Assessing
Pressure Wave Components
for
Aortic
Stiffness
Monitoring
through Spectral Regression
Learning
Arian Aghilinejad, PhD
1
, Morteza Gharib, PhD
1
1
Division of Engineering and Applied Science, California Institute of Technology, Pasadena, CA,
United States
Corresponding Author:
Arian Aghilinejad, PhD
1200 E California Blvd, Pasadena, CA, 91125
Email:
aghili@caltech.edu
Phone: 302 252 8325
The supplementary material has been provided by the
authors to give the readers additional
information about their work.
This file include:
-
Supplemental Methods
-
Figure
s
S1
and S2
-
Tables S1, S2, and S3
Supplemental Methods
Spectral Regression Learning Method
The first step to implement the
spectral regression learning method is to transform d
ata from a
high
-
dimensional input signal space to a low
-
dimensional space
. To do so,
we employed Fourier
-
based decomposition. The Fourier series is a method of combining harmonically related sinusoids
an
d cosines to synthesize a periodic function. Any periodic pressure function
P
(t) can be expressed
as a Fourier series with
N
oscillatory components. A typical representation of the Fourier series
decomposition is as follows:
푃
(
푡
)
=
!
!
"
+
∑
(
푎
#
푐표푠
(
"
$
%
푛푡
/
+
푏
#
푠푖푛
(
"
$
%
푛푡
/
/
&
#
'
(
,
(
1
)
where
T
is the period of the pressure function
P
(t)
(i.e., the cardiac cycle or inverse of HR for
blood pressure waveform). In conducting the Fourier decomposition here, we used the
sinusoi
d
-
cosinusoide
form rather than an Amplitude
-
Phase form to make the inputs independent from each
other. This is because the system of all sine and cosine functions at different frequencies builds a
complete orthonormal set. Coefficients
푎
#
and
푏
#
are as
sociated with each individual harmonics
(
cosine and sine
) corresponding to different frequencies
푓
#
=
#
%
, and can be calculated by the
Fourier transform given by:
푎
#
=
"
%
∫
푃
(
푡
)
푐표푠
(
"
$
%
푛푡
/
푑푡
%
)
,
(
2
)
푏
#
=
"
%
∫
푃
(
푡
)
푠푖푛
(
"
$
%
푛푡
/
푑푡
%
)
,
푛휖
(
0
,
∞
)
.
(
3
)
Obtaining the coefficients using Eq.
(2) and
(3), the represented pressure waveform
푃
:
(
푡
)
with
finite selected frequency (i.e.,
n = 0 to N
) is achieved by adding up each individual frequency
component (Eq. (
1
)). In this study, the features of the pressure wave were extracted as the first
푁
*
+
low frequency components of the waveforms using the Fast Fourier Transform (FFT).
Previous
studies
suggested that to fully recover the shape of the pressure waveform using Fourier mode
decomposition, the first 20 modes are enough and
the error between the measured pressure and the
reconstructed
after 20 modes is marginal.
The selected input features in
this study (depending on
the chosen number of Fourier modes;
FN
) consists of
FN
cosine coefficients and (
FN
-
1)
sine
coefficients (sine
푏
)
is always equal to zero).
In the next step, we use appropriate regression models to build the spectral regression
learning
model for estimating the Fourier modes of the flow waveform. In this study, the support vector
regressor was selected to train and test the regression model due to its proven applicability and
performance in previous studies.
As a brief summary of
this model, a Support Vector Regressor is
a machine learning tool utilized for predicting numerical values, such as the amplitude of the
sinusoidal
-
cosinusoidal components of the flow waveform in this study. It operates by identifying
a "support vector,"
essentially a subset of data points crucial for accurate predictions. The Support
Vector Regressor aims to create a model with a margin of tolerance on either side of the predicted
values by establishing a hyperplane that maximizes the margin between the p
redicted values and
the line. Some points are considered support vectors, indicating their significance in determining
the position of this line. SVR is particularly useful when dealing with data that doesn't follow a
simple linear pattern, requiring a mor
e flexible approach to capture underlying relationships.
This
model is trained on features derived from the Fourier decomposition of the pressure waveform, as
illustrated in
the graphical abstract
. Initially, the clinical dataset is divided into training a
nd testing
data for all regression learning analyses. The models are exclusively trained on the training
population based on the Fourier decomposition of the central pressure waveform and central flow
waveform. The models are trained with the Fourier modes
of the pressure waveforms as input and
the Fourier modes of the flow waveforms as output.
After training, the testing dataset is used only
once to assess the model's accuracy. The pressure waveform in the testing dataset is first
decomposed into the Fouri
er modes and fed into the predictive model. The output of the predictive
models, representing the estimated modes of the corresponding flow waveform, is then inverse
Fourier transformed into the time domain using the computed modes and the length of the si
gnal.
The combination of the predictive model and the inverse Fourier transform is referred to as the
spectral regression learning model.
To implement this model, Python's scikit
-
learn (sklearn) and
TensorFlow packages are employed for training and testing
the algorithms. The data split for
training and testing is 70% and 30%, respectively. The pandas and numpy packages are also
utilized for data processing. The hyperparameters in the models are determined using a ten
-
fold
cross
-
validation (CV) scheme with
the GridSearchCV library.
Forward and Backward Wave Separation Components
In the arterial system, the pressure waveform comprises forward (Pf) and backward (Pb)
components. The forward component (Pf) signifies the contribution of forward waves to measured
pressure, while the backward component (Pb) represents the contribution of
backward waves,
resulting in the equation P = Pf + Pb. Pressure wave separation, initially described by Westerhof
et al., can be applied to identify the forward and backward components of the measured pressure
waveform:
푃푓
=
(
"
(
푃
+
푍
,
푄
)
(4)
푃푏
=
(
"
(
푃
−
푍
,
푄
)
(5)
Here, Q is the measured flow profile, and
푍
,
is the characteristic impedance, representing the
intrinsic relationship between pressure and flow when waves travel in one direction only. It can be
estimated from measured pressure and f
low signals historically achieved in the frequency domain
(via Fourier transform). This involves calculating the average ratio of pressure and flow harmonics
within a specific frequency range, such as high frequency with the result appearing relatively
ins
ensitive to the chosen range (Dujardin and Stone, 1981). Following the computation of forward
and backward waves, the maximum amplitude of these waves can define other wave separation
parameters. The reflection index (RI) is a significant parameter, define
d as:
푅퐼
=
|
./
|
|
./
0
.1
|
(6)
To compute the time delay between the forward and backward waves, half the time of the
maximum cross
-
correlation between Pf and Pb was utilized, following the works of Qasem and
Avolio.
Supplemental
Figures
Figure S
1
d
emonstrates scatter and Bland
-
Altman plots indicating the agreement between the
measured and estimated
uncalibrated
averages of the mean flow profile, revealing the method's
ability to capture the shape of the flow profile.
Figure S
1
Scatter and Bland
-
Altman plots indicating the agreement between the measured and estimated uncalibrated
averages of the mean flow profile using spectral regression learning method.
푅
2
=
0
.
39
Figure S
2
demonstrates
t
he distributions of the residuals between refer
ence and estimated variables
with respect to age
.
Figure S
2
The distributions of the residuals between reference and estimated variables with respect to age
for all four
variables derived from wave separation analysis including
forward wave amplitude (Pf), backward wave amplitude
(Pb), reflection index (RI), and ti
me delay (TD) between forward and backward waves
.
Supplemental
Tables
Table
S1
Mean values, errors, and correlations between estimates of
uncalibrated (norm) flow parameters
using spectral regression learning
and reference values (N=753).
Variable
Measured Flow
with the
Ultrasound
Estimated Flow
with
SRL
푡
!
!"#
$%&!
(ms)
193
±
23
191
±
22
d
푄
"#$
%&'"
0.018
±
0.002
0.0017
±
0.002
mean(
푄
%&'"
)
0.23
±
0.03
0.25
±
0.03
푅푀푆퐸
[
푄
()*
%&'"
vs
푄
"(#)
%&'"
] (%)
-
3.00
푟
[mean(
푄
()*
%&'"
)
vs
mean(
푄
"(#)
%&'"
)]
-
0.6
2
푙푖푚푖푡
표푓
퐴푔푟푒푒푚푒푛푡
[mean(
푄
()*
%&'"
)
vs
mean(
푄
"(#)
%&'"
)]
-
0.094
푀푒푎푛
퐷푖푓푓푒푟푒푛푐푒
[mean(
푄
()*
%&'"
)
vs
mean(
푄
"(#)
%&'"
)]
-
0.018
All values are (mean ± SD) except as noted.
푄
"#$%
denotes the scaled flow waveform from zero to one. SRL
indicates spectral regression learning, propose
d
in this study.
Table
S2
Wave separation and hemodynamic parameters per age group.
Parameters
Q1
40
-
52 years
Q2
53
-
65
years
Q3
66
-
78
years
Q4
79
-
91
years
Pf, mmHg
80
± 10
88
±
14
95
±
16
105
±
20
Pb, mmHg
51
± 6
55
±
8
57
±
8
60
±
8
TD, ms
78
± 19
65
±
18
62
±
19
57
±
16
RI
0.39
± 0.02
0.39
±
0.02
0.38
±
0.03
0.36
±
0.03
PWV
7.6
± 1.6
9.1
±
2.5
11.6
±
3.5
14.5
±
5.3
Zc, mmHg.s/ml
0.14
± 0.03
0.17
±
0.06
0.19
±
0.07
0.24
±
0.09
Heart Rate, bpm
61
± 9
62
±
10
62
±
10
62
±
11
*
Q1 includes 44 subjects, Q2 includes 324 subjects, Q3 includes 308 subjects, and Q4 includes 77 subjects.
Table
S3
Physiological correlates of the Carotid
-
femoral pulse wave velocity as a base model (
N=753
).
Variables
훽
SE
(
훽
)
CI
(
훽
)
P
Model for Carotid
-
femoral PWV in the absence of WSA, adjusted
퐑
ퟐ
=
0.383
Sex
1.026
0.324
(0.389, 1.663)
<0.01
+
Age, years
0.231
0.013
(0.187, 0.239)
<0.001
Mean Blood Pressure, mmHg
0.069
0.009
(0.051, 0.088)
<0.001
Heart Rate, bpm
3.517
0.663
(2.216, 4.818)
<0.001
+
Indicates a non
-
significant parameter. PWV and WSA stand for Pulse Wave Velocity and Wave Separation
Analysis, respectively.