A computational diffusion MRI and parametric dictionary learning framework for modeling the diffusion signal and its features

Highlights

• We develop an parametric dictionary learning algorithm to recover the dMRI signal with a reduced number of measurements.

• We propose a computational framework to model continuous dMRI signals and analytically recover the EAP, the ODF.

• This approach indicates a much better accuracy in terms of reconstruction error compared to state-of-the-art approaches.

Abstract

In this work, we first propose an original and efficient computational framework to model continuous diffusion MRI (dMRI) signals and analytically recover important diffusion features such as the Ensemble Average Propagator (EAP) and the Orientation Distribution Function (ODF). Then, we develop an efficient parametric dictionary learning algorithm and exploit the sparse property of a well-designed dictionary to recover the diffusion signal and its features with a reduced number of measurements. The properties and potentials of the technique are demonstrated using various simulations on synthetic data and on human brain data acquired from 7T and 3T scanners. It is shown that the technique can clearly recover the dMRI signal and its features with a much better accuracy compared to state-of-the-art approaches, even with a small and reduced number of measurements. In particular, we can accurately recover the ODF in regions of multiple fiber crossing, which could open new perspectives for some dMRI applications such as fiber tractography.

Introduction

Diffusion MRI (dMRI) assesses the integrity of brain anatomical connectivity and is very useful for examining and quantifying white matter (WM) microstructure and organization not available with other imaging modalities. dMRI determines the WM structure by exploiting the way the water molecules diffuse. The first diffusion images were obtained in the mid-1980s (Le Bihan et al., 1986), and was based on the pioneering work of Stejskal and Tanner (1965), who introduced the pulsed gradient spin-echo (PGSE) sequence. It allows the quantification of the water diffusion by estimating the displacement of water particles from the phase change that occurs during the acquisition process. More importantly, under the so called narrow pulse assumption, we can show that the normalized signal attenuation E(q) is written as the Fourier transform of the Ensemble Average Propagator (EAP) P(R)

$$E(\mathbf{q})={\int }_{\mathbf{R}\in {\mathcal{R}}^3}P(\mathbf{R})\exp (-2\pi i\mathbf{q}·\mathbf{R})d\mathbf{R}\text{,}$$

where q and R are both 3D-vectors that respectively represent the effective gradient direction and the displacement direction. We can decompose them as q = qu and R = Rr, where u and r are 3D unit vectors.

Diffusion Tensor Imaging (DTI) (Basser et al., 1994b, Basser et al., 1994a) method characterizes the diffusion by a Gaussian distribution, and is known to be a limited model. In particular DTI is not able to resolve crossing fibers. Resolving crossing fibers helps to disambiguate between several possible tracts in regions of crossing fibers and reconstruct more accurate anatomical connectivity through fiber tractography. Recently, more complex models appeared to overcome this limitation. Nevertheless, these techniques often require many acquisitions in particular when High Angular Resolution Diffusion Imaging (HARDI) (Tuch, 2004, Anderson, 2005, Tristan-Vega et al., 2009, Jian et al., 2007, Aganj et al., 2010) or Diffusion Spectrum Imaging (DSI) (Wedeen et al., 2005) are used. HARDI techniques allow the estimation of the Orientation Distribution Function (ODF) (Tuch, 2004, Anderson, 2005, Tristan-Vega et al., 2009, Aganj et al., 2010), which gives the probability that a water molecule diffuses in a given direction. Several authors (Aganj et al., 2010, Tristan-Vega et al., 2009) express the ODF ϒ(r) as the integration of the EAP over a solid angle, i.e.

$$\mathrm{\Upsilon }(\mathbf{r})={\int }_0^{\infty }P(R·\mathbf{r})R^2\mathit{dR}\text{.}$$

In Tournier et al., 2007, Alexander, 2005, the authors propose to estimate the fiber orientation distribution called the fiber ODF (fODF). The fODF is able to resolve up to 30 degrees crossings consistently (Tournier et al., 2008), which makes this model a promising ressource to estimate fiber orientation. In our work, we reconstruct the ODF as described in Eq. (2) and we do not aim to compare our approach with the family of method estimating the fODF. A review of method reconstructing the ODF and the fODF can be found in Johansen-Berg and Behrens (2009). Another HARDI technique has been proposed in Jian et al. (2007), where the authors characterize the diffusion signal by a Wishart distribution. Jian et al. (2007) shows improvements over the classical DTI technique and present an estimation scheme for the fiber orientation and EAP. Among the HARDI techniques (Zhang et al., 2012), introduces Neurite Orientation Dispersion and Density Imaging (NODDI), which allows the estimation of the microstructural complexity of dendrites and axons. Diffusion Spectrum Imaging (DSI) was developed in parallel to the HARDI techniques (Wedeen et al., 2005). In DSI, the EAP P(R) is directly obtained by taking the inverse Fourier transform of the normalized signal E(q) measured in the q-space (see Eq. (1)). However, the high resolution EAP obtained with DSI requires many measurements. HARDI and DSI are impractical for clinical use in MRI systems commonly found in hospital. Accelerated acquisitions, relying on a smaller number of sampling points, are thus very welcome to efficiently estimate the complex features of the diffusion process.

Sparse reconstruction approaches were found to successfully reduce the number of acquisitions in dMRI (Merlet and Deriche, 2010, Menzel et al., 2011, Michailovich et al., 2011, Rathi et al., 2011, Tristan-Vega and Westin, 2011, Bilgic et al., 2012, Gramfort et al., 2012, Merlet et al., 2012a, Merlet and Deriche, 2013, Ye et al., 2012). These techniques are usually based on a l₁ minimization of the diffusion signal with respect to a sparse representation. Merlet and Deriche, 2010, Menzel et al., 2011 (Merlet and Deriche, 2010, Menzel et al., 2011) combine the Compressive Sensing (CS) theory and DSI to accelerate the acquisition. Merlet and Deriche (2013) (Merlet and Deriche, 2013) use orthonormal bases to sparsely describe the diffusion signal. In Michailovich et al. (2011) and in Tristan-Vega and Westin (2011), the authors elegantly design dictionaries for sparse modeling in dMRI. They provide an overcomplete dictionary computed from a discretized version of predefined functions, i.e. the Spherical Ridgelets in Michailovich et al. (2011) (see Rathi et al. (2011) for the multiple shells version) and the Spherical Wavelets in Tristan-Vega and Westin (2011). Learning a dictionary provides an alternative way to design sparse dictionaries (Bilgic et al., 2012, Gramfort et al., 2012, Merlet et al., 2012a, Ye et al., 2012).

Some approaches have been recently proposed in order to design dictionaries that enable sparse representations (A good overview can be found in Aharon et al. (2006)). For instance, Bilgic et al., 2012, Gramfort et al., 2012 (Bilgic et al., 2012, Gramfort et al., 2012) learn dictionaries from DSI like acquisitions and use it to either denoise full DSI data or to perform undersampled DSI acquisitions and reconstructions. In particular, Gramfort et al. (2012) nicely exploit the symmetry of the signal in order to assess free parameters of the dictionary learning problem. However, these two latter works lead to non-parametric dictionaries, which does not provide continuous representations of the diffusion signal nor allow the determination of analytical formulae for diffusion features. The strength of the parametric dictionary learning approach, as the one we propose in this article, lies in its ability to address these weaknesses. A work regarding parametric dictionary learning was published in Ye et al. (2012), in which the dictionary atoms are formed by a weighted combination of 3rd order B-splines. It proved that the method is efficient on synthetic data simulated with 81 gradient directions. The work of Ye et al. (2012) appears promising in reconstructing the diffusion signals, and further enhancement could be done regarding the development of analytical formulae to estimate other diffusion features. This would make this work a good resource in the context of dictionary learning. More recently, we proposed in Merlet et al. (2012a) to learn a dictionary where each atom is constrained to be a parametric function. In Merlet et al. (2012a), this parametric function is a combination of a radial part and an angular part represented by the symmetric and real Spherical Harmonics (SH) (Descoteaux et al., 2007). The radial part is a polynomial weighted by an exponential. 50 measurements were sufficient to reconstruct very good quality diffusion signals, ODFs and EAPs. However, this approach essentially handles the learning of the radial part, i.e. the polynomial coefficients and a scale parameter in the exponential, whereas we observed (see Merlet et al., 2012b) that the angular part could make the dictionary much sparser if we adequately combine several SH functions instead of only one.

In this work, we present a method, which exploits the sparse property of a well designed dictionary based on a computational dMRI framework, in order to recover the diffusion signal with a reduced number of measurements. This framework enables a continuous modeling of the diffusion signal and leads to analytical formulae to estimate important diffusion features, namely the ODF and the EAP. To improve our previous work in Merlet et al. (2012a), we modify the parametric function, describing the atoms, to learn both the radial and the angular part, which provide a very sparse representation of diffusion signals and further reduce the number of measurements (15 measurements are found to be sufficient to start recovering the EAP and some derived diffusion features whereas 50 measurements are used in Merlet et al. (2012a)). Furthermore, we extend the experimental part of Merlet et al. (2012a) by learning and validating our approach on the synthetic data proposed in the HARDI contest at ISBI 2012,¹ and on real data acquired from both 3T and 7T scanners. A preliminary work (Merlet et al., 2012b) regarding the learning of both the radial part and the angular part of the diffusion signal was published in the proceedings of the HARDI contest at ISBI 2012 and we obtained the best results in our category. Our approach presented in this paper indicates an increase in terms of reconstruction accuracy compared to the results presented in Merlet et al. (2012b).

The article is structured as follows: we start by introducing the dMRI framework together with the proposed dictionary, then we focus on the parametric dictionary learning algorithm and finally we conclude with an experimental part illustrating the added-value of our approach with promising results showing how our approach allows the accurate reconstruction of the diffusion signal and some of its features. This experimental part is completed by a comparison with state of the art approaches, and is performed on synthetic and real data from 3T and 7T scanners.