Synopsis

We present a multiscale SMV implementation (MSMV) for background field removal in QSM. We use a combined redundant Laplacian decomposition and Laplacian pyramid approach with fuzzy masks to remove background fields and reconstruct the local field. We tested this algorithm against PDF, LBV, ESHARP and VSHARP in analytic phantom and in vivo experiments. Experiments show MSMV’s accuracy to be in par with VSHARP, with an order of magnitude speed gain. MSMV achieved results with less harmonic remnants in our in vivo test. This multiscale approach may also be extended to the susceptibility inversion problem, with adaptive preconditioners and weighting.

PURPOSE

Dipole deconvolution needed for Quantitative Susceptibility Mapping (QSM) requires an estimate of the local magnetization field from the phase of gradient-recalled echo (GRE) acquisitions. External magnetization fields often dominate but can be decoupled by exploiting the harmonic properties of the external field^1,2. Several algorithms have been proposed to this end, such as the Laplacian-based^3,4 methods, the projection onto dipole fields (PDF)⁵ method and methods that exploit the spherical mean value (SMV) property of harmonics fields^6,7. However, such approaches have caveats, e.g. the latter erodes the ROI to a smaller trustworthy region and they are all prone to contaminate the local field estimation with remnants affecting the robustness and accuracy in QSM⁸. A popular choice is the Variable Sophisticated Harmonic Artifact Reduction for Phase data⁹ (VSHARP) algorithm, which merges several SMV filters with different spherical kernel sizes. While the accuracy of the SMV operation is improved by increasing the kernel size, this reduces the effective field-of-view (FOV). Hence, by changing the kernel size,the loss of FOV is reduced to only 1 voxel at the boundary, with increased accuracy in deep-brain regions. This time-consuming operation can be performed more efficiently by using conventional convolutions and variable FOV masks.

Here we present a more computationally efficient multiscale implementation of SMV filtering using Gaussian instead of spherical kernels to build a Laplacian pyramid.

METHODS

For ϕ, the acquired phase, a Laplacian decomposition can be performed by the following operations:

$${ϕ}_{i}={κ}_{i}*{ϕ}_{i-1}, {L}_{i}={ϕ}_{i-1}-{ϕ}_{i}$$

where ϕ_i is a low-pass filtered ϕ_i-1 phase (ϕ₀=ϕ) and L_i it’s high-pass counterpart. κ_i is a Gaussian kernel with increasing size, for N number of maximum layers. Each L_i layer in this decomposition reveals structures in the phase at different scales. Since the SMV property is valid for any radially symmetric kernel, these layers contain only information of local susceptibility sources. A harmonic-free reconstruction can be calculated by combination of all the layers and discarding the residual layer ϕ_N. If large-scale susceptibility features are present, a subsequent deconvolution step will be required to recover such features, as in VSHARP.

An alternative multiscale approach is to subsample the low-pass ϕ_i phase, thus creating a Laplacian pyramid without increasing the filter size. This represents a more memory efficient decomposition.

Our algorithm uses a combined approach, with a redundant decomposition for the first 3 layers, and kernel sizes following a dyadic sequence with standard deviation σ_i=2^i-1. A Laplacian pyramid is built for layers 4 to 7, with a downsampling factor of 2 and kernel with σ=2 (Fig.1).

To avoid contributions of the noise-corrupted phase outside the FOV, fuzzy masks were created based on subsequent erosion of the FOV and convolutions with the kernel. Since the first layer uses a narrow filter (σ₁=0.5), the mask was not eroded at this stage (Fig.1).

The proposed algorithm was compared with PDF, Laplacian boundary value³ (LBV), ESHARP¹⁰ and VSHARP in the following experiments:

A) COSMOS ground-truth¹¹ (3D spoiled-GRE on a 3T Siemens Tim Trio, 1.06-mm isotropic voxels, 240×196×120, TE/TR=25/35ms, 12 head orientations), from which a noisy field map was forward simulated with a widespread distribution of external sources (Fig. 1). Results are provided for an in-house spatial and laplacian-based¹² unwrapping methods.

B) In vivo data (3D spoiled-GRE on a 3T Siemens Tim Trio, 352×352×170, 0.6×0.6×1.0 mm3, TE/TR=[7.2,13.4,19.6,25.8,32]/44ms, α=17°) processed with in-house temporal unwrapping and multi-echo combination¹³. In the absence of a ground-truth, the estimation of field remnants was carried out with the weak harmonics-QSM method¹⁴.

RESULTS

A) Results are shown in Fig.2 and Fig.3, for each unwrapping method.

B) Background removal and susceptibility reconstruction results are shown in Fig.4. Harmonic residual phases estimated by WH-QSM are shown, with the associated L2-norm calculation to measure the amplitude of these remnants.

Processing times are shown in Fig.5.

DISCUSSION & CONCLUSION

Acknowledgements

References

1. Li L, Leigh JS. Quantifying arbitrary magnetic susceptibility distributions with MR. Magn. Reson. Med. 2004; 51: 1077–1082.

2. Li L, Leigh JS. High-precision mapping of the magnetic field utilizing the harmonic function mean value property. J. Magn. Reson. 2001; 148: 442–448.

3. Zhou D, Liu T, Spincemaille P, Wang Y. Background field removal by solving the Laplacian boundary value problem. NMR in Biomedicine. 2014;27(3):312–319.

4. Li W, Avram AV, Wu B, Xiao X, Liu C. Integrated Laplacian-based phase unwrapping and background phase removal for quantitative susceptibility mapping. NMR Biomed. 2014;27(2):219-227.

5. Liu T, Khalidov I, de Rochefort L, Spincemaille P, Liu J, Tsiouris AJ, Wang Y. A novel background field removal method for MRI using projection onto dipole fields (PDF) NMR Biomed. 2011;24(9):1129–1136.

6. Schweser F, Deistung A, Lehr BW, Reichenbach JR. Quantitative imaging of intrinsic magnetic tissue properties using MRI signal phase: an approach to in vivo brain iron metabolism? Neuroimage 2011;54:2789–2807.

7. Wu B, Li W, Guidon A, Liu C. Whole brain susceptibility mapping using compressed sensing. Magn Reson Med. 2012;67(1):137–147.

8. Schweser F, Robinson SD, de Rochefort L, Li W, Bredies K. An illustrated comparison of processing methods for phase mri and qsm: removal of background field contributions from sources outside the region of interest. NMR Biomed, 2017;30:e3604. doi: 10.1002/nbm.3604

9. Fang, J., Bao, L., Li, X., van Zijl, P. C., & Chen, Z. (2017). Background field removal using a region adaptive kernel for quantitative susceptibility mapping of human brain. Journal of Magnetic Resonance, 281, 130 - 140. doi:10.1016/j.jmr.2017.05.004

10. Topfer R, Schweser F, Deistung A, Reichenbach JR, Wilman AH. SHARP edges: Recovering cortical phase contrast through harmonic extension. Magn Reson Med. 2015;73(2):851–856. doi: 10.1002/mrm.25148

11. Langkammer C, Schweser F, Shmueli K, Kames C, Li X, Guo L, Milovic C, Kim J, Wei H, Bredies K, Buch K, Guo Y, Liu Z, Meineke J, Rauscher A, Marques JP and Bilgic B. Quantitative Susceptibility Mapping: Report from the 2016 Reconstruction Challenge. Magn. Reson. Med. doi:10.1002/mrm.26830

12. Schofield MA, Zhu Y. Fast phase unwrapping algorithm for interferometric applications. Opt Lett. 2003;28(14):1194–1196.

13. Acosta-Cabronero J, Milovic C, Mattern H, Tejos C, Speck O and Callaghan MF. A robust multi-scale approach to quantitative susceptibility mapping . Neuroimage, 2018;183:7-24. doi:10.1016/j.neuroimage.2018.07.065

14. Milovic C, Bilgic B, Zhao B, Langkammer C, Tejos C and Acosta-Cabronero J. Weak-harmonic regularization for quantitative susceptibility mapping (WH-QSM); Magn Reson Med, 2018;00:1–13. doi: 10.1002/mrm.27483