Apparent Fibre Density: A novel measure for the analysis of diffusion-weighted magnetic resonance images
Introduction
Diffusion-weighted magnetic resonance imaging (DWI) uses the diffusive motion of water to perform macroscopic in vivo investigations of white matter tissue micro-structure. The interaction of diffusing water and coherently ordered cellular structures, such as axon membranes, results in an anisotropic profile of diffusion which forms the basis of many aspects of DWI. In recent years, the diffusion tensor (DT) has played a prominent role in modelling data acquired from DWI (Basser et al., 1994). Rotationally invariant indices of the diffusion tensor such as fractional anisotropy (FA) (Basser and Pierpaoli, 1996) have been used extensively for investigating white matter in both normal and diseased conditions. These include ageing, stroke, dementia, Motor Neurone Disease, multiple sclerosis and schizophrenia (amongst others; see Assaf and Pasternak(2008) for a review). FA is often used as a measure of so-called white matter ‘integrity’; however it is difficult to attribute differences in FA to specific structural changes at the microscopic level. This is because the degree of anisotropy is influenced by a number of confounding factors including axon diameter, inter-axon spacing, membrane permeability, myelination and particularly the coherence of axon orientations (Beaulieu, 2002).
A particular confound to the interpretation of FA differences is the fact that many voxels may contain partial volume fractions from two or more distinct fibre populations (a situation often referred to as ‘crossing fibres’). A recent study suggests that at the current DWI resolution, multiple fibre orientations can be detected in up to 90% of white matter voxels (Jeurissen et al., 2010). The presence of crossing fibres is a concern for DT studies because the standard rank-2 tensor contains only a single peak and therefore cannot adequately model more than a single fibre population. In regions with crossing fibres, the DT principal eigenvector (assumed to correspond with fibre orientation) and the FA are unlikely to be representative of any of the underlying fibre populations (Alexander et al., 2001b, Wheeler-Kingshott and Cercignani, 2009). For this reason, it can be difficult to interpret results obtained using the DT in these regions. For example axon degeneration, which is commonly associated with a decrease in FA (Song et al., 2003), may also cause an increase in FA if the degeneration occurs in a secondary fibre within a region containing two fibre populations (Pierpaoli et al., 2001, Douaud et al., Apr. 2011).
With the introduction of high angular resolution diffusion-weighted imaging (HARDI) (Tuch et al., 2002), more advanced methods have emerged that better characterise regions with crossing fibre populations. Spherical deconvolution is one such method, enabling the estimation of the so-called fibre orientation distribution (FOD), a continuous distribution representing the partial volume of the underlying fibres as a function of orientation (Tournier et al., 2004, Anderson, 2005, Alexander, 2005, Tournier et al., 2007, Dell'Acqua et al., 2007, Descoteaux et al., 2009).
Higher order information has significantly improved the performance of tractography algorithms (Behrens et al., 2007, Jeurissen et al., Mar. 2011, Fillard et al., May 2011). However the use of higher order models for whole-brain voxel-based analysis remains largely unexplored. In the work by Assaf and Basser(2005), the CHARMED model was proposed which can resolve multiple fibres by estimating the partial volume fractions (PVF) of one or more restricted (intra-cellular) components, as well as a hindered (extra-cellular) component. It was suggested that the PVF of the restricted component might be sensitive to various pathologies of white matter that involve the number of fibres, intra-axonal composition or myelin (Assaf and Basser, 2005). However, a limitation of CHARMED is the need for multiple b-values which leads to long scan times that are not practical for large scale population studies investigating the entire brain, particularly with patients. In other recent work, Jbabdi et al.(2010) propose voxel-wise comparisons of fibre PVFs computed using the higher-order model outlined in Behrens et al.(2007). Their proposed method permits differences to be detected in different fibre tracts within a multiple fibre voxel. However, this approach establishes spatial correspondence using an FA-based skeletonisation procedure (Smith et al., 2006), and thus only takes a relatively small amount of the white matter into consideration, with no explicit attempt to match actual white matter tracts across individuals. Another potential limitation of this method is that the total volume of a given fibre bundle (as represented by the sum of the voxel-wise PVF) may not be the same before and after spatial normalisation. During spatial normalisation, fibres may undergo non-linear volume expansion or contraction which should be accounted for when performing voxel-based comparisons of PVFs in a common template space. In other related work, higher-order information provided by Q-ball diffusion orientation distribution functions was used to investigate the heritability of white matter structure using a cohort of mono- and di-zygotic twins (Brun et al., 2010). By including higher-order information in a multi-variate analysis of ODF complexity, an increase in heritability was determined compared to a univariate analysis performed on generalised fractional anisotropy. While this study demonstrated the benefits of multi-variate statistics on higher-order models, the result is a single test statistic per voxel, and therefore it is difficult to attribute significant differences to a specific fibre bundle in regions with crossing fibres.
As part of this work, we propose a new measure, which we call the Apparent Fibre Density (AFD). The AFD is based on the widely-used assumption that intra-axonal water is restricted in the radial direction (Stanisz et al., Jan. 1997, Alexander, Aug. 2008, Alexander et al., 2010, Assaf and Basser, 2005, Assaf et al., 2008, Barazany et al., May 2009). It has been shown that when using typical human DW gradient pulse durations (~ 30 ms), the DW signal emanating from a restricted compartment is almost fully preserved (Hall and Alexander, 2006, Yeh et al., 2010). Under these conditions, the radial DW signal emanating from the intra-axonal compartment is therefore independent of axonal diameter, and hence proportional to the intra-axonal water content (Figs. 1Aand B). Since the extra-axonal water signal is strongly attenuated at high b-values (≥ 3000 s/mm2), it follows that the total radial DW signal (Fig. 1C) is approximately proportional to the volume of the intra-axonal compartment, or in other words, the density of the fibres (measured as the proportion of space occupied by the fibres, rather than fibre count). Since the FOD amplitude (Fig. 1D) is also approximately proportional to the radial DW signal for the corresponding fibre orientation (particularly at high b-values), it provides a relative measure of the intra-axonal volume occupied by fibres aligned with that direction: the Apparent Fibre Density.
Using numerical simulations as well as in vivo data, we then demonstrate that this new interpretation of the FOD amplitude as the AFD is sound. Building on this and our previous work on HARDI registration (Raffelt et al., 2011, Raffelt et al., in press), we present a novel methodology to perform biologically meaningful, voxel-wise group comparisons over both spatial and orientation domains. Using this approach, differences in AFD can be attributed to a particular fibre population even within voxels containing multiple fibre orientations. We also present a novel FOD modulation step, required during spatial normalisation to preserve a fibre bundle's total AFD (across the width of the bundle) under transformations that involve volume expansion or contraction. We then present a novel approach for statistical analysis of AFD using cluster-based inference of differences extended throughout space and orientation. Finally, we demonstrate the ability of the proposed AFD measure to detect differences between a group of Motor Neurone Disease patients and healthy control subjects.
Section snippets
Spherical deconvolution
The spherical deconvolution method provides an estimate of the distribution of fibres within each imaging voxel (Tournier et al., 2004). The method is based on the assumption that the measured HARDI signal can be expressed as the convolution over spherical coordinates of a single canonical fibre response function with the fibre orientation distribution (FOD). The response function is assumed to correspond to the DWI signal (as a function of orientation) that would be measured for a voxel with a
Qualitative analytical simulations
To aid the understanding of Apparent Fibre Density and spherical deconvolution in general, simulations were included to qualitatively demonstrate how the FOD amplitude (AFD) is affected by fibre partial volume fractions and deviations in the DW signal from the assumed single fibre response function. Given the shape of the fibre response function, we hypothesised that AFD is primarily affected by deviations along radial orientations.
We performed simulations by analytically modelling the
Analytical simulations
Figs. 7A–C illustrates the spherical deconvolution process and resulting FOD for a voxel with varying fibre volume fractions (where f1 = 0.5, 0.7, 1 respectively). As shown, the FOD amplitude is very sensitive to the relative fibre volume fractions of the underlying fibre populations. A comparison of Figs. 7C–E demonstrates the change in FOD observed when the actual fibre response deviates from the assumed response function. As shown in Fig. 7D a decrease in fibre axial diffusivity, (λ||),
Discussion
Interpreting population differences in DTI-derived scalar measures such as FA or tensor eigenvalues is challenging in regions with crossing fibres (Pierpaoli et al., 2001, Tuch et al., 2005, Douaud et al., Apr. 2011). In this work we have presented a new measure, called Apparent Fibre Density, that uses higher-order information provided by FODs to investigate population differences not only in space but also over orientation. This enables differences to be attributed to a single fibre bundle in
Conclusion
In this work we have presented a novel measure for the analysis of HARDI data that is robust in voxels with crossing fibres. Analytical and numerical simulations suggest that the FOD amplitude (AFD) can be used as a quantitative measure for population and longitudinal analysis. To enable robust voxel-based comparisons we presented a novel approach to modulate the FOD and capture differences in fibre bundle cross-sectional area that may occur during atrophy or abnormal development. We
Acknowledgments
J-DT and AC are grateful to the National Health and Medical Research Council (NHMRC) of Australia, Austin Health, and by the Victorian Government's Operational Infrastructure Support Program for their support.
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These two authors contributed equally to this work.