Computational Anatomy: VBM and

Alternatives

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

Overview* Volumetric differences

* Serial Scans* Jacobian Determinants

* Voxel-based Morphometry* Multivariate Approaches* Difference Measures* Another approach

TemplateTemplateWarpedOriginal

Deformation Field

Deformation field

Jacobians

Jacobian Matrix (or just “Jacobian”)

Jacobian Determinant (or just “Jacobian”) - relative volumes

Early

Late

Difference

Data from the Dementia Research Group, Queen Square.

Serial Scans

Regions of expansion and contraction

* Relative volumes encoded in Jacobian determinants.

Rigid Registration Software Packages* AIR: Automated Image Registration

http://bishopw.loni.ucla.edu/AIR5/

* FLIRT: FMRIB’s Linear Image Registration Toolhttp://www.fmrib.ox.ac.uk/fsl/flirt/

* MNI_AutoReghttp://www.bic.mni.mcgill.ca/users/louis/MNI_AUTOREG_home/readme/

* SPMhttp://www.fil.ion.ucl.ac.uk/spm

* VTK CISG Registration Toolkithttp://www.image-registration.com/

...and many others...

Nonlinear Registration SoftwareOnly listing public software that can (probably) estimate detailed

warps suitable for longitudinal analysis.

* HAMMERhttp://oasis.rad.upenn.edu/sbia/

* MNI_ANIMAL Software Packagehttp://www.bic.mni.mcgill.ca/users/louis/MNI_ANIMAL_home/readme/

* SPM2http://www.fil.ion.ucl.ac.uk/spm

* VTK CISG Registration Toolkithttp://www.image-registration.com/

…there is much more software that is less readily available...

Late Early

Warped early Difference

Early CSFLate CSF

Relative volumesCSF “modulated” by

relative volumes

Late CSF - Early CSF Late CSF - modulated CSF

Smoothed

Smoothing

Before convolution Convolved 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).

Overview* Volumetric differences* Voxel-based Morphometry

* Method* Interpretation Issues

* Multivariate Approaches* Difference Measures* Another approach

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).

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/

Pre-processing for Voxel-Based Morphometry (VBM)

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

Mixture of Gaussiansy1c1

y2

y3

c2

c3

C

CyIcI

Bias Field

()y y ()

y1c1

y2

y3

c2

c3

C

CyIcI

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.

“Mixing Proportions”

y1c1

y2

y3

c2

c3

C

CyIcI

Deforming the Tissue Probability Maps* Tissue probability

maps are deformed according to parameters .

y1c1

y2

y3

c2

c3

C

CyIcI

Warped, Modulated Grey Matter 12mm FWHM Smoothed Version

SPM5b Pre-processed data for four subjects

Statistical Parametric Mapping…

gCBF

rCBF

x

o

o

o

o

o

o

x

x

x

x

x

g..

k1

k2

k

group 1 group 2

voxel by voxelmodelling

–

parameter estimate standard error

=

statistic imageor

SPM

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.

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

Some Explanations of the Differences

ThickeningThinning

Folding

Mis-classify

Mis-classify

Mis-register

Mis-register

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).

Overview* Volumetric differences* Voxel-based Morphometry* Multivariate Approaches

* Scan Classification* Cross-Validation

* Difference Measures* Another approach

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.

“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?

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

Training and Classifying

ControlTraining Data

PatientTraining Data

?

?

??

Classifying

Controls

Patients

?

?

??

y=f(wTx+w0)

Difference between means

Does not take account of variances and covariances

m1

m2

w m2-m1

Fisher’s Linear Discriminant

Curse of dimensionality !

w S-1(m2-m1

)

Support Vector Classifier (SVC)

Support Vector Classifier (SVC)

SupportVector

SupportVector

Support

Vector

w is a weighted linear combination of the support vectors

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 = i ai xi

* So y = f(i ai xiTx + w0)

* Nonlinear classification is by

y = f(i ai (xi,x) + w0)

* where (xi,x) is some function of xi and x.

* e.g. RBF classification (xi,x) = exp(-||xi-x||2/(22))

Nonlinear SVC

Over-fitting

Test data

A simpler model can often do better...

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

Two-fold Cross-validation

Use half the data for training.

and the other half for testing.

Two-fold Cross-validation

Then swap around the training and test data.

Leave One Out Cross-validation

Use all data except one point for training.

The one that was left out is used for testing.

Leave One Out Cross-validation

Then leave another point out.

And so on...

Regression (e.g. against age)

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

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

* Derived from Deformations* Derived from Deformations + Residuals

* Another approach

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.

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, 4:375-405 (2003) plus supplement• Beg, Miller, Trouvé, L. Younes. “Computing Large Deformation Metric Mappings via Geodesic Flows of Diffeomorphisms”. Int. J. Comp. Vision, 61:1573-1405 (2005)

Tilak Ratnanather gave me the following two slides…

Computing the geodesic: problem statement

2

0 1

0 1

11

0

11

Problem Statement: Given and , compute such that

arg min ( , ), ( , ) d d ( ( ,1)) d

where ( , ) ( , ) ( , )

v

ty

I I v

Lv y t Lv y t y t I y I y

y t y t v y tt

I0: Template I1:Target

1.386Young 2.541 3.696 4.620 Schizophrenia

3D Hippocampus: Young to Schizophrenia

3D Hippocampus: Young to Alzheimer’s

Young4.766 Alzheimer’s

Data from the lab. of Dr. Csernansky, Washington University, St Louis.

Metrics on 3D Hippocampus in Neuro-psychiatric Disorders.

3.8132.6211.430

Accuracy of Automated Volumetric Inter-subject Registration

Sulcal misregistration

0

2

4

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12

A D M P R SPM2Method

Dis

tanc

e (

mm

)

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)

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

http://www.messybeast.com/freak-face.htm

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.

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

Anatomist/BrainVISA Framework* Free software available from:

http://brainvisa.info/* 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):968-982 (2004)

Design of an artificial neuroanatomist

3Dretina

Bottom-upflow

Fields ofview of

neural nets

Elementaryfolds

Sulci

Correlates of handedness14 subjects 128 subjects

Central sulcussurface is larger

in dominant hemisphere

Handedness correlates : localization after affine normalization

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