Sebastian Rosenzweig^{1} and Martin Uecker^{1,2}

^{1}Department of Diagnostic and Interventional Radiology, University Medical Center Göttingen, Göttingen, Germany, ^{2}German Centre for Cardiovascular Research (DZHK), University Medical Center Göttingen, Göttingen, Göttingen, Germany

### Synopsis

**Multiband
MRI can be used to acquire
several slices at the same time. Here, we propose a new method
of multiband MRI based on Regularized Nonlinear Inversion (NLINV).
This method does not require a priori knowledge
about the coil sensitivities. Simultaneous estimation of
images and coil sensitivities of two slices is demonstrated from
six-fold undersampled data for a simulated multi-band acquisition.**### Introduction
and Purpose

Reducing
image
acquisition time
is
of
utmost importance
in in-vivo MRI. Lately,
multiband MRI has attracted increased interest among researchers
since it allows the simultaneous acquisition of several slices of an
object. In
multiband MRI, the
spatial encoding information inherent in a multi-coil receiver system
can be used to dis-entangle
the information
of
multiple slices, while only
the time of a single slice
measurement is needed to
acquire the data. In
the common reconstruction
approach
a
linear equation is solved by
making use of previously
estimated
coil sensitivities. Here,
we propose a
new method
of multiband MRI based
on Regularized
Nonlinear
Inversion (NLINV).

^{1} This
method does not require
a priori knowledge
about the coil sensitivities
and is attractive
especially for
real-time
imaging where coil sensitivities may change due to motion or
interactive changes to the slice position.

### Method

Phantom:
All measurements
are performed on a phantom
of 6 bottles arranged in a triangle. To
create two distinguishable transverse slices, the
peripheral
bottles are completely
filled with water whereas
the centered ones
are only half-full,
see
Fig. (1).

Simulation
of multiband
MRI data: To mimic the simultaneous and undersampled
acquisition of two slices (multiband MRI) we first measure the full
k-spaces of each slice (B_{0} = 3T, 2D FLASH, 20-channel head coil). We
then add or subtract the k-spaces of the two slices (Hadamard
encoding) and apply k-space masks P_{A} or P_{B}
respectively to simulate undersampling for all channels. We
obtain: Y_{A}
= P_{A} (k_{0}+k_{1}),
Y_{B}
= P_{B} (k_{0}-k_{1}).
A visualization of this process can be found in Fig. (1). Both
k-space masks contain the center of k-space. In this work, only
k-space Y_{A} contains samples outside the center where it is
further undersampled by a factor of three. This corresponds to a
total acceleration factor of six (ignoring center lines).

Reconstruction
of multiband MRI data with Regularized Nonlinear Inversion: With the
definitions **X**=(X_{0},X_{1}) and **Y**=(Y_{A},Y_{B}),
we can set up the nonlinear equation F(**X**)=**Y**,
that need to be solved.
Here X_{i}=(r_{i},**c**_{i})
with r_{i}(x) the object function (spin
density) of slice i and **c**_{i}(x)
an N-dimensional vector containing the coil sensitivities of
the N=20 coils for slice i. F is the non-linear encoding
function. As in NLINV, the system is jointly solved for image and
coil sensitivities using the Iteratively Regularized Gauss-Newton
Method: Using an initial estimate **X**_{n} we
obtain **X**_{n+1}=**X**_{n}+d**X**
by solving the linearized equation DF(**X**_{n})d**X**+F(**X**_{n})=**Y**
with suitable
regularization terms.
The solution to this linear system is then computed using the
method of Conjugate Gradients.

### Results
and Discussion

Fig.
(2) shows the reconstructed object function of the lower slice (a)
and upper slice (b) after n=7 Newton steps. Both
slices
can be reproduced without significant artifacts. As a
reference, we perform a coil-wise inverse Fast Fourier Transform of
the full k-spaces for each slice and use the Sum-of-Squares method to
reconstruct the image from the multiple channels. The result is shown
in Fig. (2c), which shows aliasing artifacts at the position of the
peripheral bottles. In Fig. (3), we show the reconstructed complex
coil sensitivities of the first three coils for (a) the lower and
(b) the upper slice. As our phantom does not cover the entire field of
view, we obtain black spaces where no information can be extracted.

### Conclusion
and Outlook

In this work, we extended the method of Regularized Nonlinear
Inversion to multiband MRI. We successfully reconstructed two slices
from six-fold undersampled simulated multiband data without prior knowledge of the coil
sensitivities. One possible next step is the application of the
proposed method for real-time MRI, which would allow the simultaneous
acquisition of multiple slices without significantly longer
acquisition times.

### Acknowledgements

No acknowledgement found.### References

1. Uecker M, Hohage T, Block KT, Frahm J. Image reconstruction by
regularized nonlinear inversion--joint estimation of coil
sensitivities and image content. Magn Reson Med. 2008
Sep;60(3):674-82.