Anders Dyhr Sandgaard^{1}, Valerij G. Kiselev^{2}, Noam Shemesh^{3}, and Sune Nørhøj Jespersen^{1,4}

^{1}Department of Clinical Medicine, Center for Functionally Integrative Neuroscience, Aarhus University, Aarhus, Denmark, ^{2}Medical Physics, Department of Radiology, Faculty of Medicine, University of Freiburg, Freiburg, Germany, ^{3}Champalimaud Research, Champalimaud Centre for the Unknown, Lisbon, Portugal, ^{4}Department of Physics and Astronomy, Aarhus University, Aarhus, Denmark

Mapping tissue magnetic properties with MRI may improve the diagnosis of diseases and enhance our understanding of their basic mechanisms. However, MRI signals are sensitive to both structure and magnetic susceptibility, the so-called “magnetic microstructure”, rendering accurate susceptibility estimation a great challenge. Here, we present an analytical expression for the Larmor frequency of a white matter model of myelinated axons with axially symmetric microscopic susceptibility anisotropy and orientation dispersion. The modelled axons are surrounded by microscopic spherical inclusions with a scalar susceptibility. This goes beyond previous models of white matter magnetic microstructure.

Figure 1 conceptualizes the magnetic microstructure. We model it as a porous medium embedded in an NMR-visible fluid

$$\chi^C=\begin{pmatrix}\chi_\parallel&0&0\\0&\chi_\perp&0\\0&0&\chi_\perp\end{pmatrix}\qquad{(1)}$$

in the lipid eigen-frame ($$$\chi_\parallel$$$ is in the radial direction), and $$$\Delta\chi=\chi_\parallel-\chi_\perp$$$ defines the microscopic susceptibility anisotropy. Cylinders occupy the fraction $$$\zeta^C$$$ of the total volume $$$V$$$. Randomly positioned microscopic spherical inclusions with scalar susceptibility $$$\chi^S$$$ (Figure 1C) reside outside cylinders according to $$$\nu^S(\mathrm{r})$$$ and occupy a volume fraction $$$\zeta^S$$$.

When exposing the sample to an external magnetic field $$$\mathbf{B}_0=\mathrm{B}_0\hat{\mathbf{B}}$$$, inclusions become magnetized and induce a magnetic field, perturbing the Larmor frequency of the measured FID signal $$$S(t)$$$. In the fast diffusion regime and in the limit of small signal phase, this perturbation, $$$\Omega=\mathrm{arg}\{S(t)\}/t$$$, becomes

$$\Omega=\gamma{\mathrm{B}_{0}}{{\mathbf{\hat{B}}}^{\text{T}}}\frac{1}{\left(1-\zeta^S-\zeta^C\right)V}\int_Vd\mathbf{r}\left(1-\nu\left(\mathbf{r}\right)\right)\int_{V}{d\mathbf{{r}'}}\mathbf{\Upsilon}\left(\mathbf{r-{r}'}\right)\nu \left({\mathbf{{r}'}}\right)\mathbf{\chi}\left({\mathbf{{r}'}}\right)\mathbf{\hat{B}}.\qquad{(2)}$$

$$$\mathbf{\Upsilon}$$$ is the Lorentz-corrected dipole field

We find $$$\Omega^\mathrm{Macro}$$$ depends on coarsed magnetic microstructure and overall sample shape

$$\Omega^\mathrm{Macro}=\gamma{\mathrm{B}_{0}}{{\mathbf{\hat{B}}}^{\text{T}}}\mathbf{\Upsilon}^{\mathrm{Macro}}\left(\zeta^C\chi_\perp+\zeta^S\chi^S+\zeta^C\Delta\chi\left(\sum_{m=-2}^2p^{2m}\mathbf{\mathcal{Y}}^{2m}-1\right)\right)\mathbf{\hat{B}},\qquad{(4)}$$

where $$\mathbf{\Upsilon}^{\mathrm{Macro}}=\frac{1}{V}\int_Vd\mathbf{r}\int_Vd\mathbf{r}'\mathbf{\Upsilon}(\mathbf{r}-\mathbf{r}')\qquad{(5)}$$

denotes the sample-averaged dipole tensor, $$$p^{2m}$$$ are $$$l=2$$$ Laplace expansion coefficients of the fiber orientation distribution (fODF), $$$\mathcal{Y}^{2m}$$$ is the $$$l=2$$$ symmetric trace-free tensor representation of SO(3)

$$\Omega^\mathrm{Meso}=-\gamma{\mathrm{B}_{0}}\left(\left(\zeta^C\chi_\perp-\tilde{\zeta}^S\chi^S\right)\sum_{m=-2}^2p^{2m}Y_2^m(\mathbf{\hat{B}})+\Delta\chi\frac{1}{3}\left(\frac{1}{6}\zeta^C-\tilde{\zeta}^C\right)\left(p^{2m}Y_2^m(\mathbf{\hat{B}})-1\right)

\right).\qquad{(6)}$$

$$$\tilde{\zeta}^C$$$ describes intra-cylindrical structure

$$\tilde{\zeta}^C=\frac{1}{1-\zeta^S-\zeta^C}\left(\sum_{m>w}\left(\zeta^C_{R_m}-\zeta^C_{r_m}\right)\mathrm{ln}\left(\frac{R_w}{r_w}\right)-\sum_w\mathrm{ln}\left(\frac{R_w}{r_w}\right)\right).\qquad{(7)}$$

$$$R_m$$$ denotes the radius of the outer layers of the mth shell with volume fraction $$$\zeta^C_{R_m}$$$, while $$$r_m$$$ denotes inner radii of each shell. $$$\tilde{\zeta}^S$$$ is an effective spherical weight defined by

$$\tilde{\zeta}^S=\zeta^S\frac{\zeta^C+\zeta^S\frac{\zeta^{EC}}{1-\zeta^{EC}}}{1-\zeta^S-\zeta^C},\qquad{(8)}$$

where $$$\zeta^{EC}$$$ denotes extra-cylindrical volume fraction.

We performed computer simulations to validate $$$\Omega^\mathrm{Meso}$$$ (Eq. (6)). $$$\Omega^\mathrm{Meso}$$$ was validated previously

We considered the predicted behavior of $$$\Omega^\mathrm{Meso}$$$ for cylinders with axially symmetric orientation dispersion surrounded by spherical inclusions (a full description can be found in Figure 4). We varied 1) Orientation dispersion from parallel to uniformly oriented; 2) susceptibility anisotropy; and 3) sphere susceptibility.

Our model is a simplification of real WM microstructure, where axons exhibit axonal caliber-variation, undulation

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Conceptualization
of a sample with a given microstructure, much smaller than the sample size. The
measured Larmor frequency of the sample, assuming fast diffusion in the low
signal phase limit, is described by two contributions. $$$\Omega^\mathrm{Meso}$$$ accounts for the Larmor frequency within a mesoscopic sphere^{8,17} with explicit microstructure. $$$\Omega^\mathrm{Macro}$$$ describes the Larmor frequency at distances longer than the
size of the mesoscopic sphere. It depends on coarse grained magnetic susceptibility,
and dipole field averaged over the sample.

Model
validation of $$$\Omega^\mathrm{Meso}$$$ for a cylinder surrounded by spheres. Simulations are
made to calculate the ground truth field by discretizing the microstructure on
an 800^{3} grid. In **A**, the cylinder consists of two shells, and one
shell in **B**. The colorbar encodes different volume fractions of spheres outside the
cylinder. The sphere radius is varied as $$$R^S=8,13,19,25,30$$$ grid units for each fraction. Eq.(6) agrees with the ground truth.

Predicted $$$\Omega^\mathrm{Meso}$$$ as function of B-field orientation for cylinders with axially
symmetric orientation dispersion consisting of 1 shell. The frequency is linear with $$$\sin(\theta)^2$$$, where
$$$\theta$$$ is angle between mean cylinder axis and external field. **A**: dispersion dampens variations while focal point remains at the magic angle, where the orientation dependent contribution is zero. **B:** anisotropy scales and shifts the zero-crossing. **C: **spheres scales the frequency. The solid black line (-)
indicates identical settings across all panels.

DOI: https://doi.org/10.58530/2022/2458