Qiuyun Fan^{1,2}, Aapo A Nummenmaa^{1,2}, Qiyuan Tian^{1,2}, Ned A Ohringer^{1,2}, Thomas Witzel^{1,2}, Lawrence L Wald^{1,2,3}, Bruce R Rosen^{1,2,3}, and Susie Y Huang^{1,2,3}

Separating out the scalar and orientation-dependent components of the diffusion MRI signal offers the possibility of increasing sensitivity to microscopic tissue features unconfounded by the fiber orientation. Recent approaches to estimating apparent axon diameter in white matter have employed spherical averaging to avoid the confounding effects of fiber crossings and dispersion at the expense of losing sensitivity to effective compartment size. Here, we investigate the feasibility and benefits of incorporating higher-order spherical harmonic (SH) components into a rotationally invariant axon diameter estimation framework and demonstrate improved precision of axon diameter estimation in the in vivo human brain.

A
healthy subject was scanned on the 3T Connectome scanner with 300mT/m maximum
gradient strength using a custom-made 64-channel head coil^{11}.
Real-valued diffusion data was acquired to avoid buildup of the noise floor^{12}. Sagittal 2-mm isotropic
resolution diffusion-weighted spin-echo EPI images were acquired with whole
brain coverage. The following parameters were used: TR/TE=4000/77ms, δ=8ms, Δ=19/49ms, 8 diffusion gradient strengths
linearly spaced from 30-290mT/m per Δ,
32-64 diffusion directions, parallel imaging (R=2) and simultaneous multislice
(MB=2). Diffusion data were corrected for susceptibility and eddy current
distortions using the TOPUP^{13} and EDDY^{14,15}
tool in FSL.

The
normalized kernel $$$c_{l}$$$ was evaluated in the 3D parameter space X=(a, Dh, fr),
and the best fit to the model was found by searching on the grid by minimizing
the “energy” function^{1} (Figure 3), i.e., $$$\widetilde{x}=\arg_{x \in X} min\sum_j^T\sum_{l=-L}^L (ns_{l}-nc_{l} (x))^{2}$$$

Simulation
data was generated by adding 100 samples of noise at SNR=20. Voxel-wise fitting
for axon diameter a, restricted fraction f_{r}, and hindered diffusivity D_{h} was
performed.

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Figure 1. Illustration of the signal representation using spherical harmonics. The white matter signal
(S) can be modeled as a mixture of restricted (S_{r}) and hindered (S_{h}) diffusion signal, which can be represented by the
spherical harmonics (Y_{lm}) and corresponding coefficients (s_{lm}). When the fiber axis is aligned with
the primary axis of the coordinate system, s_{lm}=0 for all m≠0, so that the signal can be
fully represented by s_{l0}, l=0,2,4,… s_{00} and s_{20} as a
function of gradient strength are plotted for the restricted and hindered
compartments. Note that the L=0 component is identical to the spherical mean signal.

Figure 2. The response kernel in the
spherical harmonic space. The restricted diffusion response kernel (c_{r}) as a function of diameter is shown on
the left (a-d), and the hindered response kernel (c_{h}) as a function of hindered diffusivity
is shown on the right (e-h).

Figure 3. Exemplary “energy” function for
different spherical harmonic orders L. The “energy” function was evaluated for
the ground truth parameter {a, D_{h}}={6μm,10^{-9} m^{2}/s} (marked with the white cross) and its
neighborhood in the parameter space X = {a, D_{h}}. The global minimum of the
“energy” appears at the ground truth value for both L=0 and L=2
components, each of which shows a slightly different pattern. By combining the
two components (L0+L2), the “low-energy” region shrinks, indicating that
incorporating complementary contrasts originating from different orders is
helpful for finding the global minimum.

Figure 4. Simulation results of estimated
axon diameters and restricted volume fractions. The *in vivo* imaging protocol
was used to generate the noise-free data (keeping f_{r}=0.7 for the left figure and keeping diameter=6 μm for the right), and 100 samples of noise were added
with an SNR=20. The mean value for the 100 trials is
plotted as solid lines, and the shading bounded by dotted lines denotes the
standard deviation. By combining L=0 and L=2 components (in green), the
standard deviation in axon diameter estimates decreases compared to using L=0
components alone (in orange), especially in the small diameter regime.

Figure 5. Estimated apparent axon
diameter map of *in vivo* human data. A sagittal slice through the right
corticospinal tract was shown for the map calculated using L=0 components only
(left) and that obtained by combining L=0 and L=2 components (right). Overall,
the two maps show similar patterns, but there is a subtle improvement in
contrast between the larger apparent axon diameter in the corticospinal tract
and surrounding white matter on the L0+L2 map, indicating that including
higher-order spherical harmonics may provide better sensitivity to axon
diameter compared to the spherical mean approach.