Susceptibility mapping has been successfully applied as a preclinical imaging tool. Susceptibility tensor imaging (STI) studies have further exploited the sensitivity of magnetic susceptibility contrast to the cellular content of healthy and diseased tissues to observe structural and chemical properties in the brain¹, kidney², and heart³. Interpretations of susceptibility image data are limited in part due to reconstruction artifacts spawning from errors in tissue frequency data. One potential source of these errors is inadequate phase data in the image background where there is typically a dearth of signal. Understanding the sources of these artifacts will aid in developing the image acquisition and processing tools needed to improve the accuracy and reliability of both STI and susceptibility mapping in general.

A 64×64×64 numerical phantom (Fig. 1A) was created with each voxel represented by a cylindrically symmetric susceptibility tensor, $$${\bfχ}$$$. Frequency map data, Δf, were calculated for each image orientation according to¹ $$Δf\tt({\bf k})=\left(\frac{1}{3}{\bf ĥ}^T{\bfχ}({\bf k}){\bfĥ}-{\bf k}^T{\bfĥ}\frac{{\bf k}^T{\bfχ}({\bf k}){\bf ĥ}}{{\bf k}^T{\bf k}}\right)\frac{γ}{2π}\it h\tt_0μ_0$$ 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, h₀ is the magnetic field strength (9.4 T), and μ₀ is the vacuum permittivity. Complex-valued image data, S, were then generated from each frequency map using $$S = S_0e^{-j2πΔfTE}$$ Here, S₀ 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 operator⁴, 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 χ₁,₂,₃ 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 ŷ are the true and reconstructed values, respectively. The angular difference, φ, between the reconstructed ($$${\bfû}$$$) and true ($$${\bfû}_t$$$) fiber direction unit eigenvectors was defined in the range from 0° to 90°: $$φ({\bfû},{\bfû}_t)=\begin{cases}cos^{-1}({\bfû}_t^{\tt T}{\bfû}) & cos^{-1}({\bfû}_t^{\tt T}{\bfû})\leq90^{\circ}\\180^{\circ}-cos^{-1}({\bfû}_t^{\tt T}{\bfû}) & cos^{-1}({\bfû}_t^{\tt T}{\bfû})>90^{\circ}\end{cases}$$

The mean susceptibility, susceptibility anisotropy, and principal eigenvector data computed from the reconstructed tensors are shown in Figs. 2-4, respectively. When the simulated exterior phase information is retained (i.e., S_0,ext=1), the reconstructed susceptibility tensor data is reasonably accurate, though Δχ was underestimated in the red and green circles and overestimated in the blue circle. Noise marginally degraded the tensor image results, but the single greatest source of error was the loss of exterior phase information that occurs due to the absence of signal outside the phantom (i.e., S_0,ext=0). These reconstruction errors were most visible in the regions near the phantom boundary. The absence of exterior phase information resulted in $$$\it{E}_{\tt\overline{χ}}$$$, $$$\it{E}_{\ttΔχ}$$$, and $$$φ({\bfû},{\bfû}_t)$$$) having distribution medians of 84.5%, –31.2%, and 10.9°, respectively. Following the addition of noise, these figures grew to 88.2%, –39.0%, and 16.0°.The phantom data suggest that the absence of phase information outside the object introduces substantial error into tensor estimates, particularly in regions near the object boundary. This error is visible in an STI study of the mouse heart⁵, and is most likely a result of the non-local properties of MR phase data. Susceptibility imaging protocols would benefit from embedding the object of interest in a susceptibility-matched medium that produces MR signal. This, of course, would not be suitable for in vivo studies or even for many ex vivo studies. A more feasible solution would be to develop phase processing algorithms that seek to estimate image phase outside the object, such as iHARPERELLA^6,7. Inadequate phase data can also result from imperfect background phase removal, frequency contributions from microstructure⁸, and the complex field pattern and ensemble signal averaging within a voxel. Each of these can complicate the relationship between susceptibility and phase, and may not be fully described by a linear relationship. Greater understanding of susceptibility mapping artifacts will aid in developing suitable remedial imaging protocols and post-processing techniques.