Introduction

Translating hepatic 3D MRE to wide-bore low-field MR systems(B₀≤1.0T) can serve as a means of accommodating liver patients with high BMI. Furthermore, the longer T2*relaxation times at low-field may be beneficial to mitigate iron overload(1, 2). Another aspect is that low-field MR system bring down the financial entry point of MR which might allow a wider spread of MRE(3).

However, low-field MR systems come with a penalty in SNR and phase-to-noise ratio(PNR). A Hadamard-motion-encoding scheme may be used to increase PNR(4), where unique combinations of MEGs are applied on all the gradient axes simultaneously; simultaneous gradients lead however to concomitant fields which are inversely proportional to the static magnetic field B₀(5).

The goal is to design an experiment that measures the effect of concomitant fields on Hadamard-encoded 3D MRE at 0.55T.

Methods

When applying a magnetic gradient field, there is a concomitant magnetic field B_c resulting from Maxwell’s equations and given to the lowest order by the following equation(5):
$$B_c(x,y,z,t)=\frac{1}{2B_0}\left(G_x^2z^2+G_y^2z^2+G_z^2\frac{x^2+y^2}{4}-G_xG_zxz-G_yG_zyz\right)$$

Where B₀ is the static field, and G_x,G_y,G_z are the applied gradients at time point t, and x,y,z are the spatial coordinates.

Let us consider a bipolar MEG G with lobes(G₂=-G₁), with a total duration T, applied on x,y,z and without overlap with any imaging gradient in the MRE sequence. Following the terminology in(5), the phase accrual $$$\phi_c$$$ due to B_c by a spin isochromat is:

$$\varphi_c=\frac{\gamma T}{2B_0}(\overbrace{G_x^2z^2+G_y^2z^2+G_z^2\frac{x^2+y^2}{4}}^{\text{"self-squared"terms}}\underbrace{-G_xG_zxz-G_yG_zyz}_{\text{"cross"terms}})$$

The following 4x4 Hadamard matrix(H) is typically used in MRE measurement:
$$H=\left[\begin{array}{llll}-1&+1&-1&+1\\+1&-1&-1&+1\\-1&-1&+1&+1\\+1&+1&+1&+1\end{array}\right]$$

The first 3 terms in the previous Eq, the “self-squared”-terms, are constant throughout the different measurements of Hadamard-encoding and are therefore encoded in the same term that encodes constant phase errors such as magnetic field inhomogeneity. However, the last 2 terms, the “cross”-terms, change signs between the different measurements of Hadamard-encoding depending on the polarity of the applied motion encoding gradients. As a result, the “cross”-terms cannot be resolved using a conventional 4x4 Hadamard-encoding matrix. We present a 6x6 encoding matrix M that resolves the “cross”-terms by accounting for the different possible sign combinations of the "cross"-terms:

$$M=\left[\begin{array}{llllll}-1&+1&-1&+1&+1&-1\\+1&-1&-1&+1&+1&+1\\-1&-1&+1&+1&-1&+1\\+1&+1&+1&+1&-1&-1\\+1&-1&+1&+1&+1&-1\\-1&+1&+1&+1&+1&+1\end{array}\right]\\$$
$$\left[\begin{array}{l}m_1\\m_2\\m_3\\m_4\\m_5\\m_6\end{array}\right]=M\left[\begin{array}{l}\varphi_{U_z}\\\varphi_{U_x}\\\varphi_{U_y}\\\varphi_{err}\\\varphi_{xz}\\\varphi_{yz}\end{array}\right]$$

Where m_k(k=1,2,..,6) is the k-th MRE phase measurement, $$$\phi_{U_x}$$$,$$$\phi_{U_y}$$$,$$$\phi_{U_z}$$$ are the phase accruals due to the 3D displacement field. $$$\phi_{err}$$$ is the phase accrual due to constant phase errors and the “self-squared”-terms. $$$\phi_{xz}$$$and$$$\phi_{yz}$$$ are the phase accrual due to the first and second “cross”-terms respectively.

The proposed encoding matrix M is incorporated into a 3D Ristretto MRE sequence(6) extending the sequence from 4to6 measurements. The MEGs are applied without any overlap with the imaging gradients.

The extended 3D MRE sequence is implemented on a 0.55T system(MAGNETOM Free.Max, Siemens Healthineers AG, Erlangen, Germany). Two phantom experiments were performed: (I)without vibration to verify that M is solving for the “cross”-terms. Note that the first “cross”-term($$$G_xG_zxz$$$)varies with x and z, and the second “cross”-term($$$G_yG_zyz$$$)varies with y and z. (II)with 60Hz vibration(7) to verify that the phase accrual due to B_c is constant with respect to the acquired wave-phase-offsets and can be filtered-out using temporal Fourier Transform. The phantom used is an ultrasound-gel phantom shifted to right of the iso-centre to increase the apparent effect of the concomitant fields.

The imaging parameters of the extended 3D MRE sequence were as follows: 8slices,4mm isotropic resolution,a 96X64 acquisition matrix,flip-angle=25°,in-plane GRAPPA acceleration factor of 2 resulting in a FOV of 386X256X32mm3,TR=18.24ms,TE=12.90ms,receiver bandwidth=180Hz/px. The motion encoding gradients followed the proposed 6x6 encoding matrix. The total acquisition time was 126seconds preformed in six measurements of 21seconds each.

Results

The results of the phantom experiment without vibration are shown in Figure1. There is no spatial variation in $$$\phi_{U_x}$$$(Figure1.B). However, we observe a spatial variation in x (left-right) in $$$\phi_{xz}$$$(Figure1.C) and in y (posterior-anterior) in $$$\phi_{yz}$$$(Figure1.D). The spatial variation observed in Figures1.C-D suggests that the proposed encoding matrix M is solving for the “cross”-terms. The amplitude of the displacement field in the three terms is very low(0.9±0.2um,1.4±0.2um,and1.6±0.2um respectively)(Figures1.E-G).

The results with vibration are in Figure2. $$$\phi_{U_x}$$$(Figure2.B) varies with the vibration, however, $$$\phi_{xz}$$$(Figure2.C) and $$$\phi_{yz}$$$(Figure2.D) only vary spatially in x and y respectively. In addition, the amplitude of the displacement field encoded in $$$\phi_{U_x}$$$(81±11um) is approximately an order of magnitude higher than the amplitudes encoded in $$$\phi_{xz}$$$(5±2um) and $$$\phi_{yz}$$$(10±2um)(Figures2.E-G).

Discussion and conclusions

In our study, we characterised the effects of concomitant fields on Hadamard-encoded 3D MRE at 0.55T using a 6x6 encoding matrix and verified in phantom experiments that the displacements encoded due to concomitant field terms are negligible. The phase errors due to concomitant fields in MRE can be considered constant across wave-offsets and therefore will mainly contribute to the DC-component of the temporal Fourier Transform.

Acknowledgements

No acknowledgement found.

References

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