Complex-valued Correlation in fMRI
Mary C. Kociuba
PhD Student in Computational Sciences
Department of Mathematics, Statistics, and Computer Science
1
MR Imaging pipeline
•  Raw unprocessed signal
directly from the scanner
Acquisition
Reconstruction
Me!
•  Complex-valued
reconstruction methods for
fast imaging techniques
Processing
Analysis
You!
2
Outline
•  Purpose
•  Background & motivation
•  Complex-valued correlation in the frequency domain
•  Low CNR simulation results - increased sensitivity
•  Experimental human results - increased specificity
•  Conclusions
3
Purpose
Motivate the use of phase information in fMRI
Representation for complex-valued correlation in the temporal
Fourier frequency domain
Demonstrate the increased sensitivity and specificity of complexvalued (CV) over magnitude-only (MO) correlation in fMRI data
analysis
4
Background
MR Image Acquisition and Reconstruction
Tip longitudinal net magnetization
into the transverse plane
Net magnetization aligned
with magnetic field
Excitation
Relaxation
MR signal detection
5
Background
MR Image Acquisition and Reconstruction
Imaginary Channel
Tip longitudinal net magnetization
into the transverse plane
Signal = Re + i Im
2D Spatial Encoding
Acquire MR complex-valued
signal in k-space
Im
θ
Re
Real Channel
ρ = √Re2 + Im2
θ = tan-1[Im/Re]
6
Background
What is k-space?
k-space is the spatial frequencies of the object in the image space
Spatial frequency defines how often some pattern occurs in space
Any image - no matter how complicated – can be represented as an
ensemble of spatial frequency components
FT
Each pixel maps
to every point in
k-space
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Background
MR Image Acquisition and Reconstruction
k-space
Tip longitudinal net magnetization
into the transverse plane
2D Spatial Encoding
Acquire MR complex-valued
signal in k-space
2D IFT
MR Image
2D complex-valued MR Image
8
Background
MR Image Reconstruction
(ΩyR+ iΩyI)
×
(SR+ iSI)
×
(ΩxR+ iΩxI)
=
(VR+ iVI)
magnitude
+i
×
+i
×
+i
=
+i
phase
9
Background
MR Image Acquisition and Reconstruction
MR Image magnitude
Tip longitudinal net magnetization
into the transverse plane
2D Spatial Encoding
Acquire MR complex-valued
signal in k-space
2D IFT
MR Image phase
2D complex-valued MR Image
Discard phase before processing and analysis
10
Background
Use only half the data- biological info in space
n
1
MR Image magnitude
MR Image phase
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Background
Susceptibility Weighted Imaging (SWI)
Exploit susceptibility differences between tissues & use phase information to
detect these differences
Combine magnitude and phase images for enhanced contrast
magnitude image
phase image
Rowe, Haacke: MRI, 27:1271-1280, 2009.
12
Background
The phase time-series in fMRI- biological info in time
Noise in the phase time-series is more pronounced, and susceptible to
dynamic changes in the magnetic field over time and physiological noise
Previous studies have shown…
task related phase arises from large non-randomly oriented blood vessels
[Hoogenraad et al., 1998; Menon, 2002; Rowe, 2005b; Nencka et al., 2007; Tomasi et al., 2007]
randomly oriented microvasculature produce a BOLD phase change
[Zhao et al., 2007; Feng et al. 2009; Arja et al., 2009]
the fMRI signal directly associated with neuronal action potentials may be
manifested to some degree in the phase
[Bandettini et al., 2005; Bodurka and Bandettini, 2002; Bodurka et al., 1999; Heller et al., 2007;
Petridou et al., 2006]
).
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Background
Task related phase change in fMRI time-series
fMRI Bilateral finger tapping experiment – with B0 field correction and
physiological noise removed
20s off+16×(8 s on 8 s off), 276 TRs
10 axial slices, 96 × 96, FOV = 24 cm
Magnitude Only
TH = 2.5 mm, TR = 1 s, TE = 42.8 ms
FA = 45◦, BW = 125 kHz, ES = 768 ms
Phase Only
Magnitude and Phase
8
0
Hahn et. al, NeuroImage 2012
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Background
Task related phase change in fMRI time-series
Time series are complex, bivariate with phase coupled means
Real
Imaginary
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Background
Task related phase change in fMRI time-series
Time series are complex, bivariate with phase coupled means
Magnitude
Phase
FT of phase time-series has task peak
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Temporal Fourier correlation
Spatial to Temporal Frequencies
n
n
1
1
Spatial frequencies
2D IFT
MR Image
1D FT
p
1
Temporal frequencies
17
Temporal Fourier correlation
Covariance
With complex-valued voxel α, vα, and voxel β, vβ, time-series demeaned and
the 1D IFT, ΩC , the covariance in terms of temporal frequencies
cov !! , !!
1
= !!
!
! (! )
!
1
=
2
!
!!!! !!!! + !!!! !!!! !
!!!
!
!! !=! Ω! + !Ω! !!! + !!!! !
!
!! !=! Ω! + !Ω! !!" + !!!" !
!
Ω! = ! Ω! + !Ω! !
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Temporal Fourier correlation
Temporal Fourier Frequencies
fMRI studies analyze the voxel time series correlated with experimental block
design of a given task performed by the subject.
1.5
20
real
1.4
18
16
1.3
14
1.2
1D FT
1.1
12
10
1
8
0.9
6
imaginary
0.8
0.7
0.6
4
2
0
50
100
150
time (s)
200
250
0
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
frequencies (Hz)
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Temporal Fourier correlation
Temporal Fourier Frequencies
fcMRI studies analyze the temporal frequency band < 0.1 Hz.
Biswal et al. MRM 1995
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Temporal Fourier correlation
Covariance
cov !! , !!
1
= !!
!
!
1
(!! ) =
2
!
!!!! !!!! + !!!! !!!! !
!!!
! such that the entry (α, β) in Σ is the
For p voxels in a p × p matrix, Σ,
!! !=! the
Ω!two
+ !Ω
! voxel α and voxel β
! !time-series
!! + !!!! of
covariance between
voxel
!
!!-1/2
!=! Ω! + !Ω! !!" + !!!" !
R = D-1/2 Σ D
!
covariance matrix is written
of b bands,
Ω! =as
! Ωa! summation
+ !Ω! !
Σ = Σ1 + … + Σb
correlation matrix is written as a summation of correlation of bands
R = D-1/2(Σ1 + … + Σb)D-1/2 = R1 + … + Rb
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Simulation
MATLAB simulation with varying degree of CNRρ and CNRϕ
The MO and CV correlations are computed between the two time-series in
each surface
Signal-to-Noise Ratio (SNR)- baseline magnitude signal over the s.d. of timeseries noise,
SNR = ρ/σ
Contrast-to-Noise Ratio (CNR)Amplitude difference between the baseline signal and the task related change
signal for the magnitude and phase, Aρ and Aϕ,
CNRρ = Aρ/σ
CNRϕ = Aϕ/(σ/ρ)
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Simulation
MATLAB simulation with varying degree of CNRρ and CNRϕ
The MO and CV correlations are computed between the two time-series in
each surface
Task related signal change in the magnitude Aρ is 1-2% signal change, and
the task related change in the phase Aϕ is approximately π/36
Aρ
ρ
Aϕ
50
0
π/36
0
1
0
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Simulation Results
Aρ
ρ
Aϕ
50
π/36
Surface
parameters
0
0
Fisher-z
transform
of correlation
z = ½ ln
!!!
!!!
!
CV
0
Difference
CNRϕ
MO
1
CNRρ
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Experimental human results
Acquisition Parameters
3.0 T Discovery MR750 MRI scanner, GE single channel quadrature head
coil, 10 interleaved 4 mm thick axial slices, 96×96 in dimension for a 24.0 cm
FOV, TR/TE = 1000/39 ms, flip angle = 25°, bandwidth = 111 kHz
bilateral finger tapping fMRI experiment performed for sixteen 22-second
periods, 720 TRs
Data corrected for magnetic B0-field, respiration
correlation as a summation of 3 bands
R1 (0.0009 - 0.024 Hz), R2 (0.026 - 0.037 Hz), R3 (0.038 - 0.08 Hz)
sum of the bands = the total correlation
task-activated frequency peak in R2
Activation computed, 2 voxels chosen based on activation locations, and hi/
low CNR, in the motor cortex and the supplementary motor cortex
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CV
data
high CNR
high CNR
MO
data
low CNR
Motor cortex
a)
Motor cortex
Supplementary motor cortex
Motor
Supplementary
Motor cortex
cortex
Supplementary motor
motor cortex
cortex
high CNR
high CNR
low CNR
low
CNR
high
CNR
high
low
CNR
low
CNR
high CNR
high CNR
CNR
low CNR
low CNR
low CNR
Supplementary motor cortex
Experimental human results
Seed voxel correlation- motor cortex
a)a)
Ra)
1
RR
1R
11
R2
R3
RR
2R
22
b)
RR
3R
33
b)b)b)
CV
MO
CV
MO
CV
MO
CV
MO
Total
Correlation
Task frequency band
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Supplementary motor cortex
high CNR
CV
R2
MO
R3a)
Supplementary motor cortex
R1
high CNR
R2
low CNR
R1
high CNR
Motor cortex
b) R
1
low CNR
a)
high CNR
a)
low CNR
Supplementary motor cortex
high CNR
high CNR
low CNR
low CNR
high CNR
Motor cortex
Motor cortex
R3
w CNR
low CNR
plementary motor cortex
Seed voxel correlation- increased specificity
R3b)
low CNR
Moto
Experimental human results
R2
b)
CV
R
3
b)
CV
MO
CV
MO
MO
Motion!
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Conclusions
Why?
Magnitude-only signal has no rotational information to leverage for greater
sensitivity
Imaginary Channel
Im
Signal = Re + i Im
θ
Re
MO signal change
CV signal change
Im
Real Channel
Re
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Conclusions
There is biological information in the phase
Start your analysis with better data, less processing required
A natural application of complex-valued correlation is to non-task
fMRI, where the correlation detects long-range connectivity
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