Ute Goerke^{1}

High acceleration is an effective tool to achieve whole brain coverage with acceptable total acquisition times. A limitation of parallel imaging is high noise amplification at high acceleration factors. In this work, an algorithm for parallel imaging of RASER in two dimensions implemented. The noise enhancement is theoretically derived and experimentally validated. Results show that even at high acceleration noise amplification remains low and high resolution whole brain images can be obtained with RASER.

Introduction

High acceleration is an effective tool to achieve whole brain coverage with acceptable total acquisition times. A limitation of parallel imaging is high noise amplification at high acceleration factors. In this work, an algorithm for parallel imaging of RASER (rapid acquisition by sequential excitation and refocusing) in two dimensions implemented. The noise enhancement is theoretically derived and experimentally validated. An example of a
RASER is ultrafast imaging method using
spatiotemporal encoding (SPEN) (Fig. 1)^{1, 2}. The signal in the SPEN-dimension
is sequentially excited by a chirp-pulse and in same order recorded during an
echo readout train. The signal is localized as result of signal dephasing in
the periphery of the quadratic phase profile generated by frequency-sweep of
the chirp-pulse. The vertex of the quadratic phase moves along the
SPEN-dimension encoding an image profile. The signal in the SPEN-dimension is,
hence, acquired in the image domain. Parallel imaging in the SPEN-dimension was
achieved by using multi-band excitation. The phase-encoded slab-selective
dimension is undersampled for acceleration.

All experiments were performed on a 7 T scanner
(Siemens, Erlangen, Germany) with Nova 8Tx 32Rx head coil. Full reference scans
with two to four bands were acquired of a spherical phantom filled with
silicone oil. Other imaging parameters were spatial resolution (2 mm)^{3},
acquired matrix size: 200 x # bands·24 x 64. *T*_{E} = 75 ms, *T*_{R} = 800 ms. Acceleration
along the SPEN-dimension was simulated by superimposing excitation bands. For the subject data, a full
reference scan with short *T*_{R} (800 ms) was acquired in addition to the
*T*_{2}-weighted accelerated scan with long *T*_{R} (2 s). Imaging parameters are spatial
resolution (1.25 mm)^{3}, *T*_{E} = 70 ms, multi-band 4, acceleration in the
phase-encoded slab-selective dimension 4.

Images were
reconstructed using GRAPPA according to the flow diagram in Fig. 2. The pseudo multiple
replica method^{3} was used to compute
g-factors from image SNR. G-factors were also theoretically derived from GRAPPA-weights.

To
perform GRAPPA in the SPEN-dimension the signal, which is acquired in image
space *I*, has to be
Fourier-transformed to generate k-space data *s*:

$$s^{acc}_{\mu}=\sum_{\nu=1}^{N_{SPEN}}\tilde{I}_\nu\exp\left\{-\frac{2\pi i}{N_{SPEN}}\mu\nu\right\}$$[1]

where
$$$\tilde{I}_\nu=\sum_{b=1}^{M_B}I_{B\nu}$$$ represents the
superimposed signals from multi-band excitation. Considering the noise *n* during acquisition one obtains

$$s^{acc}_{\mu}+n'^{acc}_{\mu}=\sum_{\nu=1}^{N_{SPEN}}\tilde{I}_\nu\exp\left\{-\frac{2\pi i}{N_{SPEN}}\mu\nu\right\}+\sum_{\nu=1}^{N_{SPEN}}n^{acc}_\nu\exp\left\{-\frac{2\pi i}{N_{SPEN}}\mu\nu\right\}$$[2]

Assuming the noise is random and uncorrelated in respect to the frequency-encoded, SPEN- and phase-encoded slab-selective dimension the noise variance of the k-space signal is

$${\sigma^{2}}\left({n'^{acc}_{c}}\right)=N_{SPEN}{\sigma^{2}}\left({n^{acc}_{c}}\right)$$[3]

For the reference scan the noise propagation is

$$s^{full}_{\mu}+n'^{full}_{\mu}=\sum_{\nu=1}^{M_{B}N_{SPEN}}I_\nu\exp\left\{-\frac{2\pi i}{M_{B}N_{SPEN}}\mu\nu\right\}+\sum_{\nu=1}^{M_{B}N_{SPEN}}n^{full}_\nu\exp\left\{-\frac{2\pi i}{M_{B}N_{SPEN}}\mu\nu\right\}$$[4]

Hence, the noise variance in the reference scan is

$${\sigma^{2}}\left({n'^{full}_{c}}\right)=N_{SPEN}{\sigma^{2}}\left({n^{full}_{c}}\right)$$ [5]

Hence, the noise ratio in k-space before applying GRAPPA-reconstruction is

$$\frac{{\sigma^{2}}\left({n'^{acc}_{c}}\right)}{{\sigma^{2}}\left({n'^{full}_{c}}\right)}=\frac{1}{M_{B}}\frac{{\sigma^{2}}\left({n^{acc}_{c}}\right)}{{\sigma^{2}}\left({n^{full}_{c}}\right)}$$[6]

Using the definition of the g-factor in reference^{4} one obtains for
multi-band acceleration

$$\frac{SNR_{full}}{SNR_{acc}}=\frac{1}{M_B}g;g^2=\frac{1}{N_z}\sum_{z=1}^{N_z}g^2_z$$[7]

where *g*_{z} is the g-factor for each
phase-encoding step in the slab-selective dimension.

1. Chamberlain R, Park JY, Corum C, Yacoub E, Ugurbil K, Jack CR, Garwood M. RASER: A new ultrafast magnetic resonance imaging method. Magn Res Med 2007;58(4):794-799.

2. Goerke U, Garwood M, Ugurbil K. Functional magnetic resonance imaging using RASER. NeuroImage 2011;54(1):350-360.

3. Robson PM, Grant AK, Madhuranthakam AJ, Lattanzi R, Sodickson DK, McKenzie CA. Comprehensive quantification of signal-to-noise ratio and g-factor for image-based and k-space-based parallel imaging reconstructions. Magn Res Med 2008;60(4):895-907.

4. Breuer FA, Kannengiesser SAR, Blaimer M, Seiberlich N, Jakob PM, Griswold MA. General Formulation for Quantitative G-factor Calculation in GRAPPA Reconstructions. Magn Res Med 2009;62(3):739-746.

RASER pulse sequence. The chirp pulse sequentially
excites magnetization along the SPEN-dimension which is refocused in the same
order during the readout echo train. Frequency-encoding is performed by
alternating readout gradients. The slab-selective dimension is phase-encoded to
achieve volume-coverage. Multi-band excitation is implemented for parallel
imaging in the SPEN dimension. Undersampling of the phase-encoded
slab-selective dimension is used for acceleration.

The flow diagram shows the order in which the two GRAPPA-reconstruction
steps for the accelerated SPEN and phase-encoded slab-selective dimension
interleaved with various phase correction procedures are implemented.

g-factor maps derived from image SNR according to
equ. (7) and from GRAPPA-weights for various numbers of bands of the
chirp-pulse.