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Computational Anatomy: VBM and Alternatives

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Motivation for Computational Anatomy *See Wednesdays symposium 13:30-15:00 *Cortical Fingerprinting: What Anatomy Can Tell Us About Functional Architecture *There are many ways of examining brain structure. Depends on: *The question you want to ask *The data you have *The available software

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Overview *Volumetric differences *Serial Scans *Jacobian Determinants *Voxel-based Morphometry *Multivariate Approaches *Difference Measures *Another approach

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TemplateWarpedOriginal Deformation Field Deformation field

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Jacobians Jacobian Matrix (or just Jacobian) Jacobian Determinant (or just Jacobian) - relative volumes

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Early Late Difference Data from the Dementia Research Group, Queen Square. Serial Scans

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Regions of expansion and contraction *Relative volumes encoded in Jacobian determinants.

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Rigid Registration Software Packages *AIR: Automated Image Registration *FLIRT: FMRIBs Linear Image Registration Tool *MNI_AutoReg *SPM *VTK CISG Registration Toolkit many others...

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Nonlinear Registration Software Only listing public software that can (probably) estimate detailed warps suitable for longitudinal analysis. *HAMMER *MNI_ANIMAL Software Package *SPM2 *VTK CISG Registration Toolkit …there is much more software that is less readily available...

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Late Early Warped earlyDifference Early CSFLate CSF Relative volumes CSF modulated by relative volumes

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Late CSF - Early CSF Late CSF - modulated CSF Smoothed

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Smoothing Before convolutionConvolved with a circleConvolved with a Gaussian Smoothing is done by convolution. Each voxel after smoothing effectively becomes the result of applying a weighted region of interest (ROI).

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Overview *Volumetric differences *Voxel-based Morphometry *Method *Interpretation Issues *Multivariate Approaches *Difference Measures *Another approach

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Voxel-Based Morphometry *I. C. Wright et al. A Voxel-Based Method for the Statistical Analysis of Gray and White Matter Density Applied to Schizophrenia. NeuroImage 2: (1995). *I. C. Wright et al. Mapping of Grey Matter Changes in Schizophrenia. Schizophrenia Research 35:1-14 (1999). *J. Ashburner & K. J. Friston. Voxel-Based Morphometry - The Methods. NeuroImage 11: (2000). *J. Ashburner & K. J. Friston. Why Voxel-Based Morphometry Should Be Used. NeuroImage 14: (2001). *C. D. Good et al. Automatic Differentiation of Anatomical Patterns in the Human Brain: Validation with Studies of Degenerative Dementias. NeuroImage 17:29-46 (2002).

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Voxel-Based Morphometry *Produce a map of statistically significant differences among populations of subjects. *e.g. compare a patient group with a control group. *or identify correlations with age, test-score etc. *The data are pre-processed to sensitise the tests to regional tissue volumes. *Usually grey or white matter. *Can be done with SPM package, or e.g. *HAMMER and FSL

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Pre-processing for Voxel-Based Morphometry (VBM)

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VBM Preprocessing in SPM5b *Segmentation in SPM5b also estimates a spatial transformation that can be used for spatially normalising images. *It uses a generative model, which involves: *Mixture of Gaussians (MOG) *Bias Correction Component *Warping (Non-linear Registration) Component

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Mixture of Gaussians y1y1 c1c1 y2y2 y3y3 c2c2 c3c3 C C yIyI cIcI

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Bias Field ( ) y y ( ) y1y1 c1c1 y2y2 y3y3 c2c2 c3c3 C C yIyI cIcI

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Tissue Probability Maps *Tissue probability maps (TPMs) are used instead of the proportion of voxels in each Gaussian as the prior. ICBM Tissue Probabilistic Atlases. These tissue probability maps are kindly provided by the International Consortium for Brain Mapping, John C. Mazziotta and Arthur W. Toga.

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Mixing Proportions y1y1 c1c1 y2y2 y3y3 c2c2 c3c3 C C yIyI cIcI

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Deforming the Tissue Probability Maps *Tissue probability maps are deformed according to parameters. y1y1 c1c1 y2y2 y3y3 c2c2 c3c3 C C yIyI cIcI

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Warped, Modulated Grey Matter 12mm FWHM Smoothed Version SPM5b Pre-processed data for four subjects

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Statistical Parametric Mapping… group 1group 2 voxel by voxel modelling – parameter estimate standard error = statistic image or SPM

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Validity of the statistical tests in SPM *Residuals are not normally distributed. *Little impact on uncorrected statistics for experiments comparing groups. *Invalidates experiments that compare one subject with a group. *Corrections for multiple comparisons. *Mostly valid for corrections based on peak heights. *Not valid for corrections based on cluster extents. *SPM makes the inappropriate assumption that the smoothness of the residuals is stationary. *Bigger blobs expected in smoother regions.

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Interpretation Problem *What do the blobs really mean? *Unfortunate interaction between the algorithm's spatial normalization and voxelwise comparison steps. *Bookstein FL. "Voxel-Based Morphometry" Should Not Be Used with Imperfectly Registered Images. NeuroImage 14: (2001). *W.R. Crum, L.D. Griffin, D.L.G. Hill & D.J. Hawkes. Zen and the art of medical image registration: correspondence, homology, and quality. NeuroImage 20: (2003). *N.A. Thacker. Tutorial: A Critical Analysis of Voxel-Based Morphometry pdf

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Some Explanations of the Differences Thickening Thinning Folding Mis-classify Mis-register

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Cortical Thickness Mapping *Direct measurement of cortical thickness may be better for studying neuro-degenerative diseases *http://surfer.nmr.mgh.harvard.edu/ *http://brainvoyager.com/ *Some example references *B. Fischl & A.M. Dale. Measuring Thickness of the Human Cerebral Cortex from Magnetic Resonance Images. PNAS 97(20): (2000). *S.E. Jones, B.R. Buchbinder & I. Aharon. Three-dimensional mapping of cortical thickness using Laplace's equation. Human Brain Mapping 11 (1): (2000). *J.P. Lerch et al. Focal Decline of Cortical Thickness in Alzheimers Disease Identified by Computational Neuroanatomy. Cereb Cortex (2004). *Narr et al. Mapping Cortical Thickness and Gray Matter Concentration in First Episode Schizophrenia. Cerebral Cortex (2005). *Thompson et al. Abnormal Cortical Complexity and Thickness Profiles Mapped in Williams Syndrome. Journal of Neuroscience 25(16): (2005).

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Overview *Volumetric differences *Voxel-based Morphometry *Multivariate Approaches *Scan Classification *Cross-Validation *Difference Measures *Another approach

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Multivariate Approaches *Z. Lao, D. Shen, Z. Xue, B. Karacali, S. M. Resnick and C. Davatzikos. Morphological classification of brains via high-dimensional shape transformations and machine learning methods. NeuroImage 21(1):46-57, *C. Davatzikos. Why voxel-based morphometric analysis should be used with great caution when characterizing group differences. NeuroImage 23(1):17-20, * K. J. Friston and J. Ashburner. Generative and recognition models for neuroanatomy. NeuroImage 23(1):21-24, 2004.

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Globals for VBM *Shape is multivariate *Dependencies among volumes in different regions *SPM is mass univariate *globals used as a compromise *Can be either ANCOVA or proportional scaling Where should any difference between the two brains on the left and that on the right appear?

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Multivariate Approaches *An alternative to mass-univariate testing (SPMs) *Generate a description of how to separate groups of subjects *Use training data to develop a classifier *Use the classifier to diagnose test data *Data should be pre-processed so that clinically relevant features are emphasised *use existing knowledge

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Training and Classifying Control Training Data Patient Training Data ? ? ? ?

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Classifying Controls Patients ? ? ? ? y=f(w T x+w 0 )

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Difference between means Does not take account of variances and covariances m1m1 m2m2 w m 2 -m 1

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Fishers Linear Discriminant Curse of dimensionality ! w S -1 (m 2 -m 1 )

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Support Vector Classifier (SVC)

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Support Vector Support Vector Support Vector w is a weighted linear combination of the support vectors

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Going Nonlinear *Linear classification is by y = f(w T x + w 0 ) *where w is a weighting vector, x is the test data, w 0 is an offset, and f(.) is a thresholding operation *w is a linear combination of SVs w = i a i x i *So y = f( i a i x i T x + w 0 ) *Nonlinear classification is by y = f( i a i (x i,x) + w 0 ) *where (x i,x) is some function of x i and x. *e.g. RBF classification (x i,x) = exp(-||x i -x|| 2 /(2 2 ))

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Nonlinear SVC

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Over-fitting Test data A simpler model can often do better...

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Cross-validation *Methods must be able to generalise to new data *Various control parameters *More complexity -> better separation of training data *Less complexity -> better generalisation *Optimal control parameters determined by cross- validation *Test with data not used for training *Use control parameters that work best for these data

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Two-fold Cross-validation Use half the data for training. and the other half for testing.

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Two-fold Cross-validation Then swap around the training and test data.

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Leave One Out Cross-validation Use all data except one point for training. The one that was left out is used for testing.

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Leave One Out Cross-validation Then leave another point out. And so on...

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Regression (e.g. against age)

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Other Considerations *Should really take account of Bayes Rule: P(sick | data) = P(data | sick) x P(sick) P(data | sick) x P(sick) + P(data | healthy) x P(healthy) Requires prior probabilities *Sometimes decisions should be weighted using Decision Theory *Utility Functions/Risk *e.g. a false negative may be more serious than a false positive

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Overview *Volumetric differences *Voxel-based Morphometry *Multivariate Approaches *Difference Measures *Derived from Deformations *Derived from Deformations + Residuals *Another approach

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Distance Measures *Kernel-based classifiers (such as SVC) use measures of distance between data points (scans). *I.e. measure of how different each scan is from each other scan. *The measure is likely to depend on the application.

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Deformation Distance Summary Deformations can be considered within a small or large deformation setting. Small deformation setting is a linear approximation. Large deformation setting accounts for the nonlinear nature of deformations. Uses Lie Group Theory. Miller, Trouvé, Younes On the Metrics and Euler-Lagrange Equations of Computational Anatomy. Annual Review of Biomedical Engineering, 4: (2003) plus supplement Beg, Miller, Trouvé, L. Younes. Computing Large Deformation Metric Mappings via Geodesic Flows of Diffeomorphisms. Int. J. Comp. Vision, 61: (2005) Tilak Ratnanather gave me the following two slides…

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Computing the geodesic: problem statement I 0 : Template I 1 :Target

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1.386 Young Schizophrenia 3D Hippocampus: Young to Schizophrenia 3D Hippocampus: Young to Alzheimers Young Alzheimers Data from the lab. of Dr. Csernansky, Washington University, St Louis. Metrics on 3D Hippocampus in Neuro-psychiatric Disorders

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Accuracy of Automated Volumetric Inter-subject Registration Hellier et al. Inter subject registration of functional and anatomical data using SPM. MICCAI'02 LNCS 2489 (2002) Hellier et al. Retrospective evaluation of inter-subject brain registration. MIUA (2001)

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One-to-One Mappings *One-to-one mappings break down beyond a certain scale *The concept of a single best mapping may become meaningless at higher resolution Pictures taken from

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A Combined Distance Measure *Exact registration may not be possible. *Base distance measures on deformations plus residuals after registration. *Could use a related framework to that used for registering/segmenting. *Distance measures should be adjusted based on user expertise. *E.g. Some brain regions may be more informative than others, so give them more weighting. *Differences may be focal or more global *Could use some sort of high- or low-pass filtering.

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Overview *Volumetric differences *Voxel-based Morphometry *Multivariate Approaches *Difference Measures *Another approach

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Anatomist/BrainVISA Framework *Free software available from: *Automated identification and labelling of sulci etc. *These could be used to help spatial normalisation etc. *Can do morphometry on sulcal areas, etc *J.-F. Mangin, D. Rivière, A. Cachia, E. Duchesnay, Y. Cointepas, D. Papadopoulos-Orfanos, D. L. Collins, A. C. Evans, and J. Régis. Object- Based Morphometry of the Cerebral Cortex. IEEE Trans. Medical Imaging 23(8): (2004)

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Design of an artificial neuroanatomist 3D retina Bottom-up flow Fields of view of neural nets Elementary folds Sulci

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Correlates of handedness 14 subjects128 subjects Central sulcus surface is larger in dominant hemisphere

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Handedness correlates : localization after affine normalization

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Some of the potentially interesting posters *(#728 T-PM ) A Matlab-based toolbox to facilitate multi-voxel pattern classification of fMRI data. *(#699 T-AM ) Pattern classification of hippocampal shape analysis in a study of Alzheimer's Disease *(#697 M-AM ) Metric distances between hippocampal shapes predict different rates of shape changes in dementia of Alzheimer type and nondemented subjects: a validation study *(#721 M-PM ) Unbiased Diffeomorphic Shape and Intensity Template Creation: Application to Canine Brain *(#171 T-AM ) A Population-Average, Landmark- and Surface-based (PALS) Atlas of Human Cerebral Cortex *(#70 M-PM ) Cortical Folding Hypotheses: What can be inferred from shape? *(#714 T-AM ) Shape Analysis of Neuroanatomical Structures Based on Spherical Wavelets

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