Hunter G. Moss^{1,2}, Emilie T. McKinnon^{1,2,3}, Joseph A. Helpern^{1,2}, and Jens H. Jensen^{1,2}

Fiber ball imaging (FBI) is a recently proposed diffusion MRI (dMRI) method for estimating fiber orientation density functions together with specific microstructural parameters in white matter. The theory underlying FBI predicts the b-value dependence for the dMRI harmonic power of any given degree as long as the b-value is sufficiently large. Good agreement between theory and experiment has been previously demonstrated for the zero-degree harmonic power. Here the predicted functional forms for higher degree harmonics are shown to also agree well with experimental measurements, providing additional support for the validity of FBI.

The spherical harmonic expansion for the dMRI signal can be written as $$S(b,{\bf n})=S_{0}\sum_{l=0}^{\infty}\sum_{m=-l}^{l}a_{l}^{m}Y_{l}^{m}(\theta,\phi),\tag{1}$$ where $$$S_{0}$$$ is the signal for the $$$b=0$$$, $$$Y_{l}^{m}$$$ are the spherical harmonics, $$$(\theta,\phi)$$$ are the spherical angles for $$${\bf n}$$$, and $$$a_{l}^{m}$$$ are the expansion coefficients of degree $$$l$$$ and order $$$m$$$. The harmonic power for each degree may then be defined as

$$p_{l}\equiv\frac{1}{2l+1}\sum_{m=-l}^{l}|a_{l}^{m}|^{2}.\tag{2}$$

All odd degree harmonics vanish by reflection symmetry, $$$S(b,-{\bf n})=S(b,{\bf n})$$$, implying $$$p_{l}>0$$$ for even degree harmonics only. Neglecting extra-axonal contributions, the theory underlying FBI predicts^{1}

$$p_{l}\approx\frac{u_{l}}{b}\left[g_{l}(bD_{a})\right]^{2},\tag{3}$$

for even degrees when $$$b\geq4000$$$ s/mm^{2} where $$$u_{l}$$$ is a scaling factor independent of $$$b$$$, and

$$g_{l}(x)=\frac{\left(\frac{l}{2}\right)!x^{\frac{l+1}{2}}}{\Gamma\left(l+\frac{3}{2}\right)} {_1}F_{1}\left(\frac{l+1}{2};l+\frac{3}{2};-x\right), \text{ for } l=0,\space2,\space4,\space 6,...\tag{4}$$

with $$${_1}F_{1}$$$ being the confluent hypergeometric function and $$$\Gamma$$$ indicating the gamma function. Equation (3) implies $$$p_{0}\approx u_{0}/b$$$ for large $$$b$$$, which is equivalent to the observed $$$b^{-\frac{1}{2}}$$$ decay of the direction-averaged signal. However, for higher even degrees, Equation (3) predicts that the harmonic power has a peak for finite $$$b$$$-values. Thus, one finds

$$b_{l}=\frac{\nu_{l}}{D_{a}},\tag{5}$$

where $$$b_{l}$$$* *is the $$$b$$$-value for the harmonic power peak of degree $$$l$$$ and $$$\nu_{l}$$$ is a constant. The first few values of $$$\nu_{l}$$$ are found numerically to be $$$\nu_{2}\approx3.969$$$, $$$\nu_{4}\approx11.040$$$, $$$\nu_{6}\approx22.023$$$, and $$$\nu_{8}\approx37.014$$$.

For one adult volunteer, HARDI data were gathered on a
Siemens 3T Prisma with* *$$$b$$$-values ranging from 1000
to 10,000 s/mm^{2 }with 64
diffusion-encoding directions and 11 $$$b=0$$$ s/mm^{2} per $$$b$$$-value shell. Other acquisition parameters were TE=110ms, TR=4400ms, FOV=222 x222mm^{2}, and voxel size = (3mm)^{3}. ^{ }

Image processing included denoising,^{5} Gibbs ringing removal,^{6} and Rician noise bias correction.^{7} In-house MATLAB scripts were used to calculate the spherical harmonic expansion of the dMRI signal up to $$$l=6$$$, as well as the corresponding $$$p_{l}$$$* _{ }*for each harmonic using Equation (2). Additionally, several diffusion and microstructural parameters were estimated from the data by applying diffusional kurtosis imaging and the fiber ball WM (FBWM) model.

A quadratic fit to $$$p_{4}$$$ at $$$b=3000,4000\text{ and }5000$$$ s/mm^{2} estimated $$$b_{4}$$$ which was used in Equation (5) to calculate $$$D_{a}$$$. This value for $$$D_{a}$$$ was fixed in Equation (3) and the harmonic power of each degree for $$$b=0-10,000$$$ s/mm^{2} was calculated. Normalization was done for each $$$p_{l}$$$ by $$$p_{2}$$$ at $$$b=1000$$$ s/mm^{2 }$$$(p_{2}^{*})$$$ and the scale factor $$$u_{l}$$$ was set so that theory and experiment coincided at $$$b=6000$$$ s/mm^{2}. Thus, for the full dataset, only four adjustable parameters were used.

1. Jensen JH, Russell Glenn G, Helpern JA. Fiber ball imaging. Neuroimage. 2016;124(Pt A):824-833.

2. Tuch DS, Reese TG, Wiegell MR, Makris N, Belliveau JW, Wedeen VJ. High angular resolution diffusion imaging reveals intravoxel white matter fiber heterogeneity. Magn Reson Med. 2002;48(4):577-582.

3. McKinnon ET, Jensen JH, Glenn GR, Helpern JA. Dependence on b-value of the direction-averaged diffusion-weighted imaging signal in brain. Magn Reson Imaging. 2017;36:121-127.

4. Veraart J, Fieremans E, Novikov DS. On the scaling behavior of water diffusion in human brain white matter. Neuroimage. 2018;185:379-387.

5. Veraart J, Novikov DS, Christiaens D, Ades-Aron B, Sijbers J, Fieremans E. Denoising of diffusion MRI using random matrix theory. Neuroimage. 2016;142:394-406.

6. Kellner E, Dhital B, Kiselev VG, Reisert M. Gibbs-ringing artifact removal based on local subvoxel-shifts. Magn Reson Med. 2016;76(5):1574-1581.

7. Gudbjartsson H, Patz S. The Rician distribution of noisy MRI data. Magn Reson Med. 1995;34(6):910-914.

8. Jensen JH, Helpern JA. MRI quantification of non-Gaussian water diffusion by kurtosis analysis. NMR Biomed. 2010;23(7):698-710.

9. McKinnon ET, Helpern JA, Jensen JH. Modeling white matter microstructure with fiber ball imaging. Neuroimage. 2018;176:11-21.

10. Yang AW, Jensen JH, Hu CC, Tabesh A, Falangola MF, Helpern JA. Effect of cerebral spinal fluid suppression for diffusional kurtosis imaging. J Magn Reson Imaging. 2013;37(2):365-371.

11. Kaden E, Kelm ND, Carson RP, Does MD, Alexander DC. Multi-compartment microscopic diffusion imaging. Neuroimage. 2016;139:346-359.

12. Veraart J, Novikov DS, Fieremans E. TE dependent Diffusion Imaging (TEdDI) distinguishes between compartmental T2 relaxation times. Neuroimage. 2018;182:360-369.

13. Dhital B, Reisert M, Kellner E, Kiselev VG. Intra-axonal diffusivity in brain white matter arXiv preprint arXiv:1712.04565

The harmonic power, $$$p_{l}$$$, as
a function of the $$$b$$$-value, $$$b$$$,
for $$$l=2,4,\text{ and }6$$$. The data points are
experimental measurements, and the solid lines are theoretical predictions. A
total of only four adjustable parameters were used to fit the theory for the
full set of experimental data. Good agreement is found for all three degrees
shown. That the experimental values for $$$p_{2}$$$ exceed the theoretical predications for $$$b\leq3000$$$ s/mm^{2} is likely due to the
contribution from extra-axonal water, which is not included in the theory.

$$$D_{a}$$$ as calculated from the $$$p_{4}$$$ peak (solid circles) and FBWM (open squares) in 10 slices. FBWM $$$D_{a}$$$ values are averages (±SD) over all slice WM voxels whereas the corresponding $$$p_{4}$$$ $$$D_{a}$$$ is calculated from the peak of the
average $$$p_{4}$$$ for the same WM voxels. SD's for $$$p_{4}$$$ $$$D_{a}$$$ are not given, since in voxels with fiber crossings $$$p_{4}$$$ is small, making voxelwise $$$D_{a}$$$ estimates unreliable. While the methods yield similar results, $$$D_{a}$$$ from $$$p_{4}$$$ is higher. As a visual aid, a coronal view shows a representative volume region (dashed red box) and slice being averaged over
(blue line).