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Incorporating mesoscopic orientation dependent R₂ from magnetic susceptibility into the Standard Model of Diffusion in White Matter

Anders Dyhr Sandgaard¹, Andrada Ianus², Noam Shemesh², Valerij G. Kiselev³, and Sune Nørhøj Jespersen^1,4
¹Center of Functionally Integrative Neuroscience, Department of Clinical Medicine, Aarhus University, Aarhus, Denmark, ²Champalimaud Research,Champalimaud Centre for the Unknown, Lisbon, Portugal, ³Division of Medical Physics, Department of Radiology, University Medical Center Freiburg, Freiburg, Germany, ⁴Department of Physics and Astronomy, Aarhus University, Aarhus, Denmark

Synopsis

Keywords: Microstructure, Microstructure

Motivation: $$$R_2$$$ in WM is orientation dependent due to microscopic magnetic anisotropy. So far, the Standard Model of diffusion (SM) has been extended to only include isotropic $$$R_2$$$ (TEdDI).

Goal(s): Our goal is to characterize $$$R_2$$$ anisotropy of a multi-echo dMRI signal for long diffusion times and incorporate $$$R_2$$$ anisotropy into TEdDI (STEdDI).

Approach: We simulate $$$R_2$$$ of PGSE signal in magnetized cylinders using Monte-Carlo, and fit TEdDI and STEdDI on ex vivo mouse multi-echo dMRI data acquired at 16.4T.

Results: $$$R_2$$$ anisotropy outside axons are non-axially-symmetric, depends on B0 direction, gradient direction and b-value. Residuals are significantly lower with STEdDI in dMRI data.

Impact: Interplay between microscopic magnetic fields and diffusion weighting affects $$$R_2$$$ in extra-axonal space. Incorporating $$$R_2$$$ anisotropy$$$\,$$$into modeling lowered the residuals and may allow rotation-free estimation of $$$R_2$$$ anisotropy, which could be useful to gain a deeper insight into brain microstructure.

Introduction

Estimating all the parameters of the Standard Model (SM) of white matter (WM) is ill-posed^1–4. It has been shown that incorporating isotropic transverse relaxation ($$$R_2$$$) to the SM^5–7 signal kernel helps estimating compartmental relaxation rates but with little improvement on SM parameters. In addition, several studies^5,8–11 have found $$$R_2\,$$$to be anisotropic presumably due to mesoscopic susceptibility effects, and when applying diffusion gradients, mesoscopic susceptibility effects can bias the apparent diffusivity from extra-axonal water^{5,8,9,12–18}. Our main question here is: How is apparent R2 affected by diffusion weighting and magnetic susceptibility in WM, and can we incorporate it into SM? We investigate $$$R_2\,$$$anisotropy of a multi-echo PGSE signal with Monte-Carlo (MC) simulations using a digital phantom made of parallel cylinders. We acquire multi-echo dMRI data of an ex-vivo mouse brain and fit our proposed model STEdDI – a minimal extension of the SM model TEdDI⁶ where S denotes adding susceptibility-related orientation dependent extra-axonal $$$R_2$$$. The effect of $$$R_2\,$$$anisotropy in the SM kernel was previously analyzed by Tax et. al⁵, but to our knowledge, fitting the parameters with R2 anisotropy added to the SM kernel has not been pursued. We also fit TEdDI and compare their Bayesian Information Criteria (BIC)¹⁹.

Methods

Theory
We propose STEdDI, a minimal extension of TEdDI, which models $$$R_2\,$$$for extra-axonal water as $$$R_{2e}(\theta_\mathrm{\hat{B}})=A+B\sin^4(\theta_\mathrm{\hat{B}})$$$, valid for long diffusion times. Here $$$\theta_\mathrm{\hat{B}}\,$$$is the angle between cylinder axis $$$\mathrm{\hat{n}}\,$$$and the scanner $$$\mathrm{{B}_0}=B_0\mathrm{\hat{B}}$$$. This $$$R_{2e}\,$$$anisotropy stems from the microscopic fields generated outside magnetized cylinders (axons) – irrespective of susceptibility anisotropy^20,21. An overview of STEdDI is shown in figure 1A.

Monte-Carlo simulations
We performed MC random walk simulations to investigate the apparent extra-axonal $$$R_2$$$ originating from the interplay between diffusion gradients $$$\mathrm{{g}}$$$ and the induced field $$$\mathrm{\Delta B}$$$, for a bundle of randomly packed parallel cylinders. The details are listed in figure 1B.

Ex vivo mouse brain
We acquired multi-echo dMRI data of a mouse brain to fit STEdDI and TEdDI (see acquisition details in figure 5 caption). We fitted the signal from three manually segmented ROIs of Corpus Callosum (see figure 5). We then compared the models by computing their difference in BIC $$$(\mathrm{\Delta BIC})$$$¹⁸.

Results

MC simulations
Figure 2-3 shows the apparent $$$R_2\,$$$from the whole extra-axonal signal^21-23 and the contribution from the n’th cumulants $$$\langle\varphi^n\rangle\equiv \langle(\varphi_\mathrm{\hat{B}}+\varphi_\mathrm{\hat{g}})^n\rangle_c - \langle\varphi^n_\mathrm{\hat{B}}\rangle_c-\langle\varphi^n_\mathrm{\hat{g}}\rangle_c$$$ due to $$$\mathrm{{g}}$$$ and $$$\mathrm{\Delta B}$$$. It is clear that the phase from diffusion gradients $$$\varphi_\mathrm{\hat{g}}$$$ interacts with the phase from internal fields $$$\varphi_\mathrm{\hat{B}}$$$, making $$$R_2\,$$$deviate from $$$\sin^4(\theta_\mathrm{\hat{B}})$$$. The 6th cumulant cross-term $$$\langle\varphi^6\rangle\,$$$produced the highest relaxation, but as the internal field variance $$$\delta\Omega=\gamma B_0\sqrt{\mathrm{\hat{B}}^\mathrm{T}\cdot\mathrm{\Delta{B}}}\,$$$increased, its contribution became negligible compared to $$$\langle\varphi^2_\mathrm{\hat{B}}\rangle_c$$$. Figure 4 shows the spins' signal contribution versus root-mean-square displacement. The total signal relaxation vs. $$$T_E\,$$$is also weighted differently depending on $$$\mathrm{{g}}$$$, which in turn modulated the signal in a $$$\mathrm{\Delta B}$$$-dependent way. The non-axially symmetric contribution comes from spins near the cylinders (see green and yellow curve in figure 3 and figure 4), where $$$\mathrm{g}\,$$$and $$$\mathrm{\Delta B}\,$$$can interfere constructively or destructively, respectively.

Ex vivo mouse brain
Figure 5 shows the estimated parameters in the CC-ROIs by fitting TEDdi and STEdDI, respectively. Adding $$$R_2\,$$$anisotropy had a substantial impact on parameter values. We found $$$(\mathrm{\Delta BIC})<-6\,$$$in all ROIs thus favoring STEdDI over TEdDI²⁵.

Discussion

Our simulations demonstrate how $$$R_2\,$$$of a multi-echo PGSE signal depend on the interaction between the induced magnetic field from cylinders and diffusion gradients in the limit of long diffusion times, especially when the applied gradients were strong compared to $$$B_0$$$. These correlations may therefore be important at 3T and/or high gradient scanners like the Connectome. We plan to investigate this analytically in the future. Here we demonstrate that it may be feasible to include $$$R_2\,$$$anisotropy in TEdDI and estimate it without having to rotate the sample. This is because orientation dispersion also makes $$$R_2\,$$$vary depending on the applied diffusion gradient²⁶. We find $$$R_2\,$$$anisotropy to be in agreement with Tax et al⁵ when scaling by the field-strength ratio$$$\,{(16.4\text{T}/3\text{T})^2}$$$. Planar encoding^27,28 may increase the sensitivity to $$$R_2\,$$$anisotropy, as it allows one to probe the signal from axons along a single axis. However, planar encoding comes at the expense of lower SNR²⁹, and potentially stronger correlations with internal magnetic fields. This will also be investigated in the future.

Conclusion

The apparent$$$\,R_2\,$$$of the PGSE signal from water outside long magnetized cylinders is anisotropic and depends on experimental settings like diffusion weighting and $$$B_0$$$. Incorporating $$$R_2\,$$$anisotropy into the Standard Model of diffusion in white mater lowered the fit residuals without adding too much model complexity and was estimated in an ex-vivo mouse brain without acquiring data from multiple sample orientations.

Acknowledgements

The study is funded by the Independent Research Fund Denmark (Grant 10.46540/3103-00144B).

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Figures

Figure 1 – Overview of STEdDI model (A) and Monte-Carlo simulation (B): STEdDI model is similar to TEdDI⁶ but includes anisotropic R₂ relaxation in extra-axonal space. Monte-Carlo simulation is performed outside parallel cylinders as described in B. The signal S(T_E=60ms)is used for further analysis.

Figure 2 – Apparent relaxation rate R₂ of extra-axonal multi-echo PGSE signal. 1st column shows R₂=Δ_TEln(S)=(ln(S(T_E1))-ln(S(T_E2)))/(T_E2 -T_E1) from the whole signal and the next only cross terms of the 2nd-6th cumulant (see result). A shows R₂ for varying sin⁴(θ_B) while colors represent different combination (b,g) for increasing b. The black dashed line shows the second cumulant Δ_TE<φ²_B>_c only from internal fields.

Figure 3 – Apparent relaxation rate R₂ of extra-axonal multi-echo PGSE signal. 1st column shows R₂=Δ_TEln(S)=(ln(S(T_E1))-ln(S(T_E2)))/(T_E2 -T_E1) from the whole signal and the next only cross terms of the 2nd-6th cumulant (see methods). B varies sin(θ_g) with B·n=0. g is varied by rotating it either in the same plane as B₀ (dashed) or in the perpendicular plane (solid). Rows indicate various fields strength via δΩ²=<Ω²(r)>.

Figure 4 - Signal contribution of a spin as a function of its root-mean-square displacement at the echo T_E=80ms: Colors represent different b-values, and line with the same color different gradient directions. Multiple peaks arise since spin diffuse over a total period of 80ms while diffusion weighting is done over a period of Δ=50ms. The yellow and green lines indicate b=5ms/µm² but at gradient directions along y (parallel to B₀) and x (perpendicular to B₀), respectively.

Figure 5 – TEdDi and STEdDI fitting of WM ROIs in ex vivo mouse brain. Acquisition details: Bruker Aeon 16.4T equipped with micro5 coils. 3D EPI with linear diffusion encoding along 30 different direction per b-value, where b=0,1,2,…,8 ms/µm² and diffusion times Δ/δ=15/6 ms. Echo times were T_E=35,41,...,65 ms. Resolution was 100 µm in plane with 500 µm thickness. Raw complex signals were denoised29,30 prior to EPI reconstruction and Gibbs unrung³¹. DTI fitting was done using b up to 1ms/µm². All three ROIs have WM roughly perpendicular to B₀.

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

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DOI: https://doi.org/10.58530/2024/3462