Streamline Flows for White Matter Fibre Pathway Segmentation in Diffusion MRI

Abstract

Abstract. We introduce a fibre tract segmentation algorithm based on the geometric coherence of fibre orientations as indicated by a streamline flow model. The inference of local flow approximations motivates a pairwise consistency measure between fibre ODF maxima. We use this measure in a recursive algorithm to cluster consistent ODF maxima, leading to the segmentation of white matter pathways. The method requires minimal seeding compared to streamline tractography-based methods, and allows multiple tracts to pass through the same voxels. We illustrate the approach with a segmentation of the corpus callosum and one of the cortico-spinal tract, with each example seeded at a single voxel.

A Geometry-Based Particle Filtering Approach to White Matter Tractography

“…The work in [8,9] argues for the representation of white matter fibres as sets of dense, locally parallel 3D curves called `streamline flows', and derives their differential geometry, which is characterized by three curvature functions: the tangential , normal and bi-normal curvatures. A local model for such flows is then proposed in [8,9] which consists of two orientation functions θ( x, y, z ) and ϕ( x, y, z ), which define the local orientation of the flow (its tangent vector) at every point ( x, y, z ) in E 3 (three-dimensional Euclidean space): $$\begin{array}{cc}right\theta (x,y,z)& left={\tan }^{-1}\left(\frac{K_{T}x+K_{N}y}{1+K_{N}x-K_{T}y}\right)+K_{B}z,\\ right\varphi (x,y,z)& left=\alpha \theta (x,y,z).\end{array}$$ Here α is a constant, and K T , K N and K B are scalar parameters that specify the values of the tangential, normal and bi-normal curvatures of the flow.…”

“…A local model for such flows is then proposed in [8,9] which consists of two orientation functions θ( x, y, z ) and ϕ( x, y, z ), which define the local orientation of the flow (its tangent vector) at every point ( x, y, z ) in E 3 (three-dimensional Euclidean space): $$\begin{array}{cc}right\theta (x,y,z)& left={\tan }^{-1}\left(\frac{K_{T}x+K_{N}y}{1+K_{N}x-K_{T}y}\right)+K_{B}z,\\ right\varphi (x,y,z)& left=\alpha \theta (x,y,z).\end{array}$$ Here α is a constant, and K T , K N and K B are scalar parameters that specify the values of the tangential, normal and bi-normal curvatures of the flow. This formulation is justified in [8,9] via minimal surface theory as a smooth local model for 3D streamline flows with a small parameter set that describes the flow geometry. In fact, the formulation for θ (and ϕ) given in (1) is that of a generalized helicoid .…”

“…The method incorporates a geometrical model for co-varying 3D curve sets, called `streamline flows' (SF) [8,9], which is infered at each position along the fibre with a particle filter, so that the estimation at each step builds upon previous estimates. With a small parameter set, the SF model captures within a 3D neighborhood $\mathrm{scriptN}$ the full 3D geometry of a collection of curves that may vary within $\mathrm{scriptN}$ in the tangential direction, as well as in the normal and bi-normal directions.…”

Classification in DTI using shapes of white matter tracts

“…DTI can provide information about the axon bundles of the white matter such as preferred orientation, information about local tissue structure using properties of tensors at each voxel. Using these tensors at each voxel fiber bundles are extracted using various techniques like solving stream-line equations [1], clustering [2], montecarlo based methods [3]. Typically fiber bundles are traced from seed points in regions with anatomical interest.…”

Clustering of the Human Skeletal Muscle Fibers Using Linear Programming and Angular Hilbertian Metrics

“…Mean-shift was also used in [11] where each fiber is first embedded in a high dimensional space using its sequence of points, and kernels with variable bandwidths are considered in the mean-shift algorithm. More recently, fibers were represented in [12] using their differential geometry and frame transportation and a consistency measure was used for clustering. Another class of methods suggested to circumvent the limitation of unsupervised clustering where the obtained segmentation may not correspond to anatomical knowledge.…”