White matter microstructure from nonparametric axon diameter distribution mapping
Graphical abstract
Introduction
The axon diameter distribution (ADD) is a key microstructural feature in the peripheral and central nervous systems. Conduction velocity scales with axon diameter (Tasaki et al., 1943, Waxman et al., 1995) and therefore provides an important functional marker that reflects information transmission in the nervous system. Conventional neuroanatomical methods applied to postmortem human or animal tissue have provided evidence that the ADD changes in neurological conditions such as amyotrophic lateral sclerosis (ALS) (Cluskey and Ramsden, 2001) and multiple sclerosis (MS) (Trapp et al., 1998, Evangelou et al., 2001, Lovas et al., 2000). In addition, several pathologies, including autism (Hughes, 2007), dyslexia (Njiokiktjien et al., 1994), schizophrenia (Randall, 1983), and even alcoholism (Livy and Elberger, 2008), have been associated with changes in the distribution of axon size. Changes in the number of axons and their diameters also take place throughout normal development accompanying the period of dynamic behavioral, cognitive, and emotional changes in childhood and adolescence (Yakovlev, 1967, Pfefferbaum et al., 1994, Gregg et al., 2007, Barnea-Goraly et al., 2005, Schlaug et al., 2009).
The biological significance of axon caliber is not limited to development or well-defined pathologies. Central nervous system neurites exhibit a beaded axonal morphology following mechanical, chemical, or metabolic insults (Ochs et al., 1997, Roediger and Armati, 2003). Beading is also seen in the peripheral nervous system when nerves are incrementally stretched and rapidly fixed by freeze-substitution or cold fixation (Ochs and Jersild, 1987). Recently, this mechanism of axonal beading was proposed to explain both the reduction in the mean apparent diffusion coefficient (ADC) during traumatic brain injury (TBI) and cerebral ischemia, and its re-elevation observed during recovery (Budde and Frank, 2010). Diffusion tensor imaging (DTI)-derived parameters, including fractional anisotropy (FA), while sensitive to these changes in axonal dimensions, do not provide information about specific microstructural or morphological changes or about their biophysical basis. Assuming that it is well-defined in the presence of beading, the ADD would directly reveal these underlying microstructural changes resulting from TBI or ischemia.
A common way to measure the ADD is by using optical or electron microscopy on fixed specimens. Measurement of postmortem histology-based ADD can be slow and tedious and is highly invasive. In addition, the examined tissue has to undergo a set of complex procedures that typically include extraction, freezing, cutting, dehydration, fixation, potting and mounting—processes known to change the cross-section area of fibers (Ohnishi et al., 1976). Moreover, cross-sectional slices from microscopy-based methods suffer from sampling only a small field of view (FOV). Measuring and mapping the ADD noninvasively using MRI presents several obvious advantages over traditional neuroanatomical methods: the procedure can potentially be performed on a live animal or human and over a large volume of interest. Although in most cases the MRI voxel resolution is on the order of millimeters, the MR signal can be sensitized to the microscopic motion of diffusing water molecules and, indirectly, to the morphology of the restricting compartments that confine them, which are on the order of a few micrometers.
To achieve this degree of sensitivity, diffusion-weighted MRI can be used. A single pair of Stejskal Tanner (Stejskal and Tanner, 1965) pulsed-field gradients (PFG) is traditionally used to estimate certain microstructural features of porous materials (Callaghan et al., 1991) or macroscopic anisotropy (Basser, 1995). Single-PFG (recently termed single diffusion encoding, SDE, (Shemesh et al., 2015)) has also been used to estimate the ADD of white matter tissue (Assaf et al., 2008), by employing a parametric method that assumed the ADD is given by a gamma distribution. While such a parametric ADD may well describe healthy, normal neural tissue, a nonparametric ADD estimation would provide a more general and objective description of any axonal tissue, whether healthy or injured.
Although not developed for biological applications, an estimation scheme for nonparametric pore size distribution was originally suggested by Hollingsworth and Johns (Hollingsworth and Johns, 2003) for emulsions. Using an s-PFG NMR experiment and subsequently solving a linear regression problem, they estimated an empirical droplet size distribution. When this linear regression problem is solved, the design matrix (i.e., a matrix of values of explanatory variables, which in this case is the response to different droplet sizes) is highly ill-conditioned as a result of its intrinsically high multicollinearity (degree of correlation between the different independent variables). A similar problem arose when Assaf and Basser attempted to estimate the nonparametric ADD using a discrete histogram to represent this distribution (Assaf and Basser, 2007). In some cases, a solution is obtained by invoking certain assumptions about the character of the distribution. For instance, smoothness of the distribution is a common and usually valid assumption in natural porous materials (Song et al., 2002, Hollingsworth and Johns, 2003).
An extension of the s-PFG MR experiment, the double-PFG (d-PFG) (Cory et al., 1990, Mitra, 1995) experiment, employs two successively applied PFG pairs with amplitudes G1 and G2, separated by a mixing time, τm, and a prescribed relative angle between them, φ (Fig. 1A). The d-PFG experiment, recently termed double diffusion encoding (DDE) (Shemesh et al., 2015), may be sensitized to the temporal correlations of molecular motions within the two diffusion periods and to the direction of the applied gradients (Feinberg and Jakab, 1990, Stapf et al., 1999). Using this experiment one can measure the average pore size (Özarslan and Basser, 2007, Koch and Finsterbusch, 2008, Komlosh et al., 2011) and the estimates of features of microscopic anisotropy (Lawrenz and Finsterbusch, 2011, Jespersen et al., 2013, Szczepankiewicz et al., 2015).
To address the problem of multicollinearity in ADD estimation with SDE MR we extended the method by using a DDE experiment (Benjamini et al., 2012). This adds an independent second dimension to the parameter space, namely the angle between the PFG pairs, which was shown to reduce the multicollinearity. The theoretical and experimental benefits of the DDE in the context of size distribution estimation were discussed and demonstrated in previous studies (Benjamini et al., 2012, Benjamini et al., 2014a) as were strategies for optimal experimental design (Katz and Nevo, 2014). As a precursor to analyzing white matter microstructure, the DDE-based method was first validated on an ADD phantom using NMR (Benjamini and Nevo, 2013, Benjamini et al., 2014b) and later MRI (along with an experimental comparison between single- and double-diffusion encoding based methods) (Benjamini et al., 2014a).
In this study, a nonparametric DDE ADD estimation framework was developed, implemented, and tested on an ex vivo ferret spinal cord model. A voxelwise nonparametric ADD estimation provided maps such as average axon diameter and extra-axonal water fraction, along with unique ADD variance and sub-micron restricted structures map. The spatially resolved ADD map was then used to segment distinct domains in white matter on the basis of their statistical similarity. This method was further validated by a comparison of the ADD estimates with ADDs obtained from optical microscopy of the same specimen from different white matter tracts.
Section snippets
White matter model
In models of the microstructure and morphology of white matter, axons are usually idealized as infinite cylinders. White matter is commonly considered to comprised two compartments, each with distinct water diffusion properties. In one water is restricted, in the other diffusion is Gaussian (or hindered) (Assaf et al., 2004). The intra-axonal space is modeled as a finite and discrete distribution of impermeable parallel cylinders, in which water diffusion is restricted; the extra-axonal space
Specimen preparation
The animal used in this study was housed and treated at the Uniformed Services University of the Health Sciences (USUHS) according to national guidelines and institutional oversight. As part of standard necropsy for an unrelated study, a healthy adult male ferret was euthanized and underwent transcardial perfusion with ice-cold 0.1 M phosphate buffered saline (PBS, pH 7.4, Quality Biological) followed by 4% paraformaldehyde (PFA, Santa Cruz Biotechnology, in PBS 0.1 M pH 7.4) at USUHS, according to
Local microstructure
The VW ADD within each white matter voxel was estimated by solving Eq. (5), along with the extra-axonal and sub-micron fractions, and the intra- and extra-axonal diffusivities. The VW ADD can be subsequently transformed into NW ADD (see Theory). These values can be used to create maps in which Din, Dex, fex, and fsm are assigned to each of the white matter voxels. Even more descriptive, moments of both VW and NW ADDs, e.g., mean, variance, etc., may be calculated and mapped with potentially
Conclusions
In this study we have used a nonparametric method to estimate on a voxel-by-voxel basis the volume-weighted ADD within a spinal cord specimen. We have shown how the VW ADD can be transformed to NW ADD, and how these two weightings produce different contrasts within the spinal cord white matter. A sub-micron compartment was incorporated in the model to account for highly restricted water molecules that result in an unattenuated DW signal above the noise floor. The new white matter model
Acknowledgments
This work was supported by funds provided by the Intramural Research Program of the Eunice Kennedy Shriver National Institute of Child Health and Human Development (grant number ZIA-HD000266), and the Center for Neuroregenerative Medicine (CNRM) under the auspices of the Henry Jackson Foundation (grant number 306135-2.01-60855). The authors thank Dr. Alexandru Avram for fruitful discussions, Dr. Kryslaine Radomski and Dr. Elizabeth B. Hutchinson for dissecting and preparing the spinal cord, Dr.
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