Diffeomorphometry and geodesic positioning systems for human anatomy

Weakly Supervised Bayesian Shape Modeling from Unsegmented Medical Images

Anatomical shape analysis plays a pivotal role in clinical research and hypothesis testing, where the relationship between form and function is paramount. Correspondence-based statistical shape modeling (SSM) facilitates population-level morphometrics but requires a cumbersome, potentially bias-inducing construction pipeline. Traditional construction pipelines require manual and computationally expensive steps, hindering their widespread use. Furthermore, such methods utilize templates or assumptions (e.g., linearity) that can bias or limit the expressivity of the variation captured by the constructed SSM. Recent advancements in deep learning have streamlined this process in inference by providing SSM prediction directly from unsegmented medical images. However, the proposed approaches are fully supervised and require utilizing a traditional SSM construction pipeline to create training data, thus inheriting the associated burdens and limitations. To address these challenges, we introduce a weakly supervised deep learning approach to predict SSM from images using point cloud supervision. Specifically, we propose reducing the supervision associated with the state-of-the-art fully Bayesian variational information bottleneck DeepSSM (BVIB-DeepSSM) model. BVIB-DeepSSM is an effective, principled framework for predicting probabilistic anatomical shapes from images with quantification of both aleatoric and epistemic uncertainties. Whereas the original BVIB-DeepSSM method requires strong supervision in the form of ground truth correspondence points, the proposed approach utilizes weak supervision via point cloud surface representations, which are more readily obtainable. Furthermore, the proposed approach learns correspondence in a completely data-driven manner without prior assumptions about the expected variability in shape cohort. Our experiments demonstrate that this approach yields similar accuracy and uncertainty estimation to the fully supervised scenario while substantially enhancing the feasibility of model training for SSM construction.

Cross-modality mapping using image varifolds to align tissue-scale atlases to molecular-scale measures with application to 2D brain sections

This paper explicates a solution to building correspondences between molecular-scale transcriptomics and tissue-scale atlases. This problem arises in atlas construction and cross-specimen/technology alignment where specimens per emerging technology remain sparse and conventional image representations cannot efficiently model the high dimensions from subcellular detection of thousands of genes. We address these challenges by representing spatial transcriptomics data as generalized functions encoding position and high-dimensional feature (gene, cell type) identity. We map onto low-dimensional atlas ontologies by modeling regions as homogeneous random fields with unknown transcriptomic feature distribution. We solve simultaneously for the minimizing geodesic diffeomorphism of coordinates through LDDMM and for these latent feature densities. We map tissue-scale mouse brain atlases to gene-based and cell-based transcriptomics data from MERFISH and BARseq technologies and to histopathology and cross-species atlases to illustrate integration of diverse molecular and cellular datasets into a single coordinate system as a means of comparison and further atlas construction.

A Universal Method for Crossing Molecular and Atlas Modalities using Simplex-Based Image Varifolds and Quadratic Programming

This paper explicates a solution to the problem of building correspondences between molecular-scale transcriptomics and tissue-scale atlases. The central model represents spatial transcriptomics as generalized functions encoding molecular position and high-dimensional transcriptomic-based (gene, cell type) identity. We map onto low-dimensional atlas ontologies by modeling each atlas compartment as a homogeneous random field with unknown transcriptomic feature distribution. The algorithm presented solves simultaneously for the minimizing geodesic diffeomorphism of coordinates and latent atlas transcriptomic feature fractions by alternating LDDMM optimization for coordinate transformations and quadratic programming for the latent transcriptomic variables. We demonstrate the universality of the algorithm in mapping tissue atlases to gene-based and cell-based MERFISH datasets as well as to other tissue scale atlases. The joint estimation of diffeomorphisms and latent feature distributions allows integration of diverse molecular and cellular datasets into a single coordinate system and creates an avenue of comparison amongst atlas ontologies for continued future development.

Learning spatiotemporal statistical shape models for non-linear dynamic anatomies

Numerous clinical investigations require understanding changes in anatomical shape over time, such as in dynamic organ cycle characterization or longitudinal analyses (e.g., for disease progression). Spatiotemporal statistical shape modeling (SSM) allows for quantifying and evaluating dynamic shape variation with respect to a cohort or population of interest. Existing data-driven SSM approaches leverage information theory to capture population-level shape variations by learning correspondence-based (landmark) representations of shapes directly from data using entropy-based optimization schemes. These approaches assume sample independence and thus are unsuitable for sequential dynamic shape observations. Previous methods for adapting entropy-based SSM optimization schemes for the spatiotemporal case either utilize a cross-sectional design (ignoring within-subject correlation) or impose other limiting assumptions, such as the linearity of shape dynamics. Here, we present a principled approach to spatiotemporal SSM that relaxes these assumptions to correctly capture population-level shape variation over time. We propose to incorporate modeling the underlying time dependency into correspondence optimization via a regularized principal component polynomial regression. This approach is flexible enough to capture non-linear temporal dynamics while encoding population-specific spatial regularity. We demonstrate our method's efficacy on synthetic data and left atrium segmented from cardiac MRI scans. Our approach better captures the population modes of variation and a statistically significant time dependency than existing methods.

Incomplete spectrum QSM using support information

Introduction

Reconstructing a bounded object from incomplete k-space data is a well posed problem, and it was recently shown that this incomplete spectrum approach can be used to reconstruct undersampled MRI images with similar quality to compressed sensing approaches. Here, we apply this incomplete spectrum approach to the field-to-source inverse problem encountered in quantitative magnetic susceptibility mapping (QSM). The field-to-source problem is an ill-posed problem because of conical regions in frequency space where the dipole kernel is zero or very small, which leads to the kernel's inverse being ill-defined. These "ill-posed" regions typically lead to streaking artifacts in QSM reconstructions. In contrast to compressed sensing, our approach relies on knowledge of the image-space support, more commonly referred to as the mask, of our object as well as the region in k-space with ill-defined values. In the QSM case, this mask is usually available, as it is required for most QSM background field removal and reconstruction methods.

Methods

We tuned the incomplete spectrum method (mask and band-limit) for QSM on a simulated dataset from the most recent QSM challenge and validated the QSM reconstruction results on brain images acquired in five healthy volunteers, comparing incomplete spectrum QSM to current state-of-the art-methods: FANSI, nonlinear dipole inversion, and conventional thresholded k-space division.

Results

Without additional regularization, incomplete spectrum QSM performs slightly better than direct QSM reconstruction methods such as thresholded k-space division (PSNR of 39.9 vs. 39.4 of TKD on a simulated dataset) and provides susceptibility values in key iron-rich regions similar or slightly lower than state-of-the-art algorithms, but did not improve the PSNR in comparison to FANSI or nonlinear dipole inversion. With added (ℓ1-wavelet based) regularization the new approach produces results similar to compressed sensing based reconstructions (at sufficiently high levels of regularization).

Discussion

Incomplete spectrum QSM provides a new approach to handle the "ill-posed" regions in the frequency-space data input to QSM.

Projective Diffeomorphic Mapping of Molecular Digital Pathology with Tissue MRI

Reconstructing dense 3D anatomical coordinates from 2D projective measurements has become a central problem in digital pathology for both animal models and human studies. Here we describe Projective Large Deformation Diffeomorphic Metric Mapping (LDDMM), a technique which projects diffeomorphic mappings of dense human magnetic resonance imaging (MRI) atlases at tissue scales onto sparse measurements at micrometre scales associated with histological and more general optical imaging modalities. We solve the problem of dense mapping surjectively onto histological sections by incorporating technologies for crossing modalities that use nonlinear scattering transforms to represent multiple radiomic-like textures at micron scales, together with a Gaussian mixture-model framework for modelling tears and distortions associated to each section. We highlight the significance of our method through incorporation of neuropathological measures and MRI, of relevance to the development of biomarkers for Alzheimer's disease and one instance of the integration of imaging data across the scales of clinical imaging and digital pathology.

Spatiotemporal Cardiac Statistical Shape Modeling: A Data-Driven Approach

Clinical investigations of anatomy's structural changes over time could greatly benefit from population-level quantification of shape, or spatiotemporal statistic shape modeling (SSM). Such a tool enables characterizing patient organ cycles or disease progression in relation to a cohort of interest. Constructing shape models requires establishing a quantitative shape representation (e.g., corresponding landmarks). Particle-based shape modeling (PSM) is a data-driven SSM approach that captures population-level shape variations by optimizing landmark placement. However, it assumes cross-sectional study designs and hence has limited statistical power in representing shape changes over time. Existing methods for modeling spatiotemporal or longitudinal shape changes require predefined shape atlases and pre-built shape models that are typically constructed cross-sectionally. This paper proposes a data-driven approach inspired by the PSM method to learn population-level spatiotemporal shape changes directly from shape data. We introduce a novel SSM optimization scheme that produces landmarks that are in correspondence both across the population (inter-subject) and across time-series (intra-subject). We apply the proposed method to 4D cardiac data from atrial-fibrillation patients and demonstrate its efficacy in representing the dynamic change of the left atrium. Furthermore, we show that our method outperforms an image-based approach for spatiotemporal SSM with respect to a generative time-series model, the Linear Dynamical System (LDS). LDS fit using a spatiotemporal shape model optimized via our approach provides better generalization and specificity, indicating it accurately captures the underlying time-dependency.

Geodesics in the TPS Space

In shape analysis, the interpolation of shapes’ trajectories is often performed by means of geodesics in an appropriate Riemannian Shape Space. Over the past several decades, different metrics and shape spaces have been proposed, including Kendall shape space, LDDMM based approaches, and elastic contour, among others. Once a Riemannian space is chosen, geodesics and parallel transports can be used to build splines or piecewise geodesics paths. In a recent paper, we introduced a new Riemannian shape space named TPS Space based on the Thin Plate Spline interpolant and characterized by an appropriate metric and parallel transport rule. In the present paper, we further explore the geometry of the TPS Space by characterizing the properties of its geodesics. Several applications show the capability of the proposed formulation to conserve important physical properties of deformation, such as local strains and global elastic energy.

Discriminative and generative models for anatomical shape analysis on point clouds with deep neural networks

We introduce deep neural networks for the analysis of anatomical shapes that learn a low-dimensional shape representation from the given task, instead of relying on hand-engineered representations. Our framework is modular and consists of several computing blocks that perform fundamental shape processing tasks. The networks operate on unordered point clouds and provide invariance to similarity transformations, avoiding the need to identify point correspondences between shapes. Based on the framework, we assemble a discriminative model for disease classification and age regression, as well as a generative model for the accruate reconstruction of shapes. In particular, we propose a conditional generative model, where the condition vector provides a mechanism to control the generative process. For instance, it enables to assess shape variations specific to a particular diagnosis, when passing it as side information. Next to working on single shapes, we introduce an extension for the joint analysis of multiple anatomical structures, where the simultaneous modeling of multiple structures can lead to a more compact encoding and a better understanding of disorders. We demonstrate the advantages of our framework in comprehensive experiments on real and synthetic data. The key insights are that (i) learning a shape representation specific to the given task yields higher performance than alternative shape descriptors, (ii) multi-structure analysis is both more efficient and more accurate than single-structure analysis, and (iii) point clouds generated by our model capture morphological differences associated to Alzheimer's disease, to the point that they can be used to train a discriminative model for disease classification. Our framework naturally scales to the analysis of large datasets, giving it the potential to learn characteristic variations in large populations.

Vestibular function and cortical and sub-cortical alterations in an aging population

While it is well known that the vestibular system is responsible for maintaining balance, posture and coordination, there is increasing evidence that it also plays an important role in cognition. Moreover, a growing number of epidemiological studies are demonstrating a link between vestibular dysfunction and cognitive deficits in older adults; however, the exact pathways through which vestibular loss may affect cognition are unknown. In this cross-sectional study, we sought to identify relationships between vestibular function and variation in morphometry in brain structures from structural neuroimaging. We used a subset of 80 participants from the Baltimore Longitudinal Study of Aging, who had both brain MRI and vestibular physiological data acquired during the same visit. Vestibular function was evaluated through the cervical vestibular-evoked myogenic potential (cVEMP). The brain structures of interest that we analyzed were the hippocampus, amygdala, thalamus, caudate nucleus, putamen, insula, entorhinal cortex (ERC), trans-entorhinal cortex (TEC) and perirhinal cortex, as these structures comprise or are connected with the putative "vestibular cortex." We modeled the volume and shape of these structures as a function of the presence/absence of cVEMP and the cVEMP amplitude, adjusting for age and sex. We observed reduced overall volumes of the hippocampus and the ERC associated with poorer vestibular function. In addition, we also found significant relationships between the shape of the hippocampus (p = 0.0008), amygdala (p = 0.01), thalamus (p = 0.008), caudate nucleus (p = 0.002), putamen (p = 0.02), and ERC-TEC complex (p = 0.008) and vestibular function. These findings provide novel insight into the multiple pathways through which vestibular loss may impact brain structures that are critically involved in spatial memory, navigation and orientation.

Structural neuroimaging of the altered brain stemming from pediatric and adolescent hearing loss-Scientific and clinical challenges

There has been a spurt in structural neuroimaging studies of the effect of hearing loss on the brain. Specifically, magnetic resonance imaging (MRI) and diffusion tensor imaging (DTI) technologies provide an opportunity to quantify changes in gray and white matter structures at the macroscopic scale. To date, there have been 32 MRI and 23 DTI studies that have analyzed structural differences accruing from pre- or peri-lingual pediatric hearing loss with congenital or early onset etiology and postlingual hearing loss in pre-to-late adolescence. Additionally, there have been 15 prospective clinical structural neuroimaging studies of children and adolescents being evaluated for cochlear implants. The results of the 70 studies are summarized in two figures and three tables. Plastic changes in the brain are seen to be multifocal rather than diffuse, that is, differences are consistent across regions implicated in the hearing, speech and language networks regardless of modes of communication and amplification. Structures in that play an important role in cognition are affected to a lesser extent. A limitation of these studies is the emphasis on volumetric measures and on homogeneous groups of subjects with hearing loss. It is suggested that additional measures of morphometry and connectivity could contribute to a greater understanding of the effect of hearing loss on the brain. Then an interpretation of the observed macroscopic structural differences is given. This is followed by discussion of how structural imaging can be combined with functional imaging to provide biomarkers for longitudinal tracking of amplification. This article is categorized under: Developmental Biology > Developmental Processes in Health and Disease Translational, Genomic, and Systems Medicine > Translational Medicine Laboratory Methods and Technologies > Imaging.

Diffeomorphic Registration With Intensity Transformation and Missing Data: Application to 3D Digital Pathology of Alzheimer's Disease

This paper examines the problem of diffeomorphic image registration in the presence of differing image intensity profiles and sparsely sampled, missing, or damaged tissue. Our motivation comes from the problem of aligning 3D brain MRI with 100-micron isotropic resolution to histology sections at 1 × 1 × 1,000-micron resolution with multiple varying stains. We pose registration as a penalized Bayesian estimation, exploiting statistical models of image formation where the target images are modeled as sparse and noisy observations of the atlas. In this injective setting, there is no assumption of symmetry between atlas and target. Cross-modality image matching is achieved by jointly estimating polynomial transformations of the atlas intensity. Missing data is accommodated via a multiple atlas selection procedure where several atlas images may be of homogeneous intensity and correspond to "background" or "artifact." The two concepts are combined within an Expectation-Maximization algorithm, where atlas selection posteriors and deformation parameters are updated iteratively and polynomial coefficients are computed in closed form. We validate our method with simulated images, examples from neuropathology, and a standard benchmarking dataset. Finally, we apply it to reconstructing digital pathology and MRI in standard atlas coordinates. By using a standard convolutional neural network to detect tau tangles in histology slices, this registration method enabled us to quantify the 3D density distribution of tauopathy throughout the medial temporal lobe of an Alzheimer's disease postmortem specimen.

Local and global energies for shape analysis in medical imaging

In a previous contribution, a new Riemannian shape space, named TPS space, was introduced to perform statistics on shape data. This space was endowed with a Riemannian metric and a flat connection, with torsion, compatible with the given metric. This connection allows the definition of a Parallel Transport of the deformation compatible with the three-fold decomposition in spherical, deviatoric, and non-affine components. Such a parallel transport also conserves the Γ-energy, strictly related to the total elastic strain energy stored by the body in the original deformation. A new approach is here presented in order to calculate the bending energy on the body alone (body bending energy) and to restrict it exclusively within physical boundaries of objects involved in the deformation analysis. The novelty of this new procedure resides in the fact that we propose a new metric to be preserved during the TPS direct transport. This allows transporting the shape change more coherently with the mechanical meaning of the deformation. The geometry of the TPS space is then further discussed in order to better represent the relationship between the Γ-energy, the strain energy, and the so-called bending energy densities.

On variational solutions for whole brain serial-section histology using a Sobolev prior in the computational anatomy random orbit model

This paper presents a variational framework for dense diffeomorphic atlas-mapping onto high-throughput histology stacks at the 20 μm meso-scale. The observed sections are modelled as Gaussian random fields conditioned on a sequence of unknown section by section rigid motions and unknown diffeomorphic transformation of a three-dimensional atlas. To regularize over the high-dimensionality of our parameter space (which is a product space of the rigid motion dimensions and the diffeomorphism dimensions), the histology stacks are modelled as arising from a first order Sobolev space smoothness prior. We show that the joint maximum a-posteriori, penalized-likelihood estimator of our high dimensional parameter space emerges as a joint optimization interleaving rigid motion estimation for histology restacking and large deformation diffeomorphic metric mapping to atlas coordinates. We show that joint optimization in this parameter space solves the classical curvature non-identifiability of the histology stacking problem. The algorithms are demonstrated on a collection of whole-brain histological image stacks from the Mouse Brain Architecture Project.

New developments in neural tracing techniques have motivated the widespread use of histology as a modality for exploring the circuitry of the brain. Automated mapping of pre-labeled atlases onto modern large datasets of histological imagery is a critical step for elucidating the brain’s neural circuitry and shape. This task is challenging as histological sections are imaged independently and the reconstruction of the unsectioned volume is nontrivial. Typically, neuroanatomists use reference volumes of the same subject (e.g. MRI) to guide reconstruction. However, obtaining reference imagery is often non-standard, as in high-throughput animal models like mouse histology. Others have proposed using anatomical atlases as guides, but have not accounted for the intrinsic nonlinear shape difference from atlas to subject. Our method addresses these limitations by jointly optimizing reconstruction informed by an atlas simultaneously with the nonlinear change of coordinates that encapsulates anatomical variation. This accounts for intrinsic shape differences and enables rigorous, direct comparisons of atlas and subject coordinates. Using simulations, we demonstrate that our method recovers the reconstruction parameters more accurately than atlas-free models and innately produces accurate segmentations from simultaneous atlas mapping. We also demonstrate our method on the Mouse Brain Architecture dataset, successfully mapping and reconstructing over 1000 brains.

ESTIMATING DIFFEOMORPHIC MAPPINGS BETWEEN TEMPLATES AND NOISY DATA: VARIANCE BOUNDS ON THE ESTIMATED CANONICAL VOLUME FORM

Anatomy is undergoing a renaissance driven by the availability of large digital data sets generated by light microscopy. A central computational task is to map individual data volumes to standardized templates. This is accomplished by regularized estimation of a diffeomorphic transformation between the coordinate systems of the individual data and the template, building the transformation incrementally by integrating a smooth flow field. The canonical volume form of this transformation is used to quantify local growth, atrophy, or cell density. While multiple implementations exist for this estimation, less attention has been paid to the variance of the estimated diffeomorphism for noisy data. Notably, there is an infinite dimensional unobservable space defined by those diffeomorphisms which leave the template invariant. These form the stabilizer subgroup of the diffeomorphic group acting on the template. The corresponding flat directions in the energy landscape are expected to lead to increased estimation variance. Here we show that a least-action principle used to generate geodesics in the space of diffeomor-phisms connecting the subject brain to the template removes the stabilizer. This provides reduced-variance estimates of the volume form. Using simulations we demonstrate that the asymmetric large deformation diffeomorphic mapping methods (LDDMM), which explicitly incorporate the asymmetry between idealized template images and noisy empirical images, provide lower variance estimators than their symmetrized counterparts (cf. ANTs). We derive Cramer-Rao bounds for the variances in the limit of small deformations. Analytical results are shown for the Jacobian in terms of perturbations of the vector fields and divergence of the vector field.

Whole-brain Segmentation and Change-point Analysis of Anatomical Brain MRI-Application in Premanifest Huntington's Disease

Recent advances in MRI offer a variety of useful markers to identify neurodegenerative diseases. In Huntington's disease (HD), regional brain atrophy begins many years prior to the motor onset (during the "premanifest" period), but the spatiotemporal pattern of regional atrophy across the brain has not been fully characterized. Here we demonstrate an online cloud-computing platform, "MRICloud", which provides atlas-based whole-brain segmentation of T1-weighted images at multiple granularity levels, and thereby, enables us to access the regional features of brain anatomy. We then describe a regression model that detects statistically significant inflection points, at which regional brain atrophy starts to be noticeable, i.e. the "change-point", with respect to a disease progression index. We used the CAG-age product (CAP) score to index the disease progression in HD patients. Change-point analysis of the volumetric measurements from the segmentation pipeline, therefore, provides important information of the order and pattern of structural atrophy across the brain. The paper illustrates the use of these techniques on T1-weighted MRI data of premanifest HD subjects from a large multicenter PREDICT-HD study. This design potentially has wide applications in a range of neurodegenerative diseases to investigate the dynamic changes of brain anatomy.

Computational anatomy and diffeomorphometry: A dynamical systems model of neuroanatomy in the soft condensed matter continuum

The nonlinear systems models of computational anatomy that have emerged over the past several decades are a synthesis of three significant areas of computational science and biological modeling. First is the algebraic model of biological shape as a Riemannian orbit, a set of objects under diffeomorphic action. Second is the embedding of anatomical shapes into the soft condensed matter physics continuum via the extension of the Euler equations to geodesic, smooth flows with inverses, encoding divergence for the compressibility of atrophy and expansion of growth. Third, is making human shape and form a metrizable space via geodesic connections of coordinate systems. These three themes place our formalism into the modern data science world of personalized medicine supporting inference of high-dimensional anatomical phenotypes for studying neurodegeneration and neurodevelopment. The dynamical systems model of growth and atrophy that emerges is one which is organized in terms of forces, accelerations, velocities, and displacements, with the associated Hamiltonian momentum and the diffeomorphic flow acting as the state, and the smooth vector field the control. The forces that enter the model derive from external measurements through which the dynamical system must flow, and the internal potential energies of structures making up the soft condensed matter. We examine numerous examples on growth and atrophy. This article is categorized under: Analytical and Computational Methods > Computational Methods Laboratory Methods and Technologies > Imaging Models of Systems Properties and Processes > Organ, Tissue, and Physiological Models.

On the Complexity of Human Neuroanatomy at the Millimeter Morphome Scale: Developing Codes and Characterizing Entropy Indexed to Spatial Scale

In this work we devise a strategy for discrete coding of anatomical form as described by a Bayesian prior model, quantifying the entropy of this representation as a function of code rate (number of bits), and its relationship geometric accuracy at clinically relevant scales. We study the shape of subcortical gray matter structures in the human brain through diffeomorphic transformations that relate them to a template, using data from the Alzheimer's Disease Neuroimaging Initiative to train a multivariate Gaussian prior model. We find that the at 1 mm accuracy all subcortical structures can be described with less than 35 bits, and at 1.5 mm error all structures can be described with less than 12 bits. This work represents a first step towards quantifying the amount of information ordering a neuroimaging study can provide about disease status.

Mapping the order and pattern of brain structural MRI changes using change-point analysis in premanifest Huntington's disease

Huntington's disease (HD) is an autosomal dominant neurodegenerative disorder that progressively affects motor, cognitive, and emotional functions. Structural MRI studies have demonstrated brain atrophy beginning many years prior to clinical onset ("premanifest" period), but the order and pattern of brain structural changes have not been fully characterized. In this study, we investigated brain regional volumes and diffusion tensor imaging (DTI) measurements in premanifest HD, and we aim to determine (1) the extent of MRI changes in a large number of structures across the brain by atlas-based analysis, and (2) the initiation points of structural MRI changes in these brain regions. We adopted a novel multivariate linear regression model to detect the inflection points at which the MRI changes begin (namely, "change-points"), with respect to the CAG-age product (CAP, an indicator of extent of exposure to the effects of CAG repeat expansion). We used approximately 300 T1-weighted and DTI data from premanifest HD and control subjects in the PREDICT-HD study, with atlas-based whole brain segmentation and change-point analysis. The results indicated a distinct topology of structural MRI changes: the change-points of the volumetric measurements suggested a central-to-peripheral pattern of atrophy from the striatum to the deep white matter; and the change points of DTI measurements indicated the earliest changes in mean diffusivity in the deep white matter and posterior white matter. While interpretation needs to be cautious given the cross-sectional nature of the data, these findings suggest a spatial and temporal pattern of spread of structural changes within the HD brain. Hum Brain Mapp 38:5035-5050, 2017. © 2017 Wiley Periodicals, Inc.

BIOMARKER CHANGE-POINT ESTIMATION WITH RIGHT CENSORING IN LONGITUDINAL STUDIES

We consider in this paper a statistical two-phase regression model in which the change point of a disease biomarker is measured relative to another point in time, such as the manifestation of the disease, which is subject to right-censoring (i.e., possibly unobserved over the entire course of the study). We develop point estimation methods for this model, based on maximum likelihood, and bootstrap validation methods. The effectiveness of our approach is illustrated by numerical simulations, and by the estimation of a change point for amygdalar atrophy in the context of Alzheimer's disease, wherein it is related to the cognitive manifestation of the disease.

Parametric Surface Diffeomorphometry for Low Dimensional Embeddings of Dense Segmentations and Imagery

In the field of Computational Anatomy, biological form (including our focus, neuroanatomy) is studied quantitatively through the action of the diffeomorphism group on example anatomies - a technique called diffeomorphometry. Here we design an algorithm within this framework to pass from dense objects common in neuromaging studies (binary segmentations, structural images) to a sparse representation defined on the surface boundaries of anatomical structures, and embedded into the low dimensional coordinates of a parametric model. Our main new contribution is to introduce an expanded group action to simultaneously deform surfaces through direct mapping of points, as well as images through functional composition with the inverse. This allows us to index the diffeomorphisms with respect to two-dimensional surface geometries like subcortical gray matter structures, but explicitly map onto cost functions determined by noisy 3-dimensional measurements. We consider models generated from empirical covariance of training data, as well as bandlimited (Laplace-Beltrami eigenfunction) models when no such data is available. We show applications to noisy or anomalous segmentations, and other typical problems in neuroimaging studies. We reproduce statistical results detecting changes in Alzheimer's disease, despite dimensionality reduction. Lastly we apply our algorithm to the common problem of segmenting subcortical structures from T1 MR images.

Surface-based vertexwise analysis of morphometry and microstructural integrity for white matter tracts in diffusion tensor imaging: With application to the corpus callosum in Alzheimer's disease

In this article, we present a unified statistical pipeline for analyzing the white matter (WM) tracts morphometry and microstructural integrity, both globally and locally within the same WM tract, from diffusion tensor imaging. Morphometry is quantified globally by the volumetric measurement and locally by the vertexwise surface areas. Meanwhile, microstructural integrity is quantified globally by the mean fractional anisotropy (FA) and trace values within the specific WM tract and locally by the FA and trace values defined at each vertex of its bounding surface. The proposed pipeline consists of four steps: (1) fully automated segmentation of WM tracts in a multi-contrast multi-atlas framework; (2) generation of the smooth surface representations for the WM tracts of interest; (3) common template surface generation on which the localized morphometric and microstructural statistics are defined and a variety of statistical analyses can be conducted; (4) multiple comparison correction to determine the significance of the statistical analysis results. Detailed herein, this pipeline has been applied to the corpus callosum in Alzheimer's disease (AD) with significantly decreased FA values and increased trace values, both globally and locally, being detected in patients with AD when compared to normal aging populations. A subdivision of the corpus callosum in both hemispheres revealed that the AD pathology primarily affects the body and splenium of the corpus callosum. Validation analyses and two multiple comparison correction strategies are provided. Hum Brain Mapp 38:1875-1893, 2017. © 2017 Wiley Periodicals, Inc.

Shape and diffusion tensor imaging based integrative analysis of the hippocampus and the amygdala in Alzheimer's disease

We analyzed, in an integrative fashion, the morphometry and structural integrity of the bilateral hippocampi and amygdalas in Alzheimer's disease (AD) using T1-weighted images and diffusion tensor images (DTIs). We detected significant hippocampal and amygdalar volumetric atrophies in AD relative to healthy controls (HCs). Shape analysis revealed significant region-specific atrophies with the hippocampal atrophy mainly being concentrated on the CA1 and CA2 while the amygdalar atrophy was concentrated on the basolateral and basomedial. In all structures, the structural integrity displayed a significantly decreased mean fractional anisotropy (FA) value and an increased mean trace value in AD. In addition to the inter-group comparisons, we systematically evaluated the discriminative power of our three types of features (volume, shape, and DTI), both individually and in their possible combinations, when differentiating between AD and HCs. We found the volume features to be redundant when the more sophisticated shape features were available. A combination of the shape and DTI features of the right hippocampus, with classification automatically performed by support vector machine, yielded the strongest classification result (overall accuracy, 94.6%; sensitivity, 95.5%; specificity, 93.3%).

Linking white matter and deep gray matter alterations in premanifest Huntington disease

Huntington disease (HD) is a fatal progressive neurodegenerative disorder for which only symptomatic treatment is available. A better understanding of the pathology, and identification of biomarkers will facilitate the development of disease-modifying treatments. HD is potentially a good model of a neurodegenerative disease for development of biomarkers because it is an autosomal-dominant disease with complete penetrance, caused by a single gene mutation, in which the neurodegenerative process can be assessed many years before onset of signs and symptoms of manifest disease. Previous MRI studies have detected abnormalities in gray and white matter starting in premanifest stages. However, the understanding of how these abnormalities are related, both in time and space, is still incomplete. In this study, we combined deep gray matter shape diffeomorphometry and white matter DTI analysis in order to provide a better mapping of pathology in the deep gray matter and subcortical white matter in premanifest HD. We used 296 MRI scans from the PREDICT-HD database. Atrophy in the deep gray matter, thalamus, hippocampus, and nucleus accumbens was analyzed by surface based morphometry, and while white matter abnormalities were analyzed in (i) regions of interest surrounding these structures, using (ii) tractography-based analysis, and using (iii) whole brain atlas-based analysis. We detected atrophy in the deep gray matter, particularly in putamen, from early premanifest stages. The atrophy was greater both in extent and effect size in cases with longer exposure to the effects of the CAG expansion mutation (as assessed by greater CAP-scores), and preceded detectible abnormalities in the white matter. Near the predicted onset of manifest HD, the MD increase was widespread, with highest indices in the deep and posterior white matter. This type of in-vivo macroscopic mapping of HD brain abnormalities can potentially indicate when and where therapeutics could be targeted to delay the onset or slow the disease progression.

Hamiltonian Systems and Optimal Control in Computational Anatomy: 100 Years Since D'Arcy Thompson

The Computational Anatomy project is the morphome-scale study of shape and form, which we model as an orbit under diffeomorphic group action. Metric comparison calculates the geodesic length of the diffeomorphic flow connecting one form to another. Geodesic connection provides a positioning system for coordinatizing the forms and positioning their associated functional information. This article reviews progress since the Euler-Lagrange characterization of the geodesics a decade ago. Geodesic positioning is posed as a series of problems in Hamiltonian control, which emphasize the key reduction from the Eulerian momentum with dimension of the flow of the group, to the parametric coordinates appropriate to the dimension of the submanifolds being positioned. The Hamiltonian viewpoint provides important extensions of the core setting to new, object-informed positioning systems. Several submanifold mapping problems are discussed as they apply to metamorphosis, multiple shape spaces, and longitudinal time series studies of growth and atrophy via shape splines.

Baseline shape diffeomorphometry patterns of subcortical and ventricular structures in predicting conversion of mild cognitive impairment to Alzheimer's disease

In this paper, we propose a novel predictor for the conversion from mild cognitive impairment (MCI) to Alzheimer's disease (AD). This predictor is based on the shape diffeomorphometry patterns of subcortical and ventricular structures (left and right amygdala, hippocampus, thalamus, caudate, putamen, globus pallidus, and lateral ventricle) of 607 baseline scans from the Alzheimer's Disease Neuroimaging Initiative database, including a total of 210 healthy control subjects, 222 MCI subjects, and 175 AD subjects. The optimal predictor is obtained via a feature selection procedure applied to all of the 14 sets of shape features via linear discriminant analysis, resulting in a combination of the shape diffeomorphometry patterns of the left hippocampus, the left lateral ventricle, the right thalamus, the right caudate, and the bilateral putamen. Via 10-fold cross-validation, we substantiate our method by successfully differentiating 77.04% (104/135) of the MCI subjects who converted to AD within 36 months and 71.26% (62/87) of the non-converters. To be specific, for the MCI-converters, we are capable of correctly predicting 82.35% (14/17) of subjects converting in 6 months, 77.5% (31/40) of subjects converting in 12 months, 74.07% (20/27) of subjects converting in 18 months, 78.13% (25/32) of subjects converting in 24 months, and 73.68% (14/19) of subject converting in 36 months. Statistically significant correlation maps were observed between the shape diffeomorphometry features of each of the 14 structures, especially the bilateral amygdala, hippocampus, lateral ventricle, and two neuropsychological test scores--the Alzheimer's Disease Assessment Scale-Cognitive Behavior Section and the Mini-Mental State Examination.

Network Neurodegeneration in Alzheimer's Disease via MRI Based Shape Diffeomorphometry and High-Field Atlasing

This paper examines MRI analysis of neurodegeneration in Alzheimer’s Disease (AD) in a network of structures within the medial temporal lobe using diffeomorphometry methods coupled with high-field atlasing in which the entorhinal cortex is partitioned into eight subareas. The morphometry markers for three groups of subjects (controls, preclinical AD, and symptomatic AD) are indexed to template coordinates measured with respect to these eight subareas. The location and timing of changes are examined within the subareas as it pertains to the classic Braak and Braak staging by comparing the three groups. We demonstrate that the earliest preclinical changes in the population occur in the lateral most sulcal extent in the entorhinal cortex (alluded to as transentorhinal cortex by Braak and Braak), and then proceeds medially which is consistent with the Braak and Braak staging. We use high-field 11T atlasing to demonstrate that the network changes are occurring at the junctures of the substructures in this medial temporal lobe network. Temporal progression of the disease through the network is also examined via changepoint analysis, demonstrating earliest changes in entorhinal cortex. The differential expression of rate of atrophy with progression signaling the changepoint time across the network is demonstrated to be signaling in the intermediate caudal subarea of the entorhinal cortex, which has been noted to be proximal to the hippocampus. This coupled to the findings of the nearby basolateral involvement in amygdala demonstrates the selectivity of neurodegeneration in early AD.

Relationship of medial temporal lobe atrophy, APOE genotype, and cognitive reserve in preclinical Alzheimer's disease

This study evaluated the utility of baseline and longitudinal magnetic resonance imaging (MRI) measures of medial temporal lobe brain regions collected when participants were cognitively normal and largely in middle age (mean age 57 years) to predict the time to onset of clinical symptoms associated with mild cognitive impairment (MCI). Furthermore, we examined whether the relationship between MRI measures and clinical symptom onset was modified by apolipoprotein E (ApoE) genotype and level of cognitive reserve (CR). MRI scans and measures of CR were obtained at baseline from 245 participants who had been followed for up to 18 years (mean follow-up 11 years). A composite score based on reading, vocabulary, and years of education was used as an index of CR. Cox regression models showed that lower baseline volume of the right hippocampus and smaller baseline thickness of the right entorhinal cortex predicted the time to symptom onset independently of CR and ApoE-ɛ4 genotype, which also predicted the onset of symptoms. The atrophy rates of bilateral entorhinal cortex and amygdala volumes were also associated with time to symptom onset, independent of CR, ApoE genotype, and baseline volume. Only one measure, the left entorhinal cortex baseline volume, interacted with CR, such that smaller volumes predicted symptom onset only in individuals with lower CR. These results suggest that MRI measures of medial temporal atrophy, ApoE-ɛ4 genotype, and the protective effects of higher CR all predict the time to onset of symptoms associated with MCI in a largely independent, additive manner during the preclinical phase of Alzheimer's disease.

The diffeomorphometry of regional shape change rates and its relevance to cognitive deterioration in mild cognitive impairment and Alzheimer's disease

We proposed a diffeomorphometry-based statistical pipeline to study the regional shape change rates of the bilateral hippocampus, amygdala, and ventricle in mild cognitive impairment (MCI) and Alzheimer's disease (AD) compared with healthy controls (HC), using sequential magnetic resonance imaging (MRI) scans of 713 subjects (3,123 scans in total). The subgroup shape atrophy rates of the bilateral hippocampus and amygdala, as well as the expansion rates of the bilateral ventricles, for a majority of vertices were found to follow the order of AD>MCI>HC. The bilateral hippocampus and the left amygdala were subsegmented into multiple functionally meaningful subregions with the help of high-field MRI scans. The largest group differences in localized shape atrophy rates on the hippocampus were found to occur in CA1, followed by subiculum, CA2, and finally CA3/dentate gyrus, which is consistent with the neurofibrillary tangle accumulation trajectory. Highly nonuniform group differences were detected on the amygdala; vertices on the core amygdala (basolateral and lateral nucleus) revealed much larger atrophy rates, whereas those on the noncore amygdala (mainly centromedial) displayed similar or even smaller atrophy rates in AD relative to HC. The temporal horns of the ventricles were observed to have the largest localized ventricular expansion rate differences; with the AD group showing larger localized expansion rates on the anterior horn and the body part of the ventricles as well. Significant correlations were observed between the localized shape change rates of each of these six structures and the cognitive deterioration rates as quantified by the Alzheimer's Disease Assessment Scale-Cognitive Behavior Section increase rate and the Mini Mental State Examination decrease rate.

Amygdalar atrophy in symptomatic Alzheimer's disease based on diffeomorphometry: the BIOCARD cohort

This article examines the diffeomorphometry of magnetic resonance imaging-derived structural markers for the amygdala, in subjects with symptomatic Alzheimer's disease (AD). Using linear mixed-effects models we show differences between those with symptomatic AD and controls. Based on template centered population analysis, the distribution of statistically significant change is seen in both the volume and shape of the amygdala in subjects with symptomatic AD compared with controls. We find that high-dimensional vertex based markers are statistically more significantly discriminating (p < 0.00001) than lower-dimensional markers and volumes, consistent with comparable findings in presymptomatic AD. Using a high-field 7T atlas, significant atrophy was found to be centered in the basomedial and basolateral subregions, with no evidence of centromedial involvement.

Inferring changepoint times of medial temporal lobe morphometric change in preclinical Alzheimer's disease

This paper uses diffeomorphometry methods to quantify the order in which statistically significant morphometric change occurs in three medial temporal lobe regions, the amygdala, entorhinal cortex (ERC), and hippocampus among subjects with symptomatic and preclinical Alzheimer's disease (AD). Magnetic resonance imaging scans were examined in subjects who were cognitively normal at baseline, some of whom subsequently developed clinical symptoms of AD. The images were mapped to a common template, using shape-based diffeomorphometry. The multidimensional shape markers indexed through the temporal lobe structures were modeled using a changepoint model with explicit parameters, specifying the number of years preceding clinical symptom onset. Our model assumes that the atrophy rate of a considered brain structure increases years before detectable symptoms.

The results demonstrate that the atrophy changepoint in the ERC occurs first, indicating significant change 8–10 years prior to onset, followed by the hippocampus, 2–4 years prior to onset, followed by the amygdala, 3 years prior to onset. The ERC is significant bilaterally, in both our local and global measures, with estimates of ERC surface area loss of 2.4% (left side) and 1.6% (right side) annually. The same changepoint model for ERC volume gives 3.0% and 2.7% on the left and right sides, respectively. Understanding the order in which changes in the brain occur during preclinical AD may assist in the design of intervention trials aimed at slowing the evolution of the disease.

•

We use diffeomorphometry to quantify the order in which statistically significant morphometric change occurs in three medial temporal lobe regions, the amygdala, entorhinal cortex (ERC), and hippocampus among subjects with symptomatic and preclinical Alzheimer's disease (AD).

•

We introduce a model on anatomical shape change in which changepoint is inferred, taking place some period of time before cognitive onset of AD.

•

The analysis uses a dataset arising from the BIOCARD study, in which all subjects were cognitively normal at baseline, some of whom subsequently developed clinical symptoms of AD.

•

The results demonstrate that the atrophy changepoint in the ERC occurs first, indicating significant change 8-10 years prior to onset, followed by hippocampus, 2-4 years prior to onset, followed by amygdala, 3 years prior to onset.

•

The ERC is significant bilaterally, in both our local and global measures, with estimates of ERC surface area loss of 2.4% (left side) and 1.6% (right side) annually.

•

Understanding the order in which changes in the brain occur during preclinical AD may assist in the design of intervention trials aimed at slowing the evolution of the disease.