Manifold modeling for brain population analysis

This paper describes a method for building efficient representations of large sets of brain images. Our hypothesis is that the space spanned by a set of brain images can be captured, to a close approximation, by a low-dimensional, nonlinear manifold. This paper presents a method to learn such a low-dimensional manifold from a given data set. The manifold model is generative-brain images can be constructed from a relatively small set of parameters, and new brain images can be projected onto the manifold. This allows to quantify the geometric accuracy of the manifold approximation in terms of projection distance. The manifold coordinates induce a Euclidean coordinate system on the population data that can be used to perform statistical analysis of the population. We evaluate the proposed method on the OASIS and ADNI brain databases of head MR images in two ways. First, the geometric fit of the method is qualitatively and quantitatively evaluated. Second, the ability of the brain manifold model to explain clinical measures is analyzed by linear regression in the manifold coordinate space. The regression models show that the manifold model is a statistically significant descriptor of clinical parameters.

Microstructural connectivity of the arcuate fasciculus in adolescents with high-functioning autism

The arcuate fasciculus is a white matter fiber bundle of great importance in language. In this study, diffusion tensor imaging (DTI) was used to infer white matter integrity in the arcuate fasciculi of a group of subjects with high-functioning autism and a control group matched for age, handedness, IQ, and head size. The arcuate fasciculus for each subject was automatically extracted from the imaging data using a new volumetric DTI segmentation algorithm. The results showed a significant increase in mean diffusivity (MD) in the autism group, due mostly to an increase in the radial diffusivity (RD). A test of the lateralization of DTI measurements showed that both MD and fractional anisotropy (FA) were less lateralized in the autism group. These results suggest that white matter microstructure in the arcuate fasciculus is affected in autism and that the language specialization apparent in the left arcuate of healthy subjects is not as evident in autism, which may be related to poorer language functioning.

Group analysis of DTI fiber tract statistics with application to neurodevelopment

Diffusion tensor imaging (DTI) provides a unique source of information about the underlying tissue structure of brain white matter in vivo including both the geometry of major fiber bundles as well as quantitative information about tissue properties represented by derived tensor measures. This paper presents a method for statistical comparison of fiber bundle diffusion properties between populations of diffusion tensor images. Unbiased diffeomorphic atlas building is used to compute a normalized coordinate system for populations of diffusion images. The diffeomorphic transformations between each subject and the atlas provide spatial normalization for the comparison of tract statistics. Diffusion properties, such as fractional anisotropy (FA) and tensor norm, along fiber tracts are modeled as multivariate functions of arc length. Hypothesis testing is performed non-parametrically using permutation testing based on the Hotelling T(2) statistic. The linear discriminant embedded in the T(2) metric provides an intuitive, localized interpretation of detected differences. The proposed methodology was tested on two clinical studies of neurodevelopment. In a study of 1 and 2 year old subjects, a significant increase in FA and a correlated decrease in Frobenius norm was found in several tracts. Significant differences in neonates were found in the splenium tract between controls and subjects with isolated mild ventriculomegaly (MVM) demonstrating the potential of this method for clinical studies.

Axon tracking in serial block-face scanning electron microscopy

Electron microscopy is an important modality for the analysis of neuronal structures in neurobiology. We address the problem of tracking axons across large distances in volumes acquired by serial block-face scanning electron microscopy (SBFSEM). Tracking, for this application, is defined as the segmentation of an axon that spans a volume using similar features between slices. This is a challenging problem due to the small cross-sectional size of axons and the low signal-to-noise ratio in our SBFSEM images. A carefully engineered algorithm using Kalman-snakes and optical flow computation is presented. Axon tracking is initialized with user clicks or automatically using the watershed segmentation algorithm, which identifies axon centers. Multiple axons are tracked from slice to slice through a volume, updating the positions and velocities in the model and providing constraints to maintain smoothness between slices. Validation results indicate that this algorithm can significantly speed up the task of manual axon tracking.

4D MAP image reconstruction incorporating organ motion

Four-dimensional respiratory correlated computed tomography (4D RCCT) has been widely used for studying organ motion. Most current algorithms use binning techniques which introduce artifacts that can seriously hamper quantitative motion analysis. In this paper, we develop an algorithm for tracking organ motion which uses raw time-stamped data and simultaneously reconstructs images and estimates deformations in anatomy. This results in a reduction of artifacts and an increase in signal-to-noise ratio (SNR). In the case of CT, the increased SNR enables a reduction in dose to the patient during scanning. This framework also facilitates the incorporation of fundamental physical properties of organ motion, such as the conservation of local tissue volume. We show in this paper that this approach is accurate and robust against noise and irregular breathing for tracking organ motion. A detailed phantom study is presented, demonstrating accuracy and robustness of the algorithm. An example of applying this algorithm to real patient image data is also presented, demonstrating the utility of the algorithm in reducing artifacts.

Particle based shape regression of open surfaces with applications to developmental neuroimaging

Shape regression promises to be an important tool to study the relationship between anatomy and underlying clinical or biological parameters, such as age. In this paper we propose a new method to building shape models that incorporates regression analysis in the process of optimizing correspondences on a set of open surfaces. The statistical significance of the dependence is evaluated using permutation tests designed to estimate the likelihood of achieving the observed statistics under numerous rearrangements of the shape parameters with respect to the explanatory variable. We demonstrate the method on synthetic data and provide a new results on clinical MRI data related to early development of the human head.

A variational image-based approach to the correction of susceptibility artifacts in the alignment of diffusion weighted and structural MRI

This paper presents a method for correcting the geometric and greyscale distortions in diffusion-weighted MRI that result from inhomogeneities in the static magnetic field. These inhomogeneities may due to imperfections in the magnet or to spatial variations in the magnetic susceptibility of the object being imaged--so called susceptibility artifacts. Echo-planar imaging (EPI), used in virtually all diffusion weighted acquisition protocols, assumes a homogeneous static field, which generally does not hold for head MRI. The resulting distortions are significant, sometimes more than ten millimeters. These artifacts impede accurate alignment of diffusion images with structural MRI, and are generally considered an obstacle to the joint analysis of connectivity and structure in head MRI. In principle, susceptibility artifacts can be corrected by acquiring (and applying) a field map. However, as shown in the literature and demonstrated in this paper, field map corrections of susceptibility artifacts are not entirely accurate and reliable, and thus field maps do not produce reliable alignment of EPIs with corresponding structural images. This paper presents a new, image-based method for correcting susceptibility artifacts. The method relies on a variational formulation of the match between an EPI baseline image and a corresponding T2-weighted structural image but also specifically accounts for the physics of susceptibility artifacts. We derive a set of partial differential equations associated with the optimization, describe the numerical methods for solving these equations, and present results that demonstrate the effectiveness of the proposed method compared with field-map correction.

The geometric median on Riemannian manifolds with application to robust atlas estimation

One of the primary goals of computational anatomy is the statistical analysis of anatomical variability in large populations of images. The study of anatomical shape is inherently related to the construction of transformations of the underlying coordinate space, which map one anatomy to another. It is now well established that representing the geometry of shapes or images in Euclidian spaces undermines our ability to represent natural variability in populations. In our previous work we have extended classical statistical analysis techniques, such as averaging, principal components analysis, and regression, to Riemannian manifolds, which are more appropriate representations for describing anatomical variability. In this paper we extend the notion of robust estimation, a well established and powerful tool in traditional statistical analysis of Euclidian data, to manifold-valued representations of anatomical variability. In particular, we extend the geometric median, a classic robust estimator of centrality for data in Euclidean spaces. We formulate the geometric median of data on a Riemannian manifold as the minimizer of the sum of geodesic distances to the data points. We prove existence and uniqueness of the geometric median on manifolds with non-positive sectional curvature and give sufficient conditions for uniqueness on positively curved manifolds. Generalizing the Weiszfeld procedure for finding the geometric median of Euclidean data, we present an algorithm for computing the geometric median on an arbitrary manifold. We show that this algorithm converges to the unique solution when it exists. In this paper we exemplify the robustness of the estimation technique by applying the procedure to various manifolds commonly used in the analysis of medical images. Using this approach, we also present a robust brain atlas estimation technique based on the geometric median in the space of deformable images.

Realizing the potential of positron emission tomography with 18F-fluorodeoxyglucose to improve the treatment of Alzheimer's disease

Positron emission tomography (PET) with 18F-fluorodeoxyglucose (FDG-PET) thus far rarely has been used to advance the development of new treatments for Alzheimer's disease (AD). Now that FDG-PET with standard acquisition protocols for dementia is widely available, change in cerebral glucose metabolism is a feasible outcome variable for clinical drug trials. Individual analysis of FDG-PET results also might prove valuable. FDG-PET can detect metabolic changes very early in the course of AD and identify subjects for earlier treatment. FDG-PET reliably distinguishes AD from frontotemporal dementia so that only those most likely to benefit are enrolled in trials. Finally, objectively identifying phenotypic variations of AD with FDG-PET might have pathogenic and prognostic implications that can be used for personalized treatment approaches. The judicious use of FDG-PET is needed to accelerate the evaluation of promising new drugs and more rationally target treatments for dementing diseases.

Interactive visualization of volumetric white matter connectivity in DT-MRI using a parallel-hardware Hamilton-Jacobi solver

In this paper we present a method to compute and visualize volumetric white matter connectivity in diffusion tensor magnetic resonance imaging (DT-MRI) using a Hamilton-Jacobi (H-J) solver on the GPU (Graphics Processing Unit). Paths through the volume are assigned costs that are lower if they are consistent with the preferred diffusion directions. The proposed method finds a set of voxels in the DTI volume that contain paths between two regions whose costs are within a threshold of the optimal path. The result is a volumetric optimal path analysis, which is driven by clinical and scientific questions relating to the connectivity between various known anatomical regions of the brain. To solve the minimal path problem quickly, we introduce a novel numerical algorithm for solving H-J equations, which we call the Fast Iterative Method (FIM). This algorithm is well-adapted to parallel architectures, and we present a GPU-based implementation, which runs roughly 50-100 times faster than traditional CPU-based solvers for anisotropic H-J equations. The proposed system allows users to freely change the endpoints of interesting pathways and to visualize the optimal volumetric path between them at an interactive rate. We demonstrate the proposed method on some synthetic and real DT-MRI datasets and compare the performance with existing methods.

Quantifying metabolic asymmetry modulo structure in Alzheimer's disease

In this paper we describe a new method for quantifying metabolic asymmetry modulo structural hemispheric differences. The study of metabolic asymmetry in Alzheimer's disease (AD) serves as a driving application. The approach is based on anatomical atlas construction by large deformation diffeomorphic metric mapping (LDDMM) first introduced in [1]. Using invariance properties of the LDDMM, we define a structurally symmetric coordinate frame in which metabolic asymmetries between the left and the right hemispheres can be studied. This structurally symmetric coordinate system of each subject provides the correspondence between left and right hemispheric structures in an individual brain. These correspondences are used for measuring metabolic asymmetry modulo structural asymmetry. Again using the atlas construction framework, we build a common symmetric coordinate system of a entire population. The metabolic asymmetry maps of individuals in a population under study are mapped into the common structurally symmetric coordinate frame, allowing for a statistical description of the populations metabolic asymmetry. In this paper we prove certain invariance properties of the LDDMM atlas construction framework that make the definition of structurally symmetric coordinate systems possible. We present results from applying the methodology to images from the Alzheimer's Disease Neuroimaging Initiative (ADNI).

Shape modeling and analysis with entropy-based particle systems

This paper presents a new method for constructing compact statistical point-based models of ensembles of similar shapes that does not rely on any specific surface parameterization. The method requires very little preprocessing or parameter tuning, and is applicable to a wider range of problems than existing methods, including nonmanifold surfaces and objects of arbitrary topology. The proposed method is to construct a point-based sampling of the shape ensemble that simultaneously maximizes both the geometric accuracy and the statistical simplicity of the model. Surface point samples, which also define the shape-to-shape correspondences, are modeled as sets of dynamic particles that are constrained to lie on a set of implicit surfaces. Sample positions are optimized by gradient descent on an energy function that balances the negative entropy of the distribution on each shape with the positive entropy of the ensemble of shapes. We also extend the method with a curvature-adaptive sampling strategy in order to better approximate the geometry of the objects. This paper presents the formulation; several synthetic examples in two and three dimensions; and an application to the statistical shape analysis of the caudate and hippocampus brain structures from two clinical studies.

A volumetric approach to quantifying region-to-region white matter connectivity in diffusion tensor MRI

In this paper we present a volumetric approach for quantitatively studying white matter connectivity from diffusion tensor magnetic resonance imaging (DT-MRI). The proposed method is based on a minimization of path cost between two regions, defined as the integral of local costs that are derived from the full tensor data along the path. We solve the minimal path problem using a Hamilton-Jacobi formulation of the problem and a new, fast iterative method that computes updates on the propagating front of the cost function at every point. The solutions for the fronts emanating from the two initial regions are combined, giving a voxel-wise connectivity measurement of the optimal paths between the regions that pass through those voxels. The resulting high-connectivity voxels provide a volumetric representation of the white matter pathway between the terminal regions. We quantify the tensor data along these pathways using nonparametric regression of the tensors and of derived measures as a function of path length. In this way we can obtain volumetric measures on white-matter tracts between regions without any explicit integration of tracts. We demonstrate the proposed method on several fiber tracts from DT-MRI data of the normal human brain.

Fiber tract-oriented statistics for quantitative diffusion tensor MRI analysis

Quantitative diffusion tensor imaging (DTI) has become the major imaging modality to study properties of white matter and the geometry of fiber tracts of the human brain. Clinical studies mostly focus on regional statistics of fractional anisotropy (FA) and mean diffusivity (MD) derived from tensors. Existing analysis techniques do not sufficiently take into account that the measurements are tensors, and thus require proper interpolation and statistics of tensors, and that regions of interest are fiber tracts with complex spatial geometry. We propose a new framework for quantitative tract-oriented DTI analysis that systematically includes tensor interpolation and averaging, using nonlinear Riemannian symmetric space. A new measure of tensor anisotropy, called geodesic anisotropy (GA) is applied and compared with FA. As a result, tracts of interest are represented by the geometry of the medial spine attributed with tensor statistics (average and variance) calculated within cross-sections. Feasibility of our approach is demonstrated on various fiber tracts of a single data set. A validation study, based on six repeated scans of the same subject, assesses the reproducibility of this new DTI data analysis framework.

A method and software for segmentation of anatomic object ensembles by deformable m-reps

Deformable shape models (DSMs) comprise a general approach that shows great promise for automatic image segmentation. Published studies by others and our own research results strongly suggest that segmentation of a normal or near-normal object from 3D medical images will be most successful when the DSM approach uses (1) knowledge of the geometry of not only the target anatomic object but also the ensemble of objects providing context for the target object and (2) knowledge of the image intensities to be expected relative to the geometry of the target and contextual objects. The segmentation will be most efficient when the deformation operates at multiple object-related scales and uses deformations that include not just local translations but the biologically important transformations of bending and twisting, i.e., local rotation, and local magnification. In computer vision an important class of DSM methods uses explicit geometric models in a Bayesian statistical framework to provide a priori information used in posterior optimization to match the DSM against a target image. In this approach a DSM of the object to be segmented is placed in the target image data and undergoes a series of rigid and nonrigid transformations that deform the model to closely match the target object. The deformation process is driven by optimizing an objective function that has terms for the geometric typicality and model-to-image match for each instance of the deformed model. The success of this approach depends strongly on the object representation, i.e., the structural details and parameter set for the DSM, which in turn determines the analytic form of the objective function. This paper describes a form of DSM called m-reps that has or allows these properties, and a method of segmentation consisting of large to small scale posterior optimization of m-reps. Segmentation by deformable m-reps, together with the appropriate data representations, visualizations, and user interface, has been implemented in software that accomplishes 3D segmentations in a few minutes. Software for building and training models has also been developed. The methods underlying this software and its abilities are the subject of this paper.

Fiber Tract-Oriented Statistics for Quantitative Diffusion Tensor MRI Analysis

Diffusion tensor imaging (DTI) has become the major modality to study properties of white matter and the geometry of fiber tracts of the human brain. Clinical studies mostly focus on regional statistics of fractional anisotropy (FA) and mean diffusivity (MD) derived from tensors. Existing analysis techniques do not sufficiently take into account that the measurements are tensors, and thus require proper interpolation and statistics based on tensors, and that regions of interest are fiber tracts with complex spatial geometry. We propose a new framework for quantitative tract-oriented DTI analysis that includes tensor interpolation and averaging, using nonlinear Riemannian symmetric space. As a result, tracts of interest are represented by the geometry of the medial spine attributed with tensor statistics calculated within cross-sections. Examples from a clinical neuroimaging study of the early developing brain illustrate the potential of this new method to assess white matter fiber maturation and integrity.

Gaussian distributions on Lie groups and their application to statistical shape analysis

The Gaussian distribution is the basis for many methods used in the statistical analysis of shape. One such method is principal component analysis, which has proven to be a powerful technique for describing the geometric variability of a population of objects. The Gaussian framework is well understood when the data being studied are elements of a Euclidean vector space. This is the case for geometric objects that are described by landmarks or dense collections of boundary points. We have been using medial representations, or m-reps, for modelling the geometry of anatomical objects. The medial parameters are not elements of a Euclidean space, and thus standard PCA is not applicable. In our previous work we have shown that the m-rep model parameters are instead elements of a Lie group. In this paper we develop the notion of a Gaussian distribution on this Lie group. We then derive the maximum likelihood estimates of the mean and the covariance of this distribution. Analogous to principal component analysis of covariance in Euclidean spaces, we define principal geodesic analysis on Lie groups for the study of anatomical variability in medially-defined objects. Results of applying this framework on a population of hippocampi in a schizophrenia study are presented.

Principal geodesic analysis for the study of nonlinear statistics of shape

A primary goal of statistical shape analysis is to describe the variability of a population of geometric objects. A standard technique for computing such descriptions is principal component analysis. However, principal component analysis is limited in that it only works for data lying in a Euclidean vector space. While this is certainly sufficient for geometric models that are parameterized by a set of landmarks or a dense collection of boundary points, it does not handle more complex representations of shape. We have been developing representations of geometry based on the medial axis description or m-rep. While the medial representation provides a rich language for variability in terms of bending, twisting, and widening, the medial parameters are not elements of a Euclidean vector space. They are in fact elements of a nonlinear Riemannian symmetric space. In this paper, we develop the method of principal geodesic analysis, a generalization of principal component analysis to the manifold setting. We demonstrate its use in describing the variability of medially-defined anatomical objects. Results of applying this framework on a population of hippocampi in a schizophrenia study are presented.

Deformable M-Reps for 3D Medical Image Segmentation

M-reps (formerly called DSLs) are a multiscale medial means for modeling and rendering 3D solid geometry. They are particularly well suited to model anatomic objects and in particular to capture prior geometric information effectively in deformable models segmentation approaches. The representation is based on figural models, which define objects at coarse scale by a hierarchy of figures - each figure generally a slab representing a solid region and its boundary simultaneously. This paper focuses on the use of single figure models to segment objects of relatively simple structure. A single figure is a sheet of medial atoms, which is interpolated from the model formed by a net, i.e., a mesh or chain, of medial atoms (hence the name m-reps), each atom modeling a solid region via not only a position and a width but also a local figural frame giving figural directions and an object angle between opposing, corresponding positions on the boundary implied by the m-rep. The special capability of an m-rep is to provide spatial and orientational correspondence between an object in two different states of deformation. This ability is central to effective measurement of both geometric typicality and geometry to image match, the two terms of the objective function optimized in segmentation by deformable models. The other ability of m-reps central to effective segmentation is their ability to support segmentation at multiple levels of scale, with successively finer precision. Objects modeled by single figures are segmented first by a similarity transform augmented by object elongation, then by adjustment of each medial atom, and finally by displacing a dense sampling of the m-rep implied boundary. While these models and approaches also exist in 2D, we focus on 3D objects. The segmentation of the kidney from CT and the hippocampus from MRI serve as the major examples in this paper. The accuracy of segmentation as compared to manual, slice-by-slice segmentation is reported.

Multiscale deformable model segmentation and statistical shape analysis using medial descriptions

This paper presents a multiscale framework based on a medial representation for the segmentation and shape characterization of anatomical objects in medical imagery. The segmentation procedure is based on a Bayesian deformable templates methodology in which the prior information about the geometry and shape of anatomical objects is incorporated via the construction of exemplary templates. The anatomical variability is accommodated in the Bayesian framework by defining probabilistic transformations on these templates. The transformations, thus, defined are parameterized directly in terms of natural shape operations, such as growth and bending, and their locations. A preliminary validation study of the segmentation procedure is presented. We also present a novel statistical shape analysis approach based on the medial descriptions that examines shape via separate intuitive categories, such as global variability at the coarse scale and localized variability at the fine scale. We show that the method can be used to statistically describe shape variability in intuitive terms such as growing and bending.