William A Grissom^{1}

Current parallel transmit pulse design methods are
based on a spatial domain formulation that has prohibitive memory and computational
requirements when the number of coils or the number of dimensions is large. We describe a k-space domain parallel transmit pulse design method
that directly solves for the columns of a sparse design matrix with a much
smaller memory footprint than existing methods, and is highly parallelizable. The
method is validated with phantom and *in vivo* 7T 8-channel spiral excitations.

The k-space domain pulse design method solves for the columns of a sparse matrix $$$\mathbf{W}$$$ that relates the discrete Fourier transform of a target pattern $$$\mathbf{d}$$$ to a vector of RF pulses, as:

$$\hat{\mathbf{b}} = \mathbf{W}\mathcal{F}\left(\mathbf{d}\right)$$

where the designed pulses are stacked end-on-end in the vector $$$\hat{\mathbf{b}}$$$. Each column of $$$\mathbf{W}$$$ represents a location in the target excitation k-space grid, and contains a set of weights that relate the desired energy at that target location to the RF pulses. These weights can be determined column-by-column, by solving the system of equations that results from discretizing the following equation over the neighborhood around each target location:

$$\delta(\vec{k}-\vec{k}_{targ}) = \sum_{i \in traj} \sum_{j \in coils} w_j(\vec{k}_i) s_j(\vec{k}-\vec{k}_i)$$

where $$$i$$$ indexes excitation trajectory locations $$$\vec{k}_i$$$ that are near the target location $$$\vec{k}_{targ}$$$. Figure 1 illustrates this equation graphically. The discretized system of equations can be recast in matrix-vector form, as:

$$\mathbf{Sw} = \mathbf{\delta}$$

where $$$\mathbf{\delta}$$$ is a vector that contains a one at the target location, and zeros elsewhere. The weights vector can be solved using regularized pseudoinverse, as:

$$\hat{\mathbf{w}} = \left(\mathbf{S}^H\mathbf{S} + \lambda \mathbf{I}\right)^{-1} \mathbf{s_{targ}}^{H}$$

where $$$\mathbf{s_{targ}}= \mathbf{S}^H\mathbf{\delta}$$$ is the row of the matrix $$$\mathbf{S}$$$ corresponding to the target location. The weights are then inserted into the sparse matrix $$$\mathbf{W}$$$ for pulse design.

Experiments
were performed to validate the k-space domain pulses and compare them to
spatial domain pulses designed using a regularized matrix pseudoinverse. First,
*B*_{1}^{+} maps were measured in a 3D-printed head phantom on a 7T scanner (Philips
Healthcare, Best, Netherlands) with 8-channel parallel transmit. *B*_{1}^{+} map processing,
*B*_{0} shimming, and RF pulse
interfacing was performed with MRCodeTool (MRCode BV, Zaltbommel, Netherlands).
Spiral-in RF pulses were designed to excite an oval region in the middle of the
brain phantom with 7.5 cm excitation-FOV, 0.7 cm resolution, 3.4 ms duration,
and one-degree flip angle. The same pulse design was repeated for a healthy
human volunteer with IRB approval, but was scaled to 90 degrees and used as a
saturation pulse, followed by a crusher and a one-degree excitation. In both
cases the excitation patterns were imaged with a gradient-recalled echo
sequence with TE/TR = 2/100 ms. The k-space domain designs used 20 x 20
neighborhoods around each target location.

- U. Katscher, P. Boernert, C. Leussler, and J. S. van den Brink. Transmit SENSE. Magn Reson Med, 49(1):144–150, Jan 2003.
- Y. Zhu. Parallel excitation with an array of transmit coils. Magn Reson Med, 51(4):775–784, Apr 2004.
- W. A. Grissom, C. Y. Yip, Z. Zhang, V. A. Stenger, J. A. Fessler, and D. C. Noll. Spatial domain method for the design of RF pulses in multicoil parallel excitation. Magn Reson Med, 56(3):620–9, Sep 2006.

Illustration of Equation 1. At each location in the target k-space grid *k*_{targ}, a system of equations is solved to determine the weights for each coil *j* at each nearby excitation trajectory location *k*_{i} that together produce a delta function at the target location. Those weights are then inserted into the design matrix **W** in the column corresponding to *k*_{targ}.

Phantom 8-channel spiral excitation patterns. The target pattern was an oval positioned to selectively excite the midbrain. The spatial domain and k-space domain excitations are nearly identical.

3D In vivo images acquired with spatial domain and k-space domain 8-channel spiral saturation pulses. Aliasing artifacts unrelated to the pulses are present due to the non-selective excitation that followed them (yellow arrows), but the shape of the saturation regions and overall intensity variations match well between the designs.