Riemannian Nonlinear Mixed Effects Models: Analyzing Longitudinal Deformations in Neuroimaging

Modeling Longitudinal Optical Coherence Tomography Images for Monitoring and Analysis of Glaucoma Progression

Glaucoma causes progressive visual field deterioration and is the leading cause of blindness worldwide. Glaucomatous damage is irreversible and greatly impacts quality of life. Therefore, it is critically important to detect glaucoma early and closely monitor progression to preserve functional vision. Glaucoma is routinely monitored in the clinical setting using optical coherence tomography (OCT) for derived measures such as the thickness of important visual structures. There is not a consensus of what measures represent the most relevant biomarkers of glaucoma progression. Further, despite the increasing availability of longitudinal OCT data, a quantitative model of 3D structural change over time associated with glaucoma does not exist. In this paper we present an algorithm that will perform hierarchical geodesic modeling at the imaging level, considering 3D OCT images as observations of structural change over time. Hierarchical modeling includes subject-wise trajectories as geodesics in the space of diffeomorphisms and population level (glaucoma vs control) trajectories are also geodesics which explain subject-wise trajectories as deviations from the mean. Our preliminary experiments demonstrate a greater magnitude of structural change associated with glaucoma compared to normal aging. Our algorithm has the potential application in patient-specific monitoring and analysis of glaucoma progression as well as a statistical model of population trends and population variability.

Hierarchical geodesic modeling on the diffusion orientation distribution function for longitudinal DW-MRI analysis

The analysis of anatomy that undergoes rapid changes, such as neuroimaging of the early developing brain, greatly benefits from spatio-temporal statistical analysis methods to represent population variations but also subject-wise characteristics over time. Methods for spatio-temporal modeling and for analysis of longitudinal shape and image data have been presented before, but, to our knowledge, not for diffusion weighted MR images (DW-MRI) fitted with higher-order diffusion models. To bridge the gap between rapidly evolving DW-MRI methods in longitudinal studies and the existing frameworks, which are often limited to the analysis of derived measures like fractional anisotropy (FA), we propose a new framework to estimate a population trajectory of longitudinal diffusion orientation distribution functions (dODFs) along with subject-specific changes by using hierarchical geodesic modeling. The dODF is an angular profile of the diffusion probability density function derived from high angular resolution diffusion imaging (HARDI) and we consider the dODF with the square-root representation to lie on the unit sphere in a Hilbert space, which is a well-known Riemannian manifold, to respect the nonlinear characteristics of dODFs. The proposed method is validated on synthetic longitudinal dODF data and tested on a longitudinal set of 60 HARDI images from 25 healthy infants to characterize dODF changes associated with early brain development.

Statistical model for dynamically-changing correlation matrices with application to brain connectivity

BACKGROUND

Recent studies have indicated that functional connectivity is dynamic even during rest. A common approach to modeling the dynamic functional connectivity in whole-brain resting-state fMRI is to compute the correlation between anatomical regions via sliding time windows. However, the direct use of the sample correlation matrices is not reliable due to the image acquisition and processing noises in resting-sate fMRI.

NEW METHOD

To overcome these limitations, we propose a new statistical model that smooths out the noise by exploiting the geometric structure of correlation matrices. The dynamic correlation matrix is modeled as a linear combination of symmetric positive-definite matrices combined with cosine series representation. The resulting smoothed dynamic correlation matrices are clustered into disjoint brain connectivity states using the k-means clustering algorithm.

RESULTS

The proposed model preserves the geometric structure of underlying physiological dynamic correlation, eliminates unwanted noise in connectivity and obtains more accurate state spaces. The difference in the estimated dynamic connectivity states between males and females is identified.

COMPARISON WITH EXISTING METHODS

We demonstrate that the proposed statistical model has less rapid state changes caused by noise and improves the accuracy in identifying and discriminating different states.

CONCLUSIONS

We propose a new regression model on dynamically changing correlation matrices that provides better performance over existing windowed correlation and is more reliable for the modeling of dynamic connectivity.

Dilated Convolutional Neural Networks for Sequential Manifold-valued Data

Efforts are underway to study ways via which the power of deep neural networks can be extended to non-standard data types such as structured data (e.g., graphs) or manifold-valued data (e.g., unit vectors or special matrices). Often, sizable empirical improvements are possible when the geometry of such data spaces are incorporated into the design of the model, architecture, and the algorithms. Motivated by neuroimaging applications, we study formulations where the data are sequential manifold-valued measurements. This case is common in brain imaging, where the samples correspond to symmetric positive definite matrices or orientation distribution functions. Instead of a recurrent model which poses computational/technical issues, and inspired by recent results showing the viability of dilated convolutional models for sequence prediction, we develop a dilated convolutional neural network architecture for this task. On the technical side, we show how the modules needed in our network can be derived while explicitly taking the Riemannian manifold structure into account. We show how the operations needed can leverage known results for calculating the weighted Fréchet Mean (wFM). Finally, we present scientific results for group difference analysis in Alzheimer's disease (AD) where the groups are derived using AD pathology load: here the model finds several brain fiber bundles that are related to AD even when the subjects are all still cognitively healthy.

Hierarchical Multi-Geodesic Model for Longitudinal Analysis of Temporal Trajectories of Anatomical Shape and Covariates

Longitudinal regression analysis for clinical imaging studies is essential to investigate unknown relationships between subject-wise changes over time and subject-specific characteristics, represented by covariates such as disease severity or a level of genetic risk. Image-derived data in medical image analysis, e.g. diffusion tensors or geometric shapes, are often represented on nonlinear Riemannian manifolds. Hierarchical geodesic models were suggested to characterize subject-specific changes of nonlinear data on Riemannian manifolds as extensions of a linear mixed effects model. We propose a new hierarchical multi-geodesic model to enable analysis of the relationship between subject-wise anatomical shape changes on a Riemannian manifold and multiple subject-specific characteristics. Each individual subject-wise shape change is represented by a univariate geodesic model. The effects of subject-specific covariates on the estimated subject-wise trajectories are then modeled by multivariate intercept and slope models which together form a multi-geodesic model. Validation was performed with a synthetic example on a S ² manifold. The proposed method was applied to a longitudinal set of 72 corpus callosum shapes from 24 autism spectrum disorder subjects to study the relationship between anatomical shape changes and the autism severity score, resulting in statistics for the population but also for each subject. To our knowledge, this is the first longitudinal framework to model anatomical developments over time as functions of both continuous and categorical covariates on a nonlinear shape space.

On Training Deep 3D CNN Models with Dependent Samples in Neuroimaging

There is much interest in developing algorithms based on 3D convolutional neural networks (CNNs) for performing regression and classification with brain imaging data and more generally, with biomedical imaging data. A standard assumption in learning is that the training samples are independently drawn from the underlying distribution. In computer vision, where we have millions of training examples, this assumption is violated but the empirical performance may remain satisfactory. But in many biomedical studies with just a few hundred training examples, one often has multiple samples per participant and/or data may be curated by pooling datasets from a few different institutions. Here, the violation of the independent samples assumption turns out to be more significant, especially in small-to-medium sized datasets. Motivated by this need, we show how 3D CNNs can be modified to deal with dependent samples. We show that even with standard 3D CNNs, there is value in augmenting the network to exploit information regarding dependent samples. We present empirical results for predicting cognitive trajectories (slope and intercept) from morphometric change images derived from multiple time points. With terms which encode dependency between samples in the model, we get consistent improvements over a strong baseline which ignores such knowledge.

A Natural Language Interface for Dissemination of Reproducible Biomedical Data Science

Computational tools in the form of software packages are burgeoning in the field of medical imaging and biomedical research. These tools enable biomedical researchers to analyze a variety of data using modern machine learning and statistical analysis techniques. While these publicly available software packages are a great step towards a multiplicative increase in the biomedical research productivity, there are still many open issues related to validation and reproducibility of the results. A key gap is that while scientists can validate domain insights that are implicit in the analysis, the analysis itself is coded in a programming language and that domain scientist may not be a programmer. Thus, there is no/limited direct validation of the program that carries out the desired analysis. We propose a novel solution, building upon recent successes in natural language understanding, to address this problem. Our platform allows researchers to perform, share, reproduce and interpret the analysis pipelines and results via natural language. While this approach still requires users to have a conceptual understanding of the techniques, it removes the burden of programming syntax and thus lowers the barriers to advanced and reproducible neuroimaging and biomedical research.

A Riemannian Framework for Longitudinal Analysis of Resting-State Functional Connectivity

Even though the number of longitudinal resting-state-fMRI studies is increasing, accurately characterizing the changes in functional connectivity across visits is a largely unexplored topic. To improve characterization, we design a Riemannian framework that represents the functional connectivity pattern of a subject at a visit as a point on a Riemannian manifold. Geodesic regression across the 'sample' points of a subject on that manifold then defines the longitudinal trajectory of their connectivity pattern. To identify group differences specific to regions of interest (ROI), we map the resulting trajectories of all subjects to a common tangent space via the Lie group action. We account for the uncertainty in choosing the common tangent space by proposing a test procedure based on the theory of latent p-values. Unlike existing methods, our proposed approach identifies sex differences across 246 subjects, each of them being characterized by three rs-fMRI scans.

A geometric framework for statistical analysis of trajectories with distinct temporal spans

Analyzing data representing multifarious trajectories is central to the many fields in Science and Engineering; for example, trajectories representing a tennis serve, a gymnast's parallel bar routine, progression/remission of disease and so on. We present a novel geometric algorithm for performing statistical analysis of trajectories with distinct number of samples representing longitudinal (or temporal) data. A key feature of our proposal is that unlike existing schemes, our model is deployable in regimes where each participant provides a different number of acquisitions (trajectories have different number of sample points or temporal span). To achieve this, we develop a novel method involving the parallel transport of the tangent vectors along each given trajectory to the starting point of the respective trajectories and then use the span of the matrix whose columns consist of these vectors, to construct a linear subspace in R ^m . We then map these linear subspaces (possibly of distinct dimensions) of R ^m on to a single high dimensional hypersphere. This enables computing group statistics over trajectories by instead performing statistics on the hypersphere (equipped with a simpler geometry). Given a point on the hypersphere representing a trajectory, we also provide a "reverse mapping" algorithm to uniquely (under certain assumptions) reconstruct the subspace that corresponds to this point. Finally, by using existing algorithms for recursive Fréchet mean and exact principal geodesic analysis on the hypersphere, we present several experiments on synthetic and real (vision and medical) data sets showing how group testing on such diversely sampled longitudinal data is possible by analyzing the reconstructed data in the subspace spanned by the first few principal components.