Applying tensor-based morphometry to parametric surfaces can improve MRI-based disease diagnosis
Highlights
► Cortical surface parameterization with slit map conformal mapping ► A sparse learning based method for surface feature selection and classification ► Surface mTBM achieved better group difference results than other surface statistics. ► Stability selection demonstrated consistent feature selection results. ► An automated and robust cortical surface registration and classification system
Introduction
Computer-assisted diagnostic classification is becoming increasingly popular in neuroimaging, especially given the vast number of features available to assist diagnosis in a 3D brain image. Early diagnosis and treatment of degenerative brain diseases, such as Alzheimer's disease, depends on the ability to identify disease in its earliest stages, when brain changes may be subtle. In addition, there is interest in understanding which brain imaging features are best for diagnostic classification, as well as biomarkers to measure the severity of disease burden. Over the last decade, many methods have been proposed to study the problem of diagnostic classification based on structural magnetic resonance imaging (MRI) (Batmanghelich et al., 2012, Cuingnet et al., 2010, Cuingnet et al., 2011, Fan et al., 2007, Golland et al., 2001, Gutman et al., 2009, Sabuncu and Van Leemput, 2011, Sun et al., 2009a, Vemuri et al., 2008, Xiang et al., 2009, Yushkevich et al., 2003), positron emission tomography (PET) (Chen et al., 2011, López et al., 2011), single photon emitting computer tomography (SPECT) (Fung and Stoeckel, 2007) or a combination of multi-source datasets (Calhoun and Adali, 2009, Chen et al., 2009, Correa et al., 2010, Groves et al., 2011, Jack et al., 2010, Kohannim et al., 2010, Sui et al., 2011, Yang et al., 2010, Yuan et al., 2012a). Surface-based modeling is useful in brain imaging to help analyze anatomical shapes, to detect abnormalities in cortical surface folding and thickness, and to statistically combine or compare 3D anatomical models across subjects (Drury et al., 1996, Fischl et al., 1999, Thompson and Toga, 1996, Vaillant et al., 2007, Wang et al., 2010c, Wang et al., 2011b, Yeo et al., 2008). Many surface-based morphometry studies describe structural differences at the group level, i.e., between different diagnostic groups. More recently, morphometric maps have also been used to classify individual subjects into diagnostic groups (Costafreda et al., 2011, Ferrarini et al., 2008, Kohannim et al., 2010, Sun et al., 2009a, Wang et al., 2010b). In one study (Sun et al., 2009a), maps of cortical gray matter density achieved 86.1% accuracy in discriminating psychotic patients from control subjects, in leave-one-out tests. In related work (Ferrarini et al., 2008), the notion of biomarker “nodes” was proposed, i.e. regions on surface meshes that contribute most to diagnostic classification; the authors tested their approach on ventricular surface models from Alzheimer's disease patients and matched controls. Overall, a set of surface-based morphometric features combined with a machine learning algorithm may offer a promising way to improve the performance of computer-assisted diagnostic systems.
An important question for diagnostic classification based on voxel-based or surface-based morphometric maps is which statistics are best to analyze. Statistics derived from anatomical surface models, such as gray matter thickness maps (Thompson et al., 2003, Thompson et al., 2005), radial distances (distances from the medial core to each surface point) (Apostolova et al., 2010a, Apostolova et al., 2010b, Carmichael et al., 2006, Carmichael et al., 2007a, Carmichael et al., 2007b, Carmichael et al., 2007c, Chou et al., 2008, Chou et al., 2009, Morra et al., 2009, Morra et al., 2010, Styner et al., 2004, Thompson et al., 2004a, Thompson et al., 2007), spherical harmonic analysis (Gutman et al., 2009, Styner et al., 2005), local area differences (related to the determinant of the Jacobian matrix) (Chung et al., 2008, Davatzikos et al., 1996, Woods, 2003), Gaussian random fields (Bansal et al., 2007), Reeb graphs (another way to compute radial distances) (Shi et al., 2009) have all been applied to analyze the shape and geometry of various brain structures. Surface tensor-based morphometry (TBM) (Chung et al., 2008, Davatzikos et al., 1996, Thompson et al., 2000a, Woods, 2003) is an intrinsic surface statistic that examines spatial derivatives of the deformation maps that register brains to common templates, and can help to detect subtle differences in local surface morphometry. In recent studies (Wang et al., 2008b, Wang et al., 2009a, Wang et al., 2010c, Wang et al., 2011b), surface multivariate TBM (mTBM) was found to be more sensitive for detecting group differences than other standard TBM-based statistics. As a result, here we decided to use mTBM statistics as the surface statistics to be included in a diagnostic classifier.
Three-dimensional statistical maps can detect consistent local differences in anatomical surfaces. But, when they are applied to classification, the feature dimension is usually much larger than the number of subjects in the sample being analyzed — the “high dimension/small sample size problem”. When a vast number of variables are measured from a small number of subjects, it is often possible to divide the subjects into groups based on the observed features, but the resulting classification rules may generalize poorly to new observations. To select the most useful features, feature reduction can be beneficial. Feature selection approaches are widely used in machine learning, (e.g. Fan et al., 2005, Guyon et al., 2002, Kuncheva and Rodríguez, 2010, Stearns, 1976). Even so, most methods still generate very large numbers of features, making it difficult to state intuitively why features are being used to make biological inferences. To address this, sparse learning methods have been proposed to select the most biologically germane features (Friedman et al., 2008, Tibshirani, 1996). Sparse learning methods enjoy strong theoretical properties (Candès and Wakin, 2008, Donoho, 2006) and are receiving increased attention in many application areas (Beck and Teboulle, 2009, Candès et al., 2006, Figueiredo et al., 2007, Wu et al., 2009). Sparse learning has also been applied in neuroimaging to study genetic influences on the brain (Hibar et al., 2011, Kohannim et al., 2011, Le Floch et al., 2011, Vounou et al., 2010, Vounou et al., 2012, Wang et al., 2012a), functional connectivity (Huang et al., 2010, Ryali et al., 2012), and for outcome predictions (Shen et al., 2010, Stonnington et al., 2010, Sun et al., 2009a, Wang et al., 2010a, Wang et al., 2010b, Wang et al., 2011a). In many computer vision, medical imaging and bioinformatics applications, using sparsity as a prior leads to state-of-the-art results (Liu and Ye, 2010, Liu et al., 2010b, Sun et al., 2009a, Wright et al., 2009).
Here we developed a new approach, based on conformal slit mapping (Wang et al., 2009a), multivariate tensor-based morphometry (mTBM), and sparse learning, to identify cortical biomarkers for classification problems. We hypothesized that mTBM might improve the accuracy for analyzing group differences in neuroimaging data, and for helping individual classification, when used with a sparse learning classifier. We tested our hypothesis on a dataset used in a prior work (Thompson et al., 2005): it consists of 42 subjects with genetically confirmed William syndrome and 40 age-matched controls. The point of using Williams syndrome data as a test is that the diagnosis can be verified using a genetic test. Despite many years of research on brain differences in Williams syndrome – finding differences widely distributed in the brain – no one known trait offers powerful group classification on its own. As such, we chose this dataset as an interesting test case, as it may also identify distinctive cortical features for further study.
Fig. 1 summarizes the steps we used to analyze cortical surface morphometry. The cortical surface data was from our prior study (Thompson et al., 2005). With 10 selected landmarks on each cortical hemispheric surface, we computed a conformal mapping from a multiply connected mesh to the so-called slit domain, which consists of a canonical rectangle or disk in which 3D curved landmarks on the original surfaces are mapped to parallel lines or concentric slits in the slit domain (Wang et al., 2008a). In this canonical parametric domain, cortical surfaces were matched by a constrained harmonic map (Wang et al., 2007). Multivariate surface statistics were computed from the registered surfaces (Wang et al., 2010c). In one experiment, they were applied to identify regions with significant differences between the two groups. In another experiment, cortical features were fed to a sparse learning method to classify each subject into one of two groups by a leave-one-out test. We also tested other possible surface morphometry statistics to compare them with our multivariate surface statistics. Although the method is illustrated on Williams syndrome data, it is intended to be useful for other disorders as well. Tests on more diverse datasets are reserved for further work.
Section snippets
Subjects
We tested our algorithm on data from a prior study by Thompson et al. (2005). Subjects and brain-scanning protocols were used exactly as in the study by (Reiss et al., 2004, Thompson et al., 2005). Exclusion criteria included a history of medical conditions not typically associated with WS, such as epilepsy or other neurological conditions. All WS participants were evaluated at the Salk Institute (La Jolla, CA) as part of a program project on genetics, neuroanatomy, neurophysiology, and
Results
We tested our method on cortical surface data from 42 subjects with genetically confirmed WS and 40 age-matched healthy controls (Thompson et al., 2005). With the slit map method, each cortical surface was conformally mapped to an annulus with concentric arcs. We then computed a constrained harmonic map to register the surfaces to a template surface, using Eq. (1). The template surface was chosen randomly from the control set. The constrained harmonic map helped us build a direct correspondence
Discussion
In this paper, our overarching goal was to detect differences in brain surface morphometry. One traditional way to do this is to set up parametric grids on surfaces, and then use differential geometry to come up with useful descriptors of surface features of interest, or to summarize the geometry as a whole. Conformal maps help to induce particularly well-organized grids on surfaces. This simplifies a number of downstream computations of derivatives and metrics. In addition, the surface metric
Conclusion
We presented an MRI-based computer-assisted diagnostic classification system that finds vertices and local features on cortical surface models to best discriminate two groups of subjects. Our system was based on mTBM and sparse learning, with the -norm based penalty. The mTBM captured the full deformation tensor information and performed better than TBM (the determinant of the Jacobian matrix) in both group difference and classification studies. The sparse learning method selected a smaller,
Acknowledgments
This work was funded by the National Institutes of Health through the NIH Roadmap for Medical Research, grant U54 RR021813 entitled Center for Computational Biology (CCB). Additional support was provided by US National Institutes of Health (NIH) (HD049653 to AR, R01 NS080655, R01 MH097268, R01 AG040060 and P41 EB015922 to PMT), and the US National Science Foundation (NSF) (IIS-0953662 to JY).
References (171)
- et al.
Subregional hippocampal atrophy predicts Alzheimer's dementia in the cognitively normal
Neurobiol. Aging
(2010) - et al.
Voxel-based morphometry — the methods
NeuroImage
(2000) - et al.
Acceleration of cerebral ventricular expansion in the Cardiovascular Health Study
Neurobiol. Aging
(2007) - et al.
Ventricular volume and dementia progression in the Cardiovascular Health Study
Neurobiol. Aging
(2007) - et al.
Linking functional and structural brain images with multivariate network analyses: a novel application of the partial least square method
NeuroImage
(2009) - et al.
Characterizing Alzheimer's disease using a hypometabolic convergence index
NeuroImage
(2011) - et al.
Local MRI analysis approach in the diagnosis of early and prodromal Alzheimer's disease
NeuroImage
(2011) - et al.
Automated ventricular mapping with multi-atlas fluid image alignment reveals genetic effects in Alzheimer's disease
NeuroImage
(2008) - et al.
Mapping correlations between ventricular expansion and CSF amyloid and tau biomarkers in 240 subjects with Alzheimer's disease, mild cognitive impairment and elderly controls
NeuroImage
(2009) - et al.
A unified statistical approach to deformation-based morphometry
NeuroImage
(2001)
Deformation-based surface morphometry applied to gray matter deformation
NeuroImage
Automated hippocampal shape analysis predicts the onset of dementia in mild cognitive impairment
NeuroImage
Inferring brain variability from diffeomorphic deformations of currents: an integrative approach
Med. Image Anal.
Classification of structural images via high-dimensional image warping, robust feature extraction, and SVM
Med. Image Comput. Comput. Assist. Interv.
Cortical surface-based analysis II: inflation, flattening, and a surface-based coordinate system
NeuroImage
Linked independent component analysis for multimodal data fusion
NeuroImage
Learning brain connectivity of Alzheimer's disease by sparse inverse covariance estimation
NeuroImage
Cortical cartography using the discrete conformal approach of circle packings
NeuroImage
Discrete conformal methods for cortical brain flattening
NeuroImage
Quantitative evaluation of three surface flattening methods
NeuroImage
Boosting power for clinical trials using classifiers based on multiple biomarkers
Neurobiol. Aging
Classifier ensembles for fMRI data analysis: an experiment
Magn. Reson. Imaging
Morphological classification of brains via high-dimensional shape transformations and machine learning methods
NeuroImage
Principal component analysis-based techniques and supervised classification schemes for the early detection of the Alzheimer's Disease
Neurocomputing
Implicit brain imaging
NeuroImage
Automated mapping of hippocampal atrophy in 1-year repeat MRI data from 490 subjects with Alzheimer's disease, mild cognitive impairment, and elderly controls
NeuroImage
Comparison of landmark-based and automatic methods for cortical surface registration
NeuroImage
Complex Analysis
Cortical differences analysis with multivariate tensor-based morphometry in 829 ADNI subjects
On the Laplace–Beltrami operator and brain surface flattening
IEEE Trans. Med. Imaging
3D comparison of low, intermediate, and advanced hippocampal atrophy in MCI
Hum. Brain Mapp.
Log-Euclidean metrics for fast and simple calculus on diffusion tensors
Magn. Reson. Med.
Identifying global anatomical differences: deformation-based morphometry
Hum. Brain Mapp.
DISCO: a coherent diffeomorphic framework for brain registration under exhaustive sulcal constraints
Landmark matching on brain surfaces via large deformation diffeomorphisms on the sphere
Statistical analyses of brain surfaces using Gaussian random fields on 2-D manifolds
IEEE Trans. Med. Imaging
Generative-discriminative basis learning for medical imaging
IEEE Trans. Med. Imaging
A fast iterative shrinkage-thresholding algorithm for linear inverse problems
SIAM J. Imag. Sci.
From sparse solutions of systems of equations to sparse modeling of signals and images
SIAM Rev.
A tensor-based morphometry study of genetic influences on brain structure using a new fluid registration method
Med. Image Comput. Comput. Assist. Interv.
Feature-based fusion of medical imaging data
IEEE Trans. Inf. Technol. Biomed.
Decoding by linear programming
IEEE Trans. Inf. Theory
An introduction to compressive sampling
IEEE Signal Process. Mag.
Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information
IEEE Trans. Inf. Theory
The detection of local shape changes via the geometry of Hotelling's T2 fields
Ann. Stat.
Mapping ventricular changes related to dementia and mild cognitive impairment in a large community-based cohort
Cerebral ventricular changes associated with transitions between normal cognitive function, mild cognitive impairment, and dementia
Alzheimer Dis. Assoc. Disord.
Tensor-based cortical surface morphometry via weighted spherical harmonic representation
IEEE Trans. Med. Imaging
Proximal splitting methods in signal processing
Canonical correlation analysis for data fusion and group inferences: examining applications of medical imaging data
IEEE Signal Process. Mag.
Cited by (38)
Predicting future cognitive decline with hyperbolic stochastic coding
2021, Medical Image AnalysisA novel pipeline leveraging surface-based features of small subcortical structures to classify individuals with autism spectrum disorder
2021, Progress in Neuro-Psychopharmacology and Biological PsychiatryCitation Excerpt :However, classification using vertex-wise high-dimensional neuroimaging features is likely to be plagued by the curse of dimensionality (Wade et al., 2017). Prior studies on surface feature-based (Sun et al., 2009; Wang et al., 2013) or voxel-based (Davatzikos et al., 2008; Uddin et al., 2011) classification mainly depended on direct feature dimension reduction or without dimension reduction, and then do the predict. Actually, such approaches do not exploit the possible relationships among features and may ignore the intrinsic properties of a structure's regional morphometry.
Tensors for neuroimaging: A review on applications of tensors to unravel the mysteries of the brain
2021, Tensors for Data Processing: Theory, Methods, and Applications