Michiel Cottaar^{1}, Filip Szczepankiewicz^{2,3}, Matteo Bastiani^{1}, Stamatios N. Sotiropoulos^{1,4}, Markus Nilsson^{2}, and Saad Jbabdi^{1}

Accurate measures of fibre dispersion might improve tractography through voxels with bending and/or fanning configurations and also provide more accurate tissue microstructure indices. When estimated from regular diffusion MRI, however, the accuracy of the dispersion estimate depends on the accuracy of the underlying microstructural model. In this work, we find that encoding water diffusion using multiple shapes of the b-tensor leads to an accurate measurement of fibre dispersion within a single b-shell. We show that this holds for complicated sub-voxel geometries using both simulations and in vivo data.

The diffusion MRI (dMRI) signal can be modelled
as the convolution of a fibre orientation distribution function (fODF) with a fibre
response function. Therefore, measuring the fODF width (fibre dispersion) requires
knowing the response function (diffusion anisotropy). This is usually done by
either estimating it from the most anisotropic voxels^{1} or by assuming an explicit microstructural model^{2-4}, whereby if the assumptions break down, estimates of fibre dispersion
can be biased (Figure 1B) due to model degeneracy^{5,}^{6}.

These models generally only consider data acquired using the
Stejskal-Tanner sequence, which is sensitive to diffusion along a single
direction (linear tensor encoding). By varying the gradient waveform more
generally one can create sequences sensitive to diffusion in a plane (planar
encoding) or to diffusion in all directions (spherical encoding)^{7,8}. Combining these sequences in
a single b-shell allows the measurement of anisotropy experienced by water at
the microscopic scale independently of the fODF (micro-anisotropy)^{8,9}. Because micro-anisotropy
determines the width of the response function, this combination of sequences
should allow us to measure fibre dispersion accurately using a single b-shell (Figure 1C)^{11} Here we show that this is
indeed the case even for voxels with complicated microstructure.

The linear, planar and spherical tensor encoding can be
characterized by a b-tensor^{12,13}:

$$\mathbf{B}_{\rm linear}=b\hat{\mathbf{g}}\cdot\hat{\mathbf{g}}^T,$$

$$\mathbf{B}_{\rm planar}=\frac{b}{2}(\mathbf{I}-\hat{\mathbf{n}}\cdot\hat{\mathbf{n}}^T),$$

$$\mathbf{B}_{\rm spherical}=\frac{b}{3}\mathbf{I},$$

where $$$b$$$** is the b-value, $$$\hat{\mathbf{g}}$$$ the diffusion gradient direction in the linear
encoding, $$$\hat{\mathbf{n}}$$$ is the normal of the plane in planar encoding,
and $$$\mathbf{I}$$$ is the 3x3 identity matrix. For a voxel containing
multiple compartments (indexed with i)
with axisymmetric Gaussian diffusion tensors the attenuation is given by**^{13}:

$$S(\mathbf{B})=\sum_{i}S_{0, i}e^{-\mathbf{B}\cdot\mathbf{D}_i}=\sum_i S_{0, i} e^{-b d_{\bot, i}} e^{-b(d_{\parallel,i}-d_{\bot,i})\hat{\mathbf{\nu}}_i\cdot \mathbf{B} \cdot \hat{\mathbf{\nu}}_i^T}$$

where $$$d_{\parallel, i}$$$/$$$d_{\bot, i}$$$ is the diffusivity along/across $$$\hat{\mathbf{\nu}}_i$$$. We assume all tensors have the same fODF described by the Bingham distribution:

$$f(\hat{\mathbf{\nu}})=\frac{1}{4\pi{}_1F_1(1/2;3/2;\mathbf{Z})}e^{-\hat{\mathbf{\nu}} \cdot \mathbf{Z} \cdot \hat{\mathbf{\nu}}^T}$$

where the Bingham matrix $$$\mathbf{Z}$$$ describes the orientation, extent, and
anisotropy of the fODF and $$${}_1 F_1$$$ is a confluent hypergeometric function.
The convolution of the Bingham distribution with the signal
attenuation gives^{2}:

$$S(\mathbf{B})=\sum_{i}S_{0,i}e^{-bd_{\bot,i}}\frac{{}_1F_1(1/2;3/2;\mathbf{Z}-b(d_{\parallel,i}-d_{\bot,i})\mathbf{B})}{{}_1F_1(1/2;3/2;\mathbf{Z})}$$

To achieve convergence in a single b-shell we can only fit a single compartment, which gives us an effective amplitude ($$$\overline{S_{0}e^{-bd_{\bot}}}$$$) and anisotropy ($$$\overline{d_{\parallel}-d_{\bot}}$$$) for all compartments:

$$S(\mathbf{B})=\overline{S_{0}e^{-bd_{\bot}}}\frac{{}_1F_1(1/2;3/2;\mathbf{Z}-b(\overline{d_{\parallel}-d_{\bot}})\mathbf{B})}{{}_1F_1(1/2;3/2;\mathbf{Z})}$$

Changing the b-value would alter the relative contributions ($$$S_{0,i}e^{-bd_{\bot,i}}$$$) of the various compartments and hence the effective anisotropy ($$$\overline{d_{\parallel}-d_{\bot}}$$$). We can avoid making assumptions about how anisotropy changes with b-value by fixing the b-value and only varying the shape and orientation of the b-tensor B, to probe the fODF (encoded in $$$\mathbf{Z}$$$). This allows us to measure the fODF even without an accurate model of how the contribution of multiple water compartments changes with b-value.

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Figure 1. Planar tensor encoding can
distinguish microstructure configurations where linear encoding cannot. We fit a biophysically plausible zeppelin + stick
model with no dispersion (orange) to the expected signal of dispersing fibers
(blue; models illustrated in panel A). B) For multi-shell linear tensor encoding
data this fit is nearly perfect, which implies that these models are degenerate. C) For linear and planar encoding data with the same b-value the best-fit zeppelin + stick model poorly matches the data from the dispersing stick model, which suggest that the inclusion of planar encoding data breaks the degeneracy between
these two models.

Figure 2. Recovered dispersion and anisotropy for three
different acquisition schemes. Red lines/stars show true parameter values. 300
datasets (120 volumes with SNR=30) were independently fitted using a dispersing
zeppelin model to data generated assuming an isotropic CSF compartment ($$$pv=0.2; bd=9$$$), an anisotropic extra-axonal compartment ($$$pv=0.32; bd_\parallel=2.7; bd_\bot=1.2$$$), and a stick-like intra-axonal compartment ($$$pv=0.48; bd_\parallel=3.6$$$).

Figure 3. Top: maps of the dispersion along the major axis
of dispersion (left) and the minor axis of dispersion (middle) and the
micro-anisotropy (right) measured at b = 2000 s/mm^{2}. Bottom: comparison of these parameters with those
measured using the shell with b = 1000 s/mm^{2}. This simplified
model does not include crossing fibres, which leads to large fibre dispersions
in most of the brain.