# A restriction spectrum sparse fascicle model for diffusion MRI

### Ariel Rokem^{1}, Christian Pötter^{2}, Robert F. Dougherty^{2}

^{1}The University of Washington eScience Institute

^{2}Center for Cognitive and Neurobiological Imaging, Stanford University

View the code on GitHub

### Example modeling one voxel:

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### Generating predictions for the WM challenge:

This model was developed as a contribution to the White Matter Challenge, ISBI 2015

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Abstract

The Sparse Fascicle Model (SFM [1]) is a member of the large family of models
that account for the diffusion MRI signal in the white matter as a combination
of signals due to compartments corresponding to different axonal fiber
populations (fascicles), and other parts of the tissue. The model proceeds in
two steps. First, an isotropic component is fit. We model the effects of both
the measurement echo time (TE), as well as the measurement b-value on the
signal. These are fit as a log(TE)-dependent decay with a low order polynomial
function, and a b-value-dependent multi-exponential decay (including also an
offset to account for the Rician noise floor). The residuals from the isotropic
component are then deconvolved with the perturbations in the signal due to a
set of fascicle kernels each modeled as a radially symmetric (λ2=λ3) diffusion
tensor. The putative kernels are distributed in a dense sampling grid on the
sphere. Furthermore, Restriction Spectrum Imaging (RSI [2]) is used to extend
the model, by adding a range of fascicle kernels in each sampling point, with
different axial and radial diffusivities, capturing diffusion at different
scales. To restrict the number of anisotropic components (fascicles) in each
voxel, and to prevent overfitting, the RS-SFM model employs the Elastic Net
algorithm (EN [3]), which applies a tunable combination of L1 and L2
regularization on the weights of the fascicle kernels. We used elements of the
SFM implemented in the dipy software library [4]
and the EN implemented in scikit-learn [5]. In addition, to account for
differences in SNR, we implemented a weighted least-squares strategy whereby
each signal’s contribution to the fit was weighted by its TE, as well as the
gradient strength used. EN has two tuning parameters determining: 1) the ratio
of L1-to-L2 regularization, and 2) the weight of the regularization relative to
the least-squares fit to the signal. To find the proper values of these
parameters, we employed k-fold cross-validation [1], leaving out one shell of
measurement in each iteration for cross-validation. We determined that the
tuning parameters with the lowest LSE [6] provide an almost-even balance of L1
and L2 penalty with weak overall regularization. Because of the combination of
a dense sampling grid (362 points distributed on the sphere), and multiple
restriction kernels (45 per sampling point), the maximal number of parameters
for the model is approximately 16300, more than the number of data
points. However, because regularization is employed, the effective number of
parameters is much smaller, resulting in an active set of approximately 20
regressors [7].

## References:

[1] Rokem et al. (2015) PLoS 1, in press.

[2] White et al. (2013) HBM 34: 327-346.

[3] Zou and Hastie (2005) J R Statist Soc B 67:301-320.

[4] Garyfallidis et al. (2014). Front Neuroinf 8:8

[5] Pedregosa et al. (2011) JMLR: 12: 2825-30

[6] Panagiotaki et al. (2012) Neuroimage 59: 2241-2254.

[7] Zou et al. (2007). Ann Statist. 35: 2173-2192.