The geometric evolution of aortic dissections: Predicting surgical success using fluctuations in integrated Gaussian curvature.

Clinical imaging modalities are a mainstay of modern disease management, but the full utilization of imaging-based data remains elusive. Aortic disease is defined by anatomic scalars quantifying aortic size, even though aortic disease progression initiates complex shape changes. We present an imaging-based geometric descriptor, inspired by fundamental ideas from topology and soft-matter physics that captures dynamic shape evolution. The aorta is reduced to a two-dimensional mathematical surface in space whose geometry is fully characterized by the local principal curvatures. Disease causes deviation from the smooth bent cylindrical shape of normal aortas, leading to a family of highly heterogeneous surfaces of varying shapes and sizes. To deconvolute changes in shape from size, the shape is characterized using integrated Gaussian curvature or total curvature. The fluctuation in total curvature (δK) across aortic surfaces captures heterogeneous morphologic evolution by characterizing local shape changes. We discover that aortic morphology evolves with a power-law defined behavior with rapidly increasing δK forming the hallmark of aortic disease. Divergent δK is seen for highly diseased aortas indicative of impending topologic catastrophe or aortic rupture. We also show that aortic size (surface area or enclosed aortic volume) scales as a generalized cylinder for all shapes. Classification accuracy for predicting aortic disease state (normal, diseased with successful surgery, and diseased with failed surgical outcomes) is 92.8±1.7%. The analysis of δK can be applied on any three-dimensional geometric structure and thus may be extended to other clinical problems of characterizing disease through captured anatomic changes.

Helicity conservation by flow across scales in reconnecting vortex links and knots.

The conjecture that helicity (or knottedness) is a fundamental conserved quantity has a rich history in fluid mechanics, but the nature of this conservation in the presence of dissipation has proven difficult to resolve. Making use of recent advances, we create vortex knots and links in viscous fluids and simulated superfluids and track their geometry through topology-changing reconnections. We find that the reassociation of vortex lines through a reconnection enables the transfer of helicity from links and knots to helical coils. This process is remarkably efficient, owing to the antiparallel orientation spontaneously adopted by the reconnecting vortices. Using a new method for quantifying the spatial helicity spectrum, we find that the reconnection process can be viewed as transferring helicity between scales, rather than dissipating it. We also infer the presence of geometric deformations that convert helical coils into even smaller scale twist, where it may ultimately be dissipated. Our results suggest that helicity conservation plays an important role in fluids and related fields, even in the presence of dissipation.

Computed tomographic measures of pulmonary vascular morphology in smokers and their clinical implications.

RATIONALE: Angiographic investigation suggests that pulmonary vascular remodeling in smokers is characterized by distal pruning of the blood vessels. OBJECTIVES: Using volumetric computed tomography scans of the chest we sought to quantitatively evaluate this process and assess its clinical associations. METHODS: Pulmonary vessels were automatically identified, segmented, and measured. Total blood vessel volume (TBV) and the aggregate vessel volume for vessels less than 5 mm(2) (BV5) were calculated for all lobes. The lobe-specific BV5 measures were normalized to the TBV of that lobe and the nonvascular tissue volume (BV5/T(issue)V) to calculate lobe-specific BV5/TBV and BV5/T(issue)V ratios. Densitometric measures of emphysema were obtained using a Hounsfield unit threshold of -950 (%LAA-950). Measures of chronic obstructive pulmonary disease severity included single breath measures of diffusing capacity of carbon monoxide, oxygen saturation, the 6-minute-walk distance, St George's Respiratory Questionnaire total score (SGRQ), and the body mass index, airflow obstruction, dyspnea, and exercise capacity (BODE) index. MEASUREMENTS AND MAIN RESULTS: The %LAA-950 was inversely related to all calculated vascular ratios. In multivariate models including age, sex, and %LAA-950, lobe-specific measurements of BV5/TBV were directly related to resting oxygen saturation and inversely associated with both the SGRQ and BODE scores. In similar multivariate adjustment lobe-specific BV5/T(issue)V ratios were inversely related to resting oxygen saturation, diffusing capacity of carbon monoxide, 6-minute-walk distance, and directly related to the SGRQ and BODE. CONCLUSIONS: Smoking-related chronic obstructive pulmonary disease is characterized by distal pruning of the small blood vessels (<5 mm(2)) and loss of tissue in excess of the vasculature. The magnitude of these changes predicts the clinical severity of disease.

DTI measures in crossing-fibre areas: increased diffusion anisotropy reveals early white matter alteration in MCI and mild Alzheimer's disease.

Though mild cognitive impairment is an intermediate clinical state between healthy aging and Alzheimer's disease (AD), there are very few whole-brain voxel-wise diffusion MRI studies directly comparing changes in healthy control, mild cognitive impairment (MCI) and AD subjects. Here we report whole-brain findings from a comprehensive study of diffusion tensor indices and probabilistic tractography obtained in a very large population of healthy controls, MCI and probable AD subjects. As expected from the literature, all diffusion indices converged to show that the cingulum bundle, the uncinate fasciculus, the entire corpus callosum and the superior longitudinal fasciculus are the most affected white matter tracts in AD. Significant differences between MCI and AD were essentially confined to the corpus callosum. More importantly, we introduce for the first time in a degenerative disorder an application of a recently developed tensor index, the "mode" of anisotropy, as well as probabilistic crossing-fibre tractography. The mode of anisotropy specifies the type of anisotropy as a continuous measure reflecting differences in shape of the diffusion tensor ranging from planar (e.g., in regions of crossing fibres from two fibre populations of similar density or regions of "kissing" fibres) to linear (e.g., in regions where one fibre population orientation predominates), while probabilistic crossing-fibre tractography allows to accurately trace pathways from a crossing-fibre region. Remarkably, when looking for whole-brain diffusion differences between MCI patients and healthy subjects, the only region with significant abnormalities was a region of crossing fibres in the centrum semiovale, showing an increased mode of anisotropy. The only white matter region demonstrating a significant difference in correlations between neuropsychological scores and a diffusion measure (mode of anisotropy) across the three groups was the same region of crossing fibres. Further examination using probabilistic tractography established explicitly and quantitatively that this previously unreported increase of mode and co-localised increase of fractional anisotropy was explained by a relative preservation of motor-related projection fibres (at this early stage of the disease) crossing the association fibres of the superior longitudinal fasciculus. These findings emphasise the benefit of looking at the more complex regions in which spared and affected pathways are crossing to detect very early alterations of the white matter that could not be detected in regions consisting of one fibre population only. Finally, the methods used in this study may have general applicability for other degenerative disorders and, beyond the clinical sphere, they could contribute to a better quantification and understanding of subtle effects generated by normal processes such as visuospatial attention or motor learning.

Local white matter geometry from diffusion tensor gradients.

We introduce a mathematical framework for computing geometrical properties of white matter fibers directly from diffusion tensor fields. The key idea is to isolate the portion of the gradient of the tensor field corresponding to local variation in tensor orientation, and to project it onto a coordinate frame of tensor eigenvectors. The resulting eigenframe-centered representation then makes it possible to define scalar indices (or measures) that describe the local white matter geometry directly from the diffusion tensor field and its gradient, without requiring prior tractography. We derive new scalar indices of (1) fiber dispersion and (2) fiber curving, and we demonstrate them on synthetic and in vivo data. Finally, we illustrate their applicability to a group study on schizophrenia.

Superquadric glyphs for symmetric second-order tensors.

Symmetric second-order tensor fields play a central role in scientific and biomedical studies as well as in image analysis and feature-extraction methods. The utility of displaying tensor field samples has driven the development of visualization techniques that encode the tensor shape and orientation into the geometry of a tensor glyph. With some exceptions, these methods work only for positive-definite tensors (i.e. having positive eigenvalues, such as diffusion tensors). We expand the scope of tensor glyphs to all symmetric second-order tensors in two and three dimensions, gracefully and unambiguously depicting any combination of positive and negative eigenvalues. We generalize a previous method of superquadric glyphs for positive-definite tensors by drawing upon a larger portion of the superquadric shape space, supplemented with a coloring that indicates the quadratic form (including eigenvalue sign). We show that encoding arbitrary eigenvalue magnitudes requires design choices that differ fundamentally from those in previous work on traceless tensors that arise in the study of liquid crystals. Our method starts with a design of 2-D tensor glyphs guided by principles of scale-preservation and symmetry, and creates 3-D glyphs that include the 2-D glyphs in their axis-aligned cross-sections. A key ingredient of our method is a novel way of mapping from the shape space of three-dimensional symmetric second-order tensors to the unit square. We apply our new glyphs to stress tensors from mechanics, geometry tensors and Hessians from image analysis, and rate-of-deformation tensors in computational fluid dynamics.

SlicerDMRI: Diffusion MRI and Tractography Research Software for Brain Cancer Surgery Planning and Visualization.

PURPOSE: We present SlicerDMRI, an open-source software suite that enables research using diffusion magnetic resonance imaging (dMRI), the only modality that can map the white matter connections of the living human brain. SlicerDMRI enables analysis and visualization of dMRI data and is aimed at the needs of clinical research users. SlicerDMRI is built upon and deeply integrated with 3D Slicer, a National Institutes of Health-supported open-source platform for medical image informatics, image processing, and three-dimensional visualization. Integration with 3D Slicer provides many features of interest to cancer researchers, such as real-time integration with neuronavigation equipment, intraoperative imaging modalities, and multimodal data fusion. One key application of SlicerDMRI is in neurosurgery research, where brain mapping using dMRI can provide patient-specific maps of critical brain connections as well as insight into the tissue microstructure that surrounds brain tumors. PATIENTS AND METHODS: In this article, we focus on a demonstration of SlicerDMRI as an informatics tool to enable end-to-end dMRI analyses in two retrospective imaging data sets from patients with high-grade glioma. Analyses demonstrated here include conventional diffusion tensor analysis, advanced multifiber tractography, automated identification of critical fiber tracts, and integration of multimodal imagery with dMRI. RESULTS: We illustrate the ability of SlicerDMRI to perform both conventional and advanced dMRI analyses as well as to enable multimodal image analysis and visualization. We provide an overview of the clinical rationale for each analysis along with pointers to the SlicerDMRI tools used in each. CONCLUSION: SlicerDMRI provides open-source and clinician-accessible research software tools for dMRI analysis. SlicerDMRI is available for easy automated installation through the 3D Slicer Extension Manager.

Resolving crossings in the corticospinal tract by two-tensor streamline tractography: Method and clinical assessment using fMRI.

An inherent drawback of the traditional diffusion tensor model is its limited ability to provide detailed information about multidirectional fiber architecture within a voxel. This leads to erroneous fiber tractography results in locations where fiber bundles cross each other. This may lead to the inability to visualize clinically important tracts such as the lateral projections of the corticospinal tract. In this report, we present a deterministic two-tensor eXtended Streamline Tractography (XST) technique, which successfully traces through regions of crossing fibers. We evaluated the method on simulated and in vivo human brain data, comparing the results with the traditional single-tensor and with a probabilistic tractography technique. By tracing the corticospinal tract and correlating with fMRI-determined motor cortex in both healthy subjects and patients with brain tumors, we demonstrate that two-tensor deterministic streamline tractography can accurately identify fiber bundles consistent with anatomy and previously not detected by conventional single-tensor tractography. When compared to the dense connectivity maps generated by probabilistic tractography, the method is computationally efficient and generates discrete geometric pathways that are simple to visualize and clinically useful. Detection of crossing white matter pathways can improve neurosurgical visualization of functionally relevant white matter areas.

Increased diffusivity in superior temporal gyrus in patients with schizophrenia: a Diffusion Tensor Imaging study.

BACKGROUND: Superior temporal gyrus (STG) volume reduction is one of the most consistent findings in schizophrenia. The goal of this study was to conduct the first Diffusion Tensor Imaging (DTI) study to investigate altered structural integrity in STG gray and white matter in patients with chronic schizophrenia compared with healthy controls. METHODS: Magnetic resonance imaging (MRI) and DTI were acquired in 21 male patients with schizophrenia and 22 age-, handedness-, and parental social economic status-matched male comparison subjects. After manual segmentation of gray and white matter, mean diffusivity and fractional anisotropy were measured within STG. Correlational analyses were also conducted to test possible associations between DTI and clinical measures, including positive and negative symptoms of schizophrenia. RESULTS: Compared with controls, patients demonstrated reduced volume, bilaterally, in STG gray matter but not in white matter. For DTI measures, patients showed increased mean diffusivity, bilaterally, in STG gray matter, and in left STG white matter. In addition, mean diffusivity in left STG white matter showed statistically significant correlations with auditory hallucinations and attentional impairments in patients. CONCLUSIONS: These findings suggest a disruption of tissue integrity in STG gray and white matter in schizophrenia. In addition, increased water diffusivity in left-side STG, which was associated with auditory hallucinations and attentional impairments, suggests the possibility of a disconnection among auditory/language processing regions in schizophrenia.

Sampling and visualizing creases with scale-space particles.

Particle systems have gained importance as a methodology for sampling implicit surfaces and segmented objects to improve mesh generation and shape analysis. We propose that particle systems have a significantly more general role in sampling structure from unsegmented data. We describe a particle system that computes samplings of crease features (i.e. ridges and valleys, as lines or surfaces) that effectively represent many anatomical structures in scanned medical data. Because structure naturally exists at a range of sizes relative to the image resolution, computer vision has developed the theory of scale-space, which considers an n-D image as an (n+1)-D stack of images at different blurring levels. Our scale-space particles move through continuous four-dimensional scale-space according to spatial constraints imposed by the crease features, a particle-image energy that draws particles towards scales of maximal feature strength, and an inter-particle energy that controls sampling density in space and scale. To make scale-space practical for large three-dimensional data, we present a spline-based interpolation across scale from a small number of pre-computed blurrings at optimally selected scales. The configuration of the particle system is visualized with tensor glyphs that display information about the local Hessian of the image, and the scale of the particle. We use scale-space particles to sample the complex three-dimensional branching structure of airways in lung CT, and the major white matter structures in brain DTI.

Invariant crease lines for topological and structural analysis of tensor fields.

We introduce a versatile framework for characterizing and extracting salient structures in three-dimensional symmetric second-order tensor fields. The key insight is that degenerate lines in tensor fields, as defined by the standard topological approach, are exactly crease (ridge and valley) lines of a particular tensor invariant called mode. This reformulation allows us to apply well-studied approaches from scientific visualization or computer vision to the extraction of topological lines in tensor fields. More generally, this main result suggests that other tensor invariants, such as anisotropy measures like fractional anisotropy (FA), can be used in the same framework in lieu of mode to identify important structural properties in tensor fields. Our implementation addresses the specific challenge posed by the non-linearity of the considered scalar measures and by the smoothness requirement of the crease manifold computation. We use a combination of smooth reconstruction kernels and adaptive refinement strategy that automatically adjust the resolution of the analysis to the spatial variation of the considered quantities. Together, these improvements allow for the robust application of existing ridge line extraction algorithms in the tensor context of our problem. Results are proposed for a diffusion tensor MRI dataset, and for a benchmark stress tensor field used in engineering research.

Looks Good To Me: Visualizations As Sanity Checks.

Famous examples such as Anscombe's Quartet highlight that one of the core benefits of visualizations is allowing people to discover visual patterns that might otherwise be hidden by summary statistics. This visual inspection is particularly important in exploratory data analysis, where analysts can use visualizations such as histograms and dot plots to identify data quality issues. Yet, these visualizations are driven by parameters such as histogram bin size or mark opacity that have a great deal of impact on the final visual appearance of the chart, but are rarely optimized to make important features visible. In this paper, we show that data flaws have varying impact on the visual features of visualizations, and that the adversarial or merely uncritical setting of design parameters of visualizations can obscure the visual signatures of these flaws. Drawing on the framework of Algebraic Visualization Design, we present the results of a crowdsourced study showing that common visualization types can appear to reasonably summarize distributional data while hiding large and important flaws such as missing data and extraneous modes. We make use of these results to propose additional best practices for visualizations of distributions for data quality tasks.

Diffusion tensor analysis with invariant gradients and rotation tangents.

Guided by empirically established connections between clinically important tissue properties and diffusion tensor parameters, we introduce a framework for decomposing variations in diffusion tensors into changes in shape and orientation. Tensor shape and orientation both have three degrees-of-freedom, spanned by invariant gradients and rotation tangents, respectively. As an initial demonstration of the framework, we create a tunable measure of tensor difference that can selectively respond to shape and orientation. Second, to analyze the spatial gradient in a tensor volume (a third-order tensor), our framework generates edge strength measures that can discriminate between different neuroanatomical boundaries, as well as creating a novel detector of white matter tracts that are adjacent yet distinctly oriented. Finally, we apply the framework to decompose the fourth-order diffusion covariance tensor into individual and aggregate measures of shape and orientation covariance, including a direct approximation for the variance of tensor invariants such as fractional anisotropy.

Delineating white matter structure in diffusion tensor MRI with anisotropy creases.

Geometric models of white matter architecture play an increasing role in neuroscientific applications of diffusion tensor imaging, and the most popular method for building them is fiber tractography. For some analysis tasks, however, a compelling alternative may be found in the first and second derivatives of diffusion anisotropy. We extend to tensor fields the notion from classical computer vision of ridges and valleys, and define anisotropy creases as features of locally extremal tensor anisotropy. Mathematically, these are the loci where the gradient of anisotropy is orthogonal to one or more eigenvectors of its Hessian. We propose that anisotropy creases provide a basis for extracting a skeleton of the major white matter pathways, in that ridges of anisotropy coincide with interiors of fiber tracts, and valleys of anisotropy coincide with the interfaces between adjacent but distinctly oriented tracts. The crease extraction algorithm we present generates high-quality polygonal models of crease surfaces, which are further simplified by connected-component analysis. We demonstrate anisotropy creases on measured diffusion MRI data, and visualize them in combination with tractography to confirm their anatomic relevance.

SlicerDMRI: Open Source Diffusion MRI Software for Brain Cancer Research.

Diffusion MRI (dMRI) is the only noninvasive method for mapping white matter connections in the brain. We describe SlicerDMRI, a software suite that enables visualization and analysis of dMRI for neuroscientific studies and patient-specific anatomic assessment. SlicerDMRI has been successfully applied in multiple studies of the human brain in health and disease, and here, we especially focus on its cancer research applications. As an extension module of the 3D Slicer medical image computing platform, the SlicerDMRI suite enables dMRI analysis in a clinically relevant multimodal imaging workflow. Core SlicerDMRI functionality includes diffusion tensor estimation, white matter tractography with single and multi-fiber models, and dMRI quantification. SlicerDMRI supports clinical DICOM and research file formats, is open-source and cross-platform, and can be installed as an extension to 3D Slicer (www.slicer.org). More information, videos, tutorials, and sample data are available at dmri.slicer.org Cancer Res; 77(21); e101-3. ©2017 AACR.

Diffusion tensor visualization with glyph packing.

A common goal of multivariate visualization is to enable data inspection at discrete points, while also illustrating larger-scale continuous structures. In diffusion tensor visualization, glyphs are typically used to meet the first goal, and methods such as texture synthesis or fiber tractography can address the second. We adapt particle systems originally developed for surface modeling and anisotropic mesh generation to enhance the utility of glyph-based tensor visualizations. By carefully distributing glyphs throughout the field (either on a slice, or in the volume) into a dense packing, using potential energy profiles shaped by the local tensor value, we remove undue visual emphasis of the regular sampling grid of the data, and the underlying continuous features become more apparent. The method is demonstrated on a DT-MRI scan of a patient with a brain tumor.

Diderot: a Domain-Specific Language for Portable Parallel Scientific Visualization and Image Analysis.

Many algorithms for scientific visualization and image analysis are rooted in the world of continuous scalar, vector, and tensor fields, but are programmed in low-level languages and libraries that obscure their mathematical foundations. Diderot is a parallel domain-specific language that is designed to bridge this semantic gap by providing the programmer with a high-level, mathematical programming notation that allows direct expression of mathematical concepts in code. Furthermore, Diderot provides parallel performance that takes advantage of modern multicore processors and GPUs. The high-level notation allows a concise and natural expression of the algorithms and the parallelism allows efficient execution on real-world datasets.

Reconstruction and visualization of planetary nebulae.

From our terrestrially confined viewpoint, the actual three-dimensional shape of distant astronomical objects is, in general, very challenging to determine. For one class of astronomical objects, however, spatial structure can be recovered from conventional 2D images alone. So-called planetary nebulae (PNe) exhibit pronounced symmetry characteristics that come about due to fundamental physical processes. Making use of this symmetry constraint, we present a technique to automatically recover the axisymmetric structure of many planetary nebulae from photographs. With GPU-based volume rendering driving a nonlinear optimization, we estimate the nebula's local emission density as a function of its radial and axial coordinates and we recover the orientation of the nebula relative to Earth. The optimization refines the nebula model and its orientation by minimizing the differences between the rendered image and the original astronomical image. The resulting model allows creating realistic 3D visualizations of these nebulae, for example, for planetarium shows and other educational purposes. In addition, the recovered spatial distribution of the emissive gas can help astrophysicists gain deeper insight into the formation processes of planetary nebulae.

An Algebraic Process for Visualization Design.

We present a model of visualization design based on algebraic considerations of the visualization process. The model helps characterize visual encodings, guide their design, evaluate their effectiveness, and highlight their shortcomings. The model has three components: the underlying mathematical structure of the data or object being visualized, the concrete representation of the data in a computer, and (to the extent possible) a mathematical description of how humans perceive the visualization. Because we believe the value of our model lies in its practical application, we propose three general principles for good visualization design. We work through a collection of examples where our model helps explain the known properties of existing visualizations methods, both good and not-so-good, as well as suggesting some novel methods. We describe how to use the model alongside experimental user studies, since it can help frame experiment outcomes in an actionable manner. Exploring the implications and applications of our model and its design principles should provide many directions for future visualization research.

The effects of noise over the complete space of diffusion tensor shape.

Diffusion tensor magnetic resonance imaging (DT-MRI) is a technique used to quantify the microstructural organization of biological tissues. Multiple images are necessary to reconstruct the tensor data and each acquisition is subject to complex thermal noise. As such, measures of tensor invariants, which characterize components of tensor shape, derived from the tensor data will be biased from their true values. Previous work has examined this bias, but over a narrow range of tensor shape. Herein, we define the mathematics for constructing a tensor from tensor invariants, which permits an intuitive and principled means for building tensors with a complete range of tensor shape and salient microstructural properties. Thereafter, we use this development to evaluate by simulation the effects of noise on characterizing tensor shape over the complete space of tensor shape for three encoding schemes with different SNR and gradient directions. We also define a new framework for determining the distribution of the true values of tensor invariants given their measures, which provides guidance about the confidence the observer should have in the measures. Finally, we present the statistics of tensor invariant estimates over the complete space of tensor shape to demonstrate how the noise sensitivity of tensor invariants varies across the space of tensor shape as well as how the imaging protocol impacts measures of tensor invariants.