The Absence of Phase Information in the Signal-Deprived Image Background is an Important Source of Error in Susceptibility Mapping

Russell Dibb^{1,2} and Chunlei Liu^{3,4}

A 64×64×64
numerical phantom (Fig. 1A) was created with each voxel represented by a cylindrically
symmetric susceptibility tensor, $$${\bfχ}$$$. Frequency
map data, Δ*f*, were calculated for
each image orientation according to^{1}
$$Δf\tt({\bf k})=\left(\frac{1}{3}{\bf ĥ}^T{\bfχ}({\bf k}){\bfĥ}-{\bf k}^T{\bfĥ}\frac{{\bf k}^T{\bfχ}({\bf k}){\bf ĥ}}{{\bf k}^T{\bf k}}\right)\frac{γ}{2π}\it h\tt_0μ_0$$ Here, $$${\bf k}$$$ is the spatial frequency vector, $$${\bf ĥ}$$$ is one of *n* = 12 magnetic field orientations (Fig. 1B), γ is the gyromagnetic ratio, *h*_{0} is the magnetic field
strength (9.4 T), and μ_{0} is the vacuum permittivity. Complex-valued
image data, *S*, were then generated from
each frequency map using $$S = S_0e^{-j2πΔfTE}$$ Here, *S*_{0 }is the initial signal magnitude (referred to as *S*_{0,int} and *S*_{0,ext} to distinguish between
the interior and the exterior of the phantom, respectively), *j* is the unit imaginary number, and *TE*
is the echo time (20 ms). Gaussian noise was added to the real and imaginary
data in **k**-space to yield a magnitude
image SNR of 30. The phase image data were unwrapped using a Laplacian operator^{4}, and normalized by TE. Using an LSQR solver to invert the susceptibility-phase relationship,
susceptibility tensor data were reconstructed with either *S*_{0,ext}=1 or *S*_{0,ext}=0 in both the presence and absence of noise.

Mean susceptibility and susceptibility anisotropy were
calculated as $$$\tt\overline{χ}={\it{tr(\ttχ)}}/3$$$, and $$$Δχ=χ_1–(χ_2+χ_3)/2$$$, respectively, where χ_{1},_{2},_{3} are the principal susceptibilities in descending
order. Percent error (*E _{ŷ}*) for these two values were calculated following
$$\it E_ŷ=\frac{ŷ–y}{|y|}×\tt100 $$ where y and

1. Liu C. Susceptibility tensor imaging. Magn Reson Med 2010;63(6):1471-1477.

2. Xie L, Dibb R, Gurley SB, Liu C, Johnson GA. Susceptibility tensor imaging reveals reduced anisotropy in renal nephropathy. 2015; Proceedings of the 23nd Annual ISMRM, Toronto, Canada. p 463. 3. Dibb R, Qi Y, Liu C. Magnetic susceptibility anisotropy of myocardium imaged by cardiovascular magnetic resonance reflects the anisotropy of myocardial filament alpha-helix polypeptide bonds. Journal of Cardiovascular Magnetic Resonance 2015;17(1):60.

4. Schofield MA, Zhu Y. Fast phase unwrapping algorithm for interferometric applications. Optics letters 2003;28(14):1194-1196.

5. Dibb R, Liu C. Whole-heart myofiber tractography derived from conjoint relaxation and susceptibility tensor imaging. 2015; Proceedings of the 23nd Annual ISMRM, Toronto, Canada. p 287.

6. Li W, Avram AV, Wu B, Xiao X, Liu C. Integrated Laplacian-based phase unwrapping and background phase removal for quantitative susceptibility mapping. NMR Biomed 2014;27(2):219-227.

7. Li W, Wu B, Liu C. iHARPERELLA: and improved method for integrated 3D phase unwrapping and background phase removal. 2015; Proceedings of the 23nd Annual ISMRM, Toronto, Canada. p 3313.

8. Wharton S, Bowtell R. Effects of white matter microstructure on phase and susceptibility maps. Magnetic Resonance in Medicine 2015;73(3):1258-1269.

Fig. 1. The simulated, spherical phantom (A) is represented
by susceptibility tensors. The color indicates the principal eigenvector orientation
within each voxel. Frequency map data were simulated for *n* = 12 magnetic field orientations (B).

Fig. 2. Missing exterior phase data leads to overestimated
mean susceptibility values. (A) Phantom mean susceptibility. (B) Image results
of the reconstructed mean susceptibility with (*S*_{0,ext }= 1) and without (*S*_{0,ext}
= 0) exterior phase data. (C) Distributions of the voxelwise percent error of
the reconstructed mean susceptibility.

Fig. 3. Inadequate exterior phase data yields underestimated
susceptibility anisotropy. (A) Phantom susceptibility anisotropy. (B) Image results
of the reconstructed susceptibility anisotropy with (S_{0,ext} = 1) and without (*S*_{0,ext }= 0) exterior phase data. (C) Distributions of the voxelwise percent error of
the reconstructed susceptibility anisotropy.

Fig. 4. Absent phase data outside the object negatively
impacts the reconstructed principal eigenvector. (A) Phantom principal eigenvector
orientation. (B) Image results of the reconstructed eigenvector data. (C) Quantitative
distributions of the voxelwise angular error of the reconstructed eigenvectors.
(D) Angular error maps with and without exterior phase data.

Proc. Intl. Soc. Mag. Reson. Med. 24 (2016)

2842