### Synopsis

**To directly estimate tissue
magnetic susceptibility distribution from the raw phase of a gradient echo
acquisition, we propose a single-step quantitative susceptibility mapping (QSM)
method that benefits from its three components: (i) the single-step processing
that prevents error propagation normally encountered
in multiple-step QSM algorithms, (ii) multiple spherical mean value
kernels that permit high fidelity background
removal, and (iii) total generalized variation regularization that promotes a piecewise-smooth solution
without staircasing artifacts. A fast
solver for the proposed method, which enables simple analytical solutions for
all of the optimization steps, is also developed. Improved image quality over conventional QSM algorithms is demonstrated using the
SNR-efficient Wave-CAIPI and 3D-EPI acquisitions.**### Purpose

Quantitative Susceptibility Mapping (QSM) estimates the
underlying tissue magnetic susceptibility

χ$\chi $ from the gradient echo
(GRE) phase signal. Conventional QSM algorithms involve sequential solution of multiple
ill-posed post-processing steps.
Recently, QSM algorithms that directly relate the unprocessed GRE
phase to the unknown

χ$\chi $ have been proposed to
improve the susceptibility estimation by preventing error propagation
through successive operations in conventional QSM

^{1-5}.
In this work, we propose a QSM model that combines the benefits from using a
single-step processing, total generalized variation regularization (TGV)

^{6},
and multiple spherical mean value (SMV) kernels

^{7-9}. A fast solver
for the proposed single-step model, which enables simple analytical solutions
for all of the optimization steps, is also developed. We demonstrate the
performance of the proposed method on three

*in vivo* experiments acquired
using the rapid 3D-EPI

^{10} and Wave-CAIPI trajectories

^{11}.
Herein, we also demonstrate the multi-echo capability of Wave-CAIPI sequence
for the first time, and introduce an automated, phase-sensitive coil
sensitivity estimation scheme.

### Methods

The underlying χ$\chi $ is
estimated from the unprocessed GRE phase ϕ$\varphi $ by solving the following optimization problem

minχ12∑i∥Mi(hi∗d∗χ)−Mi(hi∗Ψ(ϕ))∥22+TGV2α(χ)$$\underset{\chi}{min}\frac{1}{2}\sum _{i}\parallel {M}_{i}({h}_{i}\ast d\ast \chi )-{M}_{i}({h}_{i}\ast \mathrm{\Psi}(\varphi )){\parallel}_{2}^{2}+TG{V}_{\alpha}^{2}(\chi )$$

where Ψ$\mathrm{\Psi}$ is the
Laplacian unwrapping operator, d$d$ is the
spatial dipole kernel, and hi${h}_{i}$ is the SMV
kernel for the reliable phase region indicated by the mask Mi${M}_{i}$. Large
SMV kernels are used to avoid residual error amplification, whereas smaller SMV
kernels are used near the object boundary to prevent eliminating cortical phase information^{7-9}. TGV2α${TGV}_{\alpha}^{2}$
is the
second-order TGV.

The first term combines phase unwrapping, background phase removal
using multiple SMV kernels, and dipole inversion. The second term imposes prior
information by promoting a piecewise-smooth solution. By using the convolution
theorem and discrete form of
TGV2α$TG{V}_{\alpha}^{2}$, we obtain

minχ,v12∑i∥MiF−1HiDFχ−MiF−1HiFΨ(ϕ)∥22+α1∥Gχ−v∥1+α0∥(v)∥1$$\underset{\chi ,v}{min}\frac{1}{2}\sum _{i}\parallel {M}_{i}{F}^{-1}{H}_{i}DF\chi -{M}_{i}{F}^{-1}{H}_{i}F\mathrm{\Psi}(\varphi ){\parallel}_{2}^{2}+{\alpha}_{1}\parallel G\chi -v{\parallel}_{1}+{\alpha}_{0}\parallel \mathcal{E}(v){\parallel}_{1}$$

where F$F$ is the discrete Fourier transform, D=1/3−k2z/(k2x+k2y+k2z)$D=1/3-{k}_{z}^{2}/({k}_{x}^{2}+{k}_{y}^{2}+{k}_{z}^{2})$ is the dipole kernel in k-space, Hi${H}_{i}$ is hi${h}_{i}$ in k-space, G$G$ is the 3D gradient operator, and $\mathcal{E}$ is a symmetrized derivative. By introducing auxiliary
variables z0${z}_{0}$,
z1${z}_{1}$
,
and z2${z}_{2}$
with consensus constraints (z0=(v)${z}_{0}=\mathcal{E}(v)$, z1=Gχ−v${z}_{1}=G\chi -v$,
and z2,i=HiDFχ${z}_{2,i}={H}_{i}DF\chi $),
applying alternating direction method of multipliers (ADMM)^{12}, and exploiting the circulant structure of the
matrices, analytical solutions are developed for all of the optimization steps.

**Experiments**

Using
in vivo 3D EPI, Multi-Echo Wave-CAIPI at 3T, and high-resolution Wave-CAIPI data at 7T, we
compared the performances of four QSM methods:

(i) SMV
background filtering, followed by the dipole inversion with l2${l}_{2}$-quadratic smoothing regularization
defined as ∥Gχ∥22$\parallel G\chi {\parallel}_{2}^{2}$, (V-SHARP L2),

(ii) SMV
background filtering, followed by TGV-regularized dipole inversion (V-SHARP
TGV),

(iii) Single-step QSM with l2${l}_{2}$-quadratic smoothing regularization
(Single-Step L2), and

(iv) Single-step QSM with TGV regularization (Proposed:
Single-Step TGV).

This work also introduces the
multi-echo extension of Wave-CAIPI that fully utilizes the repetition time to
sample multiple echoes, which are combined to improve both magnitude and phase signal-to-noise ratios (Figure 1).
In addition, we develop a novel fully-automated coil sensitivity estimation
based on ESPIRiT^{13} and virtual body coil concept^{14} (described
in Figure 2).

For
all the reconstruction methods, the GRE phase was unwrapped using
Laplacian-based phase unwrapping

^{15-16}. For V-SHARP L2 and Single-Step
L2, the MATLAB pcg function was used as a solver with the residual tolerance
of 0.1%, and 14 SMV kernels were used with their radii ranging from 1 to 14
voxels. For V-SHARP TGV and Single-Step TGV, ADMM was used with the stopping
criterion that iterations were terminated when the change in the solution
between consecutive iterations was less than 1%, and 5 SMV kernels were used
with their radii ranging from 1 to 5 voxels.

### Results

Figure 3 through 5 display the reconstructed susceptibility maps from the
total field map using the different QSM algorithms. The proposed method had
lowest level of dipole artifacts and background phase contamination while preserving the edges.

### Discussion and Conclusion

The proposed method better mitigated the dipole
artifacts and improved homogeneity of image contrast compared to other QSM
methods evaluated thanks to its three main components: (i) the single-step
processing that prevents error propagation through
successive operations, (ii) multiple SMV kernels that permit high fidelity background removal while retaining the cortical phase information, and (iii) TGV regularization that promotes a piecewise-smooth solution
without staircasing artifacts. The reconstruction time was minimized by using
ADMM with variable splitting in both the data and regularization terms

^{17} and exploiting
the structure of the matrices. We also introduced an automated
sensitivity estimation scheme that relies on a rapid calibration acquisition
for phase-sensitive coil combination. This permitted high quality parallel
imaging with Wave-CAIPI, which was extended to acquire multiple echoes for
improved SNR. Combination of Multi-Echo Wave-CAIPI and single-step
QSM with TGV regularization thus enabled high-quality susceptibility mapping
with efficient acquisition and reconstruction.

### Acknowledgements

NIH R01EB017219, P41EB015896,
1U01MH093765, R24MH106096, 1R01EB017337-01### References

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