Synopsis

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.

Introduction

Theory

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 predicts¹

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

for even degrees when $b\geq4000$ s/mm² 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 $\nu_{8}\approx37.014$.

Method

For one adult volunteer, HARDI data were gathered on a Siemens 3T Prisma with $b$-values ranging from 1000 to 10,000 s/mm² with 64 diffusion-encoding directions and 11 $b=0$ s/mm² per $b$-value shell. Other acquisition parameters were TE=110ms, TR=4400ms, FOV=222 x222mm², 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.^8,9 In a WM mask defined as all brain voxels with mean kurtosis>1.0 and mean diffusivity<1.5 μm²/ms,¹⁰ excluding the cerebellum, the harmonic powers were either averaged over all voxels within the mask or within an individual anatomical slice of the mask.

A quadratic fit to $p_{4}$ at $b=3000,4000\text{ and }5000$ s/mm² 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/mm² was calculated. Normalization was done for each $p_{l}$ by $p_{2}$ at $b=1000$ s/mm² $(p_{2}^{*})$ and the scale factor $u_{l}$ was set so that theory and experiment coincided at $b=6000$ s/mm². Thus, for the full dataset, only four adjustable parameters were used.

Results

Discussion

Acknowledgements

References

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