FADTTS: Functional Analysis of Diffusion
Tensor Tract Statistics
Hongtu Zhu, Ph.D.
Department of Biostatistics and
Biomedical Research Imaging Center,
University of North Carolina at Chapel Hill
The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Outline
Motivation
Multivariate Varying Coefficient Models
Simulation Studies
Real Data Analysis
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Motivation
Functional
Connectivity
EEG, fMRI, resting fMRI
Structural
Connectivity
Anatomical MRI, DTI (HARDI)
group 1 group 2
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Motivation
Neonatal Brain Development
PI: John H. Gilmore.
www.google.com
Knickmeyer RC, et al. J Neurosci, 2008 28: 12176-12182.
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Motivation
Early Brain Development
2 week
1 year
2 year
Knickmeyer RC, et al. J Neurosci, 2008 28: 12176-12182.
The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Motivation
Diffusion Tensor Tract Statistics
FA
2 week
1 year
Tensor
2 year
2 week
1 year
2 year
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Motivation
Macaque Brain Development
Casey, B.J. et al. TRENDS in Cognitive Sciences, 2005 9(3): 104-110.
PI: Martin Styner
& Marc Niethammer.
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Motivation
Casey, B.J. et al. TRENDS in Cognitive Sciences, 2005 9(3): 104-110.
The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Motivation
Casey, B.J. et al. TRENDS in Cognitive Sciences, 2005 9(3): 104-110.
The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Functional Analysis of Diffusion Tensor Tract Statistics
Data
• Diffusion properties (e.g., FA, RA)
Yi (s j )  (yi,1(s j ),L , y i,m (s j ))T
• Grids {s1,L ,snG }
(e)
• Covariates (e.g., age, gender, diagnostic)
FA

x1,L , x n

MD


1

2

3

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FADTTS
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
Multivariate Varying Coefficient Model
Decomposition:
y i,k (s)  x Ti Bk (s)  i,k (s)  i,k (s)
Varying Coefficients
x1,L , x n
Low Frequency Signal
i,k () ~ SP(0, )
High Frequency Noise
 i,k () ~ SP(0, ),
 (s,s')   (s,s)1(s  s')
Covariance operator:
 y (s,s')   (s,s')   (s,s)1(s  s')

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Weighted Least Squares Estimate
n
nG
minBk (s)  K h (s  s j )[y i,k (s j )  x Ti Bk (s j )]2
i1 j 1

L
n{vec(Bˆ (s)  B(s)  0.5O(H 2 )): s [0,L0 ]} 

G(0, (s,s') 1
X )
Key Advantage
Low Frequency Signal
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Functional Principal Component Analysis
Smooth individual functions
nG
min i ,k (s) K h (s j  s)[y i,k (s j )  x Ti Bˆ k (s j )  i,k (s j )]2
j 1
Estimated covariance operator
n
ˆ ( s,t)  
ˆ i (s)
ˆ i (t)T


i1
Estimated eigenfunctions
ˆ ,
ˆ k,l (s)): l 1,L ,}
{(
k,l


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Statistical Inferences
Testing Linear Hypotheses
H 0 : Cvec( B(s)) = b0 (s) versus H1 : Cvec( B(s))  b0 (s)
Grid Point
Whole Tract
T 1
Sn (s j ) nd(s j )T [C( (s j ,s j ) 1
)C
] d(s j ) Global Test Statistics
X
L0
T 1
Sn  n  d(s)T [C( (s,s)  1
)C
] d(s)ds
X
Local Test Statistics
0
K
2
Sn (s j )   (m)
and
S

w



n
k k (1),
2
k
k 1

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Confidence Band
Asymptotics
n[bˆk,l (s) - bk,l (s) - bias(bˆk,l (s))] Gk,l ()
Critical point

P(sups[0,L0 ] | Gk,l (s) | Ck,l ( )) =1- 
Confidence band

Ck,l ( ) ˆ
Ck,l ( )
ˆ
(bk,l (s) , bk,l (s) +
)
n
n

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Comparisons
Pros
•
•
•
•
•
Directly smooth varying coefficient functions
Explicitly account for functional nature of tract statistics
Characterize low frequency signal
Drop high frequency noise
Increase statistical power
Cons
•
•
Complicated asymptotic results
Computationally intensive
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Simulation Studies
Model
Setting (13 (s), 23 (s))  c(ˆ13 (s), ˆ23 (s))
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Simulation Studies
Testing
H 0 : (13 (s), 23 (s))  (0,0)
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Power Comparison between GLM and FADTTS
n  64,   0.01

n  64,   0.05

n 128,   0.01

n 128,   0.05

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Real Data Analysis
Early Brain Development
•
Casey, B.J. et al. TRENDS in Cognitive Sciences, 2005 9(3): 104-110.
The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Real Data Analysis
Splenium


2
1
128 subjects 
MD
FA

3

Diffusion properties = Gender + Gestational age
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Real Data Analysis
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Local P-values
FA

MD

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Confidence Bands
Intercept
Gender
Age
FA
MD
1

2

3
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Functional Principal Component Analysis
Eigenvalues
1


FA
MD
2
3

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FADTTS GUI Toolbox
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FADTTS GUI Toolbox
Input: Raw data and test data.
•
Raw data include tract data, design data and diffusion
data.
•
Test data include test matrix and vector.
•
All data is in .mat format.
Output: Basic plots and P-value plots
•
Basic plots include diffusion plot, coefficient plot,
eigenvalue and eigenfunction plot, confidence band plot.
•
P-value plot include local p-value (in –log10 scale) plot
with global p-value.
Download: FADTTS GUI Toolbox with related documents and
sample data is free to download from
http://www.nitrc.org/projects/fadtts/
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Summary
• From the statistical end, we have developed a new
functional analysis pipeline for delineating the structure of
the variability of multiple diffusion properties along major
white matter fiber bundles and their association with a set of
covariates of interest.
• From the application end, FADTTS is demonstrated in a
clinical study of neurodevelopment for revealing the
complex inhomogeneous spatiotemporal maturation
patterns as the apparent changes in fiber bundle diffusion
properties.
• We developed a GUI Tool box to facilitate the application of
FADTTS.
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Future Research
• extend FADTTS to the analysis of high angular resolution
diffusion image (HARDI).
• extend FADTTS to principal directions and full diffusion
tensors on fiber bundles.
• extend to more complex fiber structures, such as the medial
manifolds of fiber tracts.
• extend FADTTS to longitudinal studies and family studies.
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References
•
•
•
•
Zhu, H.T., Kong, L.L., Li, R.Z., Styner, M., Gerig, G., Lin, W.L.,
Gilmore, J. H. (2011). FADTTS: Functional Analysis of Diffiusion
Tensor Tract Statistics varying coefficient models for DTI tract statistics.
Neuroimage, in press.
Zhu, H.T., Li, R. Z., Kong, L.L. (2011). Multivariate varying coefficient
models for functional responses. Submitted.
Zhu, H., Styner, M., Li, Y., Kong, L., Shi, Y., Lin, W., Coe, C., and
Gilmore, J. (2010). Multivariate varying coefficient models for DTI tract
statistics. In Jiang, T., Navab, N., Pluim, J., and Viergever, M., editors,
Medical Image Computing and Computer-Assisted Intervention MICCAI
2010, volume 6361 of Lecture Notes in Computer Science, pages 690697. Springer Berlin / Heidelberg.
NICTR Toolbox (2011). FADTTS: Functional Analysis of Diffusion
Tensor Tract Statistics. http://www.nitrc.org/projects/fadtts/
The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL