Geodesic distance on a Grassmannian for monitoring the progression of Alzheimer's disease

Abstract We propose a geodesic distance on a Grassmannian manifold that can be used to quantify the shape progression patterns of the bilateral hippocampi, amygdalas, and lateral ventricles in healthy control (HC), mild cognitive impairment (MCI), and Alzheimer's disease (AD). Longitudinal magnetic resonance imaging (MRI) scans of 754 subjects (3092 scans in total) were used in this study. Longitudinally, the geodesic distance was found to be proportional to the elapsed time separating the two scans in question. Cross‐sectionally, utilizing a linear mixed‐effects statistical model, we found that each structure’s annualized rate of change in the geodesic distance followed the order of AD>MCI>HC, with statistical significance being reached in every case. In addition, for each of the six structures of interest, within the same time interval (e.g., from baseline to the 6th month), we observed significant correlations between the geodesic distance and the cognitive deterioration as quantified by the ADAS‐cog increase and the MMSE decrease. Furthermore, as the disease progresses over time, this linkage between the inter‐shape geodesic distance and the cognitive decline becomes considerably stronger and more significant.

[1]  Nick C Fox,et al.  Volumetric MRI and cognitive measures in Alzheimer disease , 2008, Journal of Neurology.

[2]  Suresh Venkatasubramanian,et al.  The geometric median on Riemannian manifolds with application to robust atlas estimation , 2009, NeuroImage.

[3]  Lior Wolf,et al.  Learning over Sets using Kernel Principal Angles , 2003, J. Mach. Learn. Res..

[4]  Alan Edelman,et al.  The Geometry of Algorithms with Orthogonality Constraints , 1998, SIAM J. Matrix Anal. Appl..

[5]  Brian Knutson,et al.  Dissociable neural representations of future reward magnitude and delay during temporal discounting , 2009, NeuroImage.

[6]  Rudrasis Chakraborty,et al.  Nonlinear Regression on Riemannian Manifolds and Its Applications to Neuro-Image Analysis , 2015, MICCAI.

[7]  Hongdong Li,et al.  A Framework for Shape Analysis via Hilbert Space Embedding , 2013, 2013 IEEE International Conference on Computer Vision.

[8]  Xiaoying Tang,et al.  APOE Affects the Volume and Shape of the Amygdala and the Hippocampus in Mild Cognitive Impairment and Alzheimer's Disease: Age Matters. , 2015, Journal of Alzheimer's disease : JAD.

[9]  L. Younes,et al.  Inferring changepoint times of medial temporal lobe morphometric change in preclinical Alzheimer's disease , 2014, NeuroImage: Clinical.

[10]  Laurent Younes,et al.  Spaces and manifolds of shapes in computer vision: An overview , 2012, Image Vis. Comput..

[11]  D. Kendall.  SHAPE MANIFOLDS, PROCRUSTEAN METRICS, AND COMPLEX PROJECTIVE SPACES , 1984 .

[12]  A. Nappi,et al.  Alzheimer ' s Disease : Cell-Specific Pathology Isolates the Hippocampal Formation , 2022 .

[13]  George Richardson,et al.  Brain development and aging: Overlapping and unique patterns of change , 2013, NeuroImage.

[14]  S. Folstein,et al.  "Mini-mental state". A practical method for grading the cognitive state of patients for the clinician. , 1975, Journal of psychiatric research.

[15]  Abraham Z. Snyder,et al.  A unified approach for morphometric and functional data analysis in young, old, and demented adults using automated atlas-based head size normalization: reliability and validation against manual measurement of total intracranial volume , 2004, NeuroImage.

[16]  Alan C. Evans,et al.  A nonparametric method for automatic correction of intensity nonuniformity in MRI data , 1998, IEEE Transactions on Medical Imaging.

[17]  Michael I. Miller,et al.  Network Neurodegeneration in Alzheimer’s Disease via MRI Based Shape Diffeomorphometry and High-Field Atlasing , 2015, Front. Bioeng. Biotechnol..

[18]  Anders M. Dale,et al.  Regional Shape Abnormalities in Mild Cognitive Impairment and Alzheimer's Disease , 2009, NeuroImage.

[19]  Olivier Salvado,et al.  Detecting global and local hippocampal shape changes in Alzheimer's disease using statistical shape models , 2012, NeuroImage.

[20]  Anuj Srivastava,et al.  2D Affine and Projective Shape Analysis , 2014, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[21]  Xiaoying Tang,et al.  The diffeomorphometry of regional shape change rates and its relevance to cognitive deterioration in mild cognitive impairment and Alzheimer's disease , 2015, Human brain mapping.

[22]  C. Jack,et al.  Medial temporal atrophy on MRI in normal aging and very mild Alzheimer's disease , 1997, Neurology.

[23]  Daniel Rueckert,et al.  Measurements of medial temporal lobe atrophy for prediction of Alzheimer's disease in subjects with mild cognitive impairment , 2013, Neurobiology of Aging.

[24]  P. Mecocci,et al.  An MRI‐based index to measure the severity of Alzheimer's disease‐like structural pattern in subjects with mild cognitive impairment , 2013, Journal of internal medicine.

[25]  K. Davis,et al.  A new rating scale for Alzheimer's disease. , 1984, The American journal of psychiatry.

[26]  Nuno Vasconcelos,et al.  Geodesic Regression on the Grassmannian , 2014, ECCV.

[27]  C. Small.  The statistical theory of shape , 1996 .

[28]  L. Younes,et al.  On the metrics and euler-lagrange equations of computational anatomy. , 2002, Annual review of biomedical engineering.

[29]  D. Selkoe.  Alzheimer's disease. , 2011, Cold Spring Harbor perspectives in biology.

[30]  Xavier Pennec,et al.  A Riemannian Framework for Tensor Computing , 2005, International Journal of Computer Vision.

[31]  A. Dale,et al.  Whole Brain Segmentation Automated Labeling of Neuroanatomical Structures in the Human Brain , 2002, Neuron.

[32]  Jean-Claude Baron,et al.  Early diagnosis of alzheimer’s disease: contribution of structural neuroimaging , 2003, NeuroImage.

[33]  Niels Holm Olsen,et al.  Linear Transformation Groups and Shape Space , 2004, Journal of Mathematical Imaging and Vision.

[34]  R. Petersen,et al.  Aging, mild cognitive impairment, and Alzheimer's disease. , 2000, Neurologic clinics.

[35]  Paul M. Thompson,et al.  Mapping hippocampal and ventricular change in Alzheimer disease , 2004, NeuroImage.

[36]  Sandra E. Black,et al.  The search for diagnostic and progression markers in AD , 1999, Neurology.

[37]  Rossana Ganzola,et al.  Mapping local hippocampal changes in Alzheimer's disease and normal ageing with MRI at 3 Tesla. , 2008, Brain : a journal of neurology.

[38]  Lok Ming Lui,et al.  Quasi-conformal statistical shape analysis of hippocampal surfaces for Alzheimer's disease analysis , 2016, Neurocomputing.

[39]  K. Mardia,et al.  Statistical Shape Analysis , 1998 .

[40]  H. Soininen,et al.  Volumes of hippocampus, amygdala and frontal lobes in the MRI-based diagnosis of early Alzheimer's disease: Correlation with memory functions , 1995, Journal of neural transmission. Parkinson's disease and dementia section.

[41]  Anuj Srivastava,et al.  Statistical shape analysis: clustering, learning, and testing , 2005, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[42]  Rama Chellappa,et al.  Statistical Computations on Grassmann and Stiefel Manifolds for Image and Video-Based Recognition , 2011, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[43]  David S. Lee,et al.  The diffeomorphometry of temporal lobe structures in preclinical Alzheimer's disease☆ , 2013, NeuroImage: Clinical.

[44]  Vittorio Murino,et al.  Kernel-Based Analysis of Functional Brain Connectivity on Grassmann Manifold , 2015, MICCAI.

[45]  Xiaoying Tang,et al.  Metric Space Structures for Computational Anatomy , 2013, MLMI.

[46]  M. Folstein,et al.  Clinical diagnosis of Alzheimer's disease: Report of the NINCDS—ADRDA Work Group under the auspices of Department of Health and Human Services Task Force on Alzheimer's Disease , 2011, Neurology.

[47]  D. Kendall,et al.  The Riemannian Structure of Euclidean Shape Spaces: A Novel Environment for Statistics , 1993 .

[48]  Daniel Rueckert,et al.  Medical Image Computing and Computer-Assisted Intervention − MICCAI 2017: 20th International Conference, Quebec City, QC, Canada, September 11-13, 2017, Proceedings, Part II , 2017, Lecture Notes in Computer Science.

[49]  Lei Wang,et al.  Metric distances between hippocampal shapes indicate different rates of change over time in nondemented and demented subjects. , 2012, Current Alzheimer research.

[50]  L. Younes,et al.  Shape abnormalities of subcortical and ventricular structures in mild cognitive impairment and Alzheimer's disease: Detecting, quantifying, and predicting , 2014, Human brain mapping.

[51]  E. Frankó,et al.  Evaluating Alzheimer’s Disease Progression Using Rate of Regional Hippocampal Atrophy , 2013, PloS one.

[52]  Paul M. Thompson,et al.  Hippocampal shape analysis in Alzheimer’s disease: A population-based study , 2007, NeuroImage.

[53]  Michael I. Miller,et al.  Preclinical detection of Alzheimer's disease: hippocampal shape and volume predict dementia onset in the elderly , 2005, NeuroImage.

[54]  K. Mardia,et al.  Affine shape analysis and image analysis , 2003 .

[55]  Vladimir Fonov,et al.  Prediction of Alzheimer's disease in subjects with mild cognitive impairment from the ADNI cohort using patterns of cortical thinning , 2013, NeuroImage.

[56]  Gene E. Alexander,et al.  Volumes of Medial Temporal Lobe Structures in Patients with Alzheimer’s Disease and Mild Cognitive Impairment (and in Healthy Controls) , 1998, Biological Psychiatry.

[57]  Y. Wong.  Differential geometry of grassmann manifolds. , 1967, Proceedings of the National Academy of Sciences of the United States of America.

[58]  N. Schuff,et al.  Magnetic resonance imaging of the entorhinal cortex and hippocampus in mild cognitive impairment and Alzheimer's disease , 2001, Journal of neurology, neurosurgery, and psychiatry.

[59]  Anders M. Dale,et al.  Reliability in multi-site structural MRI studies: Effects of gradient non-linearity correction on phantom and human data , 2006, NeuroImage.

[60]  Michael Werman,et al.  Affine Invariance Revisited , 2006, 2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'06).

[61]  José M. F. Moura,et al.  Affine-permutation symmetry: invariance and shape space , 2003, IEEE Workshop on Statistical Signal Processing, 2003.

[62]  Wei Zeng,et al.  Shape Classification Using Wasserstein Distance for Brain Morphometry Analysis , 2015, IPMI.

[63]  Ivo D Dinov,et al.  3D comparison of hippocampal atrophy in amnestic mild cognitive impairment and Alzheimer's disease. , 2006, Brain : a journal of neurology.

[64]  Anqi Qiu,et al.  CSF and Brain Structural Imaging Markers of the Alzheimer's Pathological Cascade , 2012, PloS one.

[65]  Norbert Schuff,et al.  Alzheimer’s Disease Neuroimaging Initiative: A one-year follow up study using Tensor-Based Morphometry correlating degenerative rates, biomarkers and cognition , 2009 .

[66]  S. Resnick,et al.  Longitudinal pattern of regional brain volume change differentiates normal aging from MCI , 2009, Neurology.

[67]  J. Price,et al.  Mild cognitive impairment represents early-stage Alzheimer disease. , 2001, Archives of neurology.

[68]  Amity E. Green,et al.  Hippocampal Atrophy and Ventricular Enlargement in Normal Aging, Mild Cognitive Impairment (MCI), and Alzheimer Disease , 2012, Alzheimer disease and associated disorders.

[69]  S. Lele,et al.  An Invariant Approach to Statistical Analysis of Shapes , 2001 .

[70]  Xiaoying Tang,et al.  Baseline shape diffeomorphometry patterns of subcortical and ventricular structures in predicting conversion of mild cognitive impairment to Alzheimer's disease. , 2015, Journal of Alzheimer's disease : JAD.