Spectral Laplace-Beltrami wavelets with applications in medical images

Characterizing the propagation pathway of neuropathological events of Alzheimer's disease using harmonic wavelet analysis

Empirical imaging biomarkers such as the level of the regional pathological burden are widely used to measure the risk of developing neurodegenerative diseases such as Alzheimer's disease (AD). However, ample evidence shows that the brain network (wirings of white matter fibers) plays a vital role in the progression of AD, where neuropathological burdens often propagate across the brain network in a prion-like manner. In this context, characterizing the spreading pathway of AD-related neuropathological events sheds new light on understanding the heterogeneity of pathophysiological mechanisms in AD. In this work, we propose a manifold-based harmonic network analysis approach to explore a novel imaging biomarker in the form of the AD propagation pattern, which eventually allows us to identify the AD-related spreading pathways of neuropathological events throughout the brain. The backbone of this new imaging biomarker is a set of region-adaptive harmonic wavelets that represent the common network topology across individuals. We conceptualize that the individual's brain network and its associated pathology pattern form a unique system, which vibrates as do all natural objects in the universe. Thus, we can computationally excite such a brain system using selected harmonic wavelets that match the system's resonance frequency, where the resulting oscillatory wave manifests the system-level propagation pattern of neuropathological events across the brain network. We evaluate the statistical power of our harmonic network analysis approach on large-scale neuroimaging data from ADNI. Compared with the other empirical biomarkers, our harmonic wavelets not only yield a new imaging biomarker to potentially predict the cognitive decline in the early stage but also offer a new window to capture the in-vivo spreading pathways of neuropathological burden with a rigorous mathematics insight.

Revisiting convolutional neural network on graphs with polynomial approximations of Laplace-Beltrami spectral filtering

This paper revisits spectral graph convolutional neural networks (graph-CNNs) given in Defferrard (2016) and develops the Laplace-Beltrami CNN (LB-CNN) by replacing the graph Laplacian with the LB operator. We define spectral filters via the LB operator on a graph and explore the feasibility of Chebyshev, Laguerre, and Hermite polynomials to approximate LB-based spectral filters. We then update the LB operator for pooling in the LB-CNN. We employ the brain image data from Alzheimer's Disease Neuroimaging Initiative (ADNI) and Open Access Series of Imaging Studies (OASIS) to demonstrate the use of the proposed LB-CNN. Based on the cortical thickness of two datasets, we showed that the LB-CNN slightly improves classification accuracy compared to the spectral graph-CNN. The three polynomials had a similar computational cost and showed comparable classification accuracy in the LB-CNN or spectral graph-CNN. The LB-CNN trained via the ADNI dataset can achieve reasonable classification accuracy for the OASIS dataset. Our findings suggest that even though the shapes of the three polynomials are different, deep learning architecture allows us to learn spectral filters such that the classification performance is not dependent on the type of the polynomials or the operators (graph Laplacian and LB operator).

Fast mesh data augmentation via Chebyshev polynomial of spectral filtering

Deep neural networks have recently been recognized as one of the powerful learning techniques in computer vision and medical image analysis. Trained deep neural networks need to be generalizable to new data that are not seen before. In practice, there is often insufficient training data available, which can be solved via data augmentation. Nevertheless, there is a lack of augmentation methods to generate data on graphs or surfaces, even though graph convolutional neural network (graph-CNN) has been widely used in deep learning. This study proposed two unbiased augmentation methods, Laplace-Beltrami eigenfunction Data Augmentation (LB-eigDA) and Chebyshev polynomial Data Augmentation (C-pDA), to generate new data on surfaces, whose mean was the same as that of observed data. LB-eigDA augmented data via the resampling of the LB coefficients. In parallel with LB-eigDA, we introduced a fast augmentation approach, C-pDA, that employed a polynomial approximation of LB spectral filters on surfaces. We designed LB spectral bandpass filters by Chebyshev polynomial approximation and resampled signals filtered via these filters in order to generate new data on surfaces. We first validated LB-eigDA and C-pDA via simulated data and demonstrated their use for improving classification accuracy. We then employed brain images of the Alzheimer's Disease Neuroimaging Initiative (ADNI) and extracted cortical thickness that was represented on the cortical surface to illustrate the use of the two augmentation methods. We demonstrated that augmented cortical thickness had a similar pattern to observed data. We also showed that C-pDA was faster than LB-eigDA and can improve the AD classification accuracy of graph-CNN.

Tetrahedral spectral feature-Based bayesian manifold learning for grey matter morphometry: Findings from the Alzheimer's disease neuroimaging initiative

Structural and anatomical analyses of magnetic resonance imaging (MRI) data often require a reconstruction of the three-dimensional anatomy to a statistical shape model. Our prior work demonstrated the usefulness of tetrahedral spectral features for grey matter morphometry. However, most of the current methods provide a large number of descriptive shape features, but lack an unsupervised scheme to automatically extract a concise set of features with clear biological interpretations and that also carries strong statistical power. Here we introduce a new tetrahedral spectral feature-based Bayesian manifold learning framework for effective statistical analysis of grey matter morphology. We start by solving the technical issue of generating tetrahedral meshes which preserve the details of the grey matter geometry. We then derive explicit weak-form tetrahedral discretizations of the Hamiltonian operator (HO) and the Laplace-Beltrami operator (LBO). Next, the Schrödinger's equation is solved for constructing the scale-invariant wave kernel signature (SIWKS) as the shape descriptor. By solving the heat equation and utilizing the SIWKS, we design a morphometric Gaussian process (M-GP) regression framework and an active learning strategy to select landmarks as concrete shape descriptors. We evaluate the proposed system on publicly available data from the Alzheimers Disease Neuroimaging Initiative (ADNI), using subjects structural MRI covering the range from cognitively unimpaired (CU) to full blown Alzheimer's disease (AD). Our analyses suggest that the SIWKS and M-GP compare favorably with seven other baseline algorithms to obtain grey matter morphometry-based diagnoses. Our work may inspire more tetrahedral spectral feature-based Bayesian learning research in medical image analysis.

Fast Polynomial Approximation of Heat Kernel Convolution on Manifolds and Its Application to Brain Sulcal and Gyral Graph Pattern Analysis

Heat diffusion has been widely used in brain imaging for surface fairing, mesh regularization and cortical data smoothing. Motivated by diffusion wavelets and convolutional neural networks on graphs, we present a new fast and accurate numerical scheme to solve heat diffusion on surface meshes. This is achieved by approximating the heat kernel convolution using high degree orthogonal polynomials in the spectral domain. We also derive the closed-form expression of the spectral decomposition of the Laplace-Beltrami operator and use it to solve heat diffusion on a manifold for the first time. The proposed fast polynomial approximation scheme avoids solving for the eigenfunctions of the Laplace-Beltrami operator, which is computationally costly for large mesh size, and the numerical instability associated with the finite element method based diffusion solvers. The proposed method is applied in localizing the male and female differences in cortical sulcal and gyral graph patterns obtained from MRI in an innovative way. The MATLAB code is available at http://www.stat.wisc.edu/~mchung/chebyshev.

An empirical assessment of appearance descriptors applied to MRI for automated diagnosis of TLE and MCI

OBJECTIVE

Differential diagnosis of mild cognitive impairment MCI and temporal lobe epilepsy TLE is a debated issue, specifically because these conditions may coincide in the elderly population. We evaluate automated differential diagnosis based on characteristics derived from structural brain MRI of different brain regions.

METHODS

In 22 healthy controls, 19 patients with MCI, and 17 patients with TLE we used scale invariant feature transform (SIFT), local binary patterns (LBP), and wavelet-based features and investigate their predictive performance for MCI and TLE.

RESULTS

The classification based on SIFT features resulted in an accuracy of 81% of MCI vs. TLE and reasonable generalizability. Local binary patterns yielded satisfactory diagnostic performance with up to 94.74% sensitivity and 88.24% specificity in the right Thalamus for the distinction of MCI vs. TLE, but with limited generalizable. Wavelet features yielded similar results as LPB with 94.74% sensitivity and 82.35% specificity but generalize better.

SIGNIFICANCE

Features beyond volume analysis are a valid approach when applied to specific regions of the brain. Most significant information could be extracted from the thalamus, frontal gyri, and temporal regions, among others. These results suggest that analysis of changes of the central nervous system should not be limited to the most typical regions of interest such as the hippocampus and parahippocampal areas. Region-independent approaches can add considerable information for diagnosis. We emphasize the need to characterize generalizability in future studies, as our results demonstrate that not doing so can lead to overestimation of classification results.

LIMITATIONS

The data used within this study allows for separation of MCI and TLE subjects using a simple age threshold. While we present a strong indication that the presented method is age-invariant and therefore agnostic to this situation, new data would be needed for a rigorous empirical assessment of this findings.

Long-term Influences of Prenatal Maternal Depressive Symptoms on the Amygdala-Prefrontal Circuitry of the Offspring From Birth to Early Childhood

BACKGROUND

Prenatal maternal depression may have long-term impacts on amygdala-cortical development. This study explored associations of prenatal maternal depressive symptoms on the amygdala-cortical structural covariance of the offspring from birth to early childhood, derived from a longitudinal birth cohort.

METHODS

Structural magnetic resonance imaging was performed to obtain the amygdala volume and cortical thickness at each time point. Prenatal maternal depressive symptoms were measured using the Edinburgh Postnatal Depression Scale at 26 weeks of pregnancy. Regression analysis was used to examine the effects of the Edinburgh Postnatal Depression Scale on a structural coupling between the amygdala volume and cortical thickness at birth (n = 167) and 4.5 years of age (n = 199).

RESULTS

Girls whose mothers had high prenatal maternal depressive symptoms showed a positive coupling between the amygdala volume and insula thickness at birth (β = .617, p = .001) but showed a negative coupling between the amygdala volume and inferior frontal thickness at 4.5 years of age (β = -.369, p = .008). No findings were revealed in boys at any time point.

CONCLUSIONS

The development of the amygdala-prefrontal circuitry is vulnerable to environmental factors related to depression. Such a vulnerability might be sex dependent.

Recent publications from the Alzheimer's Disease Neuroimaging Initiative: Reviewing progress toward improved AD clinical trials

INTRODUCTION

The Alzheimer's Disease Neuroimaging Initiative (ADNI) has continued development and standardization of methodologies for biomarkers and has provided an increased depth and breadth of data available to qualified researchers. This review summarizes the over 400 publications using ADNI data during 2014 and 2015.

METHODS

We used standard searches to find publications using ADNI data.

RESULTS

(1) Structural and functional changes, including subtle changes to hippocampal shape and texture, atrophy in areas outside of hippocampus, and disruption to functional networks, are detectable in presymptomatic subjects before hippocampal atrophy; (2) In subjects with abnormal β-amyloid deposition (Aβ+), biomarkers become abnormal in the order predicted by the amyloid cascade hypothesis; (3) Cognitive decline is more closely linked to tau than Aβ deposition; (4) Cerebrovascular risk factors may interact with Aβ to increase white-matter (WM) abnormalities which may accelerate Alzheimer's disease (AD) progression in conjunction with tau abnormalities; (5) Different patterns of atrophy are associated with impairment of memory and executive function and may underlie psychiatric symptoms; (6) Structural, functional, and metabolic network connectivities are disrupted as AD progresses. Models of prion-like spreading of Aβ pathology along WM tracts predict known patterns of cortical Aβ deposition and declines in glucose metabolism; (7) New AD risk and protective gene loci have been identified using biologically informed approaches; (8) Cognitively normal and mild cognitive impairment (MCI) subjects are heterogeneous and include groups typified not only by "classic" AD pathology but also by normal biomarkers, accelerated decline, and suspected non-Alzheimer's pathology; (9) Selection of subjects at risk of imminent decline on the basis of one or more pathologies improves the power of clinical trials; (10) Sensitivity of cognitive outcome measures to early changes in cognition has been improved and surrogate outcome measures using longitudinal structural magnetic resonance imaging may further reduce clinical trial cost and duration; (11) Advances in machine learning techniques such as neural networks have improved diagnostic and prognostic accuracy especially in challenges involving MCI subjects; and (12) Network connectivity measures and genetic variants show promise in multimodal classification and some classifiers using single modalities are rivaling multimodal classifiers.

DISCUSSION

Taken together, these studies fundamentally deepen our understanding of AD progression and its underlying genetic basis, which in turn informs and improves clinical trial design.

Local Spectral Analysis of the Cerebral Cortex: New Gyrification Indices

Gyrification index (GI) is an appropriate measure to quantify the complexity of the cerebral cortex. There is, however, no universal agreement on the notion of surface complexity and there are various methods in literature that evaluate different aspects of cortical folding. In this paper, we give two intuitive interpretations on folding quantification based on the magnitude and variation of the mean curvature of the cortical surface. We then present a local spectral analysis of the mean curvature to introduce two local gyrification indices that satisfy our interpretations. For this purpose, the graph windowed Fourier transform is extended to the framework of surfaces discretized with triangular meshes. An adaptive window function is also proposed to deal with the intersubject cortical size variability. The intrinsic nature of the method allows us to compute the degree of folding at different spatial scales. Our experiments show that while more classical surface area-based GIs may fail at differentiating deep folds from very convoluted ones, our spectral GIs overcome this issue. The method is applied to the cortical surfaces of 124 healthy adult subjects of OASIS database and average gyrification maps are computed and compared with other GI definitions. In order to illustrate the capacity of our method to capture and quantify important aspects of gyrification, we study the relationship between brain volume and cortical complexity, and design a scaling analysis with a power law model. Results indicate an allometric relation and confirm the well-known observations that larger brains are more folded. We also perform the scaling analysis at the vertex level to investigate how the degree of folding varies locally with the brain volume. Results reveal that in our healthy adult brain database, cortical regions which are the least folded on average show an increased folding complexity when brain size increases.

Large Deformation Multiresolution Diffeomorphic Metric Mapping for Multiresolution Cortical Surfaces: A Coarse-to-Fine Approach

Brain surface registration is an important tool for characterizing cortical anatomical variations and understanding their roles in normal cortical development and psychiatric diseases. However, surface registration remains challenging due to complicated cortical anatomy and its large differences across individuals. In this paper, we propose a fast coarse-to-fine algorithm for surface registration by adapting the large diffeomorphic deformation metric mapping (LDDMM) framework for surface mapping and show improvements in speed and accuracy via a multiresolution analysis of surface meshes and the construction of multiresolution diffeomorphic transformations. The proposed method constructs a family of multiresolution meshes that are used as natural sparse priors of the cortical morphology. At varying resolutions, these meshes act as anchor points where the parameterization of multiresolution deformation vector fields can be supported, allowing the construction of a bundle of multiresolution deformation fields, each originating from a different resolution. Using a coarse-to-fine approach, we show a potential reduction in computation cost along with improvements in sulcal alignment when compared with LDDMM surface mapping.