Multi-shell diffusion signal recovery from sparse measurements
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 -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 in the q-space is related to the EAP via the Fourier transform as given by Stejskal and Tanner (1965)where is the normalized diffusion signal, with and being the measured diffusion signal and its corresponding value, respectively. Alternatively, E can be written as a function of b-value and a unit vector , such that , where
Methods
In this work, it is assumed that the dMRI data is measured along the same K gradient directions 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 (, where the signal is assumed to be normalized by the signal at i.e. ) using a monotonically decreasing function given bywhich 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 , 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).
References (57)
- et al.
Recent advances in diffusion MRI modeling: angular and radial reconstruction
Med. Image Anal.
(2011) - et al.
Compressed sensing with coherent and redundant dictionaries
Appl. Comput. Harmon. Anal.
(2011) - et al.
Bessel fourier orientation reconstruction (bfor): an analytical diffusion propagator reconstruction for hybrid diffusion imaging and computation of q-space indices
NeuroImage
(2013) - et al.
Resolution of crossing fibers with constrained compressed sensing using diffusion tensor MRI
NeuroImage
(2012) - et al.
Continuous diffusion signal, EAP and ODF estimation via compressive sensing in diffusion MRI
Med. Image Anal.
(2013) - et al.
On high b-value diffusion imaging in the human brain: ruminations and experimental insights
Magnet. Reson. Imag.
(2009) - et al.
Biexponential apparent diffusion coefficient parametrization in adult vs newborn brain
Magnet. Reson. Imag.
(2001) - et al.
Mean apparent propagator (map) MRI: a novel diffusion imaging method for mapping tissue microstructure
NeuroImage
(2013) - et al.
Diffusion MRI of complex neural architecture
Neuron
(2003) - et al.
Hybrid diffusion imaging
NeuroImage
(2007)
NODDI: practical in vivo neurite orientation dispersion and density imaging of the human brain
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
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