Axon diameter distribution influences diffusion-derived axonal density estimation in the human spinal cord: in silico and in vivo evidence

Francesco Grussu^{1}, Torben Schneider^{1,2}, Ferran Prados^{1,3}, Carmen Tur^{1}, Sébastien Ourselin^{3}, Hui Zhang^{4}, Daniel C. Alexander^{4}, and Claudia Angela Michela Gandini Wheeler-Kingshott^{1,5}

Several diffusion-weighted (DW) MRI methods provide estimates
of the density of neurites, i.e. axons in white matter (WM) and dendrites in
grey matter (GM), modelling them as *sticks*^{1,2} (zero-radius
cylinders). This simplification is reasonable in the long diffusion time limit^{3}
and long gradient pulse duration^{4}, a condition practically achievable
in most brain WM.

However, in spinal cord WM, myelinated axons can have
diameters^{5} as large as
10-15 μm, and the diffusion times employed to minimise the echo time T_{E} of DW spin echo acquisitions may not be sufficiently long to support the *stick* model.

Here, we investigated whether
in clinical settings sub-optimal diffusion times can affect neurite density (ND)
estimation in the spinal cord, assessing diffusion time dependency of ND *in vivo* and
corroborating results with simulations.

*Simulated signal
synthesis* The Camino Monte Carlo (MC) simulator^{6} was used
to synthesise DW signals from two substrates describing WM characterised by big
and small axons (figure 1: *BigAxons/SmallAxons*,
representative of spinal cord^{5} and corpus callosum^{7}),
made of cylinders with gamma-distributed radii (cylinder volume fraction (CVF)
of 0.65, diffusivity inside/outside 1.70 μm^{2}/ms). Four clinically feasible
two-shell protocols (*b*=711 s/mm^{2} and *b*=2855 s/mm^{2};
30 and 60 directions; gradient duration δ=18 ms) were synthesised, each
characterised by a diffusion time T_{d}=Δ – δ/3 among {24, 41.3,
58.7, 76} ms (gradient separation Δ was varied).

*Simulated signal analysis* CVF was estimated with NODDI^{2} on: i) 1000 unique
sub-protocols of 25 and 50 directions at *b*=711 s/mm^{2} and *b*=2855
s/mm^{2} for infinite signal-to-noise ratio (SNR); ii) 1000
protocols with unique realisations of Rician noise (SNR of 13, 10.25, 8.09, 6.38 for
increasing T_{d} to model the effect of longer T_{E}, assuming T_{2}=72
ms and SNR=13 at T_{d}=24 ms). Signals from substrates *BigAxons/SmallAxons* were averaged to simulate a mixture of
their radius distributions.

In vivo *acquisition* Three
subjects (all aged 27) were scanned axially on a 3T Philips Achieva system
(cardiac gated ZOOM-EPI sequence^{8}, cervical level C1-C3, resolution 1×1×5mm^{3},
field-of-view 64×48×60mm^{3},
T_{R} 12RR repeats). Three two-shell protocols (*b*=711 s/mm^{2} and *b*=2855 s/mm^{2}; 20 and 40
directions; δ=22 ms) were acquired with T_{d} ∈ {21.17, 44.67, 68.67} ms, varying Δ. Subjects
were scanned twice, with minimum T_{E} for each T_{d} in the
first session (T_{E} ∈ {67,
87.20, 111} ms) and fixing T_{E}=111 ms in the second session, to control
for the different T_{2}-weighting and SNR implied by a variable T_{E}.

In
vivo *analysis* ND
was estimated fitting NODDI after slice-wise motion correction and spinal cord
segmentation. We created an average DW image with good GM/WM contrast^{9},
where we segmented GM manually and determined WM as the portion of the cord not
containing GM. ND was characterised in WM calculating median and interquartile
range.

*Simulations* Figure 2
shows the estimated CVF. At infinite SNR, CVF is underestimated for the *BigAxons* and mixture *BigAxons/SmallAxons* substrates, and increasing
T_{d} decreases this bias. CVF shows neither bias nor T_{d}-dependency
for the *SmallAxons* substrate.
Adding noise provides similar trends, but decreases precision.

In vivo ND maps for
one slice per volunteer are shown in figures 3 and 4 for the two MRI sessions.
In both sessions, ND increases as T_{d} increases, especially in
certain areas as those illustrated. Table 1 reports descriptive statistics of
ND in WM and the percentage relative increase of the median ND with respect to
the median at T_{d}=21.17 ms. A trend of higher ND for longer T_{d}
is measured in all subjects and sessions (increase of about 10%).

Here we investigated whether in clinical scenarios sub-optimal diffusion times affect ND estimation in the spinal cord.

MC simulations run with a relatively long gradient pulse duration demonstrated that ND can be underestimated in microstructural geometries plausible in the spinal cord, if inadequately short diffusion times are employed. The underestimation is stronger at shorter diffusion time, but also occurs at longer diffusion time, because of the finite diameter of axons.

*In
vivo* experiments
showed a trend of increasing ND in spinal cord WM as the diffusion time
increased, not confounded by changes of echo time. This suggests that at
the shortest diffusion time we probed, the assumptions of the *stick* model were not fully met in areas
with large axons.

We conclude that careful design of the diffusion time is required for the estimation of ND in the spinal cord. Short diffusion times lead to biased estimates, whereas long diffusion times lead to more accurate but less precise values. A compromise between accuracy and precision needs to be evaluated according to the application.

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[5] Makino M. et al. Morphometric study of myelinated fibers in human cervical spinal cord white matter. Spine 1996; 21: 1010-1016.

[6] Hall M. and Alexander D. Convergence and parameter choice for Monte-Carlo simulations of diffusion MRI. IEEE Transactions on Medical Imaging 2009; 28(9): 1354-1364.

[7] Alexander D. et al. Orientationally invariant indices of axon diameter and density from diffusion MRI. NeuroImage 2010; 52(4): 1374-1389.

[8] Wheeler-Kingshott C. et al. ADC mapping of the human optic nerve: increased resolution, coverage, and reliability with CSF-suppressed ZOOM-EPI. Magnetic Resonance in Medicine 2002; 47(1): 24-31.

[9] Kearney H. et al. Spinal cord grey matter abnormalities are associated with secondary progression and physical disability in multiple sclerosis. Journal of Neurology, Neurosurgery and Psychiatry, 2015; 86(6): 608-614.

Figure 1: substrates employed for MC simulations. Top: detail of 50
µm×50 µm of substrate *BigAxons* (to
the left, with axons in green) and *SmallAxons*
(to the right, with axons in blue). Bottom: radius distribution of substrates *BigAxons* (green), *SmallAxons* (blue) and a mixture of the two (red). For all
substrates, the cylinder volume fraction (CVF) was set to 0.65. In
substrates *BigAxons* and *SmallAxons*, radii follow a gamma
distribution controlled by a scale and a shape parameter.

Figure 2: cylinder volume fraction (CVF) estimated with NODDI from
synthetic signals. To the left, results from fitting performed at infinite SNR.
To the right, results from fitting with SNR at *b*=0 decreasing with
increasing diffusion time, to account for the longer echo time required.
Medians and 95% confidence interval (CI) over the 1000 repetitions of the fitting
are reported for all simulated substrates. Ground truth CVF is also shown, in
yellow.

Figure 3: examples of neurite density (ND) maps
from the first scanning session (the minimum echo time achievable for each diffusion time was used), overlaid onto the mean *b*=0 image. Different subjects are illustrated along different rows, whereas
maps corresponding to different diffusion times are illustrated along different
columns.

Figure 4: examples of neurite density (ND) maps
from the second scanning session (echo time fixed to 111 ms for all diffusion
times), overlaid onto the mean *b*=0 image. Different subjects are illustrated along different rows, whereas maps
corresponding to different diffusion times are illustrated along different
columns. The same MRI slices shown in figure 3 are reported.

Table 1: medians and interquartile range of
neurite density (ND) distributions in subjects' white matter (WM). Values from
both MRI sessions are reported (1st session: echo time increasing for
increasing diffusion time; second session: echo time fixed to 111 ms for all diffusion
times). The percentage increase of the median ND with respect to the median ND
at the shortest diffusion time (21.17
ms) is also reported for reference.

Proc. Intl. Soc. Mag. Reson. Med. 24 (2016)

2009