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