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Diffusion-weighted phase imaging: towards a tract-specific myelin measure
Michiel Cottaar1, Benjamin C. Tendler1, Wenchuan Wu1, Karla L. Miller1, and Saad Jbabdi1
1WIN@FMRIB, University of Oxford, Oxford, United Kingdom

Synopsis

We propose a novel sequence that adds a second asymmetric spin echo after a standard Stejskal-Tanner sequence. This allows the estimation of the off-resonance frequency of the diffusion-weighted signal due to the myelin magnetic susceptibility. Varying the orientation of the diffusion-weighting gradient dephases different fibre populations. In simulations we show that for a sufficiently high b-value (>~3 ms/μm2), the intra-axonal water will dominate leading to a simple relation between the myelin-induced frequency shift and the log g-ratio. This allows the difference in log g-ratio between crossing fibres to be measured and hence estimate the myelination of individual crossing tracts.

Introduction

While a wide variety of myelin-sensitive MRI contrasts1,2 have been proposed, these only give voxel-averaged estimates. Resolving this into myelin estimates for individual crossing fibres would allow more precise estimates of the myelination of individual tracts in health and disease3.
Here we propose to measure the myelin-induced frequency offset after diffusion weighting. Diffusion-weighting makes the MRI sequence sensitive to fibre orientations. However, due to its short T2, myelin water becomes invisible after diffusion weighting on most scanners1,2 This invalidates combining diffusion-weighting with those myelin MRI contrasts that rely on detecting the myelin water properties directly (i.e., its short T16,7, T2*8, T24,9, or magnetisation transfer10–12). However, the magnetic field shift induced by the magnetic susceptibility of myelin also affects the intra- and extra-axonal water13, which are still visible after diffusion weighting. We measure this field shift as a phase offset in an asymmetric-spin echo acquisition after diffusion-weighting.
Here we present and test with Monte Carlo simulations the novel sequence and an analysis method to extract the difference in myelination between crossing fibres.

Sequence

Up to the first echo planar imaging (EPI) readout, the proposed sequence is identical to a Stejskal-Tanner14 sequence (although planar diffusion encoding could also be used). After the first readout an extra refocusing pulse is added followed by a second EPI readout, which is delayed with a time $$$t_{\rm{}phase}$$$ from the second spin echo (Figure 1A). The phase difference between the first and second readout is determined by the off-resonance frequency, but unaffected by phase offsets caused by bulk motion during the diffusion weighting15,16.
Imaging the off-resonance frequency after diffusion-weighting has two distinct advantages:

  1. In a crossing fibre configuration, different gradient orientations can be used to distinguish between the myelin-induced off-resonance frequency for each fibre population.
  2. The diffusion weighting increases the relative contribution of intra-axonal water compared with extra-axonal water17 (with its less restricted diffusion). This simplifies the analysis, because the frequency offset due to myelin susceptibility ($$$f_{\rm{}myelin}$$$) has a flat profile within the axons (Figure 2B)13 and has a simple relation with the g-ratio (i.e., ratio of inner over outer axon diameter): $$\frac{f_{\rm{}myelin}}{f_{\rm{}Larmor}}=-\frac{3}{4}\chi_{\rm{}A}\langle{}\log{}g\rangle{}\sin^2\theta,\tag{1}$$ where $$$\theta$$$ is the angle between the axons and the main magnetic field orientation, $$$\chi_{\rm{}A}$$$ is the anisotropic susceptibility of myelin (-100 ppb13), and $$$\langle{}\log{}g\rangle$$$ is the average log g-ratio.

Simulations

We ran Monte Carlo simulations18 of crossing fibres in Camino19 (see geometry in Figure 2). The signal magnitude is attenuated by the diffusion-weighted gradients, T2 (80 ms20), and T2’ (50 ms21) dephasing (Figure 1B). Ideally, the phase would only be affected by the myelin-induced off-resonance frequency (Figure 1C), but in practice we also expect off-resonance frequency due to non-myelin sources ($$$f_{\rm{}other}$$$) and random phase offsets due to even small motion during the diffusion-weighting ($$$\phi_{{\rm{}DW}}(\hat{\nu})$$$)15,16 (Figure 1D). We minimize the echo time for each b-value assuming an EPI readout time of 50 ms and the maximum gradient strength (80 mT/m) and slew rate (200 T/m/s) of a 3T Siemens Prisma scanner.

Fitting the fibre myelination

For diffusion data with a single b-value, the complex signal at the first spin-echo readout ($$$S_{\rm{}SE}$$$) is fitted by:
$$S_{\rm{}SE}(b,\hat{\nu})=\sum_kA_{{\rm{}SE};k}e^{-b\hat{\nu}\mathbf{B}_k\hat{\nu}}e^{i\phi_{{\rm{}DW}}(\hat{\nu})}\tag{2},$$
where we estimate for each crossing fibre $$$k$$$ the signal amplitude ($$$A_{{\rm{}SE};k}$$$), the demeaned diffusion tensor ($$$\mathbf{B}_k=\mathbf{D}_k-{\rm{}MD}_k\mathbf{I}$$$)22, and the phase from movement during the diffusion-weighting ($$$\phi_{{\rm{}DW}}(\hat{\nu}$$$). Because this signal is at a spin-echo, there is no phase accumulation due to off-resonance.
Simultaneously, we fit the second asymmetric spin echo signal ($$$S_{\rm{}ASE}$$$) using:
$$S_{\rm{}ASE}(b,\hat{\nu})=\sum_kA_{{\rm{}ASE};k}e^{-b\hat{\nu}\mathbf{B}_k\hat{\nu}}e^{i[\phi_{{\rm{}DW}}(\hat{\nu})+2\pi{}f_kt_{\rm{}phase}]}\tag{3},$$
Both readouts are fitted with a common demeaned diffusion tensor ($$$\mathbf{B}_k$$$), but a different amplitude ($$$A_{{\rm{}SE};k}$$$ and $$$A_{{\rm{}ASE};k}$$$). The off-resonance frequency for each crossing fibre ($$$f_k$$$) is fitted to the phase offset between readouts (Figure 3).
For each fibre population the frequency ($$$f_k$$$) will have a contribution from myelin ($$$f_{{\rm{}myelin};k}$$$; eq. 1) and other susceptibility sources ($$$f_{\rm{}other}$$$). For interleaving fibres, non-myelin susceptibility sources should affect all fibre orientations equally, so we can compare this frequency between two crossing fibres (labeled $$$k$$$ and $$$l$$$) to get:
$$f_l-f_k=f_{{\rm{}myelin};l}-f_{{\rm{}myelin};k}=\frac{3}{4}\chi_{\rm{}A}f_{\rm{}Larmor}\left(\langle{}\log{}g\rangle_k\sin^2\theta_k-\langle{}\log{}g\rangle_l\sin^2\theta_l\right)\tag{4}$$
Estimates of the g-ratio per fibre orientation can be obtained by repeating the measure for different head orientations (i.e., varying $$$\theta$$$) or by adding an extra constraint by estimating the average g-ratio across the whole voxel through different means23,24.

Accuracy and precision

A biased estimate of the frequency difference was found in the simulations due to the extra-axonal water contribution at low b-values (Figure 4A) and phase wrapping errors at long $$$t_{\rm{}phase}$$$ (Figure 4B). When selecting a b-value and $$$t_{\rm{}phase}$$$, these inaccuracies have to be traded off with the low SNR at high b-values (Figure 4C) and the lower phase accumulation at low $$$t_{\rm{}phase}$$$ (Figure 4D).

Discussion

While the details of the simulation shown here depend on the assumed microstructural geometry, relaxation properties, and scanner properties, the general features of the proposed sequence illustrated should generalise more broadly. For fast readouts (i.e., $$$<t_{\rm{}phase}$$$) the sequence can be shortened by removing the second refocusing pulse.

For interleaving crossing fibres, the off-resonance frequency difference will be purely driven by the difference in the g-ratio between the fibre populations (modulated by angle with the main magnetic field), which allows this difference to be estimated. This represents a first step to not just estimating a voxel-averaged g-ratio, but a fibre-population specific g-ratio.

Acknowledgements

We thank Matt Hall, Matteo Bastiani, and Stamatios Sotiropoulos for helpful discussions.

References

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18. Hall, M. G. & Alexander, D. C. Convergence and parameter choice for Monte-Carlo simulations of diffusion MRI. IEEE Trans Med Imaging 28, 1354–64 (2009).

19. Cook, P. et al. Camino: open-source diffusion-MRI reconstruction and processing. in 14th scientific meeting of the international society for magnetic resonance in medicine vol. 2759 2759 (Seattle WA, USA, 2006).

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Figures

Figure 1: Proposed sequence to image the myelin-induced off-resonance frequency after diffusion weighting (A). Simulated signal magnitude (B), phase evolution due to myelin (C), and the full phase evolution (D) for myelinated and unmyelinated axons crossing at right angles (see geometry in Figure 2). If the water in the myelinated axons is dephased due to diffusion-weighted gradients (green), we only expect phase accumulation due to non-myelin sources (fother). If the water in unmyelinated axons is dephased (blue) the frequency will be altered by the myelin (fother + fmyelin).

Figure 2: Assumed geometry in the Monte Carlo simulations throughout this abstract. A) Myelinated axons cross at right angles with unmyelinated axons in interleaving planes perpendicular to the main magnetic field (orange). The axons are modelled as infinite cylinders (radius of 0.4 μm) spaced 1 μm apart. The frequency offset induced by myelin magnetic susceptibility is shown in a plane perpendicular to the myelinated fibres (B) assuming an anisotropic myelin susceptibility with isotropic component of χI = 100 ppb and anisotropic component of χA = -100 ppb13 for a 3T magnetic field.

Figure 3: Best fit (gray) to the Monte Carlo signal (black) without noise added (b = 3 ms/μm2; tphase = 40 ms). For illustration purposes we only consider gradients within the plane containing the crossing fibres (see sketch on upper right). The difference in phase accumulation between the two fibres populations ((f2-f1) tphase; eq. 4) remains the same whether we only consider the off-resonance frequency due to myelin magnetic susceptibility (middle right) or also consider other susceptibility sources and movement-induced phase offsets during the diffusion weighting (lower right).

Figure 4: Accuracy (top; ground truth in blue from eq. 4) and precision (bottom) of the estimated off-resonance frequency difference of myelinated axons crossing unmyelinated axons at right angles (and at right angle with main magnetic field) as a function of b-value (left for tphase = 40 ms) and tphase (right for b = 3 ms/μm2). Each violin plot represents fits to 300 noise iterations (complex Gaussian noise added at 1% of proton density signal) for data simulated for 50 gradient directions (uniformly distributed on whole sphere).

Proc. Intl. Soc. Mag. Reson. Med. 28 (2020)
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