Jin Gao^{1,2}, Mingchen Jiang^{3}, Richard Magin^{4}, Andrew Larson^{5}, and Weiguo Li^{2,4,5}

Amyotrophic lateral sclerosis (ALS) is a progressive disease of motor neuron degeneration in brain and spinal cord with an unknown etiology. Diffusion MRI has potential to track the disease progression in ALS due to the technique’s intrinsic advantages in detecting structure changes and non-invasive nature. In this study, we investigated the feasibility of analyzing multiple high b-value diffusion-weighted images using a non-negative least squares method (requiring no prior assumptions about components) and a bi-compartment model with restricted and hindered diffusion components. Both methods were able to detect alterations of spinal cord in the G93A-SOD1 mouse model of ALS.

Introduction

Amyotrophic lateral sclerosis (ALS) is a fatal disease characterized by degeneration of motor neurons in spinal cord [1, 2]. The microstructure changes of the damaged nerves in ALS may potentially be reflected in altered diffusion properties, thus detectable with diffusion-weighted (DW) MRI. The classical mono-exponential model fails to precisely depict DW signal decay at high b values [3-7]. Here, we aim to investigate signal decay behaviors at ultra-high b values for the non-invasive assessment of spinal cord alterations in the transgenic G93A-SOD1 mouse model of ALS.Methods

Two groups of G93A-SOD1 mice (n=5) and wild-type mice
(n=4) were euthanized (approved by IACUC) on postnatal days 100 and MRI scans
performed. All MRI data were acquired with 31cm bore 9.4
T Agilent MRI scanner at $$$16^{\circ}C$$$. A DW stimulated echo sequence was applied with
following acquisition parameters: TR/TE 2000/30.5 ms, mixing time 382 ms, $$$\delta$$$/$$$\triangle$$$ 11/400
ms, slice thickness 1.5 mm, field of view (FOV) 36 × 50 mm^{2}, matrix
64 × 96 and 30 b values ranging from 0 to 858,022 s/mm^{2} with a
maximal diffusion gradient strength of 50 Gauss/cm and direction perpendicular
to long axis of mouse spinal cord. T2 weighted images were acquired
using a fast spin echo sequence with parameters: TR/TE 1000/12 ms, echo train
length 8, matrix 192 × 192, FOV 36 × 50 mm^{2}, slice thickness 1.5 mm.
Image post-processing was performed in
MATLAB (MathWorks). Normalized signal intensities were calculated from regions
of interest (ROIs) at lumbar level that were manually drawn in DW images for
each mouse in both groups. Non-negative least squares (NNLS) was employed to
analyze normalized signal intensities on 200 possible diffusion coefficients (D_{ps})
that were logarithmically spaced over the
interval of [$$$1\times10^{-8}$$$, $$$3\times10^{-3}$$$]
mm^{2}/s. To minimize
noise influence in analysis, a regularization term based on ‘energy’ in
spectrum was introduced

$$argmin_{s}\frac{1}{2}||A\vec{S}-\vec{y}||_{2}+\mu\vec{S}^{H}\vec{S}$$

, subject to $$$\vec{S}\geq0$$$, where A is a 200 x 30 matrix containing kernels exp(−b, D_{ps}), $$$\vec{S}$$$ is weights of D_{ps}, $$$\vec{y}$$$ is normalized signal intensities, $$$\mu$$$ is a regularizer
computed from $$$\chi^{2}$$$ distribution condition when regularized fit is 101% of the
non-regularized $$$\chi^{2}$$$[8].
Additionally, we proposed a bi-compartment model to evaluate the DW data $$\vec{y}=(1-f)E_{\alpha}(-(\vec{b}D_{1})^\alpha)+fe^{-\vec{b}D_{2}}$$, where $$$D_{1}$$$, $$$D_{2}$$$ are diffusion coefficients, $$$f$$$ is the fraction of the
mono-exponential compartment, $$$\vec{b}D_{1}$$$is raised to $$$\alpha$$$ power. In this model, a Mittag-Leffler function ($$$E_{\alpha}$$$) was employed to represent restricted diffusion and a
mono-exponential function to depict the hindered diffusion components [9,10]. A
Student’s t test was applied to compare parameters ( $$$D_{1}$$$, $$$D_{2}$$$, $$$\alpha$$$ and $$$f$$$) extracted from the bi-compartment
model.

Results

In DW images of a wild type representative mouse, signal intensities in spinal cord persisted while intensities in other tissues decayed away with increased b values (Fig. 1B-D). At high b values (b > $$$1\times10^{4}$$$ s/mmDiscussion

Novel diffusion-weighted MRI techniques can be beneficial to non-invasively monitor disease progression and to evaluate treatment outcomes. In this study, the signal differences with the DConclusion

We demonstrated that spinal cord alternations in a symptomatic mouse model of ALS can be detected by both NNLS and a bi-compartment model analysis of a series of DW images with b-value extended to extremely high values. Further studies are necessary to validate these methods for tracking progression in ALS or other neurodegenerative diseases.1. B. R. Foerster, et al., "25 years of neuroimaging in amyotrophic lateral sclerosis," Nat Rev Neurol, vol. 9, pp. 513-524, 2013.

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Fig. 1 MR images of the representative *ex vivo* mice, (A) anatomic sagittal T_{2} weighted image (Red region shows the lumbar spinal
cord), (B) diffusion weighted
image with b = 2.15 × 10^{3} s/mm^{2}, (C) diffusion weighted image with b = 4.20 × 10^{5} s/mm^{2},
(D) diffusion weighted image
with b = 8.58 × 10^{5} s/mm^{2}. The diffusion gradient direction is perpendicular to the long axis of the spinal cord in (B) - (D).

Fig. 2 Normalized signal intensity vs b-value in
log-scale, (A) log-scaled normalized signal intensity vs b-value, (B)
log-scaled normalized signal intensity vs log-scaled b-value.

Fig. 3 D_{ps} Fraction vs log-scaled D_{ps} values in NNLS
analysis.

Fig. 4 Parameter statistics in the bi-compartment model, (A)
parameter α and diffusion
coefficient D_{1}, (B) diffusion coefficient D_{2}, (C) fraction *f*, for wild type group (averaged *f* = 0.32) and mutant group (averaged *f* = 0.45)