2464

How far away from anisotropic microstructure do you need to be for QSM to be faithful?

Anders Dyhr Sandgaard¹ and Sune Nørhøj Jespersen^1,2
¹Center of Functionally Integrative Neuroscience, Department of Clinical Medicine, Aarhus University, Aarhus, Denmark, ²Department of Physics and Astronomy, Aarhus University, Aarhus, Denmark

Synopsis

Keywords: Susceptibility/QSM, Quantitative Susceptibility mapping

Motivation: White matter (WM) microstructure can affect estimation of WM susceptibility with QSM. However, as QSM fits all voxels at once, it is less understood how it affects estimation of surrounding tissue.

Goal(s): Our goal is to demonstrate the effect of a cylindrical microstructure on surrounding tissue in a digital phantom.

Approach: We synthesize a digital phantom with parallel rods surrounded by a rim with random dots. We estimate susceptibility with (QSM+) and without (QSM) account of microstructure.

Results: QSM is biased inside the rim, and this error spreads to the surrounding tissue characterized by a power law. QSM+ improved susceptibility fitting.

Impact: It may be important to account for microstructure in WM even though one may only be interested in analyzing surrounding tissue like gray matter. Failing to do so could lead to misinterpretation of tissue magnetic susceptibility.

Introduction

Quantitative Susceptibility Mapping¹ (QSM) is an increasingly popular modality to study chemical composition of brain tissue due to its sensitivity to both iron and myelin². However, QSM does not take microstructural frequency shifts into account. It is well-known how WM microstructure biases susceptibility estimation in WM itself^3,4 but less is known about how it affects the estimation of susceptibility in the surrounding tissue, e.g., in gray matter (GM). Here we investigate the effect of microstructure on the accuracy of QSM. Our main question is: How far away from a macroscopic region with structurally anisotropic microstructure do you need to be for the estimated QSM values to be faithful? We address this question by estimating the susceptibility with QSM of a simple digital phantom consisting of a sphere filled with axons modelled as diamagnetic parallel rods, with the sphere surrounded by a rim containing small paramagnetic spheres. We also consider the impact of accounting for the microstructure in QSM, a framework we call QSM+⁴.

Methods

We simulated QSM fitting on a very simple digital phantom. The overall shape of the phantom was either a long cylinder (P1) or a sphere (P2). It consisted of a rim (R) made up of randomly positioned paramagnetic dots. In the sphere below the rim (A), diamagnetic parallel rods were randomly positioned. An overview of the simulation is shown in Figure 1. To restrict focus to the susceptibility estimation effect of the phantom itself, no media was outside (E) the phantom. We performed QSM and QSM+ fitting⁴ using LSQR⁵ by fitting the sampled frequency shift $$$\overline\Omega(\mathbf{R})$$$ at discrete positions $$$\mathbf{R}$$$ using either 1, 2 or 3 unique sample orientations⁶. Both algorithms are listed in Figure 1. $$$\overline\chi_\mathrm{QSM+}$$$ denotes the fitted susceptibility that includes sub-voxel frequency shifts^4,7-11, denoted $$$\overline\Omega^\mathrm{Meso}(\mathbf{R})$$$, while shifts induced across voxels (i.e. disregarding the voxel inclusions’ effect on its own average shift) are called $$$\overline\Omega^\mathrm{Macro}(\mathbf{R})$$$. We optimized the number of LSQR iterations that minimized the RMSE compared to the ground truth bulk susceptibility $$$\overline\chi_\mathrm{GT}(\mathbf{R})$$$ made by averaging and discretely sampling the local “microscopic” susceptibility $$$\overline\chi_\mathrm{GT}(\mathbf{r})$$$ (cf. Figure 1), where $$$\mathbf{r}$$$ denotes positions at the “microscopic” scale.

Results

Figure 2 shows the sampled frequency shift $$$\overline\Omega(\mathbf{R})$$$ for phantoms P1 and P2, for all directions and the frequency shift estimated from either QSM (2nd column) or QSM+ (3rd column). It is clear that $$$\overline\Omega^\mathrm{Meso}(\mathbf{R})$$$ has a substantial effect inside (A) but negligible effect outside. Figure 3 shows a cross-section of the susceptibility fits from QSM and QSM+. QSM underestimated the susceptibility in (R), while overestimating inside (A). For P1, the error with QSM was around 3-17% of the value $$$\overline\chi_\mathrm{GT}(A)$$$, when including 3, 2 or 1 orientations, respectively. This error was reduced to around 3-5% with QSM+. For P2, the error with QSM was around 20-60% of $$$\overline\chi_\mathrm{GT}(A)$$$, when including 3, 2 or 1 orientations, respectively, and around 8% with QSM+. Figure 4 shows the RMSE as a function of LSQR iterations, and the optimal RMSE and mean error as a function of distance from the center.

Discussion

Our results show that sub-voxel frequency shifts can bias QSM if not accounted for. The error is not local but permeates into the surrounding tissue as a power law with distance. For P1, the exponent was distance^-3 and in P2 it was distance^-2. Our findings indicate the error in every individual voxel inside (A) decays with exponent -3, but the overall decay outside the macrostructure of (A), depends on its macroscopic shape. Since GM and WM can have a susceptibility of similar magnitude^12,13, our findings could be relevant in the vicinity of e.g. a macroscopic tract of WM with radius R. At the interface between this WM and neighboring GM, the susceptibility in GM can then appear 20-60% lower. At a distance R from the WM, the error goes down to around 3-10%. In reality, WM axons are orientationally dispersed^14-17 and have susceptibility anisotropy^3,18-20. Such realistic features will lower the effect on neighboring GM roughly by the dispersion factor^7,21 $$$p_2$$$. Nevertheless, as a rule of thumb, we believe our findings show that one should be cautious with analyzing QSM values near WM, at distances below the width of coherent WM tracts.

Conclusion

We demonstrate how sub-voxel frequency shifts from tissue microstructure affect the sampled frequency shifts in a simple digital phantom. We found the estimated susceptibility using QSM, which does not take these sub-voxel effects into account, was biased and this error eroded susceptibility estimation in neighboring tissue. This may jeopardize a truthful interpretation of GM susceptibility close to WM.

Acknowledgements

This study is funded by the Independent Research Fund (grant 8020‐00158B). We thank Prof. Dmitriy A.Yablonskiy from Mallinckrodt Institute of Radiology for insightful discussions.

References

1. Reichenbach JR, Schweser F, Serres B, Deistung A. Quantitative Susceptibility Mapping: Concepts and Applications. Clin Neuroradiol. 2015;25(2):225-230. doi:10.1007/S00062-015-0432-9/FIGURES/3

2. Wang Y, Liu T. Quantitative susceptibility mapping (QSM): Decoding MRI data for a tissue magnetic biomarker. Magn Reson Med. 2015;73(1):82-101. doi:10.1002/MRM.25358

3. Wharton S, Bowtell R. Effects of white matter microstructure on phase and susceptibility maps. Magn Reson Med. 2015;73(3):1258-1269. doi:10.1002/mrm.25189

4. Sandgaard AD, Kiselev VG, Henriques RN, Shemesh N, Jespersen SN. Incorporating the effect of white matter microstructure in the estimation of magnetic susceptibility in ex vivo mouse brain. Magn Reson Med. Published online September 29, 2023. doi:10.1002/MRM.29867

5. PAIGE McGill University CC, Saunders MA. LSQR: An Algorithm for Sparse Linear Equations and Sparse Least Squares. Published online 1982.

6. Liu T, Spincemaille P, De Rochefort L, Kressler B, Wang Y. Calculation of susceptibility through multiple orientation sampling (COSMOS): A method for conditioning the inverse problem from measured magnetic field map to susceptibility source image in MRI. Magn Reson Med. 2009;61(1):196-204. doi:10.1002/mrm.21828

7. Sandgaard AD, Shemesh N, Kiselev VG, Jespersen SN. Larmor frequency shift from magnetized cylinders with arbitrary orientation distribution. NMR Biomed. 2023;36(3):e4859. doi:10.1002/nbm.4859

8. He X, Yablonskiy DA. Biophysical mechanisms of phase contrast in gradient echo MRI. Proc Natl Acad Sci U S A. 2009;106(32):13558-13563. doi:10.1073/pnas.0904899106

9. Yablonskiy DA, Sukstanskii AL. Generalized Lorentzian Tensor Approach (GLTA) as a biophysical background for quantitative susceptibility mapping. Magn Reson Med. 2015;73(2):757-764. doi:10.1002/mrm.25538

10. Kiselev VG. Larmor frequency in heterogeneous media. J Magn Reson. 2019;299:168-175. doi:10.1016/j.jmr.2018.12.008

11. Ruh A, Scherer H, Kiselev VG. The larmor frequency shift in magnetically heterogeneous media depends on their mesoscopic structure. Magn Reson Med. 2018;79(2):1101-1110. doi:10.1002/mrm.26753

12. Marques JP, Meineke J, Milovic C, et al. QSM reconstruction challenge 2.0: A realistic in silico head phantom for MRI data simulation and evaluation of susceptibility mapping procedures. Magn Reson Med. 2021;86(1):526-542. doi:10.1002/MRM.28716

13. Bilgic B, Langkammer C, Marques JP, Meineke J, Milovic C, Schweser F. QSM reconstruction challenge 2.0: Design and report of results. Magn Reson Med. 2021;86(3):1241-1255. doi:10.1002/MRM.28754

14. Ronen I, Budde M, Ercan E, Annese J, Techawiboonwong A, Webb A. Microstructural organization of axons in the human corpus callosum quantified by diffusion-weighted magnetic resonance spectroscopy of N-acetylaspartate and post-mortem histology. Brain Struct Funct. 2014;219(5):1773-1785. doi:10.1007/s00429-013-0600-0

15. Andersson M, Kjer HM, Rafael-Patino J, et al. Axon morphology is modulated by the local environment and impacts the noninvasive investigation of its structure–function relationship. Proc Natl Acad Sci U S A. 2021;117(52):33649-33659. doi:10.1073/PNAS.2012533117

16. Lee HHH, Yaros K, Veraart J, et al. Along-axon diameter variation and axonal orientation dispersion revealed with 3D electron microscopy: implications for quantifying brain white matter microstructure with histology and diffusion MRI. Brain Struct Funct. 2019;224(4):1469-1488. doi:10.1007/s00429-019-01844-6 17. Abdollahzadeh A, Belevich I, Jokitalo E, Tohka J, Sierra A. Automated 3D Axonal Morphometry of White Matter. Sci Reports 2019 91. 2019;9(1):1-16. doi:10.1038/s41598-019-42648-2

18. Van Gelderen P, Mandelkow H, De Zwart JA, Duyn JH. A torque balance measurement of anisotropy of the magnetic susceptibility in white matter. Magn Reson Med. 2015;74(5):1388-1396. doi:10.1002/mrm.25524

19. Lee J, Shmueli K, Kang BT, et al. The contribution of myelin to magnetic susceptibility-weighted contrasts in high-field MRI of the brain. Neuroimage. 2012;59(4):3967-3975. doi:10.1016/J.NEUROIMAGE.2011.10.076

20. Lounila J, Ala-Korpela M, Jokisaari J, Savolainen MJ, Kesäniemi YA. Effects of orientational order and particle size on the NMR line positions of lipoproteins. Phys Rev Lett. 1994;72(25):4049. doi:10.1103/PhysRevLett.72.4049

21. Novikov DS, Veraart J, Jelescu IO, Fieremans E. Rotationally-invariant mapping of scalar and orientational metrics of neuronal microstructure with diffusion MRI. Neuroimage. 2018;174:518-538. doi:10.1016/J.NEUROIMAGE.2018.03.006

22. Ruh A, Kiselev VG. Calculation of Larmor precession frequency in magnetically heterogeneous media. Concepts Magn Reson Part A. 2018;47A(1):e21472. doi:10.1002/cmr.a.21472

23. Sandgaard AD, Shemesh N, Jespersen SN, Kiselev VG. To mask or not to mask? Investigating the impact of accounting for spatial frequency distributions and susceptibility sources on QSM quality. Magn Reson Med. 2023;90(1):353-362. doi:10.1002/MRM.29627

Figures

Figure 1 - Ground truth susceptibility χ_GT(r) and the calculated local frequency shift²² Ω(r). Ω(r) was coarse-grained and sampled outside the inclusions to yield the average shift \barΩ(R) The susceptibility \barχ_QSM(+)(R) was estimated using QSM and QSM+. M^A(R) is a mask^4,23 of the space with 2D microstructure. Both fits were subsequently compared to the averaged \barχ_GT(R) corresponding to χ_sphζ_sph in Rim and χ_cylζ_cyl inside. Phantom shown here is P1 (sphere).

Figure 2 - Each row corresponds to different B₀. 1st column shows the sample averaged shift \barΩ. 2nd column shows \barΩ^Macro induced across voxels. This is only considered in QSM. 3rd column includes sub-voxel shifts \barΩ^Meso which is also included in QSM+ (cf. figure 1). The last two columns show the averaged frequency profiles along different orientations as a function of distance from the center. A shows phantom P1 (sphere), while B shows phantom P2 (cylinder).

Figure 3 - A,C shows QSM fit using 1, 2 or 3 numbers of directions, and the ground truth susceptibility . The lower panel shows the residual to the ground truth. B,D shows QSM+ fits. A-B is spherical phantom while C-D is cylindrical phantom. QSM overestimates the susceptibility inside the sphere or cylinder with 2D microstructure and underestimates the rim with 3D microstructure, while QSM+ mainly suffers from partial volume effects near the interface, which were not accounted for in the sub-voxel frequency shifts.

Figure 4 - A,C show the optimal RMSE in different regions for QSM and QSM+ when fitting using LSQR for either phantom P1 or P2, respectively. ^B,D show RMSE and absolute mean error for the optimal QSM and QSM+ fits as a function of distance from the center. Dashed lines indicate the rim. QSM+ is clearly lowest, but QSM has a comparable RMSE inside when using 3 directions. Microstructure inside affects the other regions and RMSE decays approximately as a power of law of distance^-3 in C and distance^-2 in D.

Proc. Intl. Soc. Mag. Reson. Med. 32 (2024)

2464

DOI: https://doi.org/10.58530/2024/2464