Robust determination of the fibre orientation distribution in diffusion MRI: Non-negativity constrained super-resolved spherical deconvolution
Introduction
The diffusion tensor model is currently the mathematical framework most widely used to relate the diffusion-weighted magnetic resonance imaging (DW-MRI) signal to the underlying diffusion process, and provides a number of useful parameters (Basser et al., 1994). Of particular significance is the major eigenvector of the diffusion tensor, corresponding to the direction of fastest diffusion (Basser, 1995). The direction of this vector has been shown to correspond well with the orientation of the fibres in a number of major white matter structures, such as the optic tract of the rat (Lin et al., 2003). Based on this premise, a number of tractography algorithms have been proposed to map the path of white matter tracts in the brain (e.g. Mori et al., 1999, Conturo et al., 1999, Parker et al., 2002, Behrens et al., 2003, Tournier et al., 2003). Most of these techniques rely on the diffusion tensor model to provide an accurate estimate of the white matter fibre orientation.
Unfortunately, the diffusion tensor model is not always adequate, particularly in voxels containing contributions from differently oriented fibre bundles (Alexander et al., 2002, Frank, 2002, Tuch et al., 2002, Behrens et al., 2007). The reason for this is two-fold. First, the diffusion tensor model is only strictly valid for free diffusion, and is therefore only an approximation for the in vivo case. Second, and more importantly, the diffusion tensor can only possess a single maximum, and is therefore unable to adequately characterise a system consisting of several distinct fibre orientations. At the resolution currently achievable with DW-MRI, a large number of voxels will contain contributions from different bundles with distinct orientations, where the diffusion tensor model will provide a poor fit to the data (Alexander et al., 2002). In particular, the major eigenvector will in general no longer correspond to the orientation of any of the fibre tracts present (Assaf et al., 2004). It should be emphasised that a significant proportion of white matter voxels are affected by this problem. A recent study estimates that one third of white matter voxels contain more than one fibre population (Behrens et al., 2007). Moreover, many of the major tracts in the brain do pass through regions containing multiple fibre populations at some point along their path, causing tractography applications that rely on the diffusion tensor model to provide unreliable results if the tracks produced happen to venture into affected voxels (Pierpaoli et al., 2001).
We recently presented a technique that is capable of estimating the distribution of fibre orientations within each voxel directly from the diffusion-weighted (DW) data, using the concept of spherical deconvolution (Tournier et al., 2004). A number of other approaches have also been proposed to address this issue, but there are some inherent limitations with each of them. Q-ball imaging (Tuch, 2004), diffusion spectrum imaging (DSI) (Wedeen et al., 2005) and the CHARMED model (Assaf et al., 2004) all require large b-values that are difficult to obtain on a clinical system. Moreover, DSI and CHARMED both require a wide range of b-values, leading to impractical scan times. PAS-MRI (Jansons and Alexander, 2003) is very computer-intensive, which limits its widespread practical use. The recently proposed diffusion orientation transform (Özarslan et al., 2006) assumes monoexponential signal decay, which has been shown to be not valid for high q-values in vivo (Cohen and Assaf, 2002). It is worth noting that these techniques are all rooted in the q-space formalism (Callaghan et al., 1988) and thus provide an approximation to the spin propagator (or more commonly just its angular dependence), rather than the fibre orientations themselves. Although the two are clearly related, there may be cases where that relationship is not straightforward (for example, fibres crossing at angles other than 90° (Zhan and Yang, 2006)).
Multiple tensor fitting algorithms (Tuch et al., 2002, Hosey et al., 2005, Behrens et al., 2007) provide discrete fibre orientations, but tend to perform poorly when more than two fibre orientations are present. FORECAST (Anderson, 2005) is a generalisation of multiple tensor fitting, and is in essence a spherical deconvolution; we anticipate that it would also benefit from the modifications proposed here.
The spherical deconvolution technique that we proposed is not affected by the problems mentioned above. However, like most deconvolution problems, this method is ill-posed and susceptible to noise. In this study, we present a novel iterative method to perform the spherical deconvolution that preserves the angular resolution while remaining robust to noise effects. This is done by placing a non-negativity constraint of the estimated FOD, as negative fibre orientation densities are physically impossible. This constraint eliminates the need for low-pass filtering, and indeed provides enough prior information to estimate the FOD with a higher resolution than would otherwise be possible from the data alone (this is known as super-resolution; Starck et al., 2002). The method is described below, and its accuracy and precision are assessed using simulations and in vivo data (including a bootstrap analysis; Jones, 2003).
Section snippets
Spherical deconvolution
The method of spherical deconvolution (Tournier et al., 2004) can be used to estimate the distribution of fibre orientations present within each imaging voxel. With this method, the signal measured during a high angular resolution DW imaging experiment can be expressed as the convolution over spherical coordinates of the response function with the fibre orientation distribution (FOD). The response function describes the DW signal intensity that would be measured as a function of orientation for
Response function estimation
As mentioned in the Theory section, spherical deconvolution requires an estimate of the response function, corresponding to the DW signal that would be measured as a function of orientation for a single white matter fibre bundle aligned with the z-axis. Although the diffusion tensor model provides a good fit to the DW signal for a single fibre population at low b-values, it has been shown to deviate significantly at higher b-values (Mulkern et al., 1999, Clark and Le Bihan, 2001). Therefore,
Optimisation of regularisation parameters
The threshold parameter τ determines which orientations are assumed to have zero fibre density. Ideally, this would be set to zero. However, with super-CSD, the algorithm may fail to converge if the number of zero-amplitude directions identified falls below the minimum required. This was found to occur frequently with τ = 0, particularly in grey matter regions. Setting τ to a value greater than zero improves the stability of the algorithm, by allowing more directions to be identified as having
Discussion
We have shown that the CSD method presented here can significantly improve the robustness of the spherical deconvolution technique, and therefore of the estimated fibre orientations present in each voxel. This is especially important for fibre-tracking applications that rely on accurate estimates of white matter orientation. With CSD, white matter fibre orientations can be estimated to within 3.5° from a typical in vivo acquisition (b = 3000 s/mm2, SNR = 30, for a 90° crossing). The benefits of the
Conclusion
We have presented a novel method to perform spherical deconvolution that includes a constraint on the presence of negative values in the fibre orientation distribution (FOD), which are physically impossible. The addition of this a priori information also permits the use of super-resolution, whereby the FOD is estimated with higher angular resolution than would otherwise be possible from the data alone. This improves the reliability of the resulting estimates of the FOD within each voxel, and
Acknowledgments
We thank David Gadian and the Radiology and Physics Unit at the UCL Institute of Child Health, University College London, London, UK, for helpful discussion and for the use of in vivo data. We also thank Olga Ciccarelli and co-workers at the UCL Institute of Neurology, University College London, London, UK, for providing in vivo data set C. We are grateful to the National Health and Medical Research Council (NHMRC) and Austin Health for support.
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