Current quantitative susceptibility mapping (QSM) reconstruction methods use a forward model proposed by Salomir et al.1 and Marques & Bowtell2, which relates a magnetic field offset with a given distribution of magnetic susceptibility sources. This forward model has an analytical solution considering an approximation of a magnetic dipole kernel and applying a continuous Fourier Transform (FT). The direct inversion of the forward model yields undesired artefacts on the reconstructed QSM, thus the inversion is typically formulated as an iterative regularised problem. Even though this inversion approach solves some problems, QSM reconstructions still have inaccuracies, aliasing and artefacts – particularly at the interface of strong local susceptibility gradients. Most of these problems are due to discretisation and limited FT bandwidth. In order to address this issue, we propose an alternative formulation of the forward model based on the discrete cosine transform and discrete derivative operators.Salomir inferred that: $$$H_0 · dX/dz = - \Delta( \phi_{obj} )$$$. By taking the derivative along z, we may rewrite it as: $$$H_0 · d^2X/dz^2 = \Delta( h_{obj,z} )$$$. To obtain $$$h_{obj,z}$$$, we propose the use of the discrete cosine transform (DCT) instead of the FT. Note that for a continuous system, both transforms are analogous; though with the discretisation of the derivative operators, the Laplacian operator ($$$ \Delta $$$) becomes a convolution with the [1 -2 1] kernel (in 1D). The DCT uncouples such operator leading to a point-by-point operation in the frequency domain: $$$[-2+2·cos(\pi·k/N)]$$$ with k = 0 : N-1. Noting the extension to 3D is trivial: $$$[-6+2·cos(\pi·kx/N)+2·cos(\pi·ky/N)+2·cos(\pi·kz/N)]$$$, we can rewrite the dipole kernel as: $$1/3 - \frac{-2+2·cos(\pi·kz/N)}{[-6+2·cos(\pi·kx/N)+2·cos(\pi·ky/N)+2·cos(\pi·kz/N)}$$. A second-order Taylor expansion of the Laplacian operator and DCT may be rewritten as: $$$[-2+2·cos(\pi·k/N)] ~= [-2+2·(1-k^2+O(k^4))] ~= [2·k^2+2·O(k^4)]$$$. Although for low frequencies FT and DCT kernels are essentially identical, significant differences arise for high frequencies. A qualitative assessment of the kernel’s impulse response (not shown) revealed that the DCT-based approach acts, in essence, as a low-pass filter, which helps to diminish the bandwidth-related aliasing artefacts and improves SNR. The QSM reconstructions with DCT and FT-based kernels were compared: (i) quantitatively, using a spherical phantom; and (ii) qualitatively, using post-mortem, formalin-fixed coronal brain slice (age at death: 65 years, fixation duration: 4 months, pathological finding: inconspicuous) embedded in a PBS solution3, which was scanned with a 7 Tesla MRI system using a multi gradient echo sequence (4 echoes, 0.36-mm3 isotropic voxel resolution). The QSM reconstruction routine consisted of a complex multi-echo fitting step, Laplacian unwrapping, variable SHARP filtering (starting radius=10 mm) and separate non-linear MEDI inversions (lambda=1,000) using both kernels.

For a large sphere, both kernels produced QSM reconstructions that differ from the analytical model close to the extremes of the field of view (FOV) (Fig. 1 top right) due to the boundary condition constraints of the FT and DCT. Both kernels produced similar errors at the sphere interface (Fig.1 top right and bottom right). For small spheres the DCT-based kernel produced a more accurate susceptibility distribution – particularly at the interface of the sphere (Fig.1 top left and bottom left).

The post-mortem experiment (Fig. 2) showed that DCT based reconstructions return slightly narrower global histograms as a result of the low-pass filtering properties of the DCT kernel whilst anatomical details are largely preserved due to a higher SNR.

In terms of complexity the DCT and the FT kernel-based QSM reconstructions are similar, however DCT-based method showed improvements for the interface of small structures. This might be relevant for ultra-high resolution QSM where the low-pass filtering properties of the DCT appear to be beneficial. Qualitatively, this translated to slightly smoother reconstructions where boundaries of smaller structures/deposits might be better resolved.The proposed modification of the kernel-based QSM reconstruction might be particularly relevant in the context of neurodegenerative brain diseases where reliability in detecting neuropathological deposits and other small-scale abnormalities is paramount to making successful diagnoses. The proposed method might also be useful for the study of the vascular system.