Tissue magnetic susceptibility (χ)-induced B₀ disturbances can be rapidly calculated by carrying out convolution of the susceptibility map with the dipolar field kernel in the Fourier domain [1,2]. Conventionally, the continuous k-space dipolar field kernel ($$$1/3-k_{z}^2/k^2$$$) is discretized in the k-space, causing Fourier aliasing artifacts in the spatial domain. This effect decreases, but is not eliminated, with zero-filled buffers [2] added in the spatial domain. Whereas a moderate buffer size is often sufficient for brain applications, disproportionately larger buffer sizes may be required in other body parts where the susceptibility source has a non-spherical, flat or elongated shape. Large buffers burden the computational memory and can slow down repeated χ-to-B₀ calculations as in certain quantitative susceptibility mapping (QSM) algorithms [3]. Here we show that discretization of the dipolar field kernel in the spatial domain provides an effective way to avoid aliasing. The process utilizes closed-form integration of the dipolar field over a rectangular volume. It can be shown that for any bounded object, buffering by a factor of two in each Cartesian direction guarantees lack of aliasing.

Figure 1 illustrates the concept of our method. The true, continuous dipolar field kernel (Fig. 1a) is contaminated when the kernel is discretized in the Fourier domain by the replicas of the dipole sources appearing at integer multiples of the domain size (L) (Fig. 1b). The effect is generally non-zero everywhere in the spatial domain due to the long range of the aliased dipolar fields. In contrast, if the true dipolar field is first discretized and truncated in the spatial domain, the subsequent (discrete) Fourier transform leads to aliasing which does not affect the true dipolar field in the original domain (L) (Fig. 1c). If this kernel is convoluted with the susceptibility of a source object with length L_obj, the aliased portion of the kernel does not affect the true field inside the object when $$$L\geq 2L_{obj}$$$. Now consider a 3D Cartesian grid where each voxel is a rectangular cuboid of volume $$${\Delta}v={\Delta}x{\times}{\Delta}y{\times}{\Delta}z$$$. The susceptibility-induced, z-directional magnetic field in the ith voxel is given by the discrete convolution:

$$\delta B_0(\vec{r}_i)=\sum_{\vec{r}_j}K(\vec{r}_i-\vec{r}_j)\cdot\chi(\vec{r}_j)$$

where K is the average dipolar field in the ith (target) voxel per unit susceptibility of the jth (source) voxel. In computing K for anisotropic voxels, it is important that the dipolar field be averaged over the target voxel rather than merely sampled at its center. The voxel-averaged dipolar field kernel K can be analytically calculated using the formula in Fig. 2. This kernel is then Fourier-transformed discretely, multiplied by the Fourier transform of χ, and inverse-transformed to give the off-resonance field,

$$\delta B_0 = FT^{-1}(FT(K)\cdot FT(\chi)).$$

As long as the spatial domain is at least twice as large as the region of interest (in all Cartesian directions), δB₀ is free from aliasing artifacts. If the source and target regions are different, a domain size that equals the sum of the sizes of the source and the target regions (in all directions) is sufficient to avoid artifacts.

Numerical simulation was conducted in Matlab (Mathworks, MA, USA) on an ellipsoid with axis lengths =(0.4, 0.2, 0.8)[m] in a Cartesian grid with voxel size 2x2x4 [mm³]. The ellipsoid was uniformly magnetized at 3T with χ = 1 ppm. δB₀ was calculated using both the proposed and the conventional methods with 3 different buffer dimensions in the shortest-axis (y) direction to assess the aliasing artifacts. Figure 3 shows the simulation models and results. For the two small buffers (a,b), the conventional method exhibited a large mismatch between the simulated and the analytical [4] δB₀ profiles. The proposed method, on the contrary, showed excellent agreement in all cases. In particular, the proposed method was free from artifacts on the coronal mid-plane (y=0) without any buffer in the y-direction (case (a)). Figure 4 compares the computation times. The proposed method is slower in calculation of the k-space dipolar kernel (Step 1). However, this is compensated by faster subsequent calculations (Step 2) resulting from the smaller domain size afforded by the new method.The proposed method adds one more operation in computing the Fourier-domain dipolar field kernel compared to the conventional method, but this is needed only once for a given computational grid. After that, reduction of the computational memory and guaranteed elimination of the Fourier aliasing effect can improve the speed and accuracy of repeated susceptibility-to-B₀ calculations. Potential applications include respiration-induced B₀ changes in the brain [2], susceptibility-induced off-resonance calculation in the body, including breast [5], and iterative high-resolution QSM outside the brain.