Riemann-DTI Geodesic Tractography Revisited

Clinical tractography is a challenging problem in diffusion tensor imaging (DTI) due to persistent validation issues. Geodesic tractography, based on a shortest path principle, is conceptually appealing, but has not produced convincing results so far. A major weakness is its rigidity with respect to candidate tracts it is capable of producing given a pair of endpoints, showing a tendency to produce false positives (such as shortcuts) and false negatives (e.g. if a shortcut supplants the correct solution). We propose a new geodesic paradigm that appears to overcome these problems, making a step towards semi-automatic clinical use. To this end we couple the DTI tensor field to a family of Riemannian metrics, governed by control parameters. In practice these parameters may allow for edits by an expert through manual selection among multiple tract suggestions, or for bringing in a priori knowledge. In this paper, however, we consider an automatic, evidence-driven procedure to determine optimal controls and corresponding tentative tracts, and illustrate the role of edits to remediate erroneous defaults.

Continuous Histograms for Anisotropy of 2D Symmetric Piece-Wise Linear Tensor Fields

In this chapter we present an accurate derivation of the distribution of scalar invariants with quadratic behavior represented as continuous histograms. The anisotropy field, computed from a two-dimensional piece-wise linear tensor field, is used as an example and is discussed in all details. Histograms visualizing an approximation of the distribution of scalar values play an important role in visualization. They are used as an interface for the design of transfer-functions for volume rendering or feature selection in interactive interfaces. While there are standard algorithms to compute continuous histograms for piece-wise linear scalar fields, they are not directly applicable to tensor invariants with non-linear, often even non-convex behavior in cells when applying linear tensor interpolation. Our derivation is based on a sub-division of the mesh in triangles that exhibit a monotonic behavior. We compare the results to a naïve approach based on linear interpolation on the original mesh or the subdivision.

Anisotropy in the Human Placenta in Pregnancies Complicated by Fetal Growth Restriction

The placenta has a unique structure, which enables the transfer of oxygen and nutrients from the mother to the developing fetus. Abnormalities in placental structure are associated with major complications of pregnancy; for instance, changes in the complex branching structures of fetal villous trees are associated with fetal growth restriction. Diffusion MRI has the potential to measure such fine placental microstructural details. Here, we present in-vivo placental diffusion MRI scans from controls and pregnancies complicated by fetal growth restriction. We find that after 30 weeks’ gestation fractional anisotropy is significantly higher in placentas associated with growth restricted pregnancies. This shows the potential of diffusion MRI derived measures of anisotropy for assessing placental function during pregnancy.

Uncertainty in the DTI Visualization Pipeline

Diffusion-Weighted Magnetic Resonance Imaging (DWI) enables the in-vivo visualization of fibrous tissues such as white matter in the brain. Diffusion-Tensor Imaging (DTI) specifically models the DWI diffusion measurements as a second order-tensor. The processing pipeline to visualize this data, from image acquisition to the final rendering, is rather complex. It involves a considerable amount of measurements, parameters and model assumptions, all of which generate uncertainties in the final result which typically are not shown to the analyst in the visualization. In recent years, there has been a considerable amount of work on the visualization of uncertainty in DWI, and specifically DTI. In this chapter, we primarily focus on DTI given its simplicity and applicability, however, several aspects presented are valid for DWI as a whole. We explore the various sources of uncertainties involved, approaches for modeling those uncertainties, and, finally, we survey different strategies to visually represent them. We also look at several related methods of uncertainty visualization that have been applied outside DTI and discuss how these techniques can be adopted to the DTI domain. We conclude our discussion with an overview of potential research directions.

Tensor Approximation for Multidimensional and Multivariate Data

Tensor decomposition methods and multilinear algebra are powerful tools to cope with challenges around multidimensional and multivariate data in computer graphics, image processing and data visualization, in particular with respect to compact representation and processing of increasingly large-scale data sets. Initially proposed as an extension of the concept of matrix rank for 3 and more dimensions, tensor decomposition methods have found applications in a remarkably wide range of disciplines. We briefly review the main concepts of tensor decompositions and their application to multidimensional visual data. Furthermore, we will include a first outlook on porting these techniques to multivariate data such as vector and tensor fields.

Variance Measures for Symmetric Positive (Semi-) Definite Tensors in Two Dimensions

Calculating the variance of a family of tensors, each represented by a symmetric positive semi-definite second order tensor/matrix, involves the formation of a fourth order tensor $$R_{abcd}$$ R abcd . To form this tensor, the tensor product of each second order tensor with itself is formed, and these products are then summed, giving the tensor $$R_{abcd}$$ R abcd the same symmetry properties as the elasticity tensor in continuum mechanics. This tensor has been studied with respect to many properties: representations, invariants, decomposition, the equivalence problem et cetera. In this paper we focus on the two-dimensional case where we give a set of invariants which ensures equivalence of two such fourth order tensors $$R_{abcd}$$ R abcd and $$\widetilde{R}_{abcd}$$ R ~ abcd . In terms of components, such an equivalence means that components $$R_{ijkl}$$ R ijkl of the first tensor will transform into the components $$\widetilde{R}_{ijkl}$$ R ~ ijkl of the second tensor for some change of the coordinate system.

Magnetic Resonance Imaging of $$T_2$$- and Diffusion Anisotropy Using a Tiltable Receive Coil

The anisotropic microstructure of white matter is reflected in various MRI contrasts. Transverse relaxation rates can be probed as a function of fibre-orientation with respect to the main magnetic field, while diffusion properties are probed as a function of fibre-orientation with respect to an encoding gradient. While the latter is easy to obtain by varying the orientation of the gradient, as the magnetic field is fixed, obtaining the former requires re-orienting the head. In this work we deployed a tiltable RF-coil to study $$T_2$$ T 2 - and diffusional anisotropy of the brain white matter simultaneously in diffusion-$$T_2$$ T 2 correlation experiments.

Magnetic Resonance Assessment of Effective Confinement Anisotropy with Orientationally-Averaged Single and Double Diffusion Encoding

Porous or biological materials comprise a multitude of micro-domains containing water. Diffusion-weighted magnetic resonance measurements are sensitive to the anisotropy of the thermal motion of such water. This anisotropy can be due to the domain shape, as well as the (lack of) dispersion in their orientations. Averaging over measurements that span all orientations is a trick to suppress the latter, thereby untangling it from the influence of the domains’ anisotropy on the signal. Here, we consider domains whose anisotropy is modeled as being the result of a Hookean (spring) force, which has the advantage of having a Gaussian diffusion propagator while still confining the spatial range for the diffusing particles. In fact, this confinement model is the effective model of restricted diffusion when diffusion is encoded via gradients of long durations, making the model relevant to a broad range of studies aiming to characterize porous media with microscopic subdomains. In this study, analytical expressions for the powder-averaged signal under this assumption are given for so-called single and double diffusion encoding schemes, which sensitize the MR signal to the diffusive displacement of particles in, respectively, one or two consecutive time intervals. The signal for one-dimensional diffusion is shown to exhibit power-law dependence on the gradient strength while its coefficient bears signatures of restricted diffusion.

Single Encoding Diffusion MRI: A Probe to Brain Anisotropy

This chapter covers anisotropy in the context of probing microstructure of the human brain using single encoded diffusion MRI. We will start by illustrating how diffusion MRI is a perfectly adapted technique to measure anisotropy in the human brain using water motion, followed by a biological presentation of human brain. The non-invasive imaging technique based on water motions known as diffusion MRI will be further presented, along with the difficulties that come with it. Within this context, we will first review and discuss methods based on signal representation that enable us to get an insight into microstructure anisotropy. We will then outline methods based on modeling, which are state-of-the-art methods to get parameter estimations of the human brain tissue.

Challenges for Tractogram Filtering

Tractography aims at describing the most likely neural fiber paths in white matter. A general issue of current tractography methods is their large false-positive rate. An approach to deal with this problem is tractogram filtering in which anatomically implausible streamlines are discarded as a post-processing step after tractography. In this chapter, we review the main approaches and methods from literature that are relevant for the application of tractogram filtering. Moreover, we give a perspective on the central challenges for the development of new methods, including modern machine learning techniques, in this field in the next few years.