Computational Anatomy: VBM and Alternatives

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 • 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. AIR: Automated Image Registrationhttp://bishopw.loni.ucla.edu/AIR5/ FLIRT: FMRIB’s Linear Image Registration Tool http://www.fmrib.ox.ac.uk/fsl/flirt/ MNI_AutoReg http://www.bic.mni.mcgill.ca/users/louis/MNI_AUTOREG_home/readme/ SPM http://www.fil.ion.ucl.ac.uk/spm VTK CISG Registration Toolkit http://www.image-registration.com/ ...and many others... Rigid Registration Software Packages

9. Only listing public software that can (probably) estimate detailed warps suitable for longitudinal analysis. HAMMER http://oasis.rad.upenn.edu/sbia/ MNI_ANIMAL Software Package http://www.bic.mni.mcgill.ca/users/louis/MNI_ANIMAL_home/readme/ SPM2 http://www.fil.ion.ucl.ac.uk/spm VTK CISG Registration Toolkit http://www.image-registration.com/ …there is much more software that is less readily available... Nonlinear Registration Software

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 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 2:244-252 (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 11:805-821 (2000). • J. Ashburner & K. J. Friston. Why Voxel-Based Morphometry Should Be Used. NeuroImage 14:1238-1243 (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 http://oasis.rad.upenn.edu/sbia/ http://www.fmrib.ox.ac.uk/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 a a0 c3 y3 b0 b Ca cI yI Cb

19. c1 y1 m g c2 y2 s2 a a0 c3 y3 b0 b Ca cI yI Cb Bias Field y yr(b) 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. c1 y1 m g c2 y2 s2 a a0 c3 y3 b0 b Ca cI yI Cb “Mixing Proportions”

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

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

24. group 1 group 2 Statistical Parametric Mapping… – parameter estimate standard error statistic image orSPM = voxel by voxelmodelling

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 14:1454-1462 (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 20:1425-1437 (2003). • N.A. Thacker.Tutorial:A Critical Analysis of Voxel-Based Morphometry.http://www.tina-vision.net/docs/memos/2003-011.pdf

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

28. 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):11050-11055 (2000). • S.E. Jones, B.R. Buchbinder & I. Aharon. Three-dimensional mapping of cortical thickness using Laplace's equation.Human Brain Mapping 11 (1): 12-32 (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):4146-4158 (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 • 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 Does not take account of variances and covariances m1

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 Support Vector

39. Going Nonlinear • 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 aixi • So y = f(Si aixiTx + w0) • Nonlinear classification is by y = f(Si ai(xi,x) + w0) • where (xi,x) is some function of xiand 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.