Physiological Noise in fMRI
A Signal Processing Perspective
Simo Särkkä
Department of Biomedical Engineering and Computational Science (BECS)
Aalto University, Espoo, Finland
September, 2013
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September, 2013
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Contents
1
What is Physiological Noise?
2
Characteristics of Physiological Noise
3
Elimination of Physiological Noise
4
Summary and References
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Noise Sources in fMRI
1
Physical/scanner noise:
Thermal noise
Scanner drift
,→ Improve the hardware
,→ Low/high pass filtering
2
Body movement:
Head motion
Movement related changes in
magnetic field
,→ Scan again with better luck
,→ Align with a template image
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Noise Sources in fMRI (cont.)
3
Physiological noise:
,→
,→
,→
,→
Cardiac
Respiration
Vascular oscillations (not considered
on this lecture)
Amplified with better hardware
Aligning with template not possible
In 3 T, causes 30% of noise!
In 7 T, it is much over 50%!
Can be eliminated via statistical signal processing.
For a good Signal-to-Noise-Ratio (SNR) we must
eliminate the physiological noise.
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Signal-to-Noise-Ratios (SNR)
Definition of signal-to-noise-ratio (SNR):
SNR =
standard deviation of signal σs
standard deviation of noise σn
Signal and noise can be defined in different ways:
In SNR0 (image/raw SNR) only physical noise is included
in σn .
In TSNR (temporal/functional SNR) the physical and
physiological noises are in σn .
We can improve TSNR by eliminating the
physiological noise.
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Origins of Physiological Noise in fMRI
Origins of cardiac noise:
Blood moves and oxygenation along with it.
Blood vessels dilate and constrict
Tissues move
Origins of respiration noise:
Lungs move, which changes static magnetic field B0 .
Head parts (cavities) move
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What Does Cardiac Signal Look Like?
Cardiac signal is a nice (almost) periodic signal:
300
Pulse sensor signal
200
100
0
−100
−200
0
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Time [s]
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What Does Respiration Signal Look Like?
Respiration signal is a nice (almost) periodic signal:
Respiration belt signal
3500
3000
2500
2000
1500
1000
0
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Time [s]
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What Do They Look Like in fMRI Data?
In fMRI data we see (can you see the signals...?):
60
fMRI voxel signal
55
50
45
40
35
30
0
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Time [s]
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Fourier Series
Any periodic signal x(t) can be represented as a
sum of sines and cosines:
x(t) =
200
0
−200
a1 cos(2π · 1 · f0 ) + b1 sin(2π · 1 · f0 )
0
1
2
3
4
5
0
1
2
3
4
5
0
1
2
3
4
5
0
1
2
3
4
5
100
0
−100
+a2 cos(2π · 2 · f0 ) + b2 sin(2π · 2 · f0 )
100
0
−100
50
+a3 cos(2π · 3 · f0 ) + b3 sin(2π · 3 · f0 )
0
−50
p
2 + b 2 is the amplitude of frequency m · f .
am
0
m
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Fourier Series and Fourier Transform
We can also plot the amplitudes of each frequency:
100
200
80
Amplitude
Signal
100
0
60
40
−100
20
−200
0
⇒
If we encode the phases as complex valued
amplitudes, we get the Fourier transform (FT).
The discrete-time version is called discrete Fourier
transform (DFT).
Fast Fourier transform (FFT) is an efficient
algorithm for computing DFT.
0
1
2
3
4
5
Time [s]
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Frequency [Hz]
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Power Spectral Density
Power spectral density (PSD) gives the amount of
power (≈ amplitude2 ) at each frequency.
PSDs of the cardiac and respiration:
6
4
8
PSD of Cardiac
x 10
3
3.5
PSD of Respiration
x 10
2.5
3
2
Power
Power
2.5
2
1.5
1.5
1
1
0.5
0.5
0
0
2
4
6
Frequency
8
10
0
0
0.2
0.4
0.6
Frequency
0.8
1
Typically some kind of weighted and smoothed FFT
– also called periodogram in this context.
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Power Spectral Density of fMRI Data
PSD of fMRI data (can you identify the peaks...?):
140
120
Power
100
80
60
40
20
0
0
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2
3
Frequency
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Spectrogram
Spectrogram is a plot of evolution of power spectral
density over time.
Spectrograms of cardiac and respiration:
5
3
2.5
Frequency [Hz]
Frequency [Hz]
4
3
2
1
0
2
1.5
1
0.5
20
40
60
Time [s]
80
100
120
0
20
40
60
Time [s]
80
100
120
Typically computed using some form of short time
Fourier Transform.
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Spectrogram of fMRI Data
Spectrogram of fMRI data (now we can see the signals!):
5
Frequency [Hz]
4
3
2
1
0
20
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Time [s]
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Amplitude map
Amplitudes of cardiac and respiration vary in brain:
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Phase/delay map
Phases of cardiac and respiration also vary in brain:
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Methods to Eliminate Physiological Noise
Here we only consider retrospective methods for
post-processing fMRI data.
,→ Band-stop filtering removes the frequency bands of
cardiac and respiration.
,→ RETROICOR fits cardiac and respiration Fourier
series to fMRI data, and subtracts them away.
,→ DRIFTER builds stochastic resonator models for
cardiac and respiration, and uses Kalman filter for
separating them.
The physiological noise elimination can also be
combined with GLM analysis.
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Band-stop filtering: Idea
Band-stop filter eliminates certain frequency range
from signal:
6
4
1.2
3.5
1
Before filtering
After filtering
2.5
0.8
0.6
2
1.5
0.4
1
0.2
0
0.5
0
⇒
The above result in time domain:
0
2
4
6
Frequency [Hz]
8
10
0
Cardiac signal before filtering
300
200
200
100
0
−100
−200
0
5
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Time [s]
15
2
4
6
Frequency
8
10
Cardiac signal after filtering
300
Cardiac signal
Cardiac signal
PSD of Cardiac
x 10
3
Power
Magnitude response
Magnitude response of band−stop filter
1.4
20
⇒
100
0
−100
−200
Physiological Noise in fMRI
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Time [s]
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Band-stop filtering: Practice
How to do it in practice:
1
Find the cardiac/respiration peaks from
(a) PSD of fMRI signal itself
(b) PSDs of reference signals (pulse sensor, respiration belt)
2
Run band-stop filters with those centers on fMRI data.
Ways to implement a band-stop filter:
Finite impulse response (FIR) filter
Infinite impulse response (IIR) filter
State space (SS) filter
Fast Fourier transform (FFT)
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Band-stop filtering: Pros and Cons
+
+
–
–
–
–
Simple to implement and fast
Works well when TR is short
Aliasing with long TRs
Time-varying frequency is a problem
The peaks are not that sharp in practice
Completely ignores the phase
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RETROICOR: Idea
RETROICOR fits (generalized) Fourier series to
cardiac xc (t) and respiration signals xr (t):
xc (t) ≈
xr (t) ≈
M
X
m=1
M
X
c
c
sin(m φc (t))
cos(m φc (t)) + bm
am
r
r
am
cos(m φr (t)) + bm
sin(m φr (t)).
m=1
Above, phases are defined as integrals of frequency:
Z
t
φc (t) = 2π
fc (t) dt
0
Z
φr (t) = 2π
t
fr (t) dt
0
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RETROICOR: Computation
Phases φc (t) and φr (t) are estimated from the
reference signals.
c
c
r
r
Coefficients am
, bm
, am
, bm
are determined via
Fourier summation over fMRI data y (tn ):
x
am
PN
=
x
bm
=
n=1 [y (tn ) − y ] cos(m φc (tn ))
PN
2
n=1 cos (m φx (tn ))
PN
n=1 [y (tn ) − y ] sin(m φc (tn ))
,
PN
2
n=1 sin (m φx (tn ))
where x ∈ {c, r } and y is the average of y .
Finally, we subtract these fits from fMRI data.
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RETROICOR: Pros and Cons
+
+
+
+
–
–
No problems with aliasing
Works well with long TRs
Works well with time-varying frequency
Can be embedded into GLM analysis.
Quick frequency changes are a problem
Cannot cope with amplitude changes
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DRIFTER: Idea
Cardiac and respiration are modeled as
superposition of stochastic resonators, e.g.:
d 2 xc,n (t)
= −(2π m fc (t))2 xc,n (t) + noise.
2
dt
The brain activation is modeled as a smooth signal.
The rest of the signal is modeled as white noise.
The full model can be written as a multivariate
state-space model:
dx(t)/dt = A(fc (t), fr (t)) x(t) + L w(t)
y (tk ) = H x(tk ) + (tk ).
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DRIFTER: Computation
The frequency trajectories fc (t) and fr (t) from
reference signals or selected voxel areas.
Frequency estimation uses IMM algorithm – an
adaptive Kalman filter.
Kalman filter and RTS-smoother used for separating
fMRI data into cardiac, respiration, brain activation
and white noise parts.
The result is the brain activation part of the
separation.
Alternatively, we can subtract the cardiac and
respiration parts from fMRI data.
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DRIFTER: Pros and Cons
+
+
+
+
–
–
–
No problems with aliasing
Works well with long TRs
Works well with time-varying frequency
Works well with amplitude changes
Computationally heavier than the other methods
More parameters to tune
Embedding into GLM analysis harder.
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Limitations in All Methods
All the methods need reference signals to work
properly.
If cardiac/respiration is synchronized with task, we
will eliminate some of the activation.
Any other signal with an overlapping spectrum will
also be partly eliminated.
Very long TRs are a problem to all the methods.
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Summary
Physiological noise in fMRI refers to cardiac and
respiration showing in data.
Dominating noise source in high field fMRI.
Characteristics of physiological noise:
Almost periodic signals – sums of sinusoids.
Can be studied via power spectral density (PSD) plots
and spectrograms.
Spatially varying amplitude and phase structure.
Methods for eliminating physiological noise:
Band-stop filtering
RETROICOR (Fourier series)
DRIFTER (Kalman filter)
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The End
Questions?
DRIFTER Toolbox for SPM
http://becs.aalto.fi/en/research/bayes/drifter/
References
Biswal, B., DeYoe, E., Hyde, J., 1996. Reduction of physiological fluctuations in fMRI using digital filters.
Magnetic Resonance in Medicine 35, 107–113
Glover, G.H., Li, T.Q., Ress, D., 2000. Image-based method for retrospective correction of physiological motion
effects in fMRI: RETROICOR. Magnetic Resonance in Medicine 44, 162–167.
Krüger, G., Glover, G.H., 2001. Physiological noise in oxygenation-sensitive magnetic resonance imaging. Magnetic
Resonance in Medicine 46, 631–637.
Huettel, S.A., Song, A.W., McCarthy, G., 2004. Functional Magnetic Resonance Imaging. Sinauer Associates.
Hutton, C., Josephs, O., Stadler, J., Featherstone, E., Reid, A., Speck, O., Bernarding, J., Weiskopf, N., 2011. The
impact of physiological noise correction on fMRI at 7 T. NeuroImage 57, 101–112
Särkkä, S., Solin, A., Nummenmaa, A., Vehtari, A., Auranen, T., Vanni, S., Lin, F.-H., 2012. Dynamic
Retrospective Filtering of Physiological Noise in BOLD fMRI: DRIFTER. NeuroImage 60(2), 1517–1527.
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