Inferring White Matter Geometry from Diffusion Tensor MRI: Application to Connectivity Mapping

Principal Curves on Riemannian Manifolds

Euclidean statistics are often generalized to Riemannian manifolds by replacing straight-line interpolations with geodesic ones. While these Riemannian models are familiar-looking, they are restricted by the inflexibility of geodesics, and they rely on constructions which are optimal only in Euclidean domains. We consider extensions of Principal Component Analysis (PCA) to Riemannian manifolds. Classic Riemannian approaches seek a geodesic curve passing through the mean that optimizes a criteria of interest. The requirements that the solution both is geodesic and must pass through the mean tend to imply that the methods only work well when the manifold is mostly flat within the support of the generating distribution. We argue that instead of generalizing linear Euclidean models, it is more fruitful to generalize non-linear Euclidean models. Specifically, we extend the classic Principal Curves from Hastie & Stuetzle to data residing on a complete Riemannian manifold. We show that for elliptical distributions in the tangent of spaces of constant curvature, the standard principal geodesic is a principal curve. The proposed model is simple to compute and avoids many of the pitfalls of traditional geodesic approaches. We empirically demonstrate the effectiveness of the Riemannian principal curves on several manifolds and datasets.

CUDA-Accelerated Geodesic Ray-Tracing for Fiber Tracking

Diffusion Tensor Imaging (DTI) allows to noninvasively measure the diffusion of water in fibrous tissue. By reconstructing the fibers from DTI data using a fiber-tracking algorithm, we can deduce the structure of the tissue. In this paper, we outline an approach to accelerating such a fiber-tracking algorithm using a Graphics Processing Unit (GPU). This algorithm, which is based on the calculation of geodesics, has shown promising results for both synthetic and real data, but is limited in its applicability by its high computational requirements. We present a solution which uses the parallelism offered by modern GPUs, in combination with the CUDA platform by NVIDIA, to significantly reduce the execution time of the fiber-tracking algorithm. Compared to a multithreaded CPU implementation of the same algorithm, our GPU mapping achieves a speedup factor of up to 40 times.

Tumor invasion margin on the Riemannian space of brain fibers

Gliomas are one of the most challenging tumors to treat or control locally. One of the main challenges is determining which areas of the apparently normal brain contain glioma cells, as gliomas are known to infiltrate for several centimeters beyond the clinically apparent lesion visualized on standard CT or MRI. To ensure that radiation treatment encompasses the whole tumour, including the cancerous cells not revealed by MRI, doctors treat a volume of brain extending 2cm out from the margin of the visible tumour. This expanded volume often includes healthy, non-cancerous brain tissue. Knowing that glioma cells preferentially spread along nerve fibers, we propose the use of a geodesic distance on the Riemannian manifold of brain fibers to replace the Euclidean distance used in clinical practice and to correctly identify the tumor invasion margin. To compute the geodesic distance we use actual DTI data from patients with glioma and compare our predicted growth with follow-up MRI scans. Results show improvement in predicting the invasion margin when using the geodesic distance as opposed to the 2cm conventional Euclidean distance.

Optical flow and advection on 2-Riemannian manifolds: a common framework

Dynamic pattern analysis and motion extraction can be efficiently addressed using optical flow techniques. This article presents a generalization of these questions to non-flat surfaces, where optical flow is tackled through the problem of evolution processes on non-Euclidian domains. The classical equations of optical flow in the Euclidian case are transposed to the theoretical framework of differential geometry. We adopt this formulation for the regularized optical flow problem, prove its mathematical well-posedness and combine it with the advection equation. The optical flow and advection problems are dual: a motion field may be retrieved from some scalar evolution using optical flow; conversely, a scalar field may be deduced from a velocity field using advection. These principles are illustrated with qualitative and quantitative evaluations from numerical simulations bridging both approaches. The proof-of-concept is further demonstrated with preliminary results from time-resolved functional brain imaging data, where organized propagations of cortical activation patterns are evidenced using our approach.

Measures for pathway analysis in brain white matter using diffusion tensor images

In this paper we discuss new measures for connectivity analysis of brain white matter, using MR diffusion tensor imaging. Our approach is based on Riemannian geometry, the viability of which has been demonstrated by various researchers in foregoing work. In the Riemannian framework bundles of axons are represented by geodesics on the manifold. Here we do not discuss methods to compute these geodesics, nor do we rely on the availability of geodesics. Instead we propose local measures which are directly computable from the local DTI data, and which enable us to preselect viable or exclude uninteresting seed points for the potentially time consuming extraction of geodesics. If geodesics are available, our measures can be readily applied to these as well. We consider two types of geodesic measures. One pertains to the connectivity saliency of a geodesic, the second to its stability with respect to local spatial perturbations. For the first type of measure we consider both differential as well as integral measures for characterizing a geodesic's saliency either locally or globally. (In the latter case one needs to be in possession of the geodesic curve, in the former case a single tangent vector suffices.) The second type of measure is intrinsically local, and turns out to be related to a well known tensor in Riemannian geometry.

Fiber bundle estimation and parameterization

Individual white matter fibers cannot be resolved by current magnetic resonance (MR) technology. Many fibers of a fiber bundle will pass through an individual volume element (voxel). Individual visualized fiber tracts are thus the result of interpolation on a relatively coarse voxel grid, and an infinite number of them may be generated in a given volume by interpolation. This paper aims at creating a level set representation of a fiber bundle to describe this apparent continuum of fibers. It further introduces a coordinate system warped to the fiber bundle geometry, allowing for the definition of geometrically meaningful fiber bundle measures.

MR diffusion-based inference of a fiber bundle model from a population of subjects

This paper proposes a method to infer a high level model of the white matter organization from a population of subjects using MR diffusion imaging. This method takes as input for each subject a set of trajectories stemming from any tracking algorithm. Then the inference results from two nested clustering stages. The first clustering converts each individual set of trajectories into a set of bundles supposed to represent large white matter pathways. The second clustering matches these bundles across subjects in order to provide a list of candidates for the bundle model. The method is applied on a population of eleven subjects and leads to the inference of 17 such candidates.

White matter fiber tractography via anisotropic diffusion simulation in the human brain

A novel approach to noninvasively tracing brain white matter fiber tracts is presented using diffusion tensor magnetic resonance imaging (DT-MRI). This technique is based on successive anisotropic diffusion simulations over the human brain, which are utilized to construct three dimensional diffusion fronts. The fiber pathways are determined by evaluating the distance and orientation from the fronts to their corresponding diffusion seeds. Synthetic and real DT-MRI data are employed to demonstrate the tracking scheme. It is shown that the synthetic tracts are accurately replicated, and several major white matter fiber pathways can be reproduced noninvasively, with the tract branching being allowed. Since simulating the diffusion process, which is truly a physical phenomenon reflecting the underlying architecture of cerebral tissues, makes full use of the diffusion tensor data, including both the magnitude and orientation information, the proposed approach is expected to enhance robustness and reliability in white matter fiber reconstruction.

Fiber tracking in q-ball fields using regularized particle trajectories

Most of the approaches dedicated to fiber tracking from diffusion-weighted MR data rely on a tensor model. However, the tensor model can only resolve a single fiber orientation within each imaging voxel. New emerging approaches have been proposed to obtain a better representation of the diffusion process occurring in fiber crossing. In this paper, we adapt a tracking algorithm to the q-ball representation, which results from a spherical Radon transform of high angular resolution data. This algorithm is based on a Monte-Carlo strategy, using regularized particle trajectories to sample the white matter geometry. The method is validated using a phantom of bundle crossing made up of haemodialysis fibers. The method is also applied to the detection of the auditory tract in three human subjects.

A Riemannian approach to diffusion tensor images segmentation

We address the problem of the segmentation of cerebral white matter structures from diffusion tensor images. Our approach is grounded on the theoretically well-founded differential geometrical properties of the space of multivariate normal distributions. We introduce a variational formulation, in the level set framework, to estimate the optimal segmentation according to the following hypothesis: Diffusion tensors exhibit a Gaussian distribution in the different partitions. Moreover, we must respect the geometric constraints imposed by the interfaces existing among the cerebral structures and detected by the gradient of the diffusion tensor image. We validate our algorithm on synthetic data and report interesting results on real datasets. We focus on two structures of the white matter with different properties and respectively known as the corpus callosum and the corticospinal tract.

A Hamilton-Jacobi-Bellman approach to high angular resolution diffusion tractography

This paper describes a new framework for white matter tractography in high angular resolution diffusion data. A direction-dependent local cost is defined based on the diffusion data for every direction on the unit sphere. Minimum cost curves are determined by solving the Hamilton-Jacobi-Bellman using an efficient algorithm. Classical costs based on the diffusion tensor field can be seen as a special case. While the minimum cost (or equivalently the travel time of a particle moving along the curve) and the anisotropic front propagation frameworks are related, front speed is related to particle speed through a Legendre transformation which can severely impact anisotropy information for front propagation techniques. Implementation details and results on high angular diffusion data show that this method can successfully take advantage of the increased angular resolution in high b-value diffusion weighted data despite lower signal to noise ratio.