Nishant Zachariah^{1}, Jason Langley^{2}, Justin Romberg^{1}, and Xiaoping P. Hu^{3}

Multiband (MB) imaging is limited in its acceleration factor
by the high correlation that exists between receiver coils. In this work, we
present a novel technique, **A**dvanced **P**seudo Fourier **I**maging (API) which
achieves parallel excitation beyond that which is currently possible using
multiband imaging. In doing so, API forms a generic framework for seamless
transition from 2D to 3D imaging. Unlike MB, API is less sensitive to the RF
excitation profile in its slice reconstruction by virtue of the introduced phase variations. We demonstrate the viability of API through 1D simulations and 3D head
phantom data acquired at 3T.

*1D Simulation:* The multiband (MB) model (in 1D) can be expressed as$$y_{k,i} = \int_z w_k(z) c_i(z) f(z) dz + n \;\; ... (\text{Eq. 1}) $$where $$$y_{k,i}$$$ is the MB signal observed at the $$$k$$$th excitation through the $$$i$$$th coil, $$$w_k(z)$$$ is the $$$k$$$th excitation window, $$$c_i(z)$$$ represents the coil sensitivity profile of the $$$i$$$th coil, $$$f(z)$$$ is the unknown signal that needs to be recovered and $$$n$$$ is the unknown noise. API introduces a continually varying phase and can be written as $$b_{k,i} = \int_z \bar{w}_k(z) c_i(z) f(z)e^{-j\omega_k z} dz + n \;\; ... (\text{Eq. 2})$$where $$$b_{k,i}$$$ is the API signal observed at the $$$k$$$th excitation through the $$$i$$$th coil, $$$\bar{w}_k(z)$$$ denotes the multi-slice $$$k$$$th excitation API window and $$$\omega_k$$$ the applied phase encode gradient (Fig 1). Excitation windows were modeled as a conglomeration of apodized sinc functions and the coil profiles were generated using Biot-Savart’s law. To demonstrate API’s insensitivity towards
excitation profile (unlike MB), the API excitation windows were larger than
their MB counterparts with 50% overlap between successive excitations (Fig 2-4).

*3D Data Acquisition*: Data from a head phantom were
acquired on a Siemens 3T Prisma Fit (Malvern, PA) using a GRE with a flip angle of 30$$$^\circ$$$, pixel bandwidth of 30Hz, TE of 18.5msec, TR of 38msec, FOV of 158x390198 mm and resolution of 1x1x2mm. The coil profiles were estimated using a pre-scan. Phase gradient and multi-slice
excitation was implemented retrospectively for ground truth comparisons with
the final reconstruction.

Tikhonov regularized least squares was used to recover in Eq1&2. Reconstruction accuracy was quantified using normalized recovery error and peak signal to noise ratio across a range of noise profiles.

*1D Simulation*: As an initial proof of concept, the
methodology was tested for an analytic 1D case across a range of noise profiles
(Fig 2). At an acceleration factor of 8, API consistently recovered the
underlying signal with greater fidelity by nearly $$$40-50\%$$$ at the higher noise
levels relative to multiband imaging (Fig 2). For the noise matched case, API recovers the underlying signal with $$$50\%$$$ less error than multiband (Fig 3). This increase in robustness can be attributed to the
spectral spread of the API encoding operator relative to MB (Fig 4). As seen in Fig 3, despite API utilizing an excitation window that is twice as large as MB with 50% overlap, API reconstructs the underlying signal with significantly lower error.

*3D Simulation*: At an acceleration factor of 8 (which causes
MB to fail on our scanner), we find that API recovers the underlying signal
with $$$50\%$$$ less recovery error and approximately $$$6$$$dB increase in reconstruction
accuracy (Fig 5). Residuals for MB are significantly higher than that for API. Examples
of specific slice reconstructions for the head phantom are shown in Fig 5.

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[4] Kadah, Yasser M., and Xiaoping Hu. "Pseudo-Fourier imaging (PFI): A technique for spatial encoding in MRI." IEEE transactions on medical imaging 16.6 (1997): 893-902.

[5] Griswold, Mark A., et al. "Generalized autocalibrating partially parallel acquisitions (GRAPPA)." Magnetic resonance in medicine 47.6 (2002): 1202-1210.

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Fig 1. Schematic of multiband (MB) Imaging (left) and Advanced Pseudo-Fourier Imaging
(API). For the MB example shown, 3 slices are simultaneously excited and are
seen through a receiver array of coils for a total of K acquisitions. In API,
twice the number of slices are excited at a given time (6 slices in this
example) but a phase encode gradient is applied at each acquisition. Thus while
the number of acquisitions stays identical at K, each location is seen more
than once.

Fig 2. Summary
of signal recovery error for multiband (MB) and advanced pseudo Fourier imaging
(API) along the slice direction for a range of noise profiles. The plotted result demonstrates that not only is API more robust than MB but it is capable of
acceleration in noise regimes where MB would not be feasible. In the
practical noise regime (standard MR noise regime 10-25 dB), API can outperform
MB by as much as 50%. It is interesting to note that API consistently recovers
signals with less error than 0.2 (black broken line) across all tested noise
profiles unlike MB.

Fig 3. Exemplar
signal recovery for Multiband (MB) and Advanced Pseudo Fourier Imaging (API)
for an identical instantiation of noise (23.51 dB). API recovers the underlying
signal with ~50% less error than MB. If the magnitude image is considered, the
peak signal to noise ratio (reconstruction accuracy) of API is ~5dB higher than
that of MB imaging. In addition, a cursory examination of the plots will reveal
that API (green) is even visually closer to the underlying signal (black) than
MB imaging (red).

Fig 4. Normalized spectral distribution of the multiband (MB) encoding operator and
the advanced pseudo-Fourier imaging (API) encoding operator. From the plot, it
becomes clear that API (red) has a higher area under the curve which engenders
API with robust recovery relative to MB imaging. Also, API is better
conditioned but it must be noted that the last few singular values are not
utilized in the reconstruction by virtue of the Tikhonov regularization used
for both MB and API.

Fig 5. Recovery
exemplars for 3D head phantom using multiband (MB) and advanced pseudo Fourier
imaging (API) for an acceleration factor of 8 (SNR of 35dB). A cursory
examination of the residuals (factor 5 magnified for viewing) demonstrate that MB recovers the underlying signal
with large errors. This is also seen in the raw reconstructed images where the
g-factor penalty is beginning to show. Both the raw API reconstructed images
and the residuals are devoid of the errors seen in the MB case. The recovery
error quantified in the bar graph . API recovers the underlying
signal with 50% lower error.