Detecting%20Sparse%20Connectivity:%20MS%20Lesions,%20Cortical%20Thickness,%20and%20the%20

1
Detecting Sparse Connectivity MS Lesions,
Cortical Thickness, and the Bubbles Task in an
fMRI Experiment

• Keith Worsley, Nicholas Chamandy, McGill
• Jonathan Taylor, Stanford and Université de
  Montréal
• Robert Adler, Technion
• Philippe Schyns, Fraser Smith, Glasgow
• Frédéric Gosselin, Université de Montréal
• Arnaud Charil, Alan Evans, Montreal Neurological
  Institute

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Three examples of spatial point data
Multiple Sclerosis (MS) lesions
Galaxies
Bubbles
I hope to show the connections
3
Astrophysics
??? ??
4
Sloan Digital Sky Survey, data release 6, Aug. 07
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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
10
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
13
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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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 Heuristic
Euler characteristic (EC) blobs - holes (in
2D) Excursion set Xt s Z(s) t, e.g. for
neutral face
EC 0 0 -7 -11
13 14 9 1 0
30

Heuristic At high thresholds t, the holes
disappear, EC 1 or 0, E(EC) P(max Z
t).
Observed
Expected
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10
EC(Xt)
0

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

-10
-20

-4
-3
-2
-1
0
1
2
3
4
Threshold, t
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The result
Lipschitz-Killing curvatures of S (Resels(S)c)
EC densities of Z above t
filter
white noise
Z(s)


FWHM
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Results, corrected for search

• Random field theory threshold Z3.92 (P0.05)
• 3.82 3.80 3.81
  3.80
• Saddle-point approx (2007) Z? (P0.05)
• Bonferroni Z4.87 (P0.05) nothing

ZN(0,1) statistic
Average face Happy Sad
Fearful Neutral
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Scale space smooth Z(s) with range of filter
widths w continuous wavelet transform adds an
extra dimension to the random field Z(s,w)
Scale space, no signal
34
8
22.7
6
4
15.2
2
10.2
0
-2
6.8
-60
-40
-20
0
20
40
60
w FWHM (mm, on log scale)
One 15mm signal
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8
22.7
6
4
15.2
2
10.2
0
-2
6.8
Z(s,w)
-60
-40
-20
0
20
40
60
s (mm)
15mm signal is best detected with a 15mm
smoothing filter
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Matched Filter Theorem ( Gauss-Markov Theorem)
to best detect signal white noise, filter
should match signal
10mm and 23mm signals
34
8
22.7
6
4
15.2
2
10.2
0
-2
6.8
-60
-40
-20
0
20
40
60
w FWHM (mm, on log scale)
Two 10mm signals 20mm apart
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8
22.7
6
4
15.2
2
10.2
0
-2
6.8
Z(s,w)
-60
-40
-20
0
20
40
60
s (mm)
But if the signals are too close together they
are detected as a single signal half way between
them
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Scale space can even separate two signals at the
same location!
8mm and 150mm signals at the same location
10
5
0
-60
-40
-20
0
20
40
60
170
20
76
15
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w FWHM (mm, on log scale)
10
15.2
5
6.8
Z(s,w)
-60
-40
-20
0
20
40
60
s (mm)
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The result
Lipschitz-Killing curvatures of S (Resels(S)c)
EC densities of Z above t
filter
white noise
Z(s)


FWHM
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Random field theory for scale-space
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Rotation spaceTry all rotated elliptical
filters
Unsmoothed fMRI data T stat for visual stimulus
Threshold Z5.25 (P0.05)
Maximum filter
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The result
Lipschitz-Killing curvatures of S (Resels(S)c)
EC densities of Z above t
filter
white noise
Z(s)


FWHM
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Theorem (1981, 1995)
Example the chi-bar random field, a special case
of a random field of test statistics for the
magnitude of an fMRI response in the presence of
unknown delay of the hemodynamic response
function.
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Example chi-bar random field
Z1N(0,1)
Z2N(0,1)
s2
s1
Rejection regions,
Excursion sets,
Threshold t
Z2
Search Region, S
Z1
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Adler Taylor (2007), Ann. Math, (submitted)
Beautiful symmetry
Steiner-Weyl Tube Formula (1930)
Taylor Gaussian Tube Formula (2003)

• Put a tube of radius r about the search region
  ?S and rejection region Rt

Z2N(0,1)
Rt
r
Tube(Rt,r)
Tube(?S,r)
r
?S
Z1N(0,1)
t
t-r

• Find volume or probability, expand as a power
  series in r, pull off coefficients

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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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Thresholding? Cross correlation random field

• Correlation between 2 fields at 2 different
  locations,
• searched over all pairs of locations, one in S,
  one in T
• Bubbles data P0.05, n3000, c0.113, T6.22

Cao Worsley, Annals of Applied Probability
(1999)
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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

5.5
5
4.5
4
Average cortical thickness (mm)
3.5
3
2.5
Correlation -0.568, T -14.20 (423 df)
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Charil et al, NeuroImage (2007)
1.5
0
10
20
30
40
50
60
70
80
Total lesion volume (cc)
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MS lesions and cortical thickness at all pairs of
points

• Dominated by total lesions and average cortical
  thickness, so remove these effects as follows
• CT cortical thickness, smoothed 20mm
• ACT average cortical thickness
• LD lesion density, smoothed 10mm
• TLV total lesion volume
• Find partial correlation(LD, CT-ACT) removing TLV
  via linear model
• CT-ACT 1 TLV LD
• test for LD
• Repeat for all voxels in 3D, nodes in 2D
• 1 billion correlations, so thresholding
  essential!
• Look for high negative correlations
• Threshold P0.05, c0.300, T6.48

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Cluster extent rather than peak height (Friston,
1994)

• Choose a lower level, e.g. t3.11 (P0.001)
• Find clusters i.e. connected components of
  excursion set
• Measure cluster
• extent by resels
• Distribution
• fit a quadratic
• to the peak
• Distribution of maximum cluster extent
• Find distribution for a single cluster
• Bonferroni on N clusters E(EC).

Z
D1
extent
t
Peak height
s
Cao and Worsley, Advances in Applied Probability
(1999)
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Three examples of spatial point data
Multiple Sclerosis (MS) lesions
Galaxies
Bubbles
I hope I have shown the connections ...
?? ?????? !