Improving Fiber Alignment in HARDI by Combining Contextual PDE Flow with Constrained Spherical Deconvolution

We propose two strategies to improve the quality of tractography results computed from diffusion weighted magnetic resonance imaging (DW-MRI) data. Both methods are based on the same PDE framework, defined in the coupled space of positions and orientations, associated with a stochastic process describing the enhancement of elongated structures while preserving crossing structures. In the first method we use the enhancement PDE for contextual regularization of a fiber orientation distribution (FOD) that is obtained on individual voxels from high angular resolution diffusion imaging (HARDI) data via constrained spherical deconvolution (CSD). Thereby we improve the FOD as input for subsequent tractography. Secondly, we introduce the fiber to bundle coherence (FBC), a measure for quantification of fiber alignment. The FBC is computed from a tractography result using the same PDE framework and provides a criterion for removing the spurious fibers. We validate the proposed combination of CSD and enhancement on phantom data and on human data, acquired with different scanning protocols. On the phantom data we find that PDE enhancements improve both local metrics and global metrics of tractography results, compared to CSD without enhancements. On the human data we show that the enhancements allow for a better reconstruction of crossing fiber bundles and they reduce the variability of the tractography output with respect to the acquisition parameters. Finally, we show that both the enhancement of the FODs and the use of the FBC measure on the tractography improve the stability with respect to different stochastic realizations of probabilistic tractography. This is shown in a clinical application: the reconstruction of the optic radiation for epilepsy surgery planning

Diffusion, convection and erosion on R3 x S2 and their application to the enhancement of crossing fibers

In this article we study both left-invariant (convection-)diffusions and left-invariant Hamilton-Jacobi equations (erosions) on the space R3 x S2 of 3D-positions and orientations naturally embedded in the group SE(3) of 3D-rigid body movements. The general motivation for these (convection-)diffusions and erosions is to obtain crossing-preserving fiber enhancement on probability densities defined on the space of positions and orientations. The linear left-invariant (convection-)diffusions are forward Kolmogorov equations of Brownian motions on R3 x S2 and can be solved by R3 x S2-convolution with the corresponding Green’s functions or by a finite difference scheme. The left-invariant Hamilton-Jacobi equations are Bellman equations of cost processes on R3 x S2 and they are solved by a morphological R3 x S2-convolution with the corresponding Green’s functions. We will reveal the remarkable analogy between these erosions/dilations and diffusions. Furthermore, we consider pseudo-linear scale spaces on the space of positions and orientations that combines dilation and diffusion in a single evolution. In our design and analysis for appropriate linear, non-linear, morphological and pseudo-linear scale spaces on R3 x S2 we employ the underlying differential geometry on SE(3), where the frame of left-invariant vector fields serves as a moving frame of reference. Furthermore, we will present new and simpler finite difference schemes for our diffusions, which are clear improvements of our previous finite difference schemes. We apply our theory to the enhancement of fibres in magnetic resonance imaging (MRI) techniques (HARDI and DTI) for imaging water diffusion processes in fibrous tissues such as brain white matter and muscles. We provide experiments of our crossing-preserving (non-linear) left-invariant evolutions on neural images of a human brain containing crossing fibers

Estimation de fonctions de densité des orientations asymétriques pour l'imagerie par résonance magnétique de diffusion

L'imagerie par résonance magnétique (IRM) de diffusion est une modalité d'acquisition permettant de mesurer le déplacement des molécules d'eau à l'intérieur d'un médium selon un ensemble de positions et de directions échantillonnées. Comme le déplacement des molécules d'eau à l'intérieur d'un élément de volume (voxel) est influencé par les configurations des axones le traversant, l'IRM de diffusion permet de sonder l'organisation structurelle du cerveau. Or, le signal de diffusion mesuré par l'IRM n'est pas aligné avec les orientations locales des axones. On doit donc d'abord le transformer en une image de fonctions de densité des orientations, décrivant la quantité apparente de fibres neuronales traversant chaque voxel selon une orientation donnée. La fonction de densité des orientations est habituellement modélisée par une fonction sphérique symétrique, assignant une même valeur scalaire à deux directions opposées sur la surface d'une sphère unitaire. Or, les trajectoires neuronales à l'intérieur d'un voxel ne sont pas nécessairement symétriques. En considérant le signal aux voxels voisins dans l'estimation de la fonction de densité des orientations pour un voxel donné, il est cependant possible d'estimer des fonctions de densité des orientations asymétriques, qui reproduisent plus fidèlement les configurations des fibres sous-jacentes. Ainsi, l'objectif de ce mémoire est de développer une nouvelle méthode de filtrage permettant de transformer une image de fonctions de densité des orientations symétriques en une image de fonctions de densités des orientations asymétriques, puis d'utiliser celle-ci afin d'étudier l'occurrence de configurations asymétriques à l'intérieur du cerveau acquis par IRM de diffusion. S'appuyant sur des mesures comme l'indice d'asymétrie, indiquant dans quelle mesure une fonction sphérique est asymétrique, et le nombre de directions de fibres, une mesure permettant la classification des fonctions sphériques asymétriques selon leur forme, des régions asymétriques sont identifiées. Ce mémoire montre que les configurations asymétriques surviennent dans au moins 40% des voxels de la matière blanche et 70% des voxels de la matière grise. Les fonctions de densité des orientations asymétriques estimées à partir de la méthode proposée capturent des trajectoires de fibres courbées, des terminaisons de trajectoires, des trajectoires en éventails, des embranchements et d'autres configurations complexes qui ne peuvent pas être représentées adéquatement en utilisant une fonction sphérique symétrique

PDE-based Group Equivariant Convolutional Neural Networks

We present a PDE-based framework that generalizes Group equivariant Convolutional Neural Networks (G-CNNs). In this framework, a network layer is seen as a set of PDE-solvers where geometrically meaningful PDE-coefficients become the layer's trainable weights. Formulating our PDEs on homogeneous spaces allows these networks to be designed with built-in symmetries such as rotation in addition to the standard translation equivariance of CNNs. Having all the desired symmetries included in the design obviates the need to include them by means of costly techniques such as data augmentation. We will discuss our PDE-based G-CNNs (PDE-G-CNNs) in a general homogeneous space setting while also going into the specifics of our primary case of interest: roto-translation equivariance. We solve the PDE of interest by a combination of linear group convolutions and non-linear morphological group convolutions with analytic kernel approximations that we underpin with formal theorems. Our kernel approximations allow for fast GPU-implementation of the PDE-solvers, we release our implementation with this article in the form of the LieTorch extension to PyTorch, available at https://gitlab.com/bsmetsjr/lietorch . Just like for linear convolution a morphological convolution is specified by a kernel that we train in our PDE-G-CNNs. In PDE-G-CNNs we do not use non-linearities such as max/min-pooling and ReLUs as they are already subsumed by morphological convolutions. We present a set of experiments to demonstrate the strength of the proposed PDE-G-CNNs in increasing the performance of deep learning based imaging applications with far fewer parameters than traditional CNNs.Comment: 27 pages, 18 figures. v2 changes: - mentioned KerCNNs - added section Generalization of G-CNNs - clarification that the experiments utilized automatic differentiation and SGD. v3 changes: - streamlined theoretical framework - formulation and proof Thm.1 & 2 - expanded experiments. v4 changes: typos in Prop.5 and (20) v5/6 changes: minor revisio

Diseases of the Brain, Head and Neck, Spine 2020–2023

This open access book offers an essential overview of brain, head and neck, and spine imaging. Over the last few years, there have been considerable advances in this area, driven by both clinical and technological developments. Written by leading international experts and teachers, the chapters are disease-oriented and cover all relevant imaging modalities, with a focus on magnetic resonance imaging and computed tomography. The book also includes a synopsis of pediatric imaging. IDKD books are rewritten (not merely updated) every four years, which means they offer a comprehensive review of the state-of-the-art in imaging. The book is clearly structured and features learning objectives, abstracts, subheadings, tables and take-home points, supported by design elements to help readers navigate the text. It will particularly appeal to general radiologists, radiology residents, and interventional radiologists who want to update their diagnostic expertise, as well as clinicians from other specialties who are interested in imaging for their patient care

Development of Shear-Thinning and Self-Healing Hydrogels Through Guest-Host Interactions for Biomedical Applications

Hydrogels have emerged as an invaluable class of materials for biomedical applications, owing in part to their utility as structural, bioinstructive, and cell-laden implants that mimic many aspects of native tissues. Despite their many positive attributes, conventional hydrogels face numerous challenges toward translational therapies, including difficulty in delivery (i.e., invasive implantation) as well as limited control over biophysical properties (i.e., porosity, degradation, and strength). To address these challenges, the overall goal of this dissertation was the development of a class of supramolecular hydrogels that can be implanted in vivo by simple injection and that have tunable properties — either innate to the system or achieved through additional modifications. Toward this, we developed guest-host (GH) hydrogels that undergo supramolecular assembly through complexation of hyaluronic acid (HA) separately modified by adamantane (Ad-HA, guest) and β-cyclodextrin (CD-HA, host). Modular modifications were made to GH hydrogels to enable tuning of biophysical properties, including the incorporation of matrix-metalloproteinase cleavable peptides between HA and Ad to form enzymatically degradable assemblies. Additionally, dual-crosslinking (DC) of methacrylated CD-HA (CD-MeHA) and thiolated Ad-HA (Ad-HA-SH) by Michael addition subsequent to GH assembly was explored to stiffen hydrogels in vivo following injection. Finally, injectable and tough double network (DN) hydrogels were fabricated, where GH hydrogels were formed in the presence of an interpenetrating covalent network (methacrylated HA, MeHA) crosslinked by Michael addition with a dithiol under cytocompatible conditions. Both GH and DC hydrogels were further explored in vivo, with application to attenuate the maladaptive left ventricular (LV) remodeling that occurs following myocardial infarction (MI) that can result in heart failure. DC hydrogels reduced stress within the infarct region, prevented early ventricular expansion and thereby ameliorated progressive LV remodeling. Moreover, the preservation of myocardial geometry reduced incidence and severity of ischemic mitral regurgitation — an undesirable and devastating consequence of LV remodeling. Overall, the body of work represents a novel approach to engineer biomaterials with unique properties toward biomedical therapies