QSM has found important applications in quantifying iron concentration and vessel oxygenation, providing detailed gray/white matter contrast and differentiating dia- and para-magnetic sources of contrast in the tissue. Its translation is made difficult by a complicated reconstruction pipeline comprising multi-channel and multi-echo signal combination, phase unwrapping, background field removal and dipole inversion steps. Each step can be performed in various ways, ultimately impacting the reconstructed maps. In this presentation, we will focus on the effect of these choices on the resulting images, strive to make recommendations where possible, and point out existing software tools and acquisition strategies that facilitate QSM reconstruction.
QSM reconstruction pipeline
While QSM provides detailed and quantitative information about in vivo tissues, it suffers from complicated image reconstruction comprising multiple steps. A crucial first step is the combination of signal from multi-channel receive arrays, which entails the estimation of the phase offset in each coil element. Eliminating the coil sensitivity phase allows for SNR-optimal, complex-valued signal combination (1,2). For multi-echo gradient echo (GRE) acquisitions, combination of the phase signal across the echoes presents another challenge. Having performed multi-channel, multi-echo signal combination, we still need to remove the 2π jumps in the phase images using unwrapping.
The air-tissue interfaces account for the majority of the signal in the obtained unwrapped phase images. This “background” phase is almost two orders of magnitude stronger than the “foreground” phase information stemming from subtle susceptibility differences within our region of interest (ROI). The background contribution can be filtered out using tailored high-pass filters. At last, we are ready for dipole inversion, where we solve for the tissue susceptibility that gave rise to the foreground phase – which is no simple task either. In the following, we focus on each of these individual steps and their implications on the reconstructed susceptibility maps and strive to make recommendations where possible.
Coil and echo combination
Estimating coil sensitivity information allows for removing the spatially varying phase offset in each coil element before combining the coils. If these phase offsets are not removed, complex signals from each coil may end up combining destructively, leading to reduced SNR and potential phase singularities. A variety of coil combination methods exists, and a comprehensive comparison is provided in (3). Using quick and low-resolution calibration scans made with body coil and multi-channel array reception separately, it is possible to obtain “ground truth” quality coil sensitivities (1,2,4). Subtracting the phase of the body coil image from the phase of the multi-channel array data eliminates any tissue phase, and provides a good approximation to the underlying coil sensitivities (plus a slowly varying phase from the body coil). This technique is also known as “Roemer combination”. Alternatively, a single calibration scan at a very short echo time can be acquired as in COMPOSER (5). Ignoring the small amount of tissue phase information present in such calibration data, the phase of this scan provides a good approximation to the coil phase offsets (6). This also obviates the need for an additional body coil reference scan, which may not be possible on e.g. ultra-high field systems. A final class of methods allow for coil combination without the need for any calibration scans. The SVD approach (7) is applicable to multi-echo acquisitions, where a voxel-wise singular value decomposition yields an approximation to the coil phase offsets. These three techniques (Roemer, COMPOSER and SVD) yield similarly high-quality coil combinations (3).
In multi-echo acquisitions, modeling the phase evolution across time as a first order polynomial allows for the estimation of a common “frequency” image. This estimation can be performed in a nonlinear fashion using magnitude data for noise weighting (8). It is also possible to simply average the echo phase images to gain SNR. This needs to be done after phase unwrapping since unwrapping is a nonlinear step. (Unwrapped) echo images could be normalized by their echo time, and could be further weighted to optimize phase SNR (9) prior to averaging.
Phase unwrapping and background phase removal
Phase unwrapping itself is an important step where multiple options are available. PRELUDE is able to provide good quality results (10), but is computationally intensive especially for large matrix sizes. SEGUE is a much faster version (11), and is also publicly available. Another fast but “approximate” unwrapping technique is Laplacian unwrapping (12), which eliminates wraps but also applies some additional filtering. Since QSM entails high pass filtering anyway, PRELUDE and Laplacian unwrapping were seen to yield similar results when foreground phase images were compared (13).
Starting from the unwrapped phase, we then need to remove the large contribution of the background susceptibility sources to the phase inside our ROI. Most popular background filtering techniques rely on the assumption that the phase created inside the ROI by outside sources is harmonic, and that it can be filtered out using spherical mean value (SMV) kernels. A recent numerical study compared a variety of background removal techniques (14), and found that popular techniques such as PDF (15), V-SHARP (16,17), LBV (18) and HARPERELLA (19) are similarly successful with minor differences in the foreground phase images.
Dipole inversion
The final step in the reconstruction pipeline is dipole inversion, where we aim to estimate the underlying susceptibility map using the foreground phase. The unknown susceptibility is modeled to be related to the estimated tissue phase via convolution with a dipole kernel. This deconvolution problem is ill-conditioned in the sense that the kernel is not invertible, and ill-posed because the dipole convolution relation is an approximate model. As such, this deconvolution often benefits from regularization. We can classify dipole inversion algorithms broadly in terms where the regularization is enforced: either in image- or in k-space. K-space methods are relatively straightforward and fast to compute. Thresholded k-space division (TKD) is a popular approach where the dipole kernel is directly inverted, and the small entries in the kernel are held at a constant value so that their inverse is numerically stable (20). Superfast Dipole Inversion (SDI) further amends the systematic underestimation due to these constant values in TKD to provide a rapid reconstruction pipeline (21).
Image-space techniques often invoke a prior assumption on the reconstructed maps in the form of piecewise constancy or smoothness. Such assumptions can be enforced using e.g. total variation (TV) (8), total generalized variation (TGV) (22) or wavelet transforms (17). Additional edge priors derived from the magnitude image can be utilized to further stabilize the reconstruction (23,24). A drawback of image-space techniques is that they are often computationally intensive. Advanced algorithms could alleviate this computational burden (25), but the efficiency gain often comes at the cost of introduction of additional parameters that need to be tuned.
A recent QSM reconstruction challenge aimed to compare the performance of various reconstruction techniques (26). While image-space techniques were more successful in terms of their accuracy with respect to error metrics such as RMSE than k-space approaches, they also appeared over-smooth. This was probably because smoothness/sparsity assumptions were enforced too strongly to minimize global error measures with respect to the ground truth.
Outlook
We will conclude by discussing some of the open problems where machine learning could be impactful. Deep learning approaches have already provided beyond state-of-the-art results in dipole inversion (27,28), and have the potential to improve background field removal (29).
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