Title: Detecting Sparse Connectivity: MS Lesions, Cortical Thickness, and the
1Detecting 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
2Three examples of spatial point data
Multiple Sclerosis (MS) lesions
Galaxies
Bubbles
I hope to show the connections
3Astrophysics
??? ??
4Sloan Digital Sky Survey, data release 6, Aug. 07
5What is bubbles?
6Nature (2005)
7Subject is shown one of 40 faces chosen at
random
Happy
Sad
Fearful
Neutral
8 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
9Your turn
Subject response Fearful CORRECT
10Your turn
Subject response Happy INCORRECT (Fearful)
11Your turn
Subject response Happy CORRECT
12Your turn
Subject response Fearful CORRECT
13Your turn
Subject response Sad CORRECT
14Your turn
Subject response Happy CORRECT
15Your turn
Subject response Neutral CORRECT
16Your turn
Subject response Happy CORRECT
17Your turn
Subject response Happy INCORRECT (Fearful)
18Bubbles 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)
19Results
- Mask average face
- But are these features real or just noise?
- Need statistics
Happy Sad
Fearful Neutral
20Statistical 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
21Results
- Thresholded at Z1.64 (P0.05)
- Multiple comparisons correction?
- Need random field theory
ZN(0,1) statistic
Average face Happy Sad
Fearful Neutral
22Euler 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
20
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
23The result
Lipschitz-Killing curvatures of S (Resels(S)c)
EC densities of Z above t
filter
white noise
Z(s)
FWHM
24Results, 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
25Scale 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
34
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
26Matched 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
34
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
27Scale 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
34
w FWHM (mm, on log scale)
10
15.2
5
6.8
Z(s,w)
-60
-40
-20
0
20
40
60
s (mm)
28The result
Lipschitz-Killing curvatures of S (Resels(S)c)
EC densities of Z above t
filter
white noise
Z(s)
FWHM
29Random field theory for scale-space
30Rotation spaceTry all rotated elliptical
filters
Unsmoothed fMRI data T stat for visual stimulus
Threshold Z5.25 (P0.05)
Maximum filter
31The result
Lipschitz-Killing curvatures of S (Resels(S)c)
EC densities of Z above t
filter
white noise
Z(s)
FWHM
32Theorem (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.
33Example chi-bar random field
Z1N(0,1)
Z2N(0,1)
s2
s1
Rejection regions,
Excursion sets,
Threshold t
Z2
Search Region, S
Z1
34Adler 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
35Bubbles 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
36Thresholding? 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)
37MS 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)
2
Charil et al, NeuroImage (2007)
1.5
0
10
20
30
40
50
60
70
80
Total lesion volume (cc)
38MS 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
39Cluster 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)
40Three examples of spatial point data
Multiple Sclerosis (MS) lesions
Galaxies
Bubbles
I hope I have shown the connections ...
?? ?????? !