Amy FD Howard1, Alexandre A Khrapitchev2, Jeroen Mollink1,3, Rogier B Mars1,4, Nicola Sibson2, Jerome Sallet5, Saad Jbabdi1, and Karla L Miller1
1FMRIB Centre, Wellcome Centre for Integrative Neuroimaging, Nuffield Department of Clinical Neurosciences, University of Oxford, Oxford, United Kingdom, 2CR-UK/MRC Oxford Institute for Radiation Oncology, Department of Oncology, University of Oxford, Oxford, United Kingdom, 3Department of Anatomy, Donders Institute for Brain, Cognition and Behaviour, Radboud University Medical Center, Nijmegen, Netherlands, 4Donders Institute for Brain, Cognition and Behaviour, Radboud University Nijmegen, Nijmegen, Netherlands, 5Wellcome Centre for Integrative Neuroimaging, Experimental Psychology, Medical Sciences Division, University of Oxford, Oxford, United Kingdom
Synopsis
Intra-axonal axial diffusivity could be interesting biomarker
of disease, yet it is often assumed constant across the white matter. Furthermore,
when intra-axonal diffusivity is estimated, few models account for fibre
orientation dispersion which (when not explicitly modelled) will greatly affect the
estimates of axial diffusion. Here we combine the stick model of intra-axonal
diffusion with a simple model of fibre dispersion to simultaneously estimate
intra-axonal axial diffusivity and fibre dispersion on a voxel-wise basis in
high b-value data. Our results demonstrate considerable variability in the
intra-axonal axial diffusivity across the white matter.
Introduction
Diffusion MRI is a powerful tool
for measuring microstructural features of the tissue (e.g. axon integrity,
diameter and orientation). However, there is often interplay between the tissue
architecture and the diffusion characteristics making signal interpretation
challenging1,2.
For example, intra-axonal axial diffusivity
could be an important biomarker of disease but fibre orientation dispersion
must be taken into account in order to estimate it properly. With even a small
amount of fibre dispersion within an imaging voxel, the intra-axonal axial diffusivity
appears to vary with b-value (Figure 1a). If dispersion is ignored, this
apparent deviation from Gaussian diffusion (a.k.a. non-zero kurtosis) could be erroneously
assigned to a complex diffusion micro-environment (a microstructure properties
viewpoint), or it could bias the estimation of the axial diffusivity (a
diffusion properties viewpoint). Popular current models presume a global axial
diffusivity which is often either pre-defined3 or estimated
from the data4. If the assumed diffusivities are inaccurate, the estimation of the
remaining model parameters will be biased.
Here we show that orientation dispersion and
intra-axonal axial diffusivity can be estimated jointly in single-fibre voxels.
The model is based on high b-value data where we assume that the diffusion signal
can be exclusively assigned to intra-axonal water1,5 with only axial
diffusivity1,5,6. We demonstrate in simulated data the effect of getting some of the
model assumptions wrong. When applied to real data, the model shows substantial
variability in both orientation dispersion and axial diffusivity across the
brain. Methods
The dispersed stick modelThe diffusion attenuation $$$A$$$
along gradient direction $$$g$$$ is given by:
$$A=\int_{S^2}\exp[\,\kappa(\mu^Tx)^2\,]\,\cdot\,\exp[\,-bd(g^Tx)^2\,]\,\text{d}x\,.\qquad\qquad\text{Eq.}\,1$$
$$$d$$$ is the intra-axonal axial diffusivity;
$$$x$$$, a unit vector on the sphere. Here the fibre orientation distribution
is described by a Watson distribution
3,7 with orientation $$$\mu$$$,
dispersion $$$\kappa$$$ and normalising constant $$$C_W$$$, where,
$$C_W(\kappa)=\sqrt{\frac{4\pi^3}{\kappa}}\,\text{erfi}(\sqrt{\kappa}))\,.\qquad\qquad\text{Eq.}\,2$$
The orientation dispersion index is defined
as $$$ODI=2/\pi\,\text{arctan}(\kappa)$$$
3.
The model is based on four main
assumptions:
-
The
fibre orientation distribution is accurately described by a Watson distribution
with a known fibre orientation.
- Water
experiences axons to be like “sticks” with no measurable radial
diffusion.
- There
is a negligible contribution from extra-cellular water.
- The
voxel contains a single dispersed fibre population.
The
latter is enforced because the volume fractions and axial diffusivities of
crossing fibre populations are degenerate.
The integral in Equation 1 has an analytic form,
two examples of which are shown in Figure 1b. We see how the combined analysis of $$$\geq2$$$ gradient directions
(which aren’t both perpendicular to the fibre) provides a unique solution for
both the within-voxel fibre dispersion and axial diffusivity.
Simulations
Simulated data (noiseless, 32 gradient directions)
was used to examine the effect of deviations from the assumptions 1-3 in turn.
Data
The model was fitted to diffusion
MRI data acquired at $$$7\,\text{T}$$$ (Agilent, $$$400\,\text{mT/m}$$$) in a
perfusion-fixed, postmortem macaque brain
8: $$$b=7,10\,\text{ms/µm}^2$$$,
$$$1000$$$ gradient directions, $$$1\,\text{mm}$$$ isotropic voxels. The
diffusion tensor
9 was fitted to the data and single-fibre voxels selected with
$$$FA>0.5$$$, linearity $$$(L_1-L_2)\,/\,L_1\geq0.4$$$, planarity $$$(L_2-L_3)\,/\,L_1\leq0.2$$$
and sphericity $$$L_3/L_1\leq0.35$$$
10,11. The fibre orientation was
assumed equal to $$$V_1$$$.
In both simulated and real data, the model was
optimised using a Markov chain Monte Carlo (MCMC) method
12.
Results and discussion
Simulations
Figure 2 considers sensitivity to modelling assumptions. If the
intra-axonal stick model holds true, the axial diffusivity can be measured with
good precision and accuracy in the case of asymmetric dispersion (Figure 2a). Violation
of the stick-model increases the estimated diffusivity (Figure 2b). Figure 2c
highlights the importance of working at high b-values where the extra-axonal
signal is negligible1,5. Interestingly, the effect of extra-axonal water was different in
single- or multi-shell analysis.
Data
In the postmortem macaque brain, the estimates of axial diffusivity and
fibre dispersion appear fairly uncorrelated (Figure 3a). The results from
single-shell data closely match those from multi-shell data (Figure 3b), though with
a slight bias in axial diffusivity. This is likely due to either non-Gaussian diffusion,
extra-axonal contributions or non-stick-like fibres.
Figures 4 and 5 show considerable variation in
axial diffusivity and fibre dispersion across the white matter. The corpus
callosum shows low axial diffusivity in the genu and high in the splenium, as
well as a low-high-low pattern of dispersion, particularly towards the midline.
These results are consistent with previous observations of both dispersion13,14,15,16 and
axon diameter15,17,18,19.
Both the inferior cortico-spinal tract and
transverse pontocerebellar fibres (Figure 5) show notably high axial
diffusivities and low dispersion. Generally lower values of axial diffusivity and
higher dispersion are seen in the middle cerebellar peduncles and cerebellar
white matter.
Finally, Figures 4 and 5 demonstrate left-right symmetry and
spatial smoothness, matching
symmetric and smoothly changing anatomy.Conclusion
Through the simultaneous analysis of
orientation dispersion and axial diffusivity we demonstrate substantial
variability in the intra-axonal axial diffusivity within single-fibre
voxels across the brain. These results question the assumption of a global
axial diffusivity3,4. Furthermore, this variability suggests that axial
diffusivity could act as an interesting biomarker of disease. Here we analyse perfusion-fixed
postmortem tissue which likely explains the low values of diffusivity reported
here when compared to those in vivo11,20,21. These ex vivo
data will provide the opportunity to compare dispersion estimates to
microscopy-derived equivalents in the same brain8,14. Future works will apply the model to in
vivo human data.Acknowledgements
This work was supported by the
Wellcome Trust (grant WT202788/Z/16/A), EPSRC and MRC (grants EP/L016052/1 and MR/L009013/1).
AK and NS were funded by Cancer Research UK (grant C5255/A15935).
The work of RBM is supported by the
Biotechnology and Biological Sciences Research Council (BBSRC) UK [BB/N019814/1]. JS was
supported by a Sir Henry Dale Wellcome Trust Fellowship (105651/Z/14/Z). The
Wellcome Centre for Integrative Neuroimaging is supported by core funding from
the Wellcome Trust (203139/Z/16/Z).
It should be noted that SJ and KM contributed equally to this work.References
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