Multi-shell diffusion signal recovery from sparse measurements

doi:10.1016/j.media.2014.06.003

Medical Image Analysis

Volume 18, Issue 7, October 2014, Pages 1143-1156

https://doi.org/10.1016/j.media.2014.06.003 Get rights and content

Highlights

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We present a novel method for reconstruction of the diffusion MRI signal from sparse measurements.
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A novel function to model the bi-exponential decay of the signal in the radial q-space.
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Novel algorithmic derivation for spatially smooth signal reconstruction.
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Extensive experiments done on phantom and in vivo data.
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Our results show very low error in the recovered signal using at-least 60 samples.

Abstract

For accurate estimation of the ensemble average diffusion propagator (EAP), traditional multi-shell diffusion imaging (MSDI) approaches require acquisition of diffusion signals for a range of b-values. However, this makes the acquisition time too long for several types of patients, making it difficult to use in a clinical setting. In this work, we propose a new method for the reconstruction of diffusion signals in the entire q-space from highly undersampled sets of MSDI data, thus reducing the scan time significantly. In particular, to sparsely represent the diffusion signal over multiple q-shells, we propose a novel extension to the framework of spherical ridgelets by accurately modeling the monotonically decreasing radial component of the diffusion signal. Further, we enforce the reconstructed signal to have smooth spatial regularity in the brain, by minimizing the total variation (TV) norm. We combine these requirements into a novel cost function and derive an optimal solution using the Alternating Directions Method of Multipliers (ADMM) algorithm. We use a physical phantom data set with known fiber crossing angle of 45° to determine the optimal number of measurements (gradient directions and b-values) needed for accurate signal recovery. We compare our technique with a state-of-the-art sparse reconstruction method (i.e., the SHORE method of Cheng et al. (2010)) in terms of angular error in estimating the crossing angle, incorrect number of peaks detected, normalized mean squared error in signal recovery as well as error in estimating the return-to-origin probability (RTOP). Finally, we also demonstrate the behavior of the proposed technique on human in vivo data sets. Based on these experiments, we conclude that using the proposed algorithm, at least 60 measurements (spread over three b-value shells) are needed for proper recovery of MSDI data in the entire q-space.

Graphical abstract

Introduction

Diffusion MRI (dMRI) is an imaging modality that is sensitive to the neural architecture and connectivity of the brain. Consequently, it is increasingly being used in clinical settings for investigating several brain disorders such as, Alzheimer’s disease, stroke, schizophrenia and mild traumatic brain injury (Thomason and Thompson, 2011, Shenton et al., 2012). Apart from more traditional Diffusion Tensor Imaging (DTI), it is nowadays standard to use High Angular Resolution Diffusion Imaging (HARDI), which involves acquiring diffusion signals at a single b-value (single q-shell) in several gradient directions spread over the unit sphere in a quasi-uniform manner (Tuch et al., 2003, Assemlal et al., 2011). While this protocol allows for resolving the complex angular structure of the neural fibers, it does not provide information about the radial signal decay, which is known to be sensitive to various anomalies of white matter (Cohen and Assaf, 2002).

To obtain accurate information about the neural architecture, diffusion spectrum imaging (DSI) was proposed by Wedeen et al. (2005). This dMRI technique involves acquiring multiple measurements over a Cartesian grid of points in the q-space, followed by application of discrete Fourier transform to obtain an estimate of the ensemble average propagator (EAP). Unfortunately, a large number of measurements required by DSI makes it impractical to use in clinical settings. Accordingly, to speed-up the acquisition of dMRI (and DSI) data, two complementary approaches have been proposed, namely: (i) the use of compressed sensing (CS) to reduce the number of measurements (Candès et al., 2006, Donoho, 2006), and (ii) the use of multi-slice acquisition sequences for faster data acquisition (Setsompop et al., 2011, Feinberg et al., 2010). This work focuses on methodology (i), i.e., CS-based reconstruction of diffusion signal from critically undersampled measurements.

Several imaging and analysis schemes, which use fewer measurements than traditional DSI, have recently been proposed in the literature (Wu and Alexander, 2007, Jensen et al., 2005, Assemlal et al., 2011, Merlet et al., 2012, Barmpoutis et al., 2008, Descoteaux et al., 2010, Zhang et al., 2012, Ye et al., 2011, Ye et al., 2012, Hosseinbor et al., 2013). Each of these techniques captures a different aspect of the underlying tissue organization, which is missed by HARDI. Traditional methods of EAP estimation that account for the non-monoexponential (radial) decay of diffusion signals, require a relatively large number of measurements at high b-values (greater than 3000 s/mm²) (Assaf et al., 2004, Mulkern et al., 2001). Consequently, their associated scan times are deemed to be too long for non-cooperative patients, which is the main motivation for reducing the number of measurements in dMRI scans.

Although not new in application to MRI, CS-based methods of signal reconstruction has gained significant attention in the diffusion imaging community over the last few years. Several works have proposed CS-based algorithms for recovering HARDI, MSDI as well as DSI data from undersampled (aka incomplete) measurements (Ye et al., 2011, Merlet et al., 2012, Landman et al., 2012, Gramfort et al., 2012, Duarte-Carvajalino et al., 2012, Freiman et al., 2013, Scherrer et al., 2013, Assemlal et al., 2011, Michailovich et al., 2011, Rathi et al., 2011). To this end, various types of signal representation bases have also been proposed, each having different sparsifying properties. For example, for HARDI data, spherical ridgelets were proposed in Michailovich et al., 2008, Michailovich and Rathi, 2010, and for MSDI data, spherical polar Fourier (SPF) and its variants (SHORE) were proposed in (Assemlal et al., 2008, Ozarslan et al., 2008, Cheng et al., 2010, Merlet et al., 2012). In the case of the SHORE basis, to optimize the accuracy of signal reconstruction, one has to choose an appropriate scaling parameter, which could potentially be different for different types of tissue. To address this issue, (Merlet et al., 2012) used a dictionary learning technique to learn the scaling parameter and the appropriate polynomial to represent the radial decay term. On the other hand, in Özarslan et al. (2013), this scaling parameter was adaptively obtained in a data driven fashion by computing the eigenvalues of a tensor at each voxel. However, at a fundamental level, both these methods extend the original SHORE basis to sparsely represent the diffusion data. In this work, we will compare our technique with the SHORE-based reconstruction (Cheng et al., 2010, Merlet and Deriche, 2013), where sparsity is enforced through the standard $l_{1}$ -norm minimization. In our earlier work (Rathi et al., 2011), we had also proposed a basis that combined the spherical ridglets with a radial term. However, the cost function used in that work was non-convex, making it quite susceptible to local minima. In this work, we propose significant modifications and address the limitations of our earlier work, as discussed in the next section.

Section snippets

Our contributions

The framework of spherical ridgelets (SR) proposed in Michailovich et al. (2011) was used to recover HARDI data on a single b-value shell from highly undersampled set of diffusion measurements. In this work, we propose a novel extension of this basis for recovering multi-shell diffusion data. Towards this end, we incorporate a novel radial decay term which is a monotonically decreasing function with its range bounded between 0 and 1. This property is quite desirable, since it is known that the

Diffusion MRI

Under the narrow pulse assumption, the diffusion signal $S (q)$ in the q-space is related to the EAP $P (r)$ via the Fourier transform as given by Stejskal and Tanner (1965) $P (r) = \int_{r \in R^{3}} E (q) \exp (- i 2 π q \cdot r) d q,$ where $E (q) ≜ S (q) / S (0) : R^{3} \to [0, 1]$ is the normalized diffusion signal, with $S (q)$ and $S (0)$ being the measured diffusion signal and its corresponding $b = 0$ value, respectively. Alternatively, E can be written as a function of b-value and a unit vector $u \in S^{2}$ , such that $E (b, u) : R^{+} \times S^{2} \to [0, 1]$ , where $b = γ^{2} δ^{2} (Δ - δ / 3) ‖ g ‖^{2} s /$

Methods

In this work, it is assumed that the dMRI data is measured along the same K gradient directions ${u_{k}}$ at several b-values. This allows for estimating the signal attenuation with increasing b-values along each gradient direction. In particular, we propose to model the radial signal decay of $E (q)$ ( $E (q) = S (q) / S (0)$ , where the signal $S (q)$ is assumed to be normalized by the signal at $b = 0$ i.e. $S (0)$ ) using a monotonically decreasing function given by $f_{α, β} (b) = {(1 + b^{α})}^{- β}, α ⩾ 0, β ⩾ 0,$ which is related to the

Experiments

To evaluate the proposed algorithm, we constructed a physical phantom with diffusion properties similar to that of human brain tissue. The phantom consisted of a spherical spindle wound with 15 μm polyfil fibers to obtain a 45° crossing angle. A detailed description of how the phantom was made is given in (Moussavi-Biugui et al., 2011). The phantom was scanned using a Siemens 3T scanner at a spatial resolution of $2 mm \times 2 mm \times 7 mm$ , so that the crossing region lay in the center of the axial slice. We

Discussion

In this work, we proposed a novel framework for recovering dMRI data in the entire q-space from very few measurements. In particular, the proposed method is designed to use diffusion samples acquired at multiple q-shells by extending the spherical ridgelets basis to be used within a MSDI framework. We proposed a new monotonically decreasing radial decay function to be used within the estimation framework, which allows accurate modeling of single and multi-exponential attenuation of diffusion

Acknowledgements

This work has been supported by NIH Grants: R01MH097979 (Rathi), R01MH074794 (Westin), P41RR013218, P41EB015902 (Kikinis, Core PI: Westin), and Swedish research Grant VR 2012-3682 (Westin).

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View full text

Multi-shell diffusion signal recovery from sparse measurements

Highlights

Abstract

Graphical abstract

Introduction

Section snippets

Our contributions

Diffusion MRI

Methods

Experiments

Discussion

Acknowledgements

Med. Image Anal.

Appl. Comput. Harmon. Anal.

NeuroImage

NeuroImage

Med. Image Anal.

Magnet. Reson. Imag.

Magnet. Reson. Imag.

NeuroImage

Neuron

NeuroImage

NeuroImage

New modeling and experimental framework to characterize hindered and restricted water diffusion in brain white matter

Magnet. Reson. Med.

A fast iterative shrinkage–thresholding algorithm for linear inverse problems

SIAM J. Imag. Sci.

Color tv: total variation methods for restoration of vector-valued images

IEEE Trans. Image Process.

Distributed optimization and statistical learning via the alternating direction method of multipliers

Found. Trends® Mach. Learn.

Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information

IEEE Trans. Inform. Theory

Enhancing sparsity by reweighted L1 minimization

J. Fourier Anal. Appl.

An algorithm for total variation minimization and applications

J. Math. Imag. Vis.

Water diffusion compartmentation and anisotropy at high b values in the human brain

MRM

High b-value q-space analyzed diffusion-weighted MRS and MRI in neuronal tissues – a technical review

NMR Biomed.

Multiple q-shell diffusion propagator imaging

Med. Image Anal.

Compressed sensing

IEEE Trans. Inform. Theory