Benjamin Zahneisen^{1}, Murat Aksoy^{1}, and Julian Maclaren^{1}

^{1}Stanford University, Stanford, CA, United States

### Synopsis

Motion and geometric distortions of EPI acquisitions remain
challenging as motion during an fMRI scan affects the stability of the time
series in two ways: Rigid motion displaces voxels by “moving” the spins. A
change in head orientation and the complex interplay of external and internal susceptibility
differences lead to a change in the off-resonance field. In combination with an
EPI readout this change in off-resonance field leads to an “apparent” voxel
displacement. Here, we propose the simultaneous acquisition of a blip-down navigator
with the original blip-up host-EPI sequence which is used to derive a snapshot
of dynamic off-resonance changes.

### Introduction

Geometric distortions of EPI acquisitions remain challenging
especially for high-field applications or in the case of strong off-resonance
fields. Subject head motion during an fMRI scan affects the stability of the
time series in two ways: 1) Rigid body motion displaces voxels by actual movement. 2) A change in head orientation and the complex interplay of external
and internal susceptibility differences lead to a change in the off-resonance
field. In combination with an EPI readout, this change in off-resonance field
leads to an “apparent” voxel displacement.
Prospective motion correction allows for correction of the rigid
component, but dynamic changes of the fieldmap remain a source for artifacts. Here, we propose the simultaneous acquisition of a blip-down
navigator within the original blip-up host-EPI sequence. The navigator and the
host-EPI sequence then have opposite geometric distortions (voxel stretching <->
compression).
Using image processing it is possible to derive an
off-resonance map from the low resolution blip-down navigator and the host-EPI sequence.
If both, the navigator and the host sequence are accelerated,
one can choose the sampling pattern in a way that the navigator fills in
missing k-space samples.
By performing a joint off-resonance corrected hybrid-SENSE
reconstruction (1) the g-factor penalty is
reduced because of the higher sampling density in k-space.### Theory

Ignoring off-resonance effects
along the readout direction, and performing an FT along the $$$k_x$$$ domain, transforms the k-space data $$$s_0(k_x,k_y,k_z)$$$ to an hybrid space $$$ s(x,k_y,k_z)
$$$.
For every position $$$
x=x_n$$$ we can write the signal equation (forward model) in matrix notation as $${\bf s=Fm} \hspace{1cm} [1]$$ with the unknown magnetization $$$
{\bf m} \in \mathbb{C}^{N_yN_x \times 1} $$$ and the signal vector $$$ {\bf s}
\in \mathbb{C}^{N_sN_c \times 1} $$$ that consists of all $$$N_s$$$ acquired k-space samples times the number of receiver coils $$$N_c$$$.
The effective encoding
matrix $$$ \bf F $$$ models phase effects due to gradient encoding and
off-resonance contributions. In order to invert the rectangular matrix $$$\bf F$$$, we use a truncated singular value decomposition approach (TSVD) (2) where we completely
suppress contributions from eigenvalues below a certain threshold $$$\lambda$$$.
The solution is then given as $${ \bf m = F}_{pinv}^{reg}{\bf s} \hspace{1cm} [2]$$ For the joint blip
up/down reconstruction we rewrite the signal model as $${\bf F_{\uparrow
\downarrow}}=\left(\begin{array}{c}{\bf F_{\uparrow}}\\ {\bf
F_{\downarrow}}\end{array}\right) \hspace{1cm} [3] $$ where $$$\bf F_{\uparrow}$$$ and $$$\bf
F_{\downarrow}$$$ are the effective
encoding matrices for the blip up and down trajectories.
Noise enhancement maps
(g-factor) are calculated as $$ g=diag(\sqrt{\frac{{\bf
F}_{pinv}^{reg}({\bf F}_{pinv}^{reg})^T}{{\bf C C}^T} })/\sqrt{R_{tot}}$$### Methods

Measurements were performed at 3T using
an 8-channel head coil array. All reconstructions are performed off-line using
MatLab.
The “blip down navigated”-EPI acquisition is
implemented by concatenating the blip-down navigator-EPI and the original
EPI-acquisition while continuously playing out the EPI readout train (see
Figure 1). The resolution of the blip-down navigator was fixed to N_{nav}=64 and
the same in-plane acceleration Ry=2 was used for both acquisitions.
The following acquisition parameters were
used:
FOV=22x22cm,
N_{x}xN_{y}=128x128, Ry=2; partial Fourier=0.75, esp=0.55ms,
slice thickness=2mm, TR=3s. The host-EPI and the navigator-EPI were
reconstructed separately and a fieldmap was estimated using FSL 5.0’s topup ( http://fsl.fmrib.ox.ac.uk/fsl ) function
with default parameters (3).### Results

Figure 3 displays separate reconstruction of the navigator (**a**) and the host sequence (**b**). The white
arrow indicates the phase encoding direction (up or down). Uncorrected, both
reconstruction experience geometric distortions in opposing directions relative
to the red lines that mark anatomical features of the image.

Performing an off-resonance corrected hybrid-SENSE reconstruction significantly reduces distortions and brings the navigator (**d**) and the full readout (**e**) into the same undistorted space that allows for a joint reconstruction (**h**) with reduced g-factor penalty (**g**).

### Discussion

We have shown that it
is feasible to acquire a blip-down navigator and the full resolution image in
one continuous readout without significantly prolonging the acquisition time.
The duration of the navigator is on the order of 10ms which means we can
neglect motion effects between both readouts. This allows for a snapshot of the
dynamically changing off-resonance field in case of head motion or breathing
fluctuations
In contrast to methods
that rely on the phase to estimate off-resonance changes, the blip up-down
approach is not affected by flow or CSF pulsatility that can lead to erroneous
phase estimations and thus incorrect off-resonance changes.
The joint
reconstruction has the additional benefit of reducing the g-factor penalty in a
parallel imaging reconstruction. ### Acknowledgements

NIH (2R01 EB002711 , 5R01 EB008706, 5R01 EB011654), the Center of Advanced MR Technology at Stanford (P41 RR009784), Lucas Foundation.### References

1. Zhu K, Dougherty R, Wu H, Middione M, Takahashi A, Zhang
T, Pauly J, Kerr A. Hybrid-Space SENSE Reconstruction for Simultaneous
Multi-Slice MRI. IEEE Trans. Med. Imaging [Internet] 2016. doi:
10.1109/TMI.2016.2531635.
2. Hansen PC. Truncated Singular Value Decomposition
Solutions to Discrete Ill-Posed Problems with Ill-Determined Numerical Rank.
SIAM J. Sci. Stat. Comput. [Internet] 1990;11:503–518. doi: 10.1137/0911028.
3. Smith SM, Jenkinson M, Woolrich MW, et al. Advances in
functional and structural MR image analysis and implementation as FSL.
Neuroimage 2004;23 Suppl 1:S208-19. doi: 10.1016/j.neuroimage.2004.07.051.