lumbar spine towards each psoas major. The data were analysed by using uni-directional filter.

However, a bi-directional filter could be applied along the directions of propagation of wave^[106].

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We presume this chapter provided the important notion of the impact of choice of usage of phase unwrapping algorithm and image filtering in wave inversion algorithm. MD and LBE are commonly used phase unwrapping algorithms. Yet, the mean amplitude value (MAV) of LBE-unwrapped data was higher than MD-unwrapped data. Higher MAV yields greater signal in MRE, that provides accurate elastogram reconstruction^[116]. Similarly, both LFE and AIDE are promising, and widely adapted wave inversion methods used in MR elastography, researchers can decide on their own which phase unwrapping and wave inversion method to use. Since both LFE and AIDE are inverting Helmholtz equation^[9], we recommend using both methods according to own requirements and preferences.

Again, the stiffness value of psoas major can also be used as a bio-imaging marker for low back pain (LBP). Since LBP is almost non-specific^[189], MR elastography of psoas major could radiate important insights for the clinical implications of LBP. Numano et al^[75] have described the MRE technique of psoas major in detail.

There are few limitations of the present study that must come under consideration. First, we cannot measure the elasticity of psoas major in vivo. It was impossible to mention which wave inversion method and which phase unwrapping algorithm yielded the correct value of elasticity of psoas major. Second, attenuation and loss modulus were not considered, since loss modulus cannot be measured from LFE and it assumes zero attenuation^[54]. Though AIDE could calculate loss modulus and an attenuation map^[53,54], it was impossible to compare LFE and AIDE based on these parameters. That being said, Manduca et al^[54] have shown that the visual clarity of the attenuation map could be enhanced by using directional filter. Third, we performed MRE experiments at 50 Hz