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Computational Anatomy: VBM and Alternatives

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1 Computational Anatomy: VBM and Alternatives

2 Motivation for Computational Anatomy
See Wednesday’s 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

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

4 Deformation Field Original Warped Template Deformation field

5 Jacobians Jacobian Matrix (or just “Jacobian”)
Jacobian Determinant (or just “Jacobian”) - relative volumes

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

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

8 Rigid Registration Software Packages
AIR: Automated Image Registration FLIRT: FMRIB’s Linear Image Registration Tool MNI_AutoReg SPM VTK CISG Registration Toolkit ...and many others...

9 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...

10 Late Early Late CSF Early CSF CSF “modulated” by relative volumes Warped early Difference Relative volumes

11 Late CSF - modulated CSF
Late CSF - Early CSF Late CSF - modulated CSF Smoothed

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

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

14 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).

15 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

16 Pre-processing for Voxel-Based Morphometry (VBM)

17 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

18 Mixture of Gaussians c1 y1 m g c2 y2 s2 a0 a c3 y3 b b0 Ca cI yI Cb

19 Bias Field y r(b) y r(b) y1 c1 g a y2 y3 c2 c3 m s2 b a0 Ca b0 Cb yI
cI y r(b) y r(b)

20 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.

21 “Mixing Proportions” y1 c1 g a y2 y3 c2 c3 m s2 b a0 Ca b0 Cb yI cI

22 Deforming the Tissue Probability Maps
Tissue probability maps are deformed according to parameters a. y1 c1 g a y2 y3 c2 c3 m s2 b a0 Ca b0 Cb yI cI

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

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

25 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.

26 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.

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

28 Cortical Thickness Mapping
Direct measurement of cortical thickness may be better for studying neuro-degenerative diseases 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 Alzheimer’s 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).

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

30 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, 2004. C. Davatzikos. Why voxel-based morphometric analysis should be used with great caution when characterizing group differences. NeuroImage 23(1):17-20, 2004. K. J. Friston and J. Ashburner. Generative and recognition models for neuroanatomy. NeuroImage 23(1):21-24, 2004.

31 “Globals” for VBM Shape is multivariate SPM is mass univariate
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?

32 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

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

34 Classifying ? Controls ? ? ? Patients y=f(wTx+w0)

35 Difference between means
w  m2-m1 m2 m1 Does not take account of variances and covariances

36 Fisher’s Linear Discriminant
w  S-1(m2-m1 ) Curse of dimensionality !

37 Support Vector Classifier (SVC)

38 Support Vector Classifier (SVC)
w is a weighted linear combination of the support vectors Support Vector Support Vector

39 Going Nonlinear y = f(Si ai (xi,x) + w0)
Linear classification is by y = f(wTx + w0) where w is a weighting vector, x is the test data, w0 is an offset, and f(.) is a thresholding operation w is a linear combination of SVs w = Si ai xi So y = f(Si ai xiTx + w0) Nonlinear classification is by y = f(Si ai (xi,x) + w0) where (xi,x) is some function of xi and x. e.g. RBF classification (xi,x) = exp(-||xi-x||2/(2s2))

40 Nonlinear SVC

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

42 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

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

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

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

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

47 Regression (e.g. against age)

48 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

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

50 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.

51 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…

52 Computing the geodesic: problem statement
I0: Template I1:Target By shifting focus from the diffeomorphism phi to the velocity field that generates it places the problem in the class of an Optimization problem where the estimate of the velocity field desired to generate the diffeomorphism phi that we are after is found at the minimum of this cost function. The first the measure the smoothness properties of the velocity field, this is necessary to ensure that the solutions to the ODE and PDE governing the dynamics of the map are diffeomorphisms. The second term measure the amount of mis-match in the given images under the transformation that this velocity field generates.

53 Metrics on 3D Hippocampus in Neuro-psychiatric Disorders.
Data from the lab. of Dr. Csernansky, Washington University, St Louis. 3D Hippocampus: Young to Schizophrenia Young 1.386 2.541 3.696 Schizophrenia 4.620 3D Hippocampus: Young to Alzheimer’s Young Alzheimer’s 1.430 2.621 3.813 4.766

54 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)

55 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

56 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.

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

58 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)

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

60 Correlates of handedness
14 subjects 128 subjects Central sulcus surface is larger in dominant hemisphere

61 Handedness correlates : localization after affine normalization

62 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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