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.

The natural history of brain growth in autism spectrum disorders remains unclear. Cross-sectional studies have identified regional abnormalities in brain volume and cortical thickness in autism, although substantial discrepancies have been reported. Preliminary longitudinal studies using two time points and small samples have identified specific regional differences in cortical thickness in the disorder. To clarify age-related trajectories of cortical development, we examined longitudinal changes in cortical thickness within a large mixed cross-sectional and longitudinal sample of autistic subjects and age- and gender-matched typically developing controls. Three hundred and forty-five magnetic resonance imaging scans were examined from 97 males with autism (mean age = 16.8 years; range 3-36 years) and 60 males with typical development (mean age = 18 years; range 4-39 years), with an average interscan interval of 2.6 years. FreeSurfer image analysis software was used to parcellate the cortex into 34 regions of interest per hemisphere and to calculate mean cortical thickness for each region. Longitudinal linear mixed effects models were used to further characterize these findings and identify regions with between-group differences in longitudinal age-related trajectories. Using mean age at time of first scan as a reference (15 years), differences were observed in bilateral inferior frontal gyrus, pars opercularis and pars triangularis, right caudal middle frontal and left rostral middle frontal regions, and left frontal pole. However, group differences in cortical thickness varied by developmental stage, and were influenced by IQ. Differences in age-related trajectories emerged in bilateral parietal and occipital regions (postcentral gyrus, cuneus, lingual gyrus, pericalcarine cortex), left frontal regions (pars opercularis, rostral middle frontal and frontal pole), left supramarginal gyrus, and right transverse temporal gyrus, superior parietal lobule, and paracentral, lateral orbitofrontal, and lateral occipital regions. We suggest that abnormal cortical development in autism spectrum disorders undergoes three distinct phases: accelerated expansion in early childhood, accelerated thinning in later childhood and adolescence, and decelerated thinning in early adulthood. Moreover, cortical thickness abnormalities in autism spectrum disorders are region-specific, vary with age, and may remain dynamic well into adulthood.

Group differences in resting state functional magnetic resonance imaging connectivity between individuals with autism and typically developing controls have been widely replicated for a small number of discrete brain regions, yet the whole-brain distribution of connectivity abnormalities in autism is not well characterized. It is also unclear whether functional connectivity is sufficiently robust to be used as a diagnostic or prognostic metric in individual patients with autism. We obtained pairwise functional connectivity measurements from a lattice of 7266 regions of interest covering the entire grey matter (26.4 million connections) in a well-characterized set of 40 male adolescents and young adults with autism and 40 age-, sex- and IQ-matched typically developing subjects. A single resting state blood oxygen level-dependent scan of 8 min was used for the classification in each subject. A leave-one-out classifier successfully distinguished autism from control subjects with 83% sensitivity and 75% specificity for a total accuracy of 79% (P = 1.1 × 10(-7)). In subjects <20 years of age, the classifier performed at 89% accuracy (P = 5.4 × 10(-7)). In a replication dataset consisting of 21 individuals from six families with both affected and unaffected siblings, the classifier performed at 71% accuracy (91% accuracy for subjects <20 years of age). Classification scores in subjects with autism were significantly correlated with the Social Responsiveness Scale (P = 0.05), verbal IQ (P = 0.02) and the Autism Diagnostic Observation Schedule-Generic's combined social and communication subscores (P = 0.05). An analysis of informative connections demonstrated that region of interest pairs with strongest correlation values were most abnormal in autism. Negatively correlated region of interest pairs showed higher correlation in autism (less anticorrelation), possibly representing weaker inhibitory connections, particularly for long connections (Euclidean distance >10 cm). Brain regions showing greatest differences included regions of the default mode network, superior parietal lobule, fusiform gyrus and anterior insula. Overall, classification accuracy was better for younger subjects, with differences between autism and control subjects diminishing after 19 years of age. Classification scores of unaffected siblings of individuals with autism were more similar to those of the control subjects than to those of the subjects with autism. These findings indicate feasibility of a functional connectivity magnetic resonance imaging diagnostic assay for autism.

The tensors produced by diffusion tensor magnetic resonance imaging (DT-MRI) represent the covariance in a Brownian motion model of water diffusion. Under this physical interpretation, diffusion tensors are required to be symmetric, positive-definite. However, current approaches to statistical analysis of diffusion tensor data, which treat the tensors as linear entities, do not take this positivedefinite constraint into account. This difficulty is due to the fact that the space of diffusion tensors does not form a vector space. In this paper we show that the space of diffusion tensors is a type of curved manifold known as a Riemannian symmetric space. We then develop methods for producing statistics, namely averages and modes of variation, in this space. We show that these statistics preserve natural geometric properties of the tensors, including the constraint that their eigenvalues be positive. The symmetric space formulation also leads to a natural definition for interpolation of diffusion tensors and a new measure of anisotropy. We expect that these methods will be useful in the registration of diffusion tensor images, the production of statistical atlases from diffusion tensor data, and the quantification of the anatomical variability caused by disease. The framework presented in this paper should also be useful in other applications where symmetric, positive-definite tensors arise, such as mechanics and computer vision.1

Background: Systematic differences in functional connectivity MRI metrics have been consistently observed in autism, with predominantly decreased cortico-cortical connectivity. Previous attempts at single subject classification in high-functioning autism using whole brain point-to-point functional connectivity have yielded about 80% accurate classification of autism vs. control subjects across a wide age range. We attempted to replicate the method and results using the Autism Brain Imaging Data Exchange (ABIDE) including resting state fMRI data obtained from 964 subjects and 16 separate international sites.Methods: For each of 964 subjects, we obtained pairwise functional connectivity measurements from a lattice of 7266 regions of interest covering the gray matter (26.4 million “connections”) after preprocessing that included motion and slice timing correction, coregistration to an anatomic image, normalization to standard space, and voxelwise removal by regression of motion parameters, soft tissue, CSF, and white matter signals. Connections were grouped into multiple bins, and a leave-one-out classifier was evaluated on connections comprising each set of bins. Age, age-squared, gender, handedness, and site were included as covariates for the classifier.Results: Classification accuracy significantly outperformed chance but was much lower for multisite prediction than for previous single site results. As high as 60% accuracy was obtained for whole brain classification, with the best accuracy from connections involving regions of the default mode network, parahippocampaland fusiform gyri, insula, Wernicke Area, and intraparietal sulcus. The classifier score was related to symptom severity, social function, daily living skills, and verbal IQ. Classification accuracy was significantly higher for sites with longer BOLD imaging times.Conclusions: Multisite functional connectivity classification of autism outperformed chance using a simple leave-one-out classifier, but exhibited poorer accuracy than for single site results. Attempts to use multisite classifiers will likely require improved classification algorithms, longer BOLD imaging times, and standardized acquisition parameters for possible future clinical utility.

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 T2 statistic. The linear discriminant embedded in the T2 metric provides an intuitive, localized interpretation of detected differences. The proposed methodology was tested on two clinical studies of neurodevelopment. In a study of one and two 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.

Abstract. Diffusion tensor magnetic resonance imaging (DT-MRI) is emerging as an important tool in medical image analysis of the brain. However, relatively little work has been done on producing statistics of diffusion tensors. A main difficulty is that the space of diffusion tensors, i.e., the space of symmetric, positivedefinite matrices, does not form a vector space. Therefore, standard linear statistical techniques do not apply. We show that the space of diffusion tensors is a type of curved manifold known as a Riemannian symmetric space. We then develop methods for producing statistics, namely averages and modes of variation, in this space. In our previous work we introduced principal geodesic analysis, a generalization of principal component analysis, to compute the modes of variation of data in Lie groups. In this work we expand the method of principal geodesic analysis to symmetric spaces and apply it to the computation of the variability of diffusion tensor data. We expect that these methods will be useful in the registration of diffusion tensor images, the production of statistical atlases from diffusion tensor data, and the quantification of the anatomical variability caused by disease.

Many important fields of basic research in medicine and biology routinely employ tools for the statistical analysis of collections of similar shapes. Biologists, for example, have long relied on homologous, anatomical landmarks as shape models to characterize the growth and development of species. Increasingly, however, researchers are exploring the use of more detailed models that are derived computationally from three-dimensional images and surface descriptions. While computationally-derived models of shape are promising new tools for biomedical research, they also present some significant engineering challenges, which existing modeling methods have only begun to address.In this dissertation, I propose a new computational framework for statistical shape modeling that significantly advances the state-of-the-art by overcoming many of the limitations of existing methods. The framework uses a particle-system representation of shape, with a fast correspondence-point optimization based on information content. The optimization balances the simplicity of the model (compactness) with the accuracy of the shape representations by using two commensurate entropy metrics and no free parameters. The idea is to maximize both the geometric accuracy and the statistical simplicity of the shape model, in accordance with the principle of parsimony in model selection. The nonparametric representation allows the method to be applied to a larger class of problems than existing methods, including nonspherical surfaces, open surfaces, and sets of multiple surfaces.The relative simplicity of the surface representation and the low number of free parameters results in a framework that is easy to use and can operate directly on image segmentations. In collaboration with scientists from several important areas of biomedicine, I have demonstrated that the proposed method is indeed an e↵ective tool for scientific research.The specific research contributions of this dissertation are as follows. First, I describe a mathematical framework and a robust numerical algorithm for computing optimized correspondence-point shape models using an entropy-based optimization and particle-system technology. Second, I develop a series of extensions of the framework to more general classes of shape analysis problems, including the analysis of multiple-object complexes, the generalization to correspondence based on generic functions of position, an extension to handle surfaces with open boundaries, and shape modeling with simple regression. Third, I describe the application of statistical hypothesis testing, regression analysis, and multiple-analysis of covariance to the proposed shape models. I also introduce new techniques for visualization and interpretation of these statistics. Finally, in cooperation with biomedical researchers, I present validation of the above research contributions by their successful application to real-world research problems.