Choong Heon Lee^{1}, Piotr Walczak^{2}, Lindsay K. Hill^{1}, Youssef Zaim Wadghiri^{1}, and Jiangyang Zhang^{1}

Microstructures in biological tissues can produce susceptibility related microscopic background gradient (μBG). Mapping the spatial distributions of μBG can help us infer tissue microstructure. In this study, we reported an improved diffusion MRI based method to detect μBG and evaluated the method using a phantom as well as normal and dysmyelinated mouse brains. We found that 3D spatial variations of μBG were greater in white matter tracts than in gray matter structures. The contrast based on μBG spatial variations was able to distinguish white matter tracts in normal and dysmyelinated mouse brains. The method may be used to study myelin injury.

Previous reports
(2,7) showed that diffusion encoding using the BGP waveform (Fig. 1) is not
affected by background gradient, whereas the BGPR
waveform is sensitive to background gradient. We chose to
compare the BGP and BGPR (BGP/R) measurements because the two waveforms only
differ in gradient polarity after the refocusing pulse, which minimizes
potential biases from timing differences and eddy currents. The signal
attenuations equations for BGP/R are: $$$\frac{S_{BGPR}}{S_{0}}=e^{-D( C_{ss}\cdot{\bf G_s^2}+C_{ll}\cdot{\bf G_l^2}+C_{sl}\cdot{\bf G_s}\cdot{\bf G_l})}$$$ and $$$\frac{S_{BGPR}}{S_{0}}=e^{-D( C_{ss}\cdot{\bf G_s^2}+C_{ll}\cdot{\bf G_l^2})}$$$, where S_{BGPR}, S_{BGP}, and S_{0} are the diffusion-weighted and non-diffusion-weighted signals measured. **G _{s}** and

In a pixel with a spatial varying μBG, the ensemble average can be approximated as

$$\frac{S_{BGPR}}{S_{BGP}}={<e^{-D( C_{sl}\cdot{\bf G_s}\cdot{\bf G_l})}>} \approx {e^{-D( C_{sl}\cdot{\bf G_s}\cdot<{\bf G_l}>)}}\cdot{e^{-\frac{D^2\cdot{C_{sl}^2}\cdot{\bf G_{s}^2}\cdot{\sigma^2}}{2}}}$$

where < **G _{l}**>
is the ensemble average of

$$\ln(\frac{S_{BGPR}^+}{S_{BGP}^+})-\ln(\frac{S_{BGPR}^-}{S_{BGP}^-})=-2D{C_{sl}}\dot{\bf G_{s}}\dot{<\bf G_{l}>}$$

$$\ln(\frac{S_{BGPR}^+}{S_{BGP}^+})+\ln(\frac{S_{BGPR}^-}{S_{BGP}^-})=-{D^2{C_{sl}^2}\dot{\bf G_{s}^2}\dot{\bf \sigma^2}}$$

We then fitted the estimated σ^{2} along multiple directions (n≥6) to a tensor model to resolve the directions of maximum and minimum μBG variations.

Fig.
2 shows BGP/R signal attenuations from the phantom. In areas near the
air-filled sample tube, where we expected macroscopic background gradient with small local variations, the +/- BGPR measurements
were symmetric to the BGP measurements (Fig. 2A). From the gel within the tube, where we expected
minimal macroscopic background gradient but strong local variations caused by superparamagnetic particles, the average of +/- BGPR measurements were no longer
symmetric to the BGP measurements, suggesting that μBG variance ( σ^{2}) was detectable (Fig. 2B), as suggested by (4).

Fig. 3A shows
T_{2}-weighted, fractional anisotropy (FA), apparent diffusion
coefficient (ADC) from a *Shiverer* mouse brain, which showed reduced white matter contrasts due to dysmyelination compared to the wild-type (WT) brains. In the μBG variance (σ^{2}) maps, white matter structures in the WT mouse brains had significantly higher σ^{2} than the WT cortex and corresponding structures in the *Shiverer* mouse brain (Table 1). Variance maps along the x (left-right), y (dorsal-ventral), and z (rostral-caudal) axes (Fig. 3B) also showed significantly larger σ^{2} in the WT mouse brain than in the *Shiverer* mouse brain along all three axes. In the corpus callosum of the WT mouse brain, the
variances along the y and z axes (perpendicular to the axons) were higher than the variance along the
x (left-right) axis (parallel to the axons). The anisotropy in μBG variance likely reflected the organization of myelin sheath surrounding the axons.

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