Elsevier

NeuroImage

Volume 52, Issue 4, 1 October 2010, Pages 1374-1389
NeuroImage

Orientationally invariant indices of axon diameter and density from diffusion MRI

https://doi.org/10.1016/j.neuroimage.2010.05.043Get rights and content

Abstract

This paper proposes and tests a technique for imaging orientationally invariant indices of axon diameter and density in white matter using diffusion magnetic resonance imaging. Such indices potentially provide more specific markers of white matter microstructure than standard indices from diffusion tensor imaging. Orientational invariance allows for combination with tractography and presents new opportunities for mapping brain connectivity and quantifying disease processes. The technique uses a four-compartment tissue model combined with an optimized multishell high-angular-resolution pulsed-gradient-spin-echo acquisition. We test the method in simulation, on fixed monkey brains using a preclinical scanner and on live human brains using a clinical 3 T scanner. The human data take about one hour to acquire. The simulation experiments show that both monkey and human protocols distinguish distributions of axon diameters that occur naturally in white matter. We compare the axon diameter index with the mean axon diameter weighted by axon volume. The index differs from this mean and is protocol dependent, but correlation is good for the monkey protocol and weaker, but discernible, for the human protocol where greater diffusivity and lower gradient strength limit sensitivity to only the largest axons. Maps of axon diameter and density indices from the monkey and human data in the corpus callosum and corticospinal tract reflect known trends from histology. The results show orientationally invariant sensitivity to natural axon diameter distributions for the first time with both specialist and clinical hardware. This demonstration motivates further refinement, validation, and evaluation of the precise nature of the indices and the influence of potential confounds.

Introduction

White matter is the cabling that supports communication throughout the brain. It consists of bundles of axons packed to densities often over 105 mm2 (Waxman et al., 1995, Aboitiz et al., 1992a). Most axons have a diameter between 0.2 and 20 μm and lengths vary from millimeters to over a meter (Waxman et al., 1995). Axon diameter determines conduction velocity (Ritchie, 1982). Large-diameter axons, for example, in the corticospinal tract (CST) or midbody of the corpus callosum, transfer information quickly and support rapid communication for synchronous processing of sensorimotor stimuli. Small-diameter axons, for example, in the genu, pack more densely allowing fibre bundles to carry more diverse information between higher processing areas that favour quantity of information over transmission speed (Ptito, 2003, Aboitiz et al., 1992a, Lamantia and Rakic, 1990). Thus, axon diameter and density provide information about the role and performance of white matter pathways.

Diffusion magnetic resonance imaging (MRI) offers unique insight into live tissue microstructure through its sensitivity to displacements of water molecules over millisecond timescales. Semipermeable barriers within tissue affect the dispersion pattern of water and, consequently, diffusion MRI measurements are sensitive to the geometry and organization of the barriers. Diffusion tensor imaging (DTI) (Basser et al., 1994) models the distribution of particle displacements with a Gaussian distribution. The apparent diffusion tensor (DT) provides statistics of mean diffusivity and diffusion anisotropy (Basser and Pierpaoli, 1996), which provide some insight into tissue microstructure and have become popular as markers of white matter integrity. However, a limitation of these markers is that they do not relate directly to features of tissue microstructure and are sensitive to a variety of different effects simultaneously. For example, the size and packing density of cells, the permeability of cell walls and membranes, and the distribution of orientations of anisotropic cells all affect the mean diffusivity and anisotropy. Hence, a change in these statistics is difficult to associate with specific changes in tissue microstructure.

More sophisticated approaches model the geometry and material properties of tissue microstructure to predict the dispersion pattern of water within and thus the diffusion MR signal. Modelling approaches can provide sensible estimates of more specific microstructural parameters. Stanisz et al. (1997) model bovine optic nerve tissue as a three-compartment system with one population of water inside elongated ellipsoidal axons, another inside spherical glial cells, and a third in the extracellular space. Each compartment has its own dimensions, volume fraction, membrane permeability, and internal diffusivity and relaxivity. Stanisz et al. (1997) fit the three-compartment model to 800 nuclear magnetic resonance (NMR) measurements of a fixed sample using a stimulated echo sequence (Merboldt et al., 1991) with 3 ms pulses and varying diffusion time and gradient strength. Estimates of the cell sizes and densities agree with manual estimates from microscope images of the same samples. Permeability, diffusivity, and relaxivity estimates are all sensible but unvalidated.

The simpler CHARMED model (Assaf et al., 2004, Assaf et al., 2008, Assaf and Basser, 2005) has only two compartments: impermeable, parallel cylindrical axons in a homogeneous extracellular space. Each compartment has its own diffusivity, but relaxation times are the same in both. The diameter of the cylindrical axons has a two-parameter gamma distribution. The original CHARMED framework (Assaf et al., 2004, Assaf and Basser, 2005) provides maps of volume fraction between the intra- and extracellular compartments. The technique uses pulsed-gradient spin-echo (PGSE) (Stejskal and Tanner, 1965) in high-angular-resolution diffusion imaging (HARDI) (Jones et al., 1999, Tuch et al., 2002) with several b-values (multishell HARDI) to allow model fitting for arbitrary fibre orientation. Assaf and Basser (2005) demonstrate the method in live humans using a clinical MR scanner. However, the technique does not attempt to estimate the axon diameter. The axon diameter parameters of the model are fixed during fitting, and the acquisition protocol uses only a single diffusion time, so it is not designed to support such an estimate. The volume fraction estimates may correlate with axon density, but a true axon density estimate requires knowledge of the axon diameters.

Later work (Assaf et al., 2008) uses the CHARMED model to estimate the axon diameter distribution, but abandons the HARDI acquisition in favour of a fixed gradient direction to allow multiple combinations of diffusion time and gradient strength for sensitivity to axon diameter. They perform a similar NMR experiment to Stanisz et al. (1997) and estimate distributions of axon diameters in bovine optic and sciatic nerve samples that agree with distributions measured by hand on microscope images of the same samples. They also combine the method with imaging and show that segmentation of a pig spinal cord image from clustering the gamma distribution parameters shows consistency with a segmentation using various histological stains. Barazany et al. (2009) test the method in the corpus callosum of a live rat. They add an isotropic–diffusion compartment to account for partial volume with cerebrospinal fluid (CSF) for in vivo imaging. Results show spatial variation in the estimated axon diameter distribution that reflects subsequent histological evaluation.

Several limitations prevent direct translation of the existing axon diameter estimation techniques (Stanisz et al., 1997, Assaf et al., 2008, Barazany et al., 2009) above to whole-brain mapping in live human subjects:

  • 1.

    The imaging protocols require orders of magnitude higher magnetic field gradient strength than current human MRI scanners can provide. The NMR experiments in Stanisz et al. (1997) and Assaf et al. (2008) use maximum gradient strength |G|max around 1 T m1 and the imaging experiments in Assaf et al. (2008) and Barazany et al. (2009) use 300 mT m1, whereas current human systems provide |G|max between 40 and 80 mT m1.

  • 2.

    The acquisition time is too long for live human volunteers or patients to tolerate. The MRI experiments in Assaf et al. (2008) require around a day of imaging time. Even the in vivo experiment in Barazany et al. (2009) requires 2 hours of imaging. Human volunteers tolerate little more than 1 hour.

  • 3.

    The methods require prior knowledge of the fibre orientation, because the gradient direction must be perpendicular to the fibre in all the measurements. This limits parameter estimation to specific targeted structures, since the orientation of white matter fibres varies widely over the brain.

The simulation study in Alexander (2008) tests feasibility of relaxing these limitations using clinically feasible HARDI combined with a simplified version of the CHARMED model that has a single-axon diameter rather than a gamma distribution. Unlike previous multishell HARDI acquisitions, all three variables of the PGSE sequence (gradient strength |G|, pulse width δ, and separation Δ) vary between different shells, which provides the sensitivity to axon diameter. The simulation study uses an optimization algorithm to identify the combination of HARDI shells that maximize sensitivity to the model parameters. The experiment design optimization considers only protocols that divide the total number of measurements into M shells of N measurements. The algorithm optimizes the M combinations of PGSE settings with the directions held fixed. The objective function is the normalized sum of Cramer–Rao lower bounds on the model parameters averaged over a range of a priori settings. The optimization also identifies the best tradeoff between N and M with the total number of measurements NM fixed. Orientational invariance proves feasible, because M is small, typically 3 or 4, in contrast to previous protocols (Stanisz et al., 1997, Assaf et al., 2008, Barazany et al., 2009), which have several hundred distinct combinations.

Simulations with N = 30, M = 4 and maximum gradient strength |G|max = 70 mT m 1 suggest feasibility of recovering axon diameter and density without knowledge of the fibre orientation. More precisely, they show that, with protocols optimized for typical constraints of a human scanner, posterior distributions on single axon-diameters have low variance when the true diameter is between 10 and 40 μm. However, posterior distributions for true diameters between 2 and 4 μm are all similar and close to uniform over the range 0 to 5 μm. The result suggests that axon diameters of about 5 μm or less are indistinguishable from one another, although they are identifiable as small compared to diameters greater than 5 μm. Diameters over 5 μm (up to around 40 μm) can be identified more precisely. Sensitivity extends to lower axon diameters as gradient strength increases and to higher diameters as T2 increases (allowing longer diffusion times). As diffusivity decreases, the window of sensitivity shifts to lower diameters.

The aim of this article is to test the ideas in Alexander (2008) for mapping orientationally invariant indices of axon diameter and density in biological tissue. An important difference between real white matter and the simple model underlying the earlier simulation experiments is that the tissue contains a distribution of axon diameters. The main contribution here is an evaluation of the method in the presence of distributed axon diameter, which includes preliminary results from brain data. The method differs from Alexander (2008) in a few ways: (i) slightly improved a priori parameter settings for the experiment design optimization; (ii) an additional CSF compartment, as in Barazany et al. (2009), and an isotropically restricting compartment, similar to Stanisz et al. (1997), in the model; and (iii) grid search and gradient descent stages in the fitting procedure to initialize the Markov Chain Monte Carlo (MCMC). Key differences of the method with Assaf et al. (2004) and Assaf and Basser (2005) are (i) simplification of the two-compartment CHARMED model and inclusion of the CSF and isotropically restricting compartments; (ii) estimation of the axon diameter; (iii) optimized, rather than ad hoc, experiment design that explores all possible PGSE combinations; (iv) the Bayesian parameter estimation. Key differences with Stanisz et al. (1997), Assaf et al. (2008), and Barazany et al. (2009) are (i) orientational invariance, (ii) the model adaptations, (iii) the optimized experiment design, (iv) the Bayesian estimation, and (v) rather than a model of the full axon diameter distribution, as in Assaf et al. (2008) and Barazany et al. (2009), we estimate a single summary statistic, which we call the axon diameter index; Stanisz et al. (1997) also estimates a single axon-diameter. The methods section details the experiment design, data acquisition, tissue model, preprocessing, and parameter estimation procedures. Experiments then evaluate the method in simulation and show parameter maps from scan–rescan experiments on human subjects and fixed monkey brains. Finally, we conclude and discuss limitations, applications, and areas for further work.

Section snippets

Methods

This section describes the samples and imaging protocols for fixed and live brain imaging. It then specifies the tissue model, details the model-fitting procedure, and defines the indices of axon diameter and density.

Experiments and results

This section tests the acquisition protocols and fitting procedures above, first in simulation and then on the human and monkey data sets.

Discussion

In summary, this paper proposes and tests orientationally invariant indices of axon diameter, a′, and density, ρ, from diffusion MRI. Orientational invariance is important, because it enables combination of axon diameter and density information with tractography to study fibre composition along pathways with different and varying orientation. Previous techniques only work in regions with known and constant orientation, such as the spinal cord or the midsagittal corpus callosum. The discussion

Acknowledgments

The EPSRC supports DCA and MGH with grants EP/E056938/1 and EP/E007748. MP is supported by the National Science and Engineering Research Council of Canada (NSERC). TD is supported by the Lundbeck Foundation. The future and emerging technologies (FET) program of the EU FP7 framework fund the CONNECT consortium www.brain-connect.eu, which also supports this work. The authors thank Yaniv Assaf for seed code and Lee Wright for artwork.

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