Continuous diffusion signal, EAP and ODF estimation via Compressive Sensing in diffusion MRI

Abstract

In this paper, we exploit the ability of Compressed Sensing (CS) to recover the whole 3D Diffusion MRI (dMRI) signal from a limited number of samples while efficiently recovering important diffusion features such as the Ensemble Average Propagator (EAP) and the Orientation Distribution Function (ODF). Some attempts to use CS in estimating diffusion signals have been done recently. However, this was mainly an experimental insight of CS capabilities in dMRI and the CS theory has not been fully exploited. In this work, we also propose to study the impact of the sparsity, the incoherence and the RIP property on the reconstruction of diffusion signals. We show that an efficient use of the CS theory enables to drastically reduce the number of measurements commonly used in dMRI acquisitions. Only 20–30 measurements, optimally spread on several b-value shells, are shown to be necessary, which is less than previous attempts to recover the diffusion signal using CS. This opens an attractive perspective to measure the diffusion signals in white matter within a reduced acquisition time and shows that CS holds great promise and opens new and exciting perspectives in diffusion MRI (dMRI).

Introduction

Diffusion MRI (dMRI) is a recent Magnetic Resonance Imaging technique introduced by (Le Bihan and Breton, 1985, Merboldt et al., 1985, Taylor and Bushell, 1985). Since the first acquisitions of diffusion-weighted images (DWIs) in vivo, dMRI has become an established research tool for the investigation of tissue structure and orientation.

In 1965, (Stejskal and Tanner, 1965) introduced the pulsed gradient spin-echo (PGSE) sequence. It allows the quantification of the diffusion by estimating the displacement of particles from the phase change that occurs during the acquisition process. When the gradient pulses are sufficiently short, it’s well known that the measured signal E(q), after normalization, 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.

Using dMRI to infer the EAP requires the acquisition of many diffusion images sensitized to different orientations in the sampling space. The number of diffusion weighted images (DWI) required depends on how the diffusion is modeled. For instance, the well known Diffusion Tensor (DT) model (Basser et al., 1994b, Basser et al., 1994a) assumes the EAP is Gaussian and requires at least 6 DWIs plus an additional unweighted image. However, the Gaussian assumption, in Diffusion Tensor Imaging (DTI), is an over-simplification of the diffusion of water molecules in the brain and, thus, has some limitations for voxels in which there are more complicated internal structures. Therefore, it is of utmost importance to develop techniques that go beyond the limitations of DTI. For this purpose, High Angular Resolution Diffusion Imaging (HARDI) has been proposed to measure the diffusion of water molecules along more directions than DTI does. Among HARDI techniques, there is Q-Ball Imaging (QBI) (Tuch, 2004, Anderson, 2005, Descoteaux et al., 2007), which estimates the Orientation Distribution Function (ODF) from measurements taken at same radii. The ODF gives the probability that a water molecule diffuses in a given direction. Both numerical (Tuch, 2004) and analytical (Descoteaux et al., 2007, Anderson, 2005) solutions have been proposed for QBI. At first, the ODF was defined as the integration of the EAP over its radius (Tuch, 2004). Only recently was considered the correct mathematical formulation (Aganj et al., 2010, Tristn-Vega et al., 2009), which results in the normalized ODF expression. (Wedeen et al., 2005, Aganj et al., 2010, Tristn-Vega et al., 2009) express the ODF ϒ(r) as the integration of the EAP over a solid angle, i.e.

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

However, the ODF only captures angular information of the diffusion process. Another HARDI technique has been proposed in (Jian et al., 2007), where the authors characterize the diffusion signal by a continuous mixture of Gaussian, resulting to 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. Considering basics of dMRI, another technique known as Diffusion Spectrum Imaging (DSI) appeared (Wedeen et al., 2005). In DSI, we obtain the EAP P(R) by directly taking the inverse Fourier transform of the normalized signal E(q) measured in the q-space (see Eq. (1)). It aims to reconstruct the EAP in a numerical way without any prior knowledge. This results in estimating the EAP in a more accurate fashion than any other methods.

However, many measurements and a long acquisition time are necessary to obtain high-resolution EAP. Therefore, it’s clear that there is a strong need for new techniques to estimate the whole EAP with fewer measurements. To this end, multiple shells HARDI methods have been used (zarslan et al., 2006, Assemlal et al., 2009, Ozarslan et al., 2009, Cheng et al., 2010b, Descoteaux et al., 2011, Wu and Alexander, 2007, Hosseinbor et al., 2011). They consist in acquiring the signal following multiple shells schemes and then, modeling it with an adequate basis. These techniques aim to catch both radial and angular information about the water diffusion process. However, an increase of the number of measurements is expected over methods as DTI or QBI. An important problem is to accurately estimate the diffusion signal and the underlying EAP with a small number of samples. A first answer has been given while using suitable bases as the Spherical Polar Fourier (SPF) basis (Assemlal et al., 2009), the SPF dual (SPFdual) basis (Merlet et al., 2011b), the Solid Harmonic (SoH) basis (Descoteaux et al., 2011) or the SHORE basis (Ozarslan et al., 2009). We give a complementary solution by using a new acquisition and reconstruction technique called Compressive Sensing (CS).

CS aims to accurately reconstruct signals from under sampled measurements (Donoho, 2006, Candes and Wakin, 2008). This method relies on several properties: The signal to recover admits a sparse representation; the basis, in which the signal is modeled, is sufficiently incoherent; a robust acquisition protocol and an efficient reconstruction scheme is used. CS has been proven useful in recovering Magnetic Resonance (MR) images by significantly undersampling their k-spaces (Lustig et al., 2007, Guo and Yin, 2010, Ganesh and Edward, 2008, Chartrand, 2009). The application of CS in diffusion MRI is recent and can be separated in two categories: discrete CS recovering and continuous CS recovering. Discrete CS recovering has been used to accelerate the DSI technique (Merlet and Deriche, 2010, Saint-Amant and Descoteaux, 2011, Menzel et al., 2011), by exploiting the Fourier relation between the diffusion signal and the EAP. However, the so called “CS-DSI” problem consists in reconstructing a discrete version of the EAP and diffusion features have to be computed numerically.

Continuous CS recovering consists in modeling a signal with a continuous framework from few measurements via a CS reconstruction. A continuous signal modeling is advantageous because it is not acquisition dependent and enables data interpolation and extrapolation. Some works have been published toward this (Michailovich and Rathi, 2010, Rathi et al., 2011, Cheng et al., 2011b, Merlet et al., 2011b, Tristán-Vega and Westin, 2011). In (Michailovich and Rathi, 2010, Tristán-Vega and Westin, 2011), the authors work with measurements taken at same radii and only estimate the ODF. (Rathi et al., 2011) generalizes the single shell spherical ridgelets basis of (Michailovich and Rathi, 2010) to a multiple shells framework for a sparse and continuous representation of the diffusion signal. In (Rathi et al., 2011), a total number of 64 measurements are used to well estimate the diffusion signal. However, it does not provide any analytical formula to estimate diffusion features. (Cheng et al., 2011b, Merlet et al., 2011b) consider a CS reconstruction combined with a continuous representation of the diffusion signal and available closed formulae to estimate the EAP and the ODF. (Merlet et al., 2011b) is about CS recovering in SPFdual basis with 80 measurements and (Cheng et al., 2011b) is about CS recovering in SPF basis from a minimum number of 60 measurements. These two papers give a first experimental insight of CS capabilities in dMRI, where analytical formulae are available to estimate the EAP and the ODF.

It is also worthwhile to note that very recent works started to handle the learning of dictionaries from a training data set (Ye et al., 2012, Merlet et al., 2012, Gramfort et al., 2012, Bilgic et al., 2012). These techniques lead to very sparse representations of diffusion signals and are worth to be minutely investigated. The analysis done in this paper considers predefined sets of functions that form orthonormal bases commonly used in the dMRI field. The question of which basis would result in the most efficient description of diffusion signals is not addressed here and is outside the scope of the paper.

In this paper, we investigate the Compressive Sensing technique in order to accurately and continuously estimate the full 3D diffusion phenomenon as well as some of its features with a very small number of samples. More precisely, we show that only 20/30 measurements are necessary to well estimate the diffusion signal. It is nearly three times less than previous studies encountered in (Merlet et al., 2011b, Cheng et al., 2011b, Rathi et al., 2011). This significant improvement over these previous works is due to a correct use and consideration of every point of the CS theory. Then, we demonstrate that it is worth using CS recovery, when CS requirements are fulfilled and we also demonstrate how to take advantage of this technique. Before starting with the central point of this paper, i.e. the CS technique, we describe, in the first section, four bases used to model the diffusion signal. Our approach consider common and continuous representations of the diffusion signal, which enable to obtain various diffusion features such the EAP and the ODF (as in (Merlet et al., 2011b, Cheng et al., 2011b)). Then, we give, in the second part (Section 3), a complete description of the CS properties. In Sections 3.1 On the incoherence property of CS bases, 3.2 On the sparsity of CS bases respectively, we study the incoherence and sparse properties of the bases described in Section 1. Section 3.3 describes the reconstruction scheme used in CS recovery and the related theoretical results. We also, handle in Section 3.3, the acquisition point by describing some theoretical tools to validate sampling protocols, i.e. a partial evaluation of the Restricted Isometry Property (RIP). These points are studied both in a theoretical and experimental way. In the last part, we present some experimental results confronting CS recovery and state of the art recovery. In this experimental part, we begin with synthetic data and focus our attention on several points: (1) the sampling protocol (Section 4.1), where a powerful technique is described to build robust sampling schemes and (2) the quality of reconstruction on noisy synthetic data (Section 4.2). Especially in Section 4.2, we demonstrate how efficient is the CS recovery in reconstructing the diffusion signal and the ODF. These synthetic experiments also enable to compare our CS-based EAP recovery with the EAP obtained via the DSI technique. Then, we give in Sections 4.3 Real human brain data, 4.4 Phantom data some results, respectively on real monkey brain data and phantom data.

From these experiments, we finally show that CS enables to accurately handle the whole diffusion process with a smaller number of samples than state-of-the-art methods (∼20/30 measurements), while modeling the diffusion signal in one of the bases described in the following.