Signal Processing for
Quantifying Autoregulation
David Simpson
Reader in Biomedical Signal Processing,
University of Southampton
ds@isvr.soton.ac.uk
Outline
• Preprocessing
• Transfer function analysis
– Gain, phase, coherence
– Bootstrap project
• Model fitting
• Extracting parameters
• Discussion
2
blood flow velocity
original
20
10
0
blood flow velocity
-10
0
10
time (s)
20
20
cm/s
cm/s
Median filter
original
median filtered
10
0
-10
10
12
time (s)
14
5
Median filter
blood flow velocity
10
0
blood flow velocity
-10
0
10
time (s)
20
20
cm/s
cm/s
20
original
median filtered
original
median filtered
10
0
• Can not remove wide spikes
• Right-shift of signal
-10
18
18.2
time (s)
18.4
6
blood flow velocity
original
20
10
filtered velocity
0
-10
0
original
median filtered
8
10
time (s)
20
cm/s
cm/s
Smoothing
6
4
• Bidirectional low-pass
(Butterworth) filter, fc=0.5Hz
2
• Ignore the beginning!
0
5
10
15
time (s)
20
25
7
Transfer function analysis (TFA)
raw signals
25
p
v
20
• Data from Bootstrap Project
• Normalized by mean
• Not adjusted for CrCP
15
%
10
5
0
-5
-10
-15
0
50
100
150
time (s)
200
250
300
Thanks: CARNet bootstrap
project for data used
8
Transfer function analysis (TFA)
raw signals
15
%
10
• Filtered 0.03-0.5
p
v
5
0
-5
0
100
200
time (s)
300
9
Relating pressure to flow
Transfer function (frequency response)
V(f)=P(f).H(f)
10
Blood Flow
Velocity
Arterial Blood
Pressure
error
Input / output +
model
End-tidal
pCO2
Fourier Series
Periodic Signals - Cosine and Sine Waves
Period T=1/f
4
Cosine wave
2
Sine wave
0
-2
-4
0
11
t
0.5
1
time (s)
1.5
2
x (t )  a. cos(2ft   )
Gain
TFA
gain
3
2
1
0
0
0.1
0.2 0.3 0.4
frequency (Hz)
12
Phase
TFA
phase
2
0
-2
0
0.1
0.2 0.3 0.4
frequency (Hz)
13
Coherence
How well are v and p correlated, at each frequency?
|coherence|
TFA
0.8
0.6
0.4
0.2
0
0.1
0.2 0.3 0.4
frequency (Hz)
14
Power spectral estimation:
Welch method
An example from EEG
x
window
x.window
signal
1
0
-1
0
0.5
1
time (s)
Detail
0
-1
0
16
0.03
PSD
signal
1
0.02
0.01
0.2
0.4
time (s)
0
20
40
frequency (Hz)
Power spectral estimation:
Welch method
x
window
x.window
signal
1
0
-1
0
0.5
1
time (s)
Detail
PSD
signal
1
0
-1
0.4
0.6
time (s)
17
0.04
0.03
0.02
0.01
0
20
40
frequency (Hz)
Power spectral estimation:
Welch method
x
window
x.window
signal
1
0
-1
0
0.5
1
time (s)
Detail
0.04
0.03
PSD
signal
1
0
-1
0.01
0.6
18
0.02
0.8
time (s)
0
20
40
frequency (Hz)
Power spectral estimation:
Welch method
signal
1
0
-1
0
0.5
1
time (s)
Detail
0.06
PSD
signal
1
0
-1
0.8
19
1
time (s)
1.2
0.04
0.02
0
20
40
frequency (Hz)
Power spectral estimation:
Welch method
signal
1
0
-1
0
0.5
1
time (s)
Detail
0.15
0
-1
1
20
PSD
signal
1
1.2
1.4
time (s)
0.1
0.05
0
20
40
frequency (Hz)
Power spectral estimation:
Welch method.
Averaging individual estimates
0.15
PSD
0.1
0.05
0
10
20
30
frequency (Hz)
TFA analysis:
21
40
Estimated cross-spectrum
between p and v
Estimated auto-spectrum
of p
TFA
3
T=100s
T=20s
2
TFA
1
2
0
0.1
0.2 0.3 0.4
frequency (Hz)
• Frequency resolution:
Δf=1/T,
T… duration of window
phase
gain
Changing window-length
0
-2
0
0.1
0.2 0.3 0.4
frequency (Hz)
22
Estimating spectrum and cross-spectrum
• Frequency resolution:
Δf=1/T, T… duration of window
• Estimation error:  with more windows
• Compromise:
Longer windows: better frequency resolution, worse
random estimation errors
• Higher sampling rate increases frequency range
• Longer FFTs: interpolation of spectrum, transfer function,
coherence …
• Window shape: probably not very important
24
Effect of windowlength (M) and
number of windows (L)
4
Signal: N=512, fs=128
PSD
With fixed N (512), type of
window (rectangular), and
overlap (50%)
3
M=128
L=?
f=?
2
1
0
True
estimates
10
6
4
M=512
L=?
f=?
1.5
PSD
PSD
8
2
0
25
1
0.5
10
20
frequency [Hz]
10
20
frequency [Hz]
30
M=64
L=?
f=?
Mean of
estimates
30
0
10
20
frequency [Hz]
30
Critical values for coherence estimates
• 3 realizations of uncorrelated white noise
Critical value (3 windows, α=5%)
TFA
coherence
|coherence|
0.8
0.8
0.6
0.4
0.4
0.2
0.2
0
0.6
0.1
0.2 0.3 0.4
frequency (Hz)
0
0.5
1
1.5
frequency (Hz)
2
26
Critical values
2
No. of
independent
windows
C crit
0.8
0.6
10%
5%
1%
0.4
0.2
0
20
40
no. windows
27
Modelling
Arterial Blood
Pressure
End-tidal
pCO2
Adaptive
Input / output
model
Blood Flow
Velocity
error
+
29
Step responses
Predicted response to step input (13 recordings,
normal subjects)
1.5
%
1
0.5
0
-0.5
-1
-2
0
2
4
time (s)
6
830
pressure pulse response
Predicted response to
change in pressure
2
1
0
-1
-10
9 April 2015
-5
0
time (s)
5
10
31
How to quantify autoregulation from
model
SDn
Inter-subject variability
mSDn
Intra-subject variability
o
120
%
100
o
o
*
o
*
80
o
60
*
*
*
o
+
+
+
40
20
0
Mx
Pha Coh ARI
H1
L
NL
FVS
L
NL
L
PCS
Autoregulatory Parameter
NL
A1.5
L
NL
A7
32
Alternative estimator: FIR filter
impulse response
TFA
FIR
1
0
-1
3
•
•
•
•
-5
0
time (s)
5
Sampling frequency (2 Hz)
Scales are not compatible
TFA: not causal
Needs pre-processing
10
gain
-10
TFA
FIR filter
2
1
0
0
0.1
0.2 0.3 0.4
frequency (Hz)
33
Change cut-off frequency (0.03-0.8Hz)
impulse response
TFA
FIR
1
0.5
3
TFA
FIR filter
-10
-5
0
time (s)
5
gain
0
2
10
1
0
0
0.1
0.2 0.3 0.4
frequency (Hz)
34
ARI
step responses
%/%
1
Increasing ARI
0.5
0
0
5
time (s)
10
35
Selecting ARI:
best estimate of measured flow
measured
estimated
v
5
0
-5
30
40
time (s)
50
60
36
Non-linear system identification
LNL Model
Pressure
Linear
NonLinear
Linear
Filter
Static
Filter
Flow
37
Summary
• Proprocessing
• TFA
– Gain, phase, coherence
– Window-length
– Critical values for coherence
• Issues
– What model?
– Frequency bands present
– How best to quantify autoregulation from model
38