Content uploaded by Robert Elton Smith

Author content

All content in this area was uploaded by Robert Elton Smith on Sep 21, 2016

Content may be subject to copyright.

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(θv,φv) 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 (θv,φv)) 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 (θv,φv) 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