NITRC Meeting 2008
Improving the Efficiency of Statistical Map Creation and
Assessment
Valerie A. Cardenas
Center for Imaging of Neurodegenerative Disease
San Francisco Veterans Affairs Medical Center
University of California, San Francisco
NITRC Meeting 2008
Challenge
• Clinical studies aim to describe effect of disease/treatment on brain structure
• Where to look for effects?• Anatomic variability• Manual methods: time consuming, rater error• Goal: automatically measure differences, look
everywhere, account for anatomic variability
NITRC Meeting 2008
Deformation Morphometry
• Automated• Suited for discerning patterns of structural change• Explore location and extent of variation• Use nonlinear registration or “warping” of images
– Within: capture changes in brain over time– Between: measure deviation from atlas brain
• Create high resolution maps of local tissue volume or tissue volume change
• Model variability using many clinical variables
NITRC Meeting 2008
Creating Deformation MapsStep 1: Nonlinear Registration Step 2: Determinant of Jacobian Matrix at
each voxel, giving the pointwise volume change at each point
T(x1,y1,z1)V2V1
1 1 1
2 2 2
1 1 1 11 1 1
2 2 2 2
1 1 1
2 2 2
( , , )
dx dx dx
dx dy dz
dy dy dy VJ x y z
dx dy dz V
dz dz dz
dx dy dz
Maps with 1-2 million voxels
NITRC Meeting 2008
Statistical Model
y11y12
y13 y14
y21y22
y23 y24
yn1yn2
yn3 yn4
Map 1;diagnosis 0
age 65 score 16
Map 2;diagnosis 1
age 68 score 8
Map n;diagnosis 1
age 73 score 4
2int
2
2
2
2
22
12
1int
1
1
1
1
21
11
14731
18681
116650
14731
18681
116650
x
x
x
x
y
y
y
x
x
x
x
y
y
y
score
age
diag
n
score
age
diag
n
xdiag1xdisg2
xdiag3xdiag4
tdiag1tdiag2
tdiag3tdiag4
coefficient maps for each variable
statistic maps for each variable
NITRC Meeting 2008
The Multiplicity Problem
• Map formed of ~1-2 million statistics
• Measurements of volume change and statistics are not independent, due to– initial image resolution– spatial transformation– smoothing
• Bonferroni procedures too stringent
NITRC Meeting 2008
Corrections for Multiple Comparisons
• Cluster analysis– Developed for PET and fMRI analyses– Stationarity/smoothness assumptions violated in
deformation morphometry– Nonstationary methods valid for some problems
• Permutation testing– Build a null distribution
• Create statistic map using permuted labels 1000-10000 times• Need efficient computation here!!
– Compare statistic to distribution to assess significance
NITRC Meeting 2008
Ordinary Least Squares
• y: n1 observations, subjects• A: np independent variables• Solution valid if ATA full-rank
• x: p1 regression coefficients• e: n1 residuals
2
-1
( ) ( ) , min min ( ) ( )
( ) ( )
Ti i i i
T Ti i
v v v v
v v
x x
y Ax e e e y - Ax
x = (A A) A y
NITRC Meeting 2008
Computation
• Compute (ATA)-1AT, solve for estimates x at each voxel
• More efficient to use matrix decomposition– Cholesky decomposition: ATA=LLT
• Lb(vi)=ATy(vi)
• b(vi)=LTx(vi)
• L lower triangular so easy to solve
– L is computed from left to right and top to bottom!
NITRC Meeting 2008
Cholesky Decomposition: Advantage with A(vi)
1 1
2 2
3 3
1 2 1 2
1 1 1 2 11 11 21 31
1 2 2 2 21 22 22 32
1 3 2 3 31 32 3
3
3 33
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
p p
p p
p p
p p p p p p p pp pp
c c L
c c L
c c L
c c c c c c L L L L
c c c c L L L L
c c c c L L L L
c c c c
L
L L L L
To calculate Lpj, need only last row of ATA and previously computed Lij. Most of L can be computed once, only update last row at each voxel.
NITRC Meeting 2008
Limited RAM: Slice at a Time
i<slices j=0 j<subj
Compute x
i=0 j++
i++
Readimage
F
T T
F
i<slices j=0 j<subj
Compute SSE, t and F statistics
i=0 j++
i++
Readimage
F
T T
F
Assume 80 subjects, 100 slices, disk accessed 16000 times!
NITRC Meeting 2008
2+ Gb of RAM: Image in Memory
i<subj
Compute x
i=0 i++Readimage
F
T
Compute SSE, t and F statistics
Assume 80 subjects, 100 slices, disk accessed 80 times!1000 permutations, many days -> 10 minutes!Within 2 Gb can run 100 subjects, 138x148x115 shorts
NITRC Meeting 2008
num_vox/2<I<num_vox
Voxel Estimates and Statisticsin Parallel
• Dual- and quad-core processors common
• Voxel estimates and statistics independent
• Also possible to run in parallel
Create statistic maps
Compute SSE, t and F statistics i++
i<num_vox/2 i++Compute
SSE, t and F statistics
T
T
F
F
CPU 1
CPU 2
NITRC Meeting 2008
Permutation Testing in Parallel• Permutations independent
• Possible to run in parallel
Compute p-valuesfrom distribution
Permutelabelsi<500
Compute permuted SSE, t and F statistics
i++; add to distribution
Permutelabelsi<500
i++; add to distribution
Compute permuted SSE, t and F statistics
T
T
F
F
CPU 1
CPU 2
NITRC Meeting 2008
Summary
• Need fast computation for morphometry
• Several easy improvements– Matrix decomposition– Images in memory– Estimates and statistics in parallel– Permutations in parallel
• Any other suggestions?
NITRC Meeting 2008
Thanks to:
• Colin Studholme
• Mike Weiner
• Clinical collaborators at CIND and UCSF
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