Detecting connectivity between images: MS lesions, cortical thickness, and the 'bubbles' task in an

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Detecting connectivity between images MS
lesions, cortical thickness, and the 'bubbles'
task in an fMRI experiment

• Keith Worsley, Math Stats,
• Arnaud Charil, Montreal Neurological Institute,
  McGill
• Philippe Schyns, Fraser Smith, Psychology,
  Glasgow
• Jonathan Taylor,
• Stanford and Université de Montréal

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What is bubbles?
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Nature (2005)
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Subject is shown one of 40 faces chosen at random

Happy
Sad
Fearful
Neutral
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but face is only revealed through random
bubbles

• First trial Sad expression
• Subject is asked the expression Neutral
  
  
• Response
  Incorrect

75 random bubble centres
Smoothed by a Gaussian bubble
What the subject sees
Sad
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Your turn

• Trial 2

Subject response Fearful CORRECT
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Your turn

• Trial 3

Subject response Happy INCORRECT (Fearful)
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Your turn

• Trial 4

Subject response Happy CORRECT
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Your turn

• Trial 5

Subject response Fearful CORRECT
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Your turn

• Trial 6

Subject response Sad CORRECT
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Your turn

• Trial 7

Subject response Happy CORRECT
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Your turn

• Trial 8

Subject response Neutral CORRECT
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Your turn

• Trial 9

Subject response Happy CORRECT
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Your turn

• Trial 3000

Subject response Happy INCORRECT (Fearful)
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Bubbles analysis

• E.g. Fearful (3000/4750 trials)

Trial 1 2 3 4
5 6 7 750
Sum
Correct trials
Thresholded at proportion of correct
trials0.68, scaled to 0,1
Use this as a bubble mask
Proportion of correct bubbles (sum correct
bubbles) /(sum all bubbles)
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Results

• Mask average face
• But are these features real or just noise?
• Need statistics

Happy Sad
Fearful Neutral
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Statistical analysis

• Correlate bubbles with response (correct 1,
  incorrect 0), separately for each expression
• Equivalent to 2-sample Z-statistic for correct
  vs. incorrect bubbles, e.g. Fearful
• Very similar to the proportion of correct
  bubbles

ZN(0,1) statistic
Trial 1 2 3 4
5 6 7 750
Response 0 1 1 0
1 1 1 1
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Comparison

• Both depend on average correct bubbles, rest is
  constant

Proportion correct bubbles Average correct
bubbles / (average all bubbles 4)

• Z(Average correct bubbles
• average incorrect bubbles)
• / pooled sd

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Results

• Thresholded at Z1.64 (P0.05)
• Multiple comparisons correction?
• Need random field theory

ZN(0,1) statistic
Average face Happy Sad
Fearful Neutral
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Euler Characteristic blobs - holes
Excursion set Z gt threshold for neutral face
EC 0 0 -7 -11
13 14 9 1 0
Heuristic At high thresholds t, the holes
disappear, EC 1 or 0, E(EC) P(max Z gt
t).

• Exact expression for E(EC) for all thresholds,
• E(EC) P(max Z gt t) is extremely accurate.

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The details
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2
Tube(S,r)
r
S
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3

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B
A
25

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6
? is big
Tube?(S,r)
S
r
? is small
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2
?
U(s1)
s1
S
Tube
S
Tube
s2
s3
U(s3)
U(s2)
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Z2
R
r
Tube(R,r)
Z1
N2(0,I)
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Tube(R,r)
R
z
t-r
t
z1
Tube(R,r)
r
R
R
z2
z3
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Summary
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(No Transcript)
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Random field theory results

• For searching in D (2) dimensions, P-value of
  max Z is (Adler, 1981 W, 1995)
• P(max Z gt z)
• E( Euler characteristic of thresholded set )
• Resels Euler characteristic density (
  boundary)
• Resels (Lipschitz-Killing curvature/c) is
• Image area / (bubble FWHM)2 146.2
• Euler characteristic density(c) is
• (4 log(2))D/2 zD-1 exp(-z2/2) / (2p)(D1)/2
• See forthcoming book Adler, Taylor (2007)

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Results, corrected for search

• Thresholded at Z3.92 (P0.05)

ZN(0,1) statistic
Average face Happy Sad
Fearful Neutral
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Bubbles task in fMRI scanner

• Correlate bubbles with BOLD at every voxel
• Calculate Z for each pair (bubble pixel, fMRI
  voxel) a 5D image of Z statistics

Trial 1 2 3 4
5 6 7 3000
fMRI
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Discussion thresholding

• Thresholding in advance is vital, since we cannot
  store all the 1 billion 5D Z values
• Resels(image resels 146.2) (fMRI resels
  1057.2)
• for P0.05, threshold is Z 6.22 (approx)
• The threshold based on Gaussian RFT can be
  improved using new non-Gaussian RFT based on
  saddle-point approximations (Chamandy et al.,
  2006)
• Model the bubbles as a smoothed Poisson point
  process
• The improved thresholds are slightly lower, so
  more activation is detected
• Only keep 5D local maxima
• Z(pixel, voxel) gt Z(pixel, 6 neighbours of voxel)
• gt Z(4 neighbours of
  pixel, voxel)

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Discussion modeling

• The random response is Y1 (correct) or 0
  (incorrect), or YfMRI
• The regressors are Xjbubble mask at pixel j, j1
  240x38091200 (!)
• Logistic regression or ordinary regression
• logit(E(Y)) or E(Y) b0X1b1X91200b91200
• But there are only n3000 observations (trials)
  
• Instead, since regressors are independent, fit
  them one at a time
• logit(E(Y)) or E(Y) b0Xjbj
• However the regressors (bubbles) are random with
  a simple known distribution, so turn the problem
  around and condition on Y
• E(Xj) c0Ycj
• Equivalent to conditional logistic regression
  (Cox, 1962) which gives exact inference for b1
  conditional on sufficient statistics for b0
• Cox also suggested using saddle-point
  approximations to improve accuracy of inference
  
• Interactions? logit(E(Y)) or E(Y)b0X1b1X91200
  b91200X1X2b1,2

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MS lesions and cortical thickness

• Idea MS lesions interrupt neuronal signals,
  causing thinning in down-stream cortex
• Data n 425 mild MS patients
• Lesion density, smoothed 10mm
• Cortical thickness, smoothed 20mm
• Find connectivity i.e. find voxels in 3D, nodes
  in 2D with high
• correlation(lesion density, cortical thickness)
• Look for high negative correlations

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n425 subjects, correlation -0.568
Average cortical thickness
Average lesion volume
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Thresholding? Cross correlation random field

• Correlation between 2 fields at 2 different
  locations, searched over all pairs of locations
• one in R (D dimensions), one in S (E dimensions)
• sample size n
• MS lesion data P0.05, c0.325

Cao Worsley, Annals of Applied Probability
(1999)
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Normalization

• LDlesion density, CTcortical thickness
• Simple correlation
• Cor( LD, CT )
• Subtracting global mean thickness
• Cor( LD, CT avsurf(CT) )
• And removing overall lesion effect
• Cor( LD avWM(LD), CT avsurf(CT) )

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Histogram
threshold
threshold
Conditional histogram scaled to same max at
each distance
threshold
threshold
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Science (2004)
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fMRI activation detected by correlation between
subjects at the same voxel
The average nonselective time course across all
activated regions obtained during the first 10
min of the movie for all five subjects. Red line
represents the across subject average time
course. There is a striking degree of
synchronization among different individuals
watching the same movie.
Voxel-by-voxel intersubject correlation between
the source subject (ZO) and the target subject
(SN). Correlation maps are shown on unfolded left
and right hemispheres (LH and RH, respectively).
Color indicates the significance level of the
intersubject correlation in each voxel. Black
dotted lines denote borders of retinotopic visual
areas V1, V2, V3, VP, V3A, V4/V8, and estimated
border of auditory cortex (A1).The face-,
object-, and building-related borders (red, blue,
and green rings, respectively) are also
superimposed on the map. Note the substantial
extent of intersubject correlations and the
extension of the correlations beyond visual and
auditory cortices.
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What are the subjects watching during high
activation? Faces
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buildings
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hands
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Thresholding? Homologous correlation random field

• Correlation between 2 equally smooth fields at
  the same location, searched over all locations in
  R (in D dimensions)
• P-values are larger than for the usual
  correlation field (correlation between a field
  and a scalar)
• E.g. resels1000, df100, threshold5, usual
  P0.051, homologous P0.139

Cao Worsley, Annals of Applied Probability
(1999)
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Detecting Connectivity between Images the
'Bubbles' Task in fMRI

• Keith Worsley, McGill
• Phillipe Schyns, Fraser Smith, Glasgow

51
Subject is shown one of 40 faces chosen at random

Happy
Sad
Fearful
Neutral
but face is only revealed through random
bubbles

• E.g. first trial Sad expression
• Subject is asked the expression
  Neutral
  
• Response
  Incorrect0

75 random bubble centres
Smoothed by a Gaussian bubble
What the subject sees
Sad