Voxel-Based Morphometry with Unified Segmentation

1
Voxel-Based Morphometry with Unified Segmentation

• Ged Ridgway
• Centre for Medical Image ComputingUniversity
  College London

Thanks to John Ashburner and the FIL Methods
Group.
2
Preprocessing in SPM

• Realignment
• With non-linear unwarping for EPI fMRI
• Slice-time correction
• Coregistration
• Normalisation
• Segmentation
• Smoothing

SPM8bs unified tissue segmentation and spatial
normalisation procedure
But first, an introduction toComputational
Neuroanatomy
3
Aims of computational neuroanatomy

• Many interesting and clinically important
  questions might relate to the shape or local size
  of regions of the brain
• For example, whether (and where) local patterns
  of brain morphometry help to
• Distinguish schizophrenics from healthy controls
• Understand plasticity, e.g. when learning new
  skills
• Explain the changes seen in development and aging
• Differentiate degenerative disease from healthy
  aging
• Evaluate subjects on drug treatments versus
  placebo

4
Alzheimers Disease example
Baseline Image Standard clinical MRI 1.5T T1
SPGR 1x1x1.5mm voxels
Repeat image 12 month follow-uprigidly
registered
Subtraction image
5
SPM for group fMRI
Group-wisestatistics
fMRI time-series
Preprocessing
Stat. modelling
Results query
spm TImage
ContrastImage
6
SPM for structural MRI
Group-wisestatistics
?
High-res T1 MRI
?
High-res T1 MRI
?
High-res T1 MRI
?
7
The need for tissue segmentation

• High-resolution MRI reveals fine structural
  detail in the brain, but not all of it reliable
  or interesting
• Noise, intensity-inhomogeneity, vasculature,
• MR Intensity is usually not quantitatively
  meaningful (in the same way that e.g. CT is)
• fMRI time-series allow signal changes to be
  analysed statistically, compared to baseline or
  global values
• Regional volumes of the three main tissue types
  gray matter, white matter and CSF, are
  well-defined and potentially very interesting

8
Examples ofsegmentation
GM and WM segmentations overlaid on original
images
Structural image, GM and WM segments, and
brain-mask (sum of GM and WM)
9
Segmentation basic approach

• Intensities are modelled by a Gaussian Mixture
  Model (AKA Mixture Of Gaussians)
• With a specified number of components
• Parameterised by means, variances and mixing
  proportions (prior probabilities for components)

10
Non-Gaussian Intensity Distributions

• Multiple MoG components per tissue class allow
  non-Gaussian distributions to be modelled
• E.g. accounting for partial volume effects
• Or possibility of deep GM differing from cortical
  GM

11
Tissue Probability Maps

• Tissue probability maps (TPMs) can be used to
  provide a spatially varying prior distribution,
  which is tuned by the mixing proportions
• These TPMs come from the segmented images of many
  subjects, done by the ICBM project

12
Class priors

• The probability of class k at voxel i, given
  weights ? is then
• Where bij is the value of the jth TPM at voxel i.

13
Aligning the tissue probability maps

• Initially affine-registered using a
  multi-dimensional form of mutual information
• Iteratively warped to improve the fit of
  theunified segmentationmodel to the data
• Familiar DCT-basisfunction concept, asused in
  normalisation

14
MRI Bias Correction

• MR Images are corupted by smoothly varying
  intensity inhomogeneity caused by magnetic field
  imperfections and subject-field interactions
• Would make intensity distribution spatially
  variable
• A smooth intensity correction can be modelled by
  a linear combination of DCT basis functions

15
Summary of the unified model

• SPM8b implements a generative model
• Principled Bayesian probabilistic formulation
• Combines deformable tissue probability maps with
  Gaussian mixture model segmentation
• The inverse of the transformation that aligns the
  TPMs can be used to normalise the original image
• Bias correction is included within the model

16
Segmentation clean-up

• Results may contain some non-brain tissue (dura,
  scalp, etc.)
• This can be removedautomatically usingsimple
  morphologicalfiltering operations
• Erosion
• Conditional dilation

Lower segmentationshave been cleaned up
17
Limitations of the current model

• Assumes that the brain consists of only GM and
  WM, with some CSF around it.
• No model for lesions (stroke, tumours, etc)
• Prior probability model is based on relatively
  young and healthy brains
• Less appropriate for subjects outside this
  population
• Needs reasonable quality images to work with
• No severe artefacts
• Good separation of intensities
• Good initial alignment with TPMs...

18
Extensions (possible or prototype)

• Multispectral modelling
• (New Segment Toolbox)
• Deeper Bayesian philosophy
• E.g. priors over means and variances
• Marginalisation of nuisance variables
• Model comparison
• Groupwise model (enormous!)
• Combination with DARTEL (see later and new seg
  tbx)
• More tissue priors e.g. deep grey, meninges, etc.
• Imaging physics
• See Fischl et al. 2004, as cited in AF
  introduction

19
Voxel-Based Morphometry

• In essence VBM is Statistical Parametric Mapping
  of segmented tissue density
• The exact interpretation of gray matter
  concentration or density is complicated, and
  depends on the preprocessing steps used
• It is not interpretable as neuronal packing
  density or other cytoarchitectonic tissue
  properties, though changes in these microscopic
  properties may lead to macro- or mesoscopic
  VBM-detectable differences

20
A brief history of VBM

• A Voxel-Based Method for the Statistical Analysis
  of Gray and White Matter Density Wright,
  McGuire, Poline, Travere, Murrary, Frith,
  Frackowiak and Friston. NeuroImage 2(4), 1995 (!)
• Rigid reorientation (by eye), semi-automatic
  scalp editing and segmentation, 8mm smoothing,
  SPM statistics, global covars.
• Voxel-Based Morphometry The Methods. Ashburner
  and Friston. NeuroImage 11(6 pt.1), 2000
• Non-linear spatial normalisation, automatic
  segmentation
• Thorough consideration of assumptions and
  confounds

21
A brief history of VBM

• A Voxel-Based Morphometric Study of Ageing Good,
  Johnsrude, Ashburner, Henson and Friston.
  NeuroImage 14(1), 2001
• Optimised GM-normalisation (a half-baked
  procedure), modulation of segments with Jacobian
  determinants
• Unified Segmentation. Ashburner and Friston.
  NeuroImage 26(3), 2005
• Principled generative model for segmentation
  usingdeformable priors
• A Fast Diffeomorphic Image Registration
  Algorithm. Ashburner. Neuroimage 38(1), 2007
• Large deformation normalisation to average shape
  templates

22
VBM overview

• Unified segmentation and spatial normalisation
• Optional modulation with Jacobian determinant
• Optional computation of tissue totals/globals
• Gaussian smoothing
• Voxel-wise statistical analysis

23
VBM in pictures
Segment Normalise
24
VBM in pictures
Segment Normalise Modulate (?) Smooth
25
VBM in pictures
Segment Normalise Modulate (?) Smooth Voxel-w
ise statistics
26
VBM in pictures
Segment Normalise Modulate (?) Smooth Voxel-w
ise statistics
27
VBM Subtleties

• Whether to modulate
• Adjusting for total GM or Intracranial Volume
• How much to smooth
• Limitations of linear correlation
• Statistical validity

28
Modulation
Native intensity tissue density

• Multiplication of the warped (normalised) tissue
  intensities so that their regional or global
  volume is preserved
• Can detect differences in completely registered
  areas
• Otherwise, we preserve concentrations, and are
  detecting mesoscopic effects that remain after
  approximate registration has removed the
  macroscopic effects
• Flexible (not necessarily perfect) registration
  may not leave any such differences

Modulated
Unmodulated
29
Globals for VBM

• Shape is really a multivariate concept
• Dependencies among volumes in different regions
• SPM is mass univariate
• Combining voxel-wise information with global
  integrated tissue volume provides a compromise
• Using either ANCOVA or proportional scaling

Above (ii) is globally thicker, but locally
thinner than (i) either of these effects may be
of interest to us.
Below The two cortices on the right both have
equal volume
Figures from Voxel-based morphometry of the
human brain Mechelli, Price, Friston and
Ashburner. Current Medical Imaging Reviews 1(2),
2005.
30
Total Intracranial Volume (TIV/ICV)

• Global integrated tissue volume may be
  correlated with interesting regional effects
• Correcting for globals in this case may overly
  reduce sensitivity to local differences
• Total intracranial volume integrates GM, WM and
  CSF, or attempts to measure the skull-volume
  directly
• Not sensitive to global reduction of GMWM
  (cancelled out by CSF expansion skull is
  fixed!)
• Correcting for TIV in VBM statistics may give
  more powerful and/or more interpretable results

31
Smoothing

• The analysis will be most sensitive to effects
  that match the shape and size of the kernel
• The data will be more Gaussian and closer to a
  continuous random field for larger kernels
• Results will be rough and noise-like if too
  little smoothing is used
• Too much will lead to distributed, indistinct
  blobs

32
Smoothing

• Between 7 and 14mm is probably best
• (lower is okay with better registration, e.g.
  DARTEL)
• The results below show two fairly extreme
  choices, 5mm on the left, and 16mm, right

33
Nonlinearity
Caution may be needed when looking for linear
relationships between grey matter concentrations
and some covariate of interest.
Circles of uniformly increasing area.
Plot of intensity at circle centres versus area
Smoothed
34
VBMs statistical validity

• Residuals are not normally distributed
• Little impact on uncorrected statistics for
  experiments comparing reasonably sized groups
• Probably invalid for experiments that compare
  single subjects or tiny groups with a larger
  control group
• Need to use nonparametric tests that make less
  assumptions, e.g. permutation testing with SnPM

35
VBMs statistical validity

• Correction for multiple comparisons
• RFT correction based on peak heights should be OK
• Correction using cluster extents is problematic
• SPM usually assumes that the smoothness of the
  residuals is spatially stationary
• VBM residuals have spatially varying smoothness
• Bigger blobs expected in smoother regions
• Toolboxes are now available for non-stationary
  cluster-based correction
• http//www.fmri.wfubmc.edu/cms/NS-General

36
VBMs statistical validity

• False discovery rate
• Less conservative than FWE
• Popular in morphometric work
• (almost universal for cortical thickness in FS)
• Recently questioned
• Topological FDR in SPM8
• See release notes for details and paper

37
Variations on VBM

• All modulation, no gray matter
• Jacobian determinant Tensor Based Morphometry
• Davatzikos et al. (1996) JCAT 2088-97
• Deformation field morphometry
• Cao and Worsley (1999) Ann Stat 27925-942
• Ashburner et al (1998) Hum Brain Mapp 6348-357
• Other variations on TBM
• Chung et al (2001) NeuroImage 14595-606

38
Deformation and shape change
Figures from Ashburner and Friston,
Morphometry, Ch.6of Human Brain Function, 2nd
Edition, Academic Press
39
Deformation fields and Jacobians
Deformation vector field
Template
Warped
Original
Determinant of Jacobian Matrixencodes voxels
volume change
Jacobian Matrix
40
Longitudinal VBM

• Intra-subject registration over time much more
  accurate than inter-subject normalisation
• Imprecise inter-subject normalisation
• Spatial smoothing required
• Different methods have been developed to reduce
  the danger of expansion and contraction
  cancelling out

41
Longitudinal VBM variations

• Voxel Compression mapping separates expansion and
  contraction before smoothing
• Scahill et al (2002) PNAS 994703-4707
• Longitudinal VBM multiplies longitudinal volume
  change with baseline or average grey matter
  density
• Chételat et al (2005) NeuroImage 27934-946

42
Longitudinal VBM variations
43
Nonrigid registration developments

• Large deformation concept
• Regularise velocity not displacement
• (syrup instead of elastic)
• Leads to concept of geodesic
• Provides a metric for distance between shapes
• Geodesic or Riemannian average mean shape
• If velocity assumed constant computation is fast
• Ashburner (2007) NeuroImage 3895-113
• DARTEL toolbox in SPM8b
• Currently initialised from unified seg_sn.mat
  files

44
DARTEL exponentiates a velocity flow field to get
a deformation field
Velocity flow field
45
Example geodesic shape average
Average on Riemannian manifold
Linear Average
(Not on Riemannian manifold)
46
DARTEL averagetemplate evolution
Grey matter average of 452 subjects affine
Iterations
471 subjects DARTEL
47
Further mathematical concepts

• Optimal transformations minimise the geodesic
• Variational problem, Euler-Lagrange equations
• Can derive a conservation of momentum law
• Initial momentum sparse deformation
  representation
• See the work of Miller, Younes, Beg, Marsland.

48
Questioning Intersubject normalisation

• Registration algorithms might find very different
  correspondences to human experts
• Crum et al. (2003) NeuroImage 201425-1437
• Higher dimensional warping improves image
  similarity but not necessarily landmark
  correspondence
• Hellier et al. (2003) IEEE TMI 221120-1130

49
Questioning Intersubject normalisation

• Subjects can have fundamentally different
  sulcal/gyral morphological variants
• Caulo et al. (2007) Am. J. Neuroradiol.
  281480-85
• Sulcal landmarks dont always match underlying
  cytoarchitectonics
• Amunts, et al. (2007) NeuroImage 37(4)1061-5

50
Intersubject normalisation opportunities

• High-field high-resolution MR may have potential
  to image cytoarchitecture
• Will registration be better or worse at higher
  resolution?
• More information to use
• More severe discrepancies?
• Need rougher deformations
• Non-diffeomorphic?

4.7T FSE
De Vita et al (2003) Br J Radiol 76631-7
51
Intersubject normalisation opportunities

• Regions of interest for fMRI can be defined from
  functional localisers or orthogonal SPM contrasts
• No obvious equivalent for single-subject
  structural MR
• Potential to include diffusion-weighted MRI
  information in registration ?
• Zhang et al. (2006) Med. Image Analysis
  10764-785

52
Summary of key points

• VBM performs voxel-wise statistical analysis on
  smoothed (modulated) normalised segments
• SPM8b performs segmentation and spatial
  normalisation in a unified generative model
• Intersubject correspondence is imperfect
• Smoothing alleviates this problem to some extent
• Also improves statistical validity
• Some current research is focussed on more
  sophisticated registration models

53
Unified segmentation in detail

• An alternative explanation to the paper and to
  Johns slides from London 07 http//www.fil.ion.u
  cl.ac.uk/spm/course/slides07/Image_registration.pp
  t

54
Unified segmentation from the GMM upwardsThe
standard Gaussian mixture model
Voxel i, class k
Assumes independence (but spatial priors later...)
Could solve with EM
(1-5)
55
Unified segmentation from the GMM
upwardsSpatially modify mean and variance with
bias field
Note spatial dependence (on voxel i),
coefficients for linear combination of DCT basis
functions
(10)
56
Unified segmentation from the GMM
upwardsAnatomical priors through mixing
coefficients
Note spatial dependence (on voxel i)
Basic idea
Implementation
prespecified
estimated
(12)
57
Unified segmentation from the GMM upwardsAside
MRF Priors (AF, Gasers VBM5 toolbox)
probable number of neighbours in class m, for
voxel i
(45)
58
Unified segmentation from the GMM
upwardsSpatially deformable priors (inverse of
normalisation)
Prior for voxel i depends on some general
transformation model, parameterised by a
Simple idea!
Optimisation is tricky
SPM8bs model is affine DCT warp With 1000 DCT
basis functions
(13)
59
Unified segmentation from the GMM
upwardsSpatially deformable priors (inverse of
normalisation)
(14, pretty much)
60
Unified segmentation from the GMM
upwardsObjective function so far
(14, I think...)
61
Unified segmentation from the GMM
upwardsObjective function with regularisation
Assumes priors independent
gives deformations bending energy
(15,16)
62
Unified segmentation from the GMM
upwardsOptimisation approach
Maximising
With respect to
is very difficult
Iterated Conditional Modes is used this
alternately optimises certain sets of parameters,
while keeping the rest fixed at their current
best solution
63
Unified segmentation from the GMM
upwardsOptimisation approach

• EM used for mixture parameters
• Levenberg Marquardt (LM) used for bias and
  warping parameters
• Note unified segmentation model with Gaussian
  assumptions has a least-squares like
  log(objective) making it ideal for Gauss-Newton
  or LM optimisation
• Local opt, so starting estimates must be good
• May need to manually reorient troublesome scans

64
Unified segmentation from the GMM
upwardsOptimisation approach
Figure from C. Gaser

• Repeat until convergence
• Hold ?, µ, s2 and a constant, and minimise E
  w.r.t. b
• Levenberg-Marquardt strategy, using dE/dß and
  d2E/dß2
• Hold ?, µ, s2 and ß constant, and minimise E
  w.r.t. a
• Levenberg-Marquardt strategy, using dE/da and
  d2E/da2
• Hold a and ß constant, and minimise E w.r.t. ?, µ
  and s2
• Expectation Maximisation

65
Note ICM steps
66
Results of the Generative model
Key flaw, lack of neighbourhood correlation
whiteness of noise Motivates (H)MRF priors,
which should encourage contiguous tissue
classes (Note, MRF prior is not equivalent to
smoothing each resultant tissue segment, but
differences in eventual SPMs may be minor)