MR-based Quantitative Susceptibility Mapping (QSM) techniques have multiple potential applications in brain [1] and body imaging [2]. QSM techniques generally rely on the removal of background field effects to obtain a “local” B₀ map, followed by “dipole inversion” to estimate the underlying susceptibility distribution. Common background field removal techniques are based on the assumption that this field is harmonic inside the region of interest. However, concomitant gradients (CG) introduce significant unanticipated phase shifts in the acquired data that manifest as errors in the measured B₀ field map [3]. It is unknown whether CG-related B₀ errors are harmonic, which may violate assumptions made by some background field removal techniques. Therefore, the purpose of this work was to characterize the effects of CGs on QSM by assessing the ability of background field removal techniques to address CG-related field map errors.

Li and Leigh [4] demonstrated that the field map induced by regions with homogeneous magnetic susceptibility is a harmonic function (i.e. the Laplacian is zero $$$\triangledown^2 B(x,y,z,t) = 0$$$). However, the effects of CG phase errors were not considered in their analysis.

In the presence of CGs, the net magnetization becomes a non-linear function of the magnetization, contrary to the assumption of [4]. A full expression for the Laplacian of the field map with CG terms can be calculated analytically. For simplicity, here we consider the case with only an x-gradient $$$\left(G_x(t)\right)$$$ present. The magnetization becomes $$$B(t)=G_x(t)z\hat{x}+B_0+G_x(t)x\hat{z}$$$, where the first term is due to CGs. The Laplacian of this expression is

$$\triangledown^2 B(x,y,z,t)=\frac{G_x(t)^2}{\sqrt{\left(B_0+G_x(t)\right)^2+\left(G_x(t)z\right)^2}}.$$

This is nonzero since we assumed $$$\left(G_x(t)\right)$$$ is non-zero and the integral of a positive function will always be nonzero. Consequently, the field map will have a nonzero Laplacian and cannot be a harmonic function. Using simulations (below), we confirmed this conclusion numerically and tested previously proposed background field removal algorithms in the presence of CG phase shifts.

We conducted two simulations of a pulse sequence (Fig. 1) with two echoes, TE₁=3.38 ms and TE₂=22ms, and FOV=35 x 35 x 40 cm (X x Y x Z cm) on a pure water signal. One simulation was performed with the effects of CGs and the other without.

Next, the Laplacian of the measured field map was calculated numerically with and without the effects of CGs. Finally, three background field removal techniques were tested in the presence of CGs, including the Projection onto Dipole Fields (PDF) [5,6], Sophisticated Harmonic Artifact Reduction for Phase data [1], and Regularization Enabled SHARP (RESHARP) [7]. Since the simulated field map does not include inhomogeneities (e.g. magnet imperfections, shims, or susceptibility sources) the estimated background field map should be 0 Hz.

CGs create a large error in the field map (Fig. 2). This error is quadratic in nature and in this case is largest along the z-direction [3].

The Laplacian of the estimated field map, using data without CG errors, was zero as expected. However, including CG errors resulted in a small non-zero Laplacian of the estimated field map (6.1e-6 (Hz/m²) at isocenter and the second echo time). This confirms that the field map is not harmonic when CG terms are considered.

The impact of this non-zero Laplacian is characterized by applying background field removal algorithms. As shown in Fig. 2, the PDF algorithm does not remove the effects of CGs on the background field map. However, SHARP significantly reduces these errors and the RESHARP algorithm effectively removes them.

CGs create significant errors in the measured field map. For the sequence shown in Fig. 1 the error was as high as 150 Hz and would be expected to have significant impact on estimated susceptibility maps. Smaller FOVs would decrease the magnitude of this error, but other pulse sequence parameters may increase the error.

We demonstrated analytically and numerically that the effect of the CGs creates a nonzero Laplacian of the field map such that it is not harmonic. Three background field removal algorithms were tested in the presence of CG shifts. PDF was unable to remove large errors in the field map since these errors are not in the subspace spanned by dipole fields. SHARP and RESHARP were able to reduce the CG effects since the Laplacian is small enough that the spherical mean value property holds, approximately. However, these errors are dependent on gradients strengths, echo train lengths, and flow compensated gradients. Our results demonstrate that CG phase corrections [3] and/or the use of a background field removal algorithm that removes this background field component are necessary for accurate QSM.