Team ATHENA

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Overall Objectives
Scientific Foundations
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New Results
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Bibliography

Section: New Results

Computational Diffusion MRI

This sub-theme is dedicated to describe our various contributions performed within the framework of Computational Diffusion MRI. In 6.1.1 , we start by presenting our contributions to optimize dMRI acquisition schemes, then, we present our contributions related to the problem of reconstructing and characterizing important Diffusion MRI features such as the Orientation Distribution Function (ODF) in 6.1.2 and the Ensemble Average Propagator (EAP) in 6.1.3 . Finally, we end up with some additional contributions related to the characterization of the relation structure-function from functional and diffusion MRI in 6.1.4 and to more general applications such as the reconstruction and the clustering of fibers and an application related to the straightening of the spinal cord, in 6.1.5 .

Improving dMRI Acquisitions

Compressed Sensing for Accelerated EAP Recovery in Diffusion MRI

Participants : Rachid Deriche, Sylvain Merlet.

Compressed Sensing (CS) or Compressive Sampling is a recent technique to accurately reconstruct sparse signals from under sampled measurements acquired below the Shannon-Nyquist rate. In this work, we presented a CS based method for accelerating the reconstruction of the Ensemble Average Propagator (EAP), also known as the Propagator in Diffusion MRI, by significantly reducing the number of measurements. Contrarily to the time consuming acquisition technique known as the Diffusion Spectrum Imaging (DSI), our method is developed and implemented to efficiently reconstruct the EAP from reduced and non uniformly under sampled Diffusion Weighted (DW) MRI images combined to an efficient and accurate l1 norm based reconstruction algorithm. In  [48] , we have illustrated in detail the artifacts occurring in a classical EAP reconstruction à la DSI, and qualitatively and quantitatively demonstrated good and better results in recovering the EAP and some of its important features such as the Orientation Distribution Function ( ODF) from non-regularly undersampled and l1 norm based reconstructed data. This opens an original and very interesting road to shorten the dMRI acquisition time and opens new opportunities to render High Angular Resolution Diffusion Imaging (HARDI) feasible in a clinical setting.

This work has been published in [48] .

Modelling, Reconstructing and Characterizing the Orientation Diffusion Function

Online orientation distribution function reconstruction in constant solid angle and its application to motion detection in high angular resolution diffusion imaging

Participants : Rachid Deriche, Emmanuel Caruyer, Iman Aganj [Department of Electrical and Computer Engineering, University of Minnesota] , Ryan Muetzel [Center for Magnetic Resonance Research, University of Minnesota] , Christophe Lenglet [Department of Electrical and Computer Engineering, University of Minnesota] , Guillermo Sapiro [Department of Electrical and Computer Engineering, University of Minnesota] .

This work was partly supported by the CD-MRI Associated Team.

The diffusion orientation distribution function (ODF) can be reconstructed from q-ball imaging (QBI) to map the complex intravoxel structure of water diffusion. As acquisition time is particularly large for high angular resolution diffusion imaging (HARDI), fast estimation algorithms have recently been proposed, as an on-line feedback on the reconstruction accuracy. Thus the acquisition could be stopped or continued on demand. In this work, we adapted our real-time algorithm  [3] to the ODF in constant solid angle (CSA), and developed a motion detection algorithm upon this reconstruction. Results of improved fiber crossing detection by CSA ODF have been obtained, and motion detection was implemented and tested in vivo.

This work has been published in [25] .

Model-Free, Regularized, Fast, and Robust Analytical Orientation Distribution Function Estimation.

Participants : Rachid Deriche, Jian Cheng [ATHENA and LIAMA, China] , Aurobrata Ghosh, Jiang Tianzi [LIAMA, China] .

High Angular Resolution Diffusion Imaging (HARDI) can better explore the complex micro-structure of white matter compared to Diffusion Tensor Imaging (DTI). Orientation Distribution Function (ODF) in HARDI is used to describe the probability of the fiber direction. There are two type definitions of the ODF, which were respectively proposed in Q-Ball Imaging (QBI) and Diffusion Spectrum Imaging (DSI). Some analytical reconstructions methods have been proposed to estimate these two type of ODFs from single shell HARDI data. However they all have some assumptions and intrinsic modeling errors. In this work, we proposed, almost without any assumption, a uniform analytical method to estimate these two ODFs from DWI signals in q space, which is based on Spherical Polar Fourier Expression (SPFE) of signals. The solution is analytical and is a linear transformation from the q-space signal to the ODF represented by Spherical Harmonics (SH). It can naturally combines the DWI signals in different Q-shells. Moreover, it can avoid the intrinsic Funk-Radon Transform (FRT) blurring error in QBI and it does not need any assumption of the signals, such as the multiple tensor model and mono/multi-exponential decay. We validated our method using synthetic data, phantom data and real data. Our method works well in all experiments, especially for the data with low SNR, low anisotropy and non-exponential decay.

This work has been published in [26] .

Riemannian Median and Its Applications for Orientation Distribution Function Computing.

Participants : Rachid Deriche, Jian Cheng [ATHENA and LIAMA, China] , Aurobrata Ghosh, Jiang Tianzi [LIAMA, China] .

The geometric median is a classic robust estimator of centrality for data in Euclidean spaces, and it has been generalized in analytical manifold. Recently, an intrinsic Riemannian framework for Orientation Distribution Function (ODF) was proposed for the calculation in ODF field [2]. In this work, we proved the unique existence of the Riemannian median in ODF space. Then we explored its two potential applications, median filtering and atlas estimation.

This work has been published in [30]

Modelling, Reconstructing and Characterizing the Ensemble Average Propagator

Multiple q-Shell Diffusion Propagator Imaging.

Participants : Rachid Deriche, Maxime Descoteaux [Sherbrooke University, Quebec] , Denis Le-Bihan [NeuroSpin, IFR 49 CEA Saclay] , Jean-François Mangin [NeuroSpin, IFR 49 CEA Saclay] , Cyril Poupon [NeuroSpin, IFR 49 CEA Saclay] .

This work was partly supported by the EADS Grant Number 2118 and the Association France Parkinson for the NucleiPark project.

Many recent high angular resolution diffusion imaging (HARDI) reconstruction techniques have been introduced to infer an orientation distribution function (ODF) of the underlying tissue structure. These methods are more often based on a single-shell (one b-value) acquisition and can only recover angular structure information contained in the ensemble average propagator (EAP) describing the three-dimensional (3D) average diffusion process of water molecules. The EAP can thus provide richer information about complex tissue microstructure properties than the ODF by also considering the radial part of the diffusion signal. In this work, we presented a novel technique for analytical EAP reconstruction from multiple q-shell acquisitions. The solution is based on a Laplace equation by part estimation between the diffusion signal for each shell acquisition. This simplifies greatly the Fourier integral relating diffusion signal and EAP, which leads to an analytical, linear and compact EAP reconstruction. An important part of this work is dedicated to validate the diffusion signal estimation and EAP reconstruction on real datasets from ex vivo phantoms. We also illustrated multiple q-shell diffusion propagator imaging (mq-DPI) on a real in vivo human brain and performed a qualitative comparison against state-of-the-art diffusion spectrum imaging (DSI) on the same subject. mq-DPI is shown to reconstruct robust EAP from only several different b-value shells and less diffusion measurements than DSI. This opens interesting perspectives for new q-space sampling schemes and tissue microstructure investigation.

This work has been published in [14] .

Model-Free and Analytical EAP Reconstruction via Spherical Polar Fourier Diffusion MRI.

Participants : Rachid Deriche, Jian Cheng [ATHENA and LIAMA, China] , Aurobrata Ghosh, Jiang Tianzi [LIAMA, China] .

How to estimate the diffusion Ensemble Average Propagator (EAP) from the DWI signals in q-space is an open problem in diffusion MRI field. Compared with ODF, EAP has the full information about the diffusion process which reflects the complex tissue micro-structure. Diffusion Orientation Transform (DOT) and Diffusion Spectrum Imaging (DSI) are two important methods to estimate the EAP from the signal. However, DOT is based on mono-exponential assumption and DSI needs a lot of samplings and very large b values. In this work, we have proposed Spherical Polar Fourier Imaging (SPFI), a novel model-free fast robust analytical EAP reconstruction method, which almost does not need any assumption of data and does not need too many samplings. SPFI naturally combines the DWI signals with different b-values. It is an analytical linear transformation from the q-space signal to the EAP profile represented by Spherical Harmonics (SH). We validated the proposed methods in synthetic data, phantom data and real data. It works well in all experiments, especially for the data with low SNR, low anisotropy, and non-exponential decay.

This work has been published in [29] , [28]

Fast and Closed-Form Ensemble-Average-Propagator Approximation from the 4th-Order Diffusion Tensor

Participants : Rachid Deriche, Aurobrata Ghosh.

Generalized Diffusion Tensor Imaging (GDTI) was developed to model complex Apparent Diffusivity Coefficient (ADC) using Higher Order Tensors (HOT) and to overcome the inherent single-peak shortcoming of DTI. However, the geometry of a complex ADC profile doesn't correspond to the underlying structure of fibers. This tissue geometry can be inferred from the shape of the Ensemble Average Propagator (EAP). Though interesting methods for estimating a positive ADC using 4th order diffusion tensors were developed, GDTI in general was overtaken by other approaches, e.g. the Orientation Distribution Function (ODF), since it is considerably difficult to recuperate the EAP from a HOT model of the ADC in GDTI. In this work, we presented a novel closed-form approximation of the EAP using Hermite Polynomials from a modified HOT model of the original GDTI-ADC. Since the solution is analytical, it is fast, differentiable, and the approximation converges well to the true EAP.

This work has been published in [33]

Analytical Q-Ball Imaging with Optimal Regularization

Participants : Rachid Deriche, Maxime Descoteaux [Sherbrooke University, Quebec] , Cheng Guan Koay [NIH, NICHD/STBB, Bethesda] .

This work was partially supported by the CD-MRI Associated team.

Several approaches such as diffusion tensor imaging, q-ball imaging (QBI), spherical deconvolution and many others high angular resolution diffusion imaging (HARDI) have been proposed to describe the angular distribution of the white matter fibers within a voxel. The analytical QBI technique  [4] uses a predetermined regularization parameter ($ \lambda$ = 0.006), which has been well adopted in many clinical studies. Although there are well-known strategies, e.g., the generalized cross-validation (GCV) or the L-curve , for selecting the optimal regularization parameter $ \lambda$ , the predetermined regularization parameter was adopted for reasons related to practical and computational efficiency based on L-curve simulations. In this work, we incorporated the GCV technique into the analytical qball formalism. We compared and contrasted the fixed $ \lambda$ -regularization parameter (“Fixed $ \lambda$ ”) and the automatic GCV-selected optimal $ \lambda$ -regularization (“GCV-based $ \lambda$ ”), for estimating diffusion MRI data. We also discussed the potential consequences of our work on quantitative HARDI anisotropy measures and tractography studies.

Challenges in Reconstructing the Propagator via a Cumulant Expansion of the One-Dimensional q-Space MR Signal

Participants : Rachid Deriche, Aurobrata Ghosh, Evren Ozarslan [NIH, NICHD/STBB, Bethesda] .

This work was partially supported by the CD-MRI Associated team.

Generalized Diffusion Tensor Imaging (GDTI) is one of the few methods that estimate the ensemble average diffusion propagator from the diffusion weighted signal. It has a statistical approach and views the signal, which under the q-space formalism is the Fourier transform of the propagator, as the characteristic function of the propagator. Instead of taking the inverse Fourier transform of the signal, GDTI estimates the cumulants of the propagator from the signal (characteristic function) and then approximates the propagator using the Gram-Charlier Type-A series, which is a series approximation of a probability density function based on its cumulants. However, it is well known that the Gram-Charlier series has a poor convergence, especially since only a truncated series is considered (order-4 usually). The Edgeworth series, which is a reordering of the terms from the Gram-Charlier series, is known to perform better since it is a true asymptotic expansion. GDTI has never been validated numerically. We proposed, in this work, to compare the Gram-Charlier and the Edgeworth series in 1D on known diffusion propagators, where the propagator, the signal and the cumulants have analytical forms. We also compared with cumulants estimated from the signal. Our experiments strongly suggest that for analytical cumulants the Edgeworth series improves on the Gram-Charlier series, and estimating the cumulants from the signal is numerically a sensitive and important problem.

This work has been published in [34]

Relation Structure-Function via fMRI and dMRI

Characterization of the relation structure-function from Functional and Diffusion MRI

Participants : Rachid Deriche, Arnaud Messé, Habib Benali [Laboratoire d'imagerie fonctionnelle INSERM : U678 – IFR14 – IFR49 – Université Pierre et Marie Curie - Paris VI] .

This work has been performed at INSERM : U678 – Université Pierre et Marie Curie - Paris V within the framework of Arnaud Messé's PhD thesis under the joint supervision of H. Benali and R. Deriche. Various concepts ans aspects on how to link function to structure using both fMRI and dMRI have been investigated among which a study that shows that small-world attributes characterize the relationship between functional and structural connectivity using fMRI and DTI [73] , a spatial autoregressive model to link structure to function through a combined fMRI and DTI approach [40] , a connectivity-based delineation of basal ganglia using hierarchical classification [39] , a comparison of functional and anatomical segregation in the basal ganglia using fMRI and fiber tracking [36] , an enhanced voxel-based morphometry method to investigate structural changes with its application to Alzheimer's disease [19] and a study on diffusion tensor imaging and white matter lesions at the subacute stage in mild traumatic brain injury with persistent neurobehavioral impairment [38] . Other related contributions can also be found in [35] , [37] , [43]

Applications

Unsupervised white matter fiber clustering and tract probability map generation: Applications of a Gaussian process framework for white matter fibers

Participant : Rachid Deriche.

With the increasing importance of fiber tracking in diffusion tensor images for clinical needs, there has been a growing demand for an objective mathematical framework to perform quantitative analysis of white matter fiber bundles incorporating their underlying physical significance.

This work presents such a novel mathematical framework that facilitates mathematical operations between tracts using an inner product between fibers. Such inner product operation, based on Gaussian processes, spans a metric space. This metric facilitates combination of fiber tracts, rendering operations like tract membership to a bundle or bundle similarity simple. Based on this framework, we have designed an automated unsupervised atlas-based clustering method that does not require manual initialization nor an a priori knowledge of the number of clusters. Quantitative analysis can now be performed on the clustered tract volumes across subjects, thereby avoiding the need for point parameterization of these fibers, or the use of medial or envelope representations as in previous work. Experiments on synthetic data demonstrate the mathematical operations. Subsequently, the applicability of the unsupervised clustering framework has been demonstrated on a 21-subject dataset.

This work has been published in [21] , [11]

Straightening the spinal cord using fiber tractography

Participants : Rachid Deriche, Demian Wassermann, Julien Cohen-Adad [Athinoula A. Martinos Center for Biomedical Imaging, Massachusetts General Hospital / Harvard Medical School, Charlestown, MA, USA] , Stéphane Lehericy [Center for NeuroImaging Research - CENIR, Department of Neuroradiology CHUPS, Paris] , Habib Benali [Laboratoire d'imagerie fonctionnelle INSERM : U678 – IFR14 – IFR49 – Université Pierre et Marie Curie - Paris VI] , Serge Rossignol [CRSN/Faculté de médecine Université de Montréal] .

Spinal Cord MRI (SC-MRI) is a challenging research field with numerous important clinical and basic research applications. Some of the SC-MRI applications strongly need to deal with a well straightened spinal cord either for appropriate methodological developments, for better visualization or diagnostic purposes. In this work, we developed an efficient and automatic method to straighten the spinal cord image and fibres. Diffusion Tensor MRI is first used to recover by tractography the bundles of fibres contained in the spinal cord white matter. An efficient Gaussian process framework is then used to automatically recover in a robust way the most representative fibre which is used to interpolate and straighten the spinal cord image and fibres. Our method is successfully tested on real images of one cat with partial spinal cord injury and two healthy volunteers. This capability to reliably reconstruct straightened animal and human spinal cord opens new opportunities for SC-MRI research.

This work has been published in [46] , [45] , [11]


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