David SimpsonReader in Biomedical Signal Processing, University of Southampton

ds@isvr.soton.ac.uk

Signal Processing for Quantifying Autoregulation

Outline• Preprocessing

• Transfer function analysis

– Gain, phase, coherence

– Bootstrap project

• Model fitting

• Extracting parameters

• Discussion

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5

Median filter

0 10 20

-10

0

10

20

time (s)

cm/s

blood flow velocity

original

10 12 14

-10

0

10

20

time (s)

cm/s

blood flow velocity

originalmedian filtered

Median filter

618 18.2 18.4

-10

0

10

20

time (s)

cm/s

blood flow velocity

originalmedian filtered

• Can not remove wide spikes• Right-shift of signal

0 10 20

-10

0

10

20

time (s)

cm/s

blood flow velocity

originalmedian filtered

Smoothing

70 5 10 15 20 25

2

4

6

8

time (s)

cm/s

filtered velocity

originalmedian filtered

0 10 20

-10

0

10

20

time (s)

cm/s

blood flow velocity

original

• Bidirectional low-pass (Butterworth) filter, fc=0.5Hz

• Ignore the beginning!

Transfer function analysis (TFA)

8

0 50 100 150 200 250 300

-15

-10

-5

0

5

10

15

20

25

time (s)

%

raw signals

pv

• Data from Bootstrap Project• Normalized by mean• Not adjusted for CrCP

Thanks: CARNet bootstrap project for data used

0 100 200 300

-5

0

5

10

15

time (s)

%

raw signals

pv

Transfer function analysis (TFA)

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• Filtered 0.03-0.5

Relating pressure to flow

10

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

Phas

e

0 0.5 1 1.5 2-4

-2

0

2

4

time (s)

Period T=1/f

Ampl

itude

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

13

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

0 0.5 1

-1

0

1

time (s)

sign

alxwindowx.window

0 0.2 0.4

-1

0

1

time (s)

sign

al

Detail

0 20 40

0.01

0.02

0.03

frequency (Hz)

PS

D

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Power spectral estimation: Welch method

0 0.5 1

-1

0

1

time (s)

sign

alxwindowx.window

0.4 0.6

-1

0

1

time (s)

sign

al

Detail

0 20 40

0.010.020.030.04

frequency (Hz)

PS

D

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Power spectral estimation: Welch method

0 0.5 1

-1

0

1

time (s)

sign

alxwindowx.window

0.6 0.8

-1

0

1

time (s)

sign

al

Detail

0 20 40

0.01

0.02

0.03

0.04

frequency (Hz)

PS

D

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Power spectral estimation: Welch method

0 0.5 1

-1

0

1

time (s)

sign

al

0.8 1 1.2

-1

0

1

time (s)

sign

al

Detail

0 20 40

0.02

0.04

0.06

frequency (Hz)

PS

D

20

Power spectral estimation: Welch method

0 0.5 1

-1

0

1

time (s)

sign

al

1 1.2 1.4

-1

0

1

time (s)

sign

al

Detail

0 20 40

0.05

0.1

0.15

frequency (Hz)

PS

D

21

Power spectral estimation: Welch method.Averaging individual estimates

0 10 20 30 40

0.05

0.1

0.15

frequency (Hz)

PS

D

TFA analysis: Estimated cross-spectrumbetween p and v

Estimated auto-spectrumof p

0 0.1 0.2 0.3 0.4

1

2

3

frequency (Hz)

gain

TFAChanging window-length

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T=100sT=20s

0 0.1 0.2 0.3 0.4

-2

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

3

4

frequency [Hz]

PS

D

25

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%)

0 10 20 30

2

4

6

8

10

frequency [Hz]

PS

D

M=512L=?f=?

M=128L=?f=?

0 10 20 30

0.5

1

1.5

frequency [Hz]

PS

D

M=64L=?f=?

Trueestimates

Mean ofestimates

Critical values for coherence estimates

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0 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

30

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 19, 2023 31

-10 -5 0 5 10

-1

0

1

2

time (s)

pres

sure

pul

se r

espo

nse

How to quantify autoregulation from model

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Mx Pha Coh ARI H1 L NL L NL L NL L NL0

20

40

60

80

100

120

++*

*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

0

1

2

3

frequency (Hz)

gain

TFAFIR filter

-10 -5 0 5 10

-1

0

1

time (s)

impulse response

TFAFIR

Change cut-off frequency (0.03-0.8Hz)

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-10 -5 0 5 10

0

0.5

1

time (s)

impulse response

TFAFIR

0 0.1 0.2 0.3 0.40

1

2

3

frequency (Hz)

gain

TFAFIR filter

ARI

35

0 5 10

0

0.5

1

time (s)

%/%

step responses

Increasing ARI

Selecting ARI: best estimate of measured flow

36

30 40 50 60

-5

0

5

time (s)

v

measuredestimated

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Non-linear system identification

LNL Model

LinearNon-

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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