Qi Liu^{1}, Yuan Zheng^{1}, Yu Ding^{1}, Jian Xu^{1}, and Weiguo Zhang^{1}

^{1}UIH America, Inc., Houston, TX, United States

A new comprehensive spiral gradient waveform correction strategy is proposed that features multipoint gradient anchoring (MGA). By measuring multiple gradient delays in spiral waveforms at different rotation angles, it effectively ‘anchors’ the waveform at a series of locations and improves gradient waveform fidelity. Application of this innovative design on phantom and volunteer imaging indicates it is an effective and promising technique.

In this study a new comprehensive spiral gradient waveform correction strategy is proposed that features multipoint gradient anchoring (MGA). By measuring multiple gradient delays in spiral waveforms at different rotation angles, it effectively ‘anchors’ the waveform at a series of known locations. MGA can measure the whole trajectory and improve waveform fidelity by only relying on several additional prescan acquisitions with limited time cost; prior information of the gradient system response is not necessary.

$${G_x}(\theta ,t) = \cos (\theta )*{G_{x0}}(t) + \sin (\theta )*{G_{y0}}(t) $$

,where t is the readout time and indexed from 1 to m.

Since k-space locations are determined by the time integral of the gradient waveform, if linear system is assumed, the k-space deviation \(\Delta {K_x}(\theta ,t)\) of any interleaf from design trajectory will have a similar relation with that of the basis interleaf:

$$\Delta {K_x}(\theta ,t) = \cos (\theta )*\Delta {K_x}_0(t) + \sin (\theta )*\Delta {K_{y0}}(t) $$

With acquisitions at multiple angles \({\theta _1},{\theta _2}, \ldots ,{\theta _n}\) we have:

$${\rm{Kx}} = {\rm{\Theta x}}*\Psi $$

,where \({\rm{Kx}}\) is a matrix with \({\rm{K}}{{\rm{x}}_{{\rm{i,j}}}}{\rm{ = }}\Delta {K_x}({\theta _i},{t_j})\), i = 1,2,…,n, and j=1,2,…,m. \({\rm{\Theta x}}\) is a rotation matrix of size n by 2, with its ith row being \([\cos (\theta ),\sin (\theta )]\). \(\Psi \) is a matrix of size 2 by m and formed by concatenating \(\Delta {K_x}_0\) and \(\Delta {K_y}_0\).

Building upon a previous approach to measure gradient delay using prescan [Ref 4], k-space shifts can be estimated at every k-space zero-crossing, instead of a same value that fits all. Specifically, by comparing spirals acquired at rotation angles of \({\theta _p}\) and \({\theta _p} + \pi \) when only enabling x-axis gradient, a shift value can be determined whenever their trajectories cross at k-space origin (Fig. 1). Subsequently each shift value is written to fill the matrix at \({\rm{K}}{{\rm{x}}_{{\rm{i,j}}}}\) where \({\theta _i} = {\theta _p}\) and \({t_j}\) is the nearest time at which the crossing occurs when design trajectory is assumed.

The objective of spiral waveform correction is equivalent to knowing \(\Delta {K_x}_0\) and \(\Delta {K_y}_0\) at every readout time. Although theoretically \(\Psi \) can be obtained by solving the above equation, in reality \({\rm{Kx}}\) is sparse with limited number of elements due to limited zero-crossings. However these elements can still serve as ‘anchors’ that fixate \(\Psi \) at multipoints in the proposed MGA technique, largely reducing its degree of freedom. A number of approaches can be further used to recover \(\Psi \), for example by constrained optimization considering the facts that \(\Psi \) should be slowly varying in time and that the echo signal should maintain its shape across various zero-crossings. Here as a proof-of-concept, simple linear interpolation was used for \(\Delta {K_x}_0\) and \(\Delta {K_y}_0\) data points away from the anchors and a simple waveform measurement was performed based on reference [7] but only covers the beginning portion of \(\Psi \).

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[5] Robison RK, Li Z, Wang D, Ooi MB, Pipe JG. Correction of B0 Eddy Current Effects in Spiral MRI. Magn Reson Med. 2019 Apr;81(4):2501-2513. doi: 10.1002/mrm.27583.

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[7] Zhang W. U.S. Patent 6,448,773, 2002.

Figure 1.
Multipoint gradient anchoring acquisition. A) Signal peaks corresponding to
k-space zero-crossing occurs at different locations at different rotation
angles in MGA prescan. B) By examining the pair of prescan data acquired with
opposite gradient polarities at its assumed locations, a k-space shift can be
calculated for each zero-crossing that serves as an ‘anchor’ for determining
waveform distortion. C) &D) Enlarged image to show k-space shifts as
demonstrated by misalignment between echoes of opposite gradient polarities,
for the highlighted zones in B).

Figure 2.
Phantom images reconstructed without (left column) and with (right column) the
proposed MGA strategy. Images in the top and bottom rows were acquired with
protocols 1 and 2, respectively, as detailed in Tab.1. Obvious artifact
reduction was achieved with the proposed technique.

Figure 3.
Typical volunteer images of the brain at two different slices. A) & C) were
reconstructed without MGA. B) & D) were reconstructed with MGA. Arrows
point to artifacts associated with uncorrected gradient waveform.

Table 1. Imaging parameters