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 improving dMRI signal and optimize dMRI acquisition schemes. Then, we present our modeling contributions related to the problem of reconstructing and characterizing important Diffusion MRI features such as the Orientation Distribution Function (ODF) and the Ensemble Average Propagator (EAP) in 6.1.2 , including contributions of the compressed sensing theory to dMRI and contributions to on line motion detection. Finally, we end up, in 6.1.3 , with some general applications such as tractography, clustering and microstructures recovery with pore size distribution estimation.

Improving dMRI Signal and Acquisitions

Optimal Design of Multiple Q-shells experiments for Diffusion MRI

Participants : Rachid Deriche, Emmanuel Caruyer, Iman Aganj [Department of Electrical and Computer Engineering, 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] , Jian Cheng [ATHENA and LIAMA, China] , Jiang Tianzi [LIAMA, China] .

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

Recent advances in diffusion MRI make use of the diffusion signal sampled on the whole Q-space, rather than on a single sphere. While much effort has been done to design uniform sampling schemes for single shell experiment, it is yet not clear how to build a strategy to sample the diffusion signal in the whole Fourier domain. In this work, we proposed a method to generate acquisition schemes for multiple Q-shells experiment in diffusion MRI. The acquisition protocols we designed are incremental, which means they remain approximately optimal when truncated before the acquisition is complete. Our method is fast, incremental, and we can generate diffusion gradients schemes for any number of acquisitions, any number of shells, and any number of points per shell. The samples arranged on different shells do not share the same directions. The method has been tested for Spherical Polar Fourier reconstruction of the diffusion signal, and is based on Monte-Carlo simulations. Several prefered acquisition parameters are identified.

This work has been published in [20] .

Incremental gradient table for multiple Q-shells diffusion MRI

Participants : Rachid Deriche, Emmanuel Caruyer, Iman Aganj [Department of Electrical and Computer Engineering, 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 Associate Team.

Most studies on sampling optimality for diffusion MRI deal with single Q-shell acquisition. For single Q-shell acquisition, incremental gradient table has proved useful in clinical setup, where the subject is likely to move, or for online reconstruction. In this work, we proposed a generalization of the electrostatic repulsion to generate gradient tables for multiple Q-shells acquisitions, designed for incremental reconstruction or processing of data prematurely aborted.

This work has been published in [21] .

Impact of radial and angular sampling on multiple shells acquisition in diffusion MRI

Participants : Rachid Deriche, Sylvain Merlet, Emmanuel Caruyer.

In this work, we evaluated the impact of radial and angular sampling on multiple shells (MS) acquisition in diffusion MRI. The validation of our results is based on a new and efficient method to accurately reconstruct the Ensemble Average Propagator (EAP) in term of the Spherical Polar Fourier (SPF) basis from very few diffusion weighted magnetic resonance images (DW-MRI). This approach nicely exploited the duality between SPF and a closely related basis in which one can respectively represent the EAP and the diffusion signal using the same coefficients. We efficiently combined this relation to the recent acquisition and reconstruction technique called Compressed Sensing (CS). Based on results of multi-tensors models reconstruction, we showed how to construct a robust acquisition scheme for both neural fibre orientation detection and attenuation signal/EAP reconstruction.

This work has been published in [32] .

Simultaneous Smoothing and Estimation of DTI via Robust Variational Non-local Means

Participants : Rachid Deriche, Meizhu Liu [Department of CISE, University of Florida, Gainesville, USA] , Baba Vemuri [Department of CISE, University of Florida, Gainesville, USA] .

Regularized diffusion tensor estimation is an essential step in DTI analysis. There are many methods proposed in literature for this task but most of them are neither statistically robust nor feature preserving denoising techniques that can simultaneously estimate symmetric positive definite (SPD) diffusion tensors from diffusion MRI. One of the most popular techniques in recent times for feature preserving scalar-valued image denoising is the non-local means filtering method that has recently been generalized to the case of diffusion MRI denoising. However, these techniques denoise the multi-gradient volumes first and then estimate the tensors rather than achieving it simultaneously in a unified approach. Moreover, some of them do not guarantee the positive definiteness of the estimated diffusion tensors. In this work, we proposed a novel and robust variational framework for the simultaneous smoothing and estimation of diffusion tensors from diffusion MRI. Our variational principle makes use of a recently introduced total Kullback-Leibler (tKL) divergence, which is a statistically robust similarity measure between diffusion tensors, weighted by a non-local factor adapted from the traditional non-local means filters. For the data fidelity, we use the nonlinear least-squares term derived from the Stejskal-Tanner model. We have performed experimental results depicting the positive performance of our method in comparison to competing methods on synthetic and real data examples.

This work has been published in [31] .

Anisotropic LMMSE denoising of MRI based on statistitcal tissue models

Participants : Rachid Deriche, Gonzalo Vegas-Sánchez-Ferrero [Universidad de Valladolid, Spain] , Santiago Aja Fernández [Universidad de Valladolid, Spain] .

Linear Minimum Mean Squared Error Estimation (LMMSE) is a simple, yet powerful denoising technique within MRI. It is based on the computation of the mean and variance of the data being filtered according to a noise model assumed, which is usually accomplished by calculating local moments over squared neighborhoods. When these neighborhoods are centered in pixels corresponding to image contours, the estimation is not accurate due to the presence of two or more tissues with different statistical properties. In this work, we overcome this limitation by introducing an anisotropic LMMSE scheme: the grey levels of each tissue in the MRI volume are modeled as a Gamma-mixture, such that we can discriminate between the different matters to construct anisotropic neighborhoods containing only one kind of tissue. The potential of the Gamma distribution relies on its ability to fit both the Rician distribution traditionally used to model the noise in MRI and the non-central Chi noise found in modern parallel MRI systems.

This work is currently under submission.

Modeling in Diffusion MRI

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 Association France Parkinson and the ANR 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 [13] .

A Riemannian Framework for Ensemble Average Propagator Computing

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

This work was partly supported by the Association France Parkinson and the ANR NucleiPark project.

In Diffusion Tensor Imaging (DTI), Riemannian framework (RF) has been proposed for processing tensors, which is based on Information Geometry theory. Many papers have shown that RF is useful in tensor estimation, interpolation, smoothing, regularization, segmentation and so on. Recently RF also has been proposed for Orientation Distribution Function (ODF) computing and it is applicable to any Probability Density Function (PDF) based on any orthonormal basis representation. Spherical Polar Fourier Imaging (SPFI) was proposed recently to fast and robustly estimate the ODF and Ensemble Average Propagator (EAP) from arbitrary sampled DWI signals. In this work, we proposed the RF for EAPs and implemented it via SPFI. We proved that the RF for EAPs is diffeomorphism invariant, which is the natural extension of affine invariant RF for tensors. It could avoid the so-called swelling effect for interpolating EAPs, just like the RF for tensors. We also proposed the Log-Euclidean framework (LEF), Affine-Euclidean framework (AEF), for fast processing EAPs, and Geometric Anisotropy (GA) for measuring the anisotropy of EAPs, which are all the extensions of previous concepts in RM for tensors respectively.

This work has been published in [22] .

Diffeomorphism Invariant Riemannian Framework for Ensemble Average Propagator Computing

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

This work was partly supported by the Association France Parkinson and the ANR NucleiPark project.

In Diffusion Tensor Imaging (DTI), Riemannian framework based on Information Geometry theory has been proposed for processing tensors on estimation, interpolation, smoothing, regularization, segmentation, statistical test and so on. Recently Riemannian framework has been generalized to Orientation Distribution Function (ODF) and it is applicable to any Probability Density Function (PDF) under orthonormal basis representation. Spherical Polar Fourier Imaging (SPFI) was proposed for ODF and Ensemble Average Propagator (EAP) estimation from arbitrary sampled signals without any assumption. Tensors only can represent Gaussian EAP and ODF is the radial integration of EAP, while EAP has full information for diffusion process. To our knowledge, so far there is no work on how to process EAP data. In this work, we presented a Riemannian framework as a mathematical tool for such task. We proposed a state-of-the-art Riemannian framework for EAPs by representing the square root of EAP, called wavefunction based on quantum mechanics, with the Fourier dual Spherical Polar Fourier (dSPF) basis. In this framework, the exponential map, logarithmic map and geodesic have closed forms, and weighted Riemannian mean and median uniquely exist. We analyzed theoretically the similarities and differences between Riemannian frameworks for EAPs and for ODFs and tensors. The Riemannian metric for EAPs is diffeomorphism invariant, which is the natural extension of the affine-invariant metric for tensors. We proposed Log-Euclidean framework to fast process EAPs, and Geodesic Anisotropy (GA) to measure the anisotropy of EAPs. With this framework, many important data processing operations, such as interpolation, smoothing, atlas estimation, Principal Geodesic Analysis (PGA), can be performed on EAP data. The proposed Riemannian framework was validated in synthetic data for interpolation, smoothing, PGA and in real data for GA and atlas estimation. Riemannian median is much robust for atlas estimation.

This work has been published in [23] .

Theoretical Analysis and Practical Insights on EAP Estimation via a Unified HARDI Framework

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

This work was partly supported by the Association France Parkinson and the ANR NucleiPark project.

Since Diffusion Tensor Imaging (DTI) cannot describe complex non-Gaussian diffusion process, many techniques, called as single shell High Angular Resolution Diffusion Imaging (sHARDI) methods, reconstruct the Ensemble Average Propagator (EAP) or its feature Orientation Distribution Function (ODF) from diffusion weighted signals only in single shell. Q-Ball Imaging (QBI) and Diffusion Orientation Transform (DOT) are two famous sHARDI methods. However, these sHARDI methods have some intrinsic modeling errors or need some unreal assumptions. Moreover they are hard to deal with signals from different q-shells. Most recently several novel multiple shell HARDI (mHARDI) methods, including Diffusion Propagator Imaging (DPI), Spherical Polar Fourier Imaging (SPFI) and Simple Harmonic Oscillator based Reconstruction and Estimation (SHORE), were proposed to analytically estimate EAP or ODF from multiple shell (or arbitrarily sampled) signals. These three methods all represent diffusion signal with some basis functions in spherical coordinate and use plane wave formula to analytically solve the Fourier transform. To our knowledge, there is no theoretical analysis and practical comparison among these sHARDI and mHARDI methods. In this work, we proposed a unified computational framework, named Analytical Fourier Transform in Spherical Coordinate (AFT-SC), to perform such theoretical analysis and practical comparison among all these five state-of-the-art diffusion MRI methods. We compared these five methods in both theoretical and experimental aspects. With respect to the theoretical aspect, some criteria are proposed for evaluation and some differences together with some similarities among the methods are highlighted. Regarding the experimental aspect, all the methods were compared in synthetic, phantom and real data. The shortcomings and advantages of each method were highlighted from which SPFI appears to be among the best because it uses an orthonormal basis that completely separates the spherical and radial information.

This work has been published in [24] .

Compressive Sensing Ensemble Average Propagator Estimation via L1 Spherical Polar Fourier Imaging

Participants : Rachid Deriche, Sylvain Merlet, Emmanuel Caruyer, Jian Cheng [ATHENA and LIAMA, China] , Jiang Tianzi [LIAMA, China] .

In diffusion MRI (dMRI) domain, many High Angular Resolution Diffusion Imaging (HARDI) methods were proposed to estimate Ensemble Average Propagator (EAP) and Orientation Distribution Function (ODF). They normally need many samples, which limits their applications. Some Compressive Sensing (CS) based methods were proposed to estimate ODF in Q-Ball Imaging (QBI) from limited samples. However EAP estimation is much more difficult than ODF in QBI. Recently Spherical Polar Fourier Imaging (SPFI) was proposed to represent diffusion signal using Spherical Polar Fourier (SPF) basis without specific assumption on diffusion signals and analytically obtain EAP and ODF via the Fourier dual SPF (dSPF) basis from arbitrarily sampled signal. Normally the coefficients of SPF basis are estimated via Least Square with weighted L2 norm regularization (L2-SPFI). However, L2-SPFI needs a truncated basis to avoid overfitting, which brings some estimation errors. By considering the Fourier relationship between EAP and signal and the Fourier basis pair provided in SPFI, we proposed a novel EAP estimation method, named L1-SPFI, to estimate EAP from limited samples using CS technique, and favorably compared it to the classical L2-SPFI method. L1-SPFI estimates the coefficients in SPFI using least square with weighted L1 norm regularization. The weights are designed to enhance the sparsity. L1-SPFI significantly accelerates the ordinary CS based Fourier reconstruction method. This is performed by using SPF basis pair in CS estimation process which avoids the numerical Fourier transform in each iteration step. By considering high order basis in L1 optimization, L1-SPFI improves EAP reconstruction especially for the angular resolution. The proposed L1-SPFI was validated by synthetic, phantom and real data. The CS EAP and ODF estimations are discussed in detail and we showed that recovering the angular information from CS EAP requires much less samples than exact CS EAP reconstruction. Various experiments on synthetic, phantom and real data validate the fact that SPF basis can sparsely represent DW-MRI signals and L1-SPFI largely improves the CS EAP reconstruction especially the angular resolution.

This work has been published in [25] , [26] .

Spherical Polar Fourier EAP and ODF Reconstruction via Compressed Sensing in Diffusion MRI

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

In diffusion magnetic resonance imaging (dMRI), the Ensemble Average Propagator (EAP), also known as the propagator, describes completely the water molecule diffusion in the brain white matter without any prior knowledge about the tissue shape. In this work, we described a new and efficient method to accurately reconstruct the EAP in terms of the Spherical Polar Fourier (SPF) basis from very few diffusion weighted magnetic resonance images (DW-MRI). This approach exploits the duality between SPF and a closely related basis in which one can respectively represent the EAP and the diffusion signal using the same coefficients, and efficiently combines it to the recent acquisition and reconstruction technique called Compressed Sensing. Our work provides an efficient analytical solution to estimate, from few measurements, the diffusion propagator at any radius. We also provide a new analytical solution to extract an important feature characterising the tissue microstructure: the Orientation Distribution Function (ODF). We illustrate and prove the effectiveness of our method in reconstructing the propagator and the ODF on both noisy multiple q-shell synthetic and phantom data.

This work has been published in [33] .

On Line Reconstruction and Motion Detection in HARDI

Participants : Rachid Deriche, Emmanuel Caruyer, Iman Aganj [Department of Electrical and Computer Engineering, 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 Associate Team.

With acquisition protocols such as high angular resolution diffusion imaging, head motion can become an issue. Although the misalignment between diffusion-weighted images (DWIs) can be corrected in a post-processing step, this might increase partial volume effects, because of the relatively low spatial resolution of DWIs and interpolation in the registration procedure. If able to detect motion online, the scanner technician could be issued a warning and make a decision accordingly. Orientation distribution functions (ODF) can be reconstructed online using a Kalman filter (KF). In this work, we presented three contributions related to the problem of online ODF reconstruction and motion detection in HARDI. First, we developed a proper error propagation accounting for the non-linear transform on the diffusion signal. Next, we developed two motion detection algorithms, based on the monitoring of residuals, and compared them using synthetic data.

This work has been published in [18] .

Online Motion Detection in High Angular Resolution Diffusion Imaging

Participants : Rachid Deriche, Emmanuel Caruyer, Iman Aganj [Department of Electrical and Computer Engineering, 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 Associate Team.

The orientation distribution function (ODF) can be reconstructed online incrementally from diffusion-weighted MRI with a Kalman filtering framework. This online reconstruction can provide real-time feedback to the practitioner, especially appreciated for long acquisition protocols typical in Q-ball imaging. On top of the Kalman filter, we proposed a method to evaluate online the reconstruction accuracy of the estimated ODF in constant solid angle. In addition, monitoring the residuals of the Kalman filter, we designed, based on statistical tests, two algorithms for online detection of subject motion. The proposed techniques, tested on real and synthetic data under various experimental conditions, can detect rotation by angle less than 3°.

This work has been published in [19] .

From DW-MRI to Fiber Pathways and Microstructures Recovery

A Polynomial Approach for Maxima Extraction and Its Application to Tractography in HARDI

Participants : Rachid Deriche, Aurobrata Ghosh, Demian Wassermann [Harvard Medical School] .

A number of non-parametrically represented High Angular Resolution Diffusion Imaging (HARDI) spherical diffusion functions have been proposed to infer more and more accurately the heterogeneous and complex tissue microarchitecture of the cerebral white-matter. These spherical functions overcome the limitation of Diffusion Tensor Imaging (DTI) at discerning crossing, merging and fanning axonal fiber bundle configurations inside a voxel. Tractography graphically reconstructs the axonal connectivity of the cerebral white-matter in vivo and non-invasively, by integrating along the direction indicated by the local geometry of the spherical diffusion functions. Tractography is acutely sensitive to the local geometry and its correct estimation. In this work, we first proposed a polynomial approach for analytically bracketing and numerically refining with high precision all the maxima, or fiber directions, of any spherical diffusion function represented non-parametrically. This permits an accurate inference of the fiber layout from the spherical diffusion function. Then we proposed an extension of the deterministic Streamline tractography to HARDI diffusion functions that clearly discern fiber crossings. We also extended the Tensorline algorithm to these HARDI functions, to improve on the extended Streamline tractography. We illustrated our proposed methods using the Solid Angle diffusion Orientation Distribution Function (ODF-SA). We presented results on multi-tensor synthetic data, and real in vivo data of the cerebral white-matter that show markedly improved tractography results.

This work has been published in [30] .

Tract-based statistical analyzes in dMRI in autism spectrum disorder

Participants : Rachid Deriche, Anne-Charlotte Philippe, Demian Wassermann [Harvard Medical School, Boston, MA] , Pablo Barttfeld [Integrative Neuroscience Laboratory, Physics Dept. University of Buenos Aires, Buenos Aires, Argentina] , Jorge Calvar [Fundación parala Luchacontralas Enfermedades Neurológicasdela Infancia, Buenos Aires, Argentina] , Ramon Leiguarda [Fundación para la Lucha contra las Enfermedades Neurológicasdela Infancia, Buenos Aires, Argentina] , Bruno Wicker [INCM CNRS, Marseille, France] , Mariano Sigman [Integrative Neuroscience Laboratory, Physics Dept. University of Buenos Aires, Buenos Aires, Argentina] .

This work was partly supported by the ECOS-Sud grant.

Abnormal face processing is one of the hallmark features of social impairments in autism spectrum disorder (ASD). Previous neuroimaging studies showed that the fusiform gyrus is involved in face perception and is not or abnormally activated in autistic subjects. The aim of this study was to quantify potential anatomical differences in the white matter tracts that traverse the fusiform gyrus, the prefrontal cortex and the superior temporal gyrus, and correlate them with ADOS scores in ASD subjects. We used Diffusion Tensor MRI (DT) images to assess the integrity of automatically segmented white matter bundles connecting these brain areas. Then, we performed statistical analysis on these fiber bundles using diffusivity measures calculated from DT to characterize tissue microstructure changes and correlate these changes with ADOS scores. 7 adults with high functioning autism or Asperger syndrome and 11 typical adults participated in the study. We found several clusters with dissimilarities between ASD and control subjects in FA measures on tracts traversing the fusiform gyrus in both hemispheres of the brain. We observed a significant reduction of FA values in a cluster on a bundle joining the superior temporal gyrus to the prefrontal.

This work has been published in [37] . A related work on large-scale network analysis reflecting big-world characteristics in ASD has been published in  [35] .

Unsupervised automatic white matter fiber clustering using a Gaussian mixture model

Participants : Rachid Deriche, Meizhu Liu [Department of CISE, University of Florida, Gainesville, USA] , Baba Vemuri [Department of CISE, University of Florida, Gainesville, USA] .

Fiber tracking from diffusion tensor images is an essential step in numerous clinical applications. There is a growing demand for an accurate and efficient framework to perform quantitative analysis of white matter fiber bundles. In this work, we proposed a robust framework for fiber clustering. This framework is composed of two parts: accessible fiber representation, and a statistically robust divergence measure for comparing fibers. Each fiber is represented using a Gaussian mixture model (GMM), which is the linear combination of Gaussian distributions. The dissimilarity between two fibers is measured using the total square loss function between their corresponding GMMs (which is statistically robust). Finally, we performed the hierarchical total Bregman soft clustering algorithm on the GMMs, yielding clustered fiber bundles. Further, our method is able to determine the number of clusters automatically. We performed experimental results depicting favorable performance of our method on both synthetic and real data examples.

This work is currently under submission.

Riemannian geometry based brain white matter fiber clustering

Participants : Rachid Deriche, Ali Demir [Sabanci University, Istanbul, Turkey] , Gozde Unal [Sabanci University, Istanbul, Turkey] .

This work was partly supported by the PHC Bosphore grant.

Clustering of reconstructed brain white matter fibers into meaningful anatomical bundles becomes an important tool for detailed analysis of brain white matter diseases via diffusion MRI. In this work we developed a Riemannian geometry based geodesic distance measure between fiber pairs, which is then utilized in fiber clustering. A second contribution is a fiber selection algorithm, which compresses the dataset and speed up the computation time. We demonstrated methods on synthetic kissing fibers dataset, and validated on a brain white matter atlas.

This work is currently under submission.

White Matter Clustering Revisited

Participants : Rachid Deriche, Alvaro-Alejandro Sanchez-Moscosa.

This work was partly supported by the Association France Parkinson and the ANR NucleiPark project.

In   [70] , we have introduced an interesting hierarchical agglomerative based algorithm that represents and clusters white matter bundles under a Gaussian Process framework. In this work, a new implementation which drastically improves the performance of the clustering algorithm before mentioned is proposed and validated. This implementation notably improves the performance of the clustering algorithm   [70] and has been validated on real data. This approach has the advantage of running in a lower abstraction level, which leads to lowered memory requirements, shorter running times and ultimately provides the possibility to process more densely seeded tractographies. The new implementation is then used to process densely seeded streamline tractographies which provide more localized fiber bundles. Additionally, as Parkinson's Disease is believed to induce changes in the axonal bundles, tract-based statistical analysis is performed on interesting fiber tracts to find significant differences in diffusion scalar measures.

Using Radial NMR Profiles to Characterize Pore Size Distributions

Participants : Rachid Deriche, John Treilhard [Queen's University, CA] .

Extracting information about axon diameter distributions in the brain is a challenging task which provides useful information for medical purposes; for example, the ability to characterize and monitor axon diameters would be useful in diagnosing and investigating diseases like amyotrophic lateral sclerosis (ALS) or autism. In [75] , three families of operators are defined, whose action upon an NMR attenuation signal extracts the moments of the pore size distribution of the ensemble under consideration; also a numerical method is proposed to continuously reconstruct a discretely sampled attenuation profile using the eigenfunctions of the simple harmonic oscillator Hamiltonian – the SHORE basis. The work we have performed here extends this method to other bases that can offer a better description of attenuation signal behaviour – in particular, we proposed the use of the radial Spherical Polar Fourier (SPF) basis. Testing is performed to contrast the efficacy of the radial SPF basis and SHORE basis in practical attenuation signal reconstruction. The robustness of the method to additive noise is tested and analyzed. We demonstrated that a low-order attenuation signal reconstruction outperforms a higher-order reconstruction in subsequent moment estimation under noisy conditions. We proposed the simulated annealing algorithm for basis function scale parameter estimation. Finally, analytic expressions are derived and presented for the action of the operators on the radial SPF basis (obviating the need for numerical integration, thus avoiding a spectrum of possible sources of error).

This work is currently under submission.