A modified damped Richardson–Lucy algorithm to reduce isotropic background effects in spherical deconvolution
Introduction
In recent years diffusion MRI techniques have become an important research tool at the forefront of clinical neurosciences (Jones, 2008, Le Bihan et al., 2001). Diffusion tensor imaging (DTI) (Basser et al., 1994) and related tractography techniques (Basser et al., 2000, Behrens et al., 2003, Conturo et al., 1999, Mori et al., 1999, Parker et al., 2003) are the best known of these techniques. With these methods it is possible to study, in vivo, the microstructural organization of brain tissue and to extract several quantitative indices of anatomical integrity (Basser and Pierpaoli, 1996, Le Bihan, 2003). These models are not without limitations however. In particular the diffusion tensor model is unable to resolve multiple white matter orientations within voxels (Alexander, 2006, von dem Hagen and Henkelman, 2002). Alternative methods, such as diffusion spectrum imaging (DSI) (Wedeen et al., 2005), Q-ball imaging (Tuch, 2004), multiple compartment modelling (Behrens et al., 2007, Behrens et al., 2003, Jbabdi et al., 2007), Bayesian random effect modelling (King et al., 2009), Hybrid (Wu and Alexander, 2007), persistent angular structure MRI (PASMRI) (Jansons and Alexander, 2003) and spherical deconvolution (SD) (Anderson, 2005, Tournier et al., 2004) have been developed to address this problem.
Among these, the SD approach has the advantage of relatively short acquisition times, which are close to standard DTI clinical protocols, reduced computational times compared to some of the other methods, and the ability to resolve crossing fibres with a good angular resolution. SD is based on the assumption that the acquired diffusion signals from a single voxel can be modelled as a spherical convolution between the fibre orientation distribution (FOD) and the fibre response function that describes the common signal profile from the white matter (WM) bundles contained in the voxel (Tournier et al., 2004). Hence, the FOD could, in theory, provide a description that is closer to the physical fibre orientations than other methods that rely on the estimation of the water molecular displacement profile or the spin propagator (Alexander, 2005a, Tournier et al., 2007).
With the spherical deconvolution approach, highly ill-conditioned solutions are likely to occur due to the ill-posed nature of the deconvolution problem however. In the absence of regularization of the solution, even small changes in the acquired signal (e.g., MR noise) can lead to non-physical results (Bertero and Boccacci, 1998, Tournier et al., 2004). The introduction of non-negative constraints to the SD approach can improve the FOD estimation (Alexander, 2005a, Dell'Acqua et al., 2007, Jian and Vemuri, 2007, Tournier et al., 2007). Despite these recent developments, other sources of instability can still significantly affect the results in SD. The signal contribution from isotropic tissue, for example, is usually not included in spherical deconvolution models. Some authors have included an isotropic compartment in their signal model but these methods either require multiple b-value acquisitions (Jespersen et al., 2007), or apply stronger constraints on the solution, such as a finite number of fibre orientations (Behrens et al., 2007, Behrens et al., 2003) or known diffusivities (Kaden et al., 2008). No studies have, to date, investigated the direct effect of isotropic contributions on the FOD estimation.
To exclude voxels with isotropic partial volume (e.g., from grey matter (GM) or cerebrospinal fluid (CSF)), a threshold on fractional anisotropy maps (Pierpaoli and Basser, 1996), or similar maps, can been used to reject voxels with low anisotropy values. This approach is not a viable solution in those regions with low anisotropy due to highly complicated fibre orientations. Alternatively, Sakaie and Lowe have recently proposed the use of an optimised level of regularization of the estimated FOD at each voxel (Sakaie and Lowe, 2007).
In this study we combine a previously published Richardson–Lucy (RL) spherical deconvolution algorithm (Dell'Acqua et al., 2007) with an adaptive regularization technique to reduce spurious and non-physical fibre orientations in regions affected by partial volume whilst, at the same time, preserving angular resolution of the main fibre orientations. This new iterative algorithm can be described as a damped version of the RL–SD algorithm (dRL) as it uses the absolute dynamic range of the recovered fibre orientation to modulate the regularization of the solution. The results of simulations and in vivo data analysis are compared to assess the performance of the damped and the standard non-negative constrained RL algorithm using different levels of isotropic partial volume. The potential of the proposed method for application to patient studies is also assessed using scanning parameters that are usually applied in conventional DTI analysis (Jones et al., 2002).
Section snippets
Modelling isotropic partial volume in spherical deconvolution
The SD approach models the signal acquired during a high angular resolution diffusion imaging (HARDI) (Frank, 2001) experiment as:where S is the normalized signal, SDWI is the diffusion weighted signal acquired along the direction v, S0 is the non-diffusion weighted signal, fod(r) is the fibre orientation distribution and h(r,v) is the fibre response model that describes the signal acquired along v from a single fibre with orientation r, where
Simulations
Fig. 3 shows the results of the simulated datasets (b-value 3000 s/mm2) for an isotropic compartment of GM (Fig. 3a) and CSF (Fig. 3b) diffusivity. The black lines describe the profile of the RL algorithm, while the blue lines represents the profile of the dRL algorithm at different values of the threshold parameter η.
Discussion and conclusions
In this work we studied the effects of isotropic partial volume on the estimation of multiple-fibre orientations using a non-negative constrained SD algorithm and developed a novel algorithm, namely the dRL, to correct these effects. The novel algorithm was tested for its ability to reduce false positive fibre orientations and preserve angular resolution on both simulated and in vivo datasets.
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