IntroductionReconstructing a bounded object from incomplete k-space data is a well posed problem, and it was recently shown that this incomplete spectrum approach can be used to reconstruct undersampled MRI images with similar quality to compressed sensing approaches. Here, we apply this incomplete spectrum approach to the field-to-source inverse problem encountered in quantitative magnetic susceptibility mapping (QSM). The field-to-source problem is an ill-posed problem because of conical regions in frequency space where the dipole kernel is zero or very small, which leads to the kernel's inverse being ill-defined. These “ill-posed” regions typically lead to streaking artifacts in QSM reconstructions. In contrast to compressed sensing, our approach relies on knowledge of the image-space support, more commonly referred to as the mask, of our object as well as the region in k-space with ill-defined values. In the QSM case, this mask is usually available, as it is required for most QSM background field removal and reconstruction methods.MethodsWe tuned the incomplete spectrum method (mask and band-limit) for QSM on a simulated dataset from the most recent QSM challenge and validated the QSM reconstruction results on brain images acquired in five healthy volunteers, comparing incomplete spectrum QSM to current state-of-the art-methods: FANSI, nonlinear dipole inversion, and conventional thresholded k-space division.ResultsWithout additional regularization, incomplete spectrum QSM performs slightly better than direct QSM reconstruction methods such as thresholded k-space division (PSNR of 39.9 vs. 39.4 of TKD on a simulated dataset) and provides susceptibility values in key iron-rich regions similar or slightly lower than state-of-the-art algorithms, but did not improve the PSNR in comparison to FANSI or nonlinear dipole inversion. With added (ℓ1-wavelet based) regularization the new approach produces results similar to compressed sensing based reconstructions (at sufficiently high levels of regularization).DiscussionIncomplete spectrum QSM provides a new approach to handle the “ill-posed” regions in the frequency-space data input to QSM.

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Simultaneous multi-slice (SMS) acquisition is increasingly used to accelerate echo planar imaging (EPI). EPI acquisitions have been used for quantitative susceptibility mapping (QSM) but, to utilise SMS, an investigation into the effect of SMS on EPI-QSM accuracy is necessary. Here, we show that SMS has no significant effect on magnetic susceptibility maps and values, and can, therefore, provide accurate QSM within a short TR. We also show, for the first time, that multi-echo phase images can be acquired using an EPI sequence (highly) accelerated using SMS and parallel imaging, leading to more accurate QSM reconstruction compared to standard single-echo EPI.

Although EPI phase images are useful (e.g. for Quantitative Susceptibility Mapping), they often contain phase inconsistencies in the slice-select direction which persist and can degrade QSM results. Here, we analysed three EPI datasets in healthy volunteers to characterise these phase inconsistencies and understand whether they occur or interact with interleaved or sequential slice acquisition order. We characterised a ~2Hz cryogen pump artifact in sequential data and slice-to-slice phase jumps in interleaved data. We modified a previously proposed QSM processing pipeline, including 2D (VSHARP) and 3D (PDF) background field removal that removed all through-slice artifacts observed.

Direct solutions of the dipole inversion problem in quantitative susceptibility mapping (QSM) are computationally efficient but plagued by streaking artifacts. Here, we have shown that non-uniform sampling of frequency space can achieve additional streaking artifact reduction compared to QSM with thresholded k-space division and state-of-the-art regularisation. By avoiding sampling areas in frequency space where the solution is not well defined, the solution of the ill-posed inverse problem is made more robust and noise amplification is reduced. This approach could be combined with compressed sensing techniques to further improve the QSM reconstruction. This research uses open-source tools from the MR community.