Structured sparsity for spatially coherent fibre orientation estimation in diffusion MRI

Abstract

We propose a novel formulation to solve the problem of intra-voxel reconstruction of the fibre orientation distribution function (FOD) in each voxel of the white matter of the brain from diffusion MRI data. The majority of the state-of-the-art methods in the field perform the reconstruction on a voxel-by-voxel level, promoting sparsity of the orientation distribution. Recent methods have proposed a global denoising of the diffusion data using spatial information prior to reconstruction, while others promote spatial regularisation through an additional empirical prior on the diffusion image at each q-space point. Our approach reconciles voxelwise sparsity and spatial regularisation and defines a spatially structured FOD sparsity prior, where the structure originates from the spatial coherence of the fibre orientation between neighbour voxels. The method is shown, through both simulated and real data, to enable accurate FOD reconstruction from a much lower number of q-space samples than the state of the art, typically 15 samples, even for quite adverse noise conditions.

Introduction

The challenge in diffusion MRI is to infer features of the local tissue anatomy, composition and microstructure from water displacement measurements. Water diffusion in living tissues is highly affected by its cellular organization (Beaulieu, 2002). In particular, water does not diffuse equally in all directions in a highly ordered organ such as the brain and this property can be exploited to study the structural neural connectivity in a non-invasive way. The estimation of fibre connectivity patterns in vivo represents a major goal in neuroscience but also in a clinical perspective, with applications for diagnosis of stroke, schizophrenia or Parkinson's disease. In order to reconstruct entire fibre pathways and hence brain connections, tractography algorithms nowadays rely on the orientations of maximal water diffusion in each voxel. Thus, an accurate reconstruction of the local fibre populations is crucial to ensure good performance of fibre-tracking.

A great variety of approaches have been proposed to tackle the problem of intra-voxel fibre orientation estimation. Diffusion Tensor Imaging (DTI) (Basser et al., 1994) is one of the simplest and fastest reconstruction techniques since it only requires sampling 6 points of the q-space. However, it is by construction unable to model multiple fibre populations within a voxel and thus it is not valid in regions with crossings. Diffusion Spectrum Imaging (DSI) (Wedeen et al., 2005), on the other hand, is a model-free imaging technique known to provide good imaging quality. Yet, it requires strong magnetic field gradients and long acquisition times, needing typically 256 samples for a good reconstruction. As a consequence, it generally becomes too time-consuming to be of real interest in a clinical perspective. Accelerated acquisitions, relying on as few sampling points as possible while still sensitive to fibre crossings represent thus a major goal in the field.

In the last years, spherical deconvolution (SD) methods (Tournier et al., 2004, Tournier et al., 2007, Alexander, 2005) have become very popular in the framework of local reconstruction since they can recover the fibre configuration with a relatively small number of points, typically from 30 up to 60. They consider that both anisotropy and magnitude of water diffusion in white matter (WM) are constant in the whole volume. Under this assumption, SD methods acknowledge the fact that the diffusion signal can be expressed as the convolution of a response function, or kernel, with the fibre orientation distribution function (FOD). The FOD is a real-valued function on the unit sphere that indicates the orientation and the volume fraction of the fibre populations in a voxel. The Constrained Spherical Deconvolution approach of Tournier et al., 2004, Tournier et al., 2007 represents the first attempt to solve the ill-posed SD problem. It applies Tikhonov regularisation, introducing a constraint on the ℓ₂ norm of the FOD, specially to ensure its positivity. Apart from the aforementioned work, most of the state-of-the-art methods to solve SD problems promote sparse regularisation based on ℓ₁ minimisation (Jian and Vermuri, 2007, Ramirez-Manzanares et al., 2007, Mani et al., 2014), where the ℓ₁ norm is defined, for any real vector, as the sum of the absolute value of its coefficients. Yet, Daducci et al. (2014b) acknowledge in recent work that ℓ₁ minimisation is formally inconsistent with the fact that the volume fraction sums up to unity, and demonstrate the superiority of ℓ₀-norm minimisation. All these local reconstruction methods solve the FOD recovery problem for each voxel independently and thus, do not exploit the spatial coherence of the fibre tracts in the brain. A number of approaches have addressed this shortcoming by formulating the problem globally (simultaneously for all voxels) to be able to exploit the correlation between the different volumes. Some of them decouple the problem and propose a global denoising of the diffusion data prior to reconstruction (Tristán-Vega and Aja-Fernández, 2010, Wiest-Daesslé et al., 2008). Another group of methods present a joint scheme for reconstruction and spatial regularisation on the diffusion images at each q-space point. For instance, Fillard et al. (2007) propose a variational formulation to jointly estimate and regularise DTI to account for the effect of Rician noise in low SNR regimes, while Mani et al. (2014); and Michailovich et al. (2011) use the standard state-of-the-art minimisation of the total variation (TV) semi-norm (Rudin et al., 1992) of the diffusion images.

In this paper, we propose a formulation that solves the fibre configuration of all voxels of interest simultaneously and imposes spatial regularisation directly on the fibre space. This reconstruction allows us to exploit information from the neighbouring voxels that cannot be taken into account by the existing state-of-the-art methods that approach fibre reconstruction independently in each voxel. The natural smoothness of the anatomical fibre tracts through the brain can be translated in a certain spatial coherence of the FOD in neighbouring voxels. Accordingly, in the aim of recovering the global FOD field in all voxels, the present work leverages a reweighted ℓ₁-minimisation scheme to promote a spatially structured sparsity prior imposing spatial coherence. While the spatial regularisation schemes proposed by Fillard et al. (2007); Mani et al. (2014); and Michailovich et al. (2011) enforce sparsity of the images at each q-space point, our spatial regularisation relates to the fundamental coherence between fibre directions – the FOD – in neighbour voxels, thus adding anatomically driven constraints. Our code is available at https://github.com/basp-group/co-dmri and it is distributed open-source.