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Constrained linear least squares estimation of anisotropic response function for spherical deconvolution

Authors:

Abstract

Estimation of an appropriate response function for spherical deconvolution is commonly performed using the image data itself, by the following steps: 1. Use some heuristic process to select a subset of white matter voxels considered to contain a single fibre population. 2. Re-orient the diffusion signal in each voxel so that the estimated fibre orientation lies along the z-axis. 3. Transform the (rotated) diffusion signal in each single-fibre voxel to the zonal spherical harmonic (ZSH) basis (spherical harmonic (SH) basis with m=0 terms only). 4. Average the ZSH coefficients across single-fibre voxels. Research to date has focused mostly on step 1; however the subsequent steps may also considerably influence the quality of response function estimation. Here we propose an alternative solution to combine steps 3 and 4, which explicitly prevents non-physical solutions and enables estimation of higher order coefficients.
Constrained linear least squares estimation of anisotropic response function for spherical deconvolution
Robert E. Smith, Thijs Dhollander, Alan Connelly
Target audience: Users of spherical deconvolution models.
Purpose: Estimation of an appropriate response function for spherical deconvolution is commonly performed using the image data
itself, by the following steps: 1. Use some heuristic process to select a subset of white matter voxels considered to contain a single
fibre population. 2. Re-orient the diffusion signal in each voxel so that the estimated fibre orientation lies along the z-axis. 3.
Transform the (rotated) diffusion signal in each single-fibre voxel to the zonal spherical harmonic (ZSH) basis (spherical harmonic
(SH) basis with m=0 terms only). 4. Average the ZSH coefficients across single-fibre voxels. Research to date has focused mostly on
step 1 (e.g. 1-3); however the subsequent steps may also considerably influence the quality of response function estimation. Here we
propose an alternative solution to combine steps 3 and 4, which explicitly prevents non-physical solutions and enables estimation of
higher order coefficients.
Methods: The existing technique for response function estimation from a single-fibre voxel selection can be expressed as follows:
 
N
vvvv DWIRT
N
RF 1
1,,
1
(1)
Here: DWIv(θ,φ) is the diffusion signal over the sphere in single-fibre voxel v, R(θvv) is a 3x3 rotation matrix calculated from the
estimated fibre direction in voxel v, in order to align the fibre direction (expressed as polar angles (θvv)) with the z-axis; T is a
transformation matrix mapping coefficients in the SH basis to samples as a function of angles θ and φ (and T-1 is its inverse); N is the
number of single-fibre voxels. Note that the diffusion signal in each single-fibre voxel v is independently transformed to the SH basis
via T-1: inversion of matrix T places an upper bound on the maximum harmonic degree lmax of this transformation.
We reformulate this process as follows:
 
 
 
N
vvvv RFTDWIR
1
2
,,minarg
s.t.
   

0,0 RFT
RFT
(2)
We seek the response function coefficients RF (in the ZSH basis) that provide the least-squares fit to the rotated diffusion image data
from all single-fibre voxels concurrently, subject to the following constraints: 1. The response function amplitude must not be
negative at any angle θ. 2. The response function amplitude must increase monotonically from the z-axis (to which each single-fibre
voxel’s estimated fibre direction is aligned) to the perpendicular plane (where the diffusion signal magnitude is maximal). Because
data from all single-fibre voxels are used at once, the maximal harmonic degree lmax of the fit is comparatively unconstrained.
Results: Figure 1 provides examples of response function estimation for two different diffusion datasets, where 300 single-fibre
voxels were selected based on relative FOD peak amplitudes within the voxel2. In the first example, the old method is limited to
lmax=6 due to the data possessing only 32 diffusion directions; the resulting response function shows considerable ringing around the
z-axis. The proposed method, when used with lmax=6, prevents the amplitude of the response function from decreasing as the angle
from the fibre orientation increases, but the ringing remains; when used with lmax=8 the proposed method yields a much more
plausible response function. In the second example, there are only small differences between the two methods; the proposed method
does however remove the small decrease in response function amplitude at the noise floor, and better traces the mean signal intensity
orthogonal to the fibre direction. Increasing lmax from 8 to 10 bears little influence, consistent with previous results2.
Fig. 1. Scatter plots showing diffusion image intensities, and response function profiles, as a function of angle from the
estimated fibre orientation in single-fibre voxels. Left: Mouse data, 32 directions. Right: Human data, 60 directions.
Discussion: The proposed approach provides response functions that accurately follow the profile of the diffusion image intensities as
a function of angle from the fibre orientation, without introducing artifactual intensity increases along the fibre direction that occur
due to use of the ZSH basis. This is achieved through use of a single least-squares solution, the physically realistic constraints
imposed, and the ability to reduce ringing artifacts through use of a greater lmax. Because diffusion image sample points are distributed
evenly over the half-sphere, the least-squares fit naturally places greater importance on fitting the response function profile
perpendicular to the fibre direction more so than along it (where the constraints are typically of greater importance).
The FODs calculated via spherical deconvolution invariably change depending on the response function used. However because this
approach focuses on response function estimation on real data, its influence on real data is currently subjective only, and therefore not
shown.
It may be possible in the future to augment this process with iterative estimation of the fibre direction (θvv) in each voxel, and/or
rejection of voxels from the single-fibre mask, without dependence on any particular diffusion model.
Conclusion: This approach provides a more robust mechanism for combining the measured diffusion signals from multiple single-
fibre voxels to estimate a white matter response function for spherical deconvolution.
References: 1. Tournier et al., NeuroImage 2004:23;1176-1185 2. Tournier et al., NMR Biomed. 2013:26;1775-1786 3. Tax et al., NeuroImage 2014:86;67-80
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