1. Voxel-Based Morphometry with Unified Segmentation
Ged Ridgway
University College London
Thanks to:
John Ashburner and the FIL Methods Group.

2. Preprocessing in SPM Realignment Slice-time correction Coregistration
With non-linear unwarping for EPI fMRI
Slice-time correction
Coregistration
Normalisation
Segmentation
Smoothing
SPM8’s unified tissue segmentation and spatial normalisation procedure
But first, a brief introduction to Computational 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. Alzheimer’s Disease example
Baseline Image
Standard clinical MRI
1.5T T1 SPGR
1x1x1.5mm voxels
Repeat image
12 month follow-up rigidly registered
Subtraction image

5. Group-wise statistics
SPM for group fMRI
Group-wise statistics
fMRI time-series
Preprocessing
Stat. modelling
Results query
“Contrast” Image
spm T Image
fMRI time-series
Preprocessing
Stat. modelling
“Contrast” Image
Results query
fMRI time-series
Preprocessing
Stat. modelling
“Contrast” Image
Results query

6. Group-wise statistics
SPM for structural MRI
Group-wise statistics ?
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 of segmentation
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 γ 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 the unified segmentation model to the data
Familiar DCT-basis function concept, as used 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
SPM8 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 removed automatically using simple morphological filtering operations
Erosion
Conditional dilation
Lower segmentations have been cleaned up

17. The new segmentation toolbox
An extended work-in-progress algorithm
Multi-spectral
New TPMs including different tissues
Reduces problems in non-brain tissue
New more flexible warping of TPMs
More precise and more “sharp/contrasty” results

18. New Segmentation – TPMs
Segment button
New Seg Toolbox

19. New Segmentation – registration
Segment button
New Seg Toolbox
9*10*9 * 3 = 2430
59*70*59 * 3 =

20. New Segmentation – results
Segment button
New Seg Toolbox

21. Limitations of the current model
Assumes that the brain consists of only the tissues modelled by the TPMs
No allowance 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...

22. Possible future extensions
Deeper Bayesian philosophy
E.g. priors over means and variances
Marginalisation of nuisance variables
Model comparison, e.g. for numbers of Gaussians
Groupwise model (enormous!)
Combination with DARTEL (see later)
More tissue priors e.g. deep grey, meninges, etc.
Imaging physics
See Fischl et al. 2004, as cited in A&F introduction

23. 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

24. 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

25. 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 using deformable priors
A Fast Diffeomorphic Image Registration Algorithm. Ashburner. Neuroimage 38(1), 2007
Large deformation normalisation to average shape templates …

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

27. VBM in pictures
Segment
Normalise

28. VBM in pictures
Segment
Normalise
Modulate (?)
Smooth

29. VBM in pictures Segment Normalise Modulate (?) Smooth
Voxel-wise statistics

30. VBM in pictures Segment Normalise Modulate (?) Smooth
Voxel-wise statistics

31. VBM Subtleties Whether to modulate
Adjusting for total GM or Intracranial Volume
How much to smooth
Limitations of linear correlation
Statistical validity

32. Modulation
1 1
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
Unmodulated
Clarify, modulation not a step (as spm2) but an option in the segment and the normalise GUIs
Modulated
2/3 1/3 1/3 2/3

33. Modulation maths (simplified)
Consider a simple case, three discrete “voxel” locations, x in the set {1, 2, 3} warped to new locations, with three warps:
Translation to y=x+1 = {2, 3, 4}
No volume changes, gradient dy/dx = d(x+1)/dx = 1
Stretching to y=2x = {2, 4, 6}
Gradient dy/dx = d(2x)/dx = 2 = volume change ratio
E.g. Consider space from 2 to 3, mapped onto 4 to 6, hence doubling in size
Nonlinear (quadratic) y = x2 = {1, 4, 9}
Gradient dy/dx = d(x2)/dx = 2x
Space from 2 to 3 mapped onto 4 to 9, factor of 5
2x at central location between 2 and 3 of 2.5 is equal to 5
In 3D, we have (x1, x2, x3) and (y1, y2, y3), the gradient is a 3x3 (Jacobian) matrix with elements dyj/dxi
The (Jacobian) determinant of this matrix is the volume change ratio

34. “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…
Note globals don’t help distinguish the thickened or folded cortex...
Figures from: Voxel-based morphometry of the human brain… Mechelli, Price, Friston and Ashburner. Current Medical Imaging Reviews 1(2), 2005.

35. 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 GM+WM (cancelled out by CSF expansion – skull is fixed!)
Correcting for TIV in VBM statistics may give more powerful and/or more interpretable results
See also Pell et al (2009) doi: /j.neuroimage

36. 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

37. 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

38. 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

39. VBM’s 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

40. VBM’s 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

41. VBM’s 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, and Justin’s papers

42. Longitudinal VBM
The simplest method for longitudinal VBM is to use cross-sectional preprocessing, but longitudinal statistical analyses
Standard preprocessing not optimal, but unbiased
Non-longitudinal statistics would be severely biased
(Estimates of standard errors would be too small)
Simplest longitudinal statistical analysis: two-stage summary statistic approach (common in fMRI)
Within subject longitudinal differences or beta estimates from linear regressions against time

43. Longitudinal VBM variations
Intra-subject registration over time is much more accurate than inter-subject normalisation
Different approaches suggested to capitalise
A simple approach is to apply one set of normalisation parameters (e.g. Estimated from baseline images) to both baseline and repeat(s)
Draganski et al (2004) Nature 427:
“Voxel Compression mapping” – separates expansion and contraction before smoothing
Scahill et al (2002) PNAS 99:

44. Longitudinal VBM variations
Can also multiply longitudinal volume change with baseline or average grey matter density
Chételat et al (2005) NeuroImage 27:
Kipps et al (2005) JNNP 76:650
Hobbs et al (2009) doi: /jnnp
Note that use of baseline (or repeat) instead of average might lead to bias
Thomas et al (2009) doi: /j.neuroimage
Unfortunately, the explanations in this reference relating to interpolation differences are not quite right... there are several open questions here...

45. Longitudinal VBM variations
Late
Early
Warped early
Difference
Early CSF
Late CSF
Relative volumes
CSF “modulated” by relative volume
Late CSF - Early CSF
Late CSF - modulated CSF
Smoothed

46. 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 38:95-113
DARTEL toolbox in SPM8
Currently initialised from unified seg_sn.mat files

47. Motivation for using DARTEL
Recent papers comparing different approaches have favoured more flexible methods
DARTEL usually outperforms DCT normalisation
Also comparable to the best algorithms from other software packages (though note that DARTEL and others have many tunable parameters...)
Klein et al. (2009) is a particularly thorough comparison, using manual segmentations
Summarised in the next slide

48. Part of Fig.1 in Klein et al.

49. DARTEL exponentiates a velocity flow field to get a deformation field
Fig.3 in DARTEL paper

50. Fig.5 in DARTEL paper

51. Example geodesic shape average
Average on Riemannian manifold
Linear Average
(Not on Riemannian manifold)

52. DARTEL average template evolution1
Rigid average (Template_0)
60 images from OASIS cross-sectional data (as in VBM practical)
Average of mwc1 using segment/DCT
Template 6

53. Summary of key points
VBM performs voxel-wise statistical analysis on smoothed (modulated) normalised segments
SPM8 performs segmentation and spatial normalisation in a unified generative model
Based on Gaussian mixture modelling, with DCT warped spatial priors, and bias field
The new segment toolbox includes non-brain priors and more flexible/precise warping of them
Subsequent (currently non-unified) use of DARTEL improves normalisation for VBM

54. Unified segmentation in detail
An alternative explanation to the paper and to John’s slides from London ‘07

55. Unified segmentation from the GMM upwards… The standard Gaussian mixture model
Voxel i, class k
Assumes independence
(but spatial priors later...)
Could solve with EM
(1-5)

56. Unified segmentation from the GMM upwards… Spatially modify mean and variance with bias field
Note spatial dependence
(on voxel i), [coefficients for linear combination of DCT basis functions]
(10)

57. Unified segmentation from the GMM upwards… Anatomical priors through mixing coefficients
Note spatial dependence
(on voxel i)
Basic idea
Implementation
prespecified:
estimated:
(12)

58. Unified segmentation from the GMM upwards… Aside: MRF Priors (A&F, Gaser’s VBM5 toolbox)
probable number of neighbours
in class m, for voxel i
(45)

59. Unified segmentation from the GMM upwards… Spatially deformable priors (inverse of normalisation)
Prior for voxel i depends on
some general transformation
model, parameterised by α
Simple idea!
Optimisation
is tricky…
SPM8’s model is affine + DCT warp
With ~1000 DCT basis functions
(13)

60. Unified segmentation from the GMM upwards… Spatially deformable priors (inverse of normalisation)
(14, pretty much)

61. Unified segmentation from the GMM upwards… Objective function so far…
(14, I think...)

62. Unified segmentation from the GMM upwards… Objective function with regularisation
Assumes priors
independent
gives deformation’s
bending energy
(15,16)

63. Unified segmentation from the GMM upwards… Optimisation 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

64. Unified segmentation from the GMM upwards… Optimisation 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

65. Unified segmentation from the GMM upwards… Optimisation approach
Figure from C. Gaser
Repeat until convergence…
Hold γ, μ, σ2 and α constant, and minimise E w.r.t. b
Levenberg-Marquardt strategy, using dE/dβ and d2E/dβ2
Hold γ, μ, σ2 and β constant, and minimise E w.r.t. α
Levenberg-Marquardt strategy, using dE/dα and d2E/dα2
Hold α and β constant, and minimise E w.r.t. γ, μ and σ2
Expectation Maximisation

66. Note ICM steps

67. 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…)