David SimpsonReader in Biomedical Signal Processing, University of Southampton
Signal Processing for Quantifying Autoregulation
Outline• Preprocessing
• Transfer function analysis
– Gain, phase, coherence– Bootstrap project
• Model fitting
• Extracting parameters
• Discussion
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Median filter
0 10 20
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time (s)
cm/s
blood flow velocity
original
10 12 14
-10
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time (s)
cm/s
blood flow velocity
originalmedian filtered
Median filter
618 18.2 18.4
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time (s)
cm/s
blood flow velocity
originalmedian filtered
• Can not remove wide spikes• Right-shift of signal
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time (s)
cm/s
blood flow velocity
originalmedian filtered
Smoothing
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time (s)
cm/s
filtered velocity
originalmedian filtered
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time (s)
cm/s
blood flow velocity
original
• Bidirectional low-pass (Butterworth) filter, fc=0.5Hz
• Ignore the beginning!
Transfer function analysis (TFA)
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0 50 100 150 200 250 300
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time (s)
%
raw signals
pv
• Data from Bootstrap Project• Normalized by mean• Not adjusted for CrCP
Thanks: CARNet bootstrap project for data used
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time (s)
%raw signals
pv
Transfer function analysis (TFA)
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• Filtered 0.03-0.5
Relating pressure to flow
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Input / outputmodel
Arterial Blood Pressure
Blood Flow Velocity
End-tidalpCO2
+-
error
V(f)=P(f).H(f)
Transfer function (frequency response)
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Fourier SeriesPeriodic Signals - Cosine and Sine Waves
)2cos(.)( ftatx
Phase
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-2
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2
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time (s)
Period T=1/f
Ampli
tude a Cosine wave
Sine wave
t
Gain
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0 0.1 0.2 0.3 0.40
1
2
3
frequency (Hz)
gain
TFA
Phase
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0 0.1 0.2 0.3 0.4
-2
0
2
frequency (Hz)
phas
eTFA
Coherence
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How well are v and p correlated, at each frequency?
0 0.1 0.2 0.3 0.4
0.2
0.4
0.6
0.8
frequency (Hz)
|coh
eren
ce|
TFA
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Power spectral estimation: Welch methodAn example from EEG
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time (s)
sign
alxwindowx.window
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time (s)
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Detail
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0.03
frequency (Hz)
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Power spectral estimation: Welch method
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time (s)
sign
alxwindowx.window
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Detail
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frequency (Hz)
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Power spectral estimation: Welch method
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time (s)
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alxwindowx.window
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Detail
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frequency (Hz)
PS
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Power spectral estimation: Welch method
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time (s)
sign
al
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time (s)
sign
al
Detail
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frequency (Hz)
PS
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Power spectral estimation: Welch method
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time (s)
sign
al
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time (s)
sign
al
Detail
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frequency (Hz)
PS
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Power spectral estimation: Welch method.Averaging individual estimates
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0.1
0.15
frequency (Hz)
PS
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TFA analysis: Estimated cross-spectrumbetween p and v
Estimated auto-spectrumof p
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frequency (Hz)
gain
TFAChanging window-length
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T=100sT=20s
0 0.1 0.2 0.3 0.4
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0
2
frequency (Hz)
phas
e
TFA
• Frequency resolution:Δf=1/T, T… duration of window
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
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0 10 20 30
1
2
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frequency [Hz]
PSD
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Effect of windowlength (M) and number of windows (L)Signal: N=512, fs=128
With fixed N (512), type of window (rectangular),
and overlap (50%)
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frequency [Hz]
PSD
M=512L=?f=?
M=128L=?f=?
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0.5
1
1.5
frequency [Hz]
PSD
M=64L=?f=?
Trueestimates
Mean ofestimates
Critical values for coherence estimates
260 0.5 1 1.5 2
0.2
0.4
0.6
0.8
frequency (Hz)
cohe
renc
e
• 3 realizations of uncorrelated white noise
Critical value (3 windows, α=5%)
0 0.1 0.2 0.3 0.4
0.2
0.4
0.6
0.8
frequency (Hz)
|coh
eren
ce|
TFA
Critical values
270 20 40
0.2
0.4
0.6
0.8
no. windows
C2 cr
it
10%5%1%No. of
independentwindows
Modelling
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AdaptiveInput / output
model
Arterial Blood Pressure
Blood Flow Velocity
End-tidalpCO2
+-
error
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Predicted response to step input (13 recordings, normal subjects)
-2 0 2 4 6 8-1
-0.5
0
0.5
1
1.5
time (s)
%
Step responses
Predicted response to change in pressure
April 24, 2023 31
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time (s)
pres
sure
pul
se re
spon
se
How to quantify autoregulation from model
32Mx Pha Coh ARI H1 L NL L NL L NL L NL
0
20
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60
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++*
*o
o *
+
o
o
A7A1.5PCSFVS
o*
*o
%
Autoregulatory Parameter
SDn Inter-subject variability
mSDn Intra-subject variability
Alternative estimator: FIR filter
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• Sampling frequency (2 Hz)• Scales are not compatible• TFA: not causal • Needs pre-processing 0 0.1 0.2 0.3 0.4
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frequency (Hz)
gain
TFAFIR filter
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time (s)
impulse response
TFAFIR
Change cut-off frequency (0.03-0.8Hz)
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0.5
1
time (s)
impulse response
TFAFIR
0 0.1 0.2 0.3 0.40
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frequency (Hz)
gain
TFAFIR filter
ARI
350 5 10
0
0.5
1
time (s)
%/%
step responses
Increasing ARI
Selecting ARI: best estimate of measured flow
3630 40 50 60
-5
0
5
time (s)
v
measuredestimated
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Non-linear system identification
LNL Model
Linear Non-Linear LinearPressure Flow
Filter FilterStatic
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
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