K-space data processing for magnetic resonance elastography (MRE)

Abstract

Objective

Magnetic resonance elastography (MRE) requires substantial data processing based on phase image reconstruction, wave enhancement, and inverse problem solving. The objective of this study is to propose a new, fast MRE method based on MR raw data processing, particularly adapted to applications requiring fast MRE measurement or high elastogram update rate.

Materials and methods

The proposed method allows measuring tissue elasticity directly from raw data without prior phase image reconstruction and without phase unwrapping. Experimental feasibility is assessed both in a gelatin phantom and in the liver of a porcine model in vivo. Elastograms are reconstructed with the raw MRE method and compared to those obtained using conventional MRE. In a third experiment, changes in elasticity are monitored in real-time in a gelatin phantom during its solidification by using both conventional MRE and raw MRE.

Results

The raw MRE method shows promising results by providing similar elasticity values to the ones obtained with conventional MRE methods while decreasing the number of processing steps and circumventing the delicate step of phase unwrapping. Limitations of the proposed method are the influence of the magnitude on the elastogram and the requirement for a minimum number of phase offsets.

Conclusion

This study demonstrates the feasibility of directly reconstructing elastograms from raw data.

Appendix

A temporal Fourier transform is applied on a set of k-space raw data acquired with varying delays between mechanical excitation and motion encoding gradients. By selecting only the temporal frequency of interest, most of the harmonics of the spatial frequency are removed.

The result is described by Eq. (12):

$$S\left( {\nu_{x} ,1 } \right) = \mathop \sum \limits_{n \in U} \mathop \sum \limits_{x = 0}^{{N_{x} - 1}} M\left( x \right)e^{{j\phi_{0} \left( x \right)}} e^{{ - \frac{{j2\pi \left( {\nu_{x} x} \right)}}{{N_{x} }}}} j^{n} N_{t} J_{n} \left( {C_{1} \left( x \right)} \right)e^{{jn\left( { - \frac{{\left( {k_{x} x} \right)}}{{N_{x} }} + C_{2} } \right)}}$$

$$U = \{ n |n = kN_{t} + 1,\quad k \in Z\} .$$

As described by the set U, the component of interest (n = 1) is not the only one component selected by the filtering because of aliasing. In most cases, the other components are negligible compared to the first one (blue signal in Fig. 8b), nevertheless the selection of other non–negligible components remains possible when the number of phase-offsets is poorly chosen as illustrated in Fig. 8c–d (red signal). It is therefore relevant to assess conditions that favor an optimal ratio between the amplitude of the 1st and the (kN _t  + 1)th, k ∊ Z ^* components of the spectrum that are susceptible to being associated to the same temporal frequency. The ratio $R$ is deduced from the previous equation:

Fig. 8

a Simulated 1D MR signal with heterogeneous magnitude and sinusoidal phase. b Plot of the temporal Fourier transform of the previous k-space performed from the data set simulating six varying delays between motion encoding gradients and mechanical excitation. c Plot of the temporal Fourier transform performed from the data set simulating 3 varying delays between motion encoding gradients and mechanical excitation. d Plot of the filtering at the temporal frequency of interest in blue for the case with six phase-offsets and in red for the case with three phase-offsets

$$R = \log \left( {\frac{{J_{1} \left( {C_{1} } \right)}}{{J_{kN + 1} \left( {C_{1} } \right)}}} \right).$$

Let us consider that a component is negligible compared to the first one when its amplitude is at least ten times inferior, i.e. when R > 1. The number of phase-offsets N _t required to satisfy this condition is plotted in Fig. 9 with respect to the C ₁ parameter. Under the assumption that R > 1, Eq. (12). simplifies to Eq. (14).

Fig. 9

Assessment of the minimal number of phase-offsets N _t required to satisfy the condition R > 1 with respect to the amplitude of the encoded wave C ₁

$$S(\nu_{x} ,1) = \mathop \sum \limits_{x = 0}^{{N_{x} - 1}} M\left( x \right)e^{{j\phi_{0} \left( x \right)}} e^{{ - \frac{{j2\pi \left( {\nu_{x} x} \right)}}{{N_{x} }}}} j N_{t} J_{1} \left( {C_{1} \left( x \right)} \right)e^{{j\left( { - \frac{{k_{x} x}}{{N_{x} }} + C_{2} } \right)}} .$$