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Sensors and Materials, Vol. 34, No. 7 (2022) 2723–2734
MYU Tokyo
S & M 3001

Separation of Aortic and Pulmonary Components from Second
Heart Sounds without an Assumption of Statistical Independence
Shun Muramatsu,* Seiichi Takamatsu, and Toshihiro Itoh
Graduate School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8654, Japan(Received November 22, 2021; accepted February 15, 2022)

Keywords: heart sound, A2–P2 splitting interval, independent component analysis, nonlinear transient
chirp signal model, demixing vector

A novel algorithm to separate aortic (A2) and pulmonary (P2) components from the second
heart sound (S2) without assuming that A2 and P2 are statistically independent, and with
optimizing demixing vectors using root-mean-square error (RMSE) between outputs and signal
models as cost function is successfully demonstrated. Conventional methods to estimate theA2–P2 splitting interval (SI) based on the separation of A2 and P2 using independent component
analysis (ICA) are subject to distortions due to the fact that A2 and P2 are not strictly statisticallyindependent. Therefore, we propose an algorithm to separate A2 and P2 without assuming their
independence. In the proposed algorithm, a nonlinear transient chirp signal model is introduced
as the proper models of A2 and P2, and the separated sound is optimized to be closest to the A2/P2-like model. To evaluate the proposed algorithm, SI estimation was performed for S2
simulated with 60 common SI patterns. The results show that the proposed algorithm canestimate SI stably regardless of the independence of A2 and P2, and can estimate SI with 95%
limits of agreement of −0.305 ± 2.15 ms, which is about 69% smaller as the error range than
ICA.

1.

Introduction

Cardiovascular disease is the most common cause of death worldwide, accounting for about
32% of all deaths in 2019.(1) Among these diseases, pulmonary arterial hypertension is a frequentand serious complication of several cardiovascular and respiratory diseases, which is difficult toassess noninvasively.(2) Therefore, there is a growing need for a non-invasive method to estimate
pulmonary arterial pressure (PAP) accurately and safely. It is known that pulmonary arterial
hypertension increases the time interval between the onset of the aortic (A2) and pulmonary(P2) components of the second heart sound (S2) and the dominant frequency of P2.(3) Therefore,
the A2–P2 splitting interval (SI) is considered to be useful for the non-invasive estimation of
PAP.
Some methods to estimate SI in the time-frequency domain have been proposed.(4–6) Xu et al.
masked the Wigner–Vile distribution on the time-frequency plane of S2, estimated the*Corresponding author: e-mail: smuramatsu@g.ecc.u-tokyo.ac.jp

https://doi.org/10.18494/SAM3738

ISSN 0914-4935 © MYU K.K.
https://myukk.org/

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instantaneous frequency and amplitude changes of A2 and P2, and reconstructed them from S2
to obtain the waveform of P2. However, this method does not give optimal results when the
instantaneous frequencies of A2 and P2 overlap significantly in the time-frequency plane,
because S2 is regarded as a monocomponent.(7) Therefore, Vivek et al. proposed a method toautomatically extract the A2 and P2 components from S2 based on the assumption that A2 andP2 are statistically independent in time.(7) This assumption of mutual independence of A2 andP2 enables the separation of A2 and P2 from S2 using the blind source separation (BSS)
technique. There are several attempts to separate A2 and P2 from S2 by applying BSS
techniques, especially independent component analysis (ICA).(7–10) However, there is no
guarantee that ICA will work properly, because A2 and P2 have overlapping timing, and they are
not considered to be strictly statistically independent.
The purpose of this study is to propose a new algorithm to extract A2 and P2 from S2 without
an assumption that A2 and P2 are statistically independent. The proposed algorithm separates
S2 into the most A2/P2-like sounds, optimizing it to be close to the A2 and P2 models.

2.

Proposed Algorithm

2.1

Overview of proposed algorithm

Figure 1 shows an overview of our proposed algorithm. The proposed method outputs the A2
and P2 by optimizing the correct demixing vector and multiplying it by the observed S2. Thisapproach is similar to that of the demixing matrix in ICA. ICA uses the kurtosis indicating a
degree of independence as a cost function for optimizing the demixing matrix.(11) In contrast,
the proposed method introduces signal models for A2 and P2, and uses the root-mean-squareerror (RMSE) between the models and the output as a cost function.
There are several attempts to model A2 and P2.(12–14) In this study, we adopt the nonlinear
transient chirp signal model.(12) According to this model, S2 is expressed as Eqs. (1)–(3).

=S2 (t ) AA2 (t )sin(ϕ A2 (t )) + AP2 (t − t0 )sin(ϕ P2 (t )) + n(t )Fig. 1.

Overview of proposed algorithm.

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t
t

−  −
πt
A
=
(t ) a0 1 − e 8  e 16 sin


60





1 







ϕ (t ) = 2π  f1t + 2 f 2t 2 

(2)

(3)

The first term of S2(t) represents the A2 component and the second term represents the P2
component. A(t) and φ(t) are the time variations of the amplitude and frequency of A2 and P2
corresponding to the subscripts. t0 is the time of occurrence of A2 and P2. n(t) is the additive
white Gaussian noise. These equations include the two parameters as characteristic frequencies( f1 and f2). These express individual differences in the waveform. Since a single valvular soundis a simple sound such as a collision sound associated with the closing of a valve, these twoparameters can absorb individual differences.
Figure 2 shows the waveforms of A2 and P2 generated by Eqs. (1)–(3) with the following
parameters also shown in Table 1. a0 is a parameter used to adjust the amplitude, so it is set to 1.0
here. f1 = 24, f2 = 226 for A2 and f1 = 22, f2 = 178 for P2 based on previous studies.(12) In practice,these two components are mixed by some mixing matrix and observed as S2 on the chest wallusing a stethoscope, for example.
2.2

Signal processing flow

Figure 3 shows the signal process flow of the proposed algorithm. S2a and S2b are two
observed S2s. The A2/P2 model list includes 12000 A2/P2 models generated using Eqs. (1)–(3).

Fig. 2.

(Color online) Simulated waveforms of A2 and P2.

Table 1
Parameters of nonlinear transient chirp signal model for A2, P2, and A2/P2 model list.
t0 (ms)a0
f1 (Hz)
A2
0
1
24
P2
0–59
1
22
A2/P2 model list
0–59
1
20

f2 (Hz)226
178
100–300

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These are fixed f1 to 20 Hz, swept f2 from 100 Hz to 300 Hz, and swept t0 from 0 to 59 ms asshown in Table 1. Originally, f1 is also variable, but since the same waveform can be expressedby only changing f2 as when f1 is changed, f1 is fixed this time. First, the demixing vector w is
optimized so that the RMSE between w[S2a S2b]T and the model is minimized. This is done forall 12000 models.
Figure 4 shows an example of the RMSE contour calculated for all models. Here, a low
RMSE indicates that the observed S2 is likely to contain that model. This is because w[S2a S2b]Twith the correct demixing vector represents one true source signal. Therefore, we select the twomodels with the lowest RMSE as the most likely A2 and P2 models, as shown by the arrows in
Fig. 4. This example is for S2, a mixture of A2 with a t0 of 0 ms and P2 with a t0 of 20 ms, andthe correct t0 is selected. There may be some errors for some S2; these errors are due to the factthat f1 is fixed, which is not a problem for the subsequent separation process in the proposed
algorithm.
Finally, wA2 and wP2 are optimized again to minimize the RMSE between the separated
sound and the selected model, and output yA2 and yP2 corresponding to the estimated A2 and P2.

Fig. 3.

Fig. 4.

Signal process flow of proposed algorithm.

Example of RMSE contour for every A2/P2 model. The two arrows point to the most A2/P2-like models.

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3.

Evaluation Method

3.1

Separation of A2 and P2 from S2

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The evaluation experiment is conducted with simulated S2 sounds. Figure 5 shows anexample of two S2s simulated under the mixing conditions shown in Fig. 6, where the aortic andpulmonary valves are 50 mm apart and the distance between the two stethoscopes on the chestwall is also 50 mm. This spatial condition is not strictly necessary; we just needed to determine
this condition to generate the mixing matrix for simulating S2. It is only important that the twostethoscopes are not in the positions where the observed sounds are the same; in other words, themixing matrix is invertible. This mixing is instantaneous, which does not take into account thereverberation and time delay in the actual propagation path from the heart valve to the chest
wall. It has been reported that the source separation algorithm for instantaneous mixing can beextended to the time-frequency domain to deal with reverberation and delay.(15) For simplicity,
we discuss the proposed algorithm under the condition of instantaneous mixing.
Sixty pairs of S2s with different SIs were simulated. The sampling frequency is 1000 Hz. As
an example, the waveforms of the simulated S2 with SIs of 0, 10, 20, and 30 ms are shown in
Fig. 7. When the SI is less than about 20 ms, it is difficult to estimate the SI from the waveform

Fig. 5.

(Color online) Two examples of simulated waveform of S2.

Fig. 6.

Spatial conditions during A2 and P2 mixing simulations.

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Fig. 7.

(Color online) Examples of S2 simulated with four different SIs.

or even to detect the split between A2 and P2. The separation of the proposed algorithm isevaluated by separating 60 pairs of S2 into A2/P2 using both ICA and the proposed algorithmand calculating the RMSE between the separated waveform and the true waveform.
3.2

Estimation of SI between A2 and P2

After separating A2 and P2 from these simulated S2, the SI is estimated. The time-centroidbased method(7) is then used to estimate the SI. The time centroid cx of N samples of a signal x(t)is defined asNN
cx = i 1 iT
abs( x (iT ))abs( x (iT )) .
=
=i 1

∑

∑

(4)

T is the sampling time interval. In other words, when cA2 and cP2 are the time centroids of theseparated A2 and P2, respectively, the SI, ts, is calculated as ts = cP2 − cA2. The accuracy of the
SI estimation was evaluated by calculating the SI from the separated signals using ICA and the
proposed algorithm and comparing it with the true value. Note that the FastICA algorithm in thepython library scikit-learn is used for ICA, and the Broyden–Fletcher–Goldfarb–Shanno
algorithm was used for optimization in the proposed algorithm.

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4.

Results and Discussion

4.1

Separation of A2 and P2

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First, conventional ICA attempted to separate the simulated S2 into A2 and P2. To evaluate
the performance, RMSE between the separated A2/P2 and the true A2/P2 is calculated; thelower the RMSE, the better the separation performance. Thirty trials were conducted for all S2s,
and 1800 data were obtained. Figure 8 shows the average value of RMSE for each SI. When the
SI is 27 ms or higher, RMSE is stable at about 0.02, which indicates successful separation. On
the other hand, when the SI is less than 26 ms, RMSE is not stable and A2/P2 is not separated
successfully with most of the SIs. This is due to the distortion caused by trying to force two
signals that are not independent from each other to become independent. As an example, Fig. 9shows separated sounds that failed to separate when the SI was 0 ms. ICA does not inherently
maintain the scale of the waveform. Therefore, we used the back projection(16) to match the scaleof the observed sound on the chosen microphone. A2 has a high distortion, and P2 has a reduced
amplitude. These waveforms have some distortions caused by the non-independence of A2 andP2.
The same process was applied to the proposed algorithm. As an example, Fig. 10 shows theseparated sounds when SI is 0 ms. Compared with the ICA separation shown in Fig. 9, theseparated A2 and P2 almost have no distortion. The box plots of the RMSE obtained by the two
methods are shown in Fig. 11. In this figure, the results when the SI is 26 ms or less are plotted,

Fig. 8.

(Color online) RMSE for A2/P2 separation using ICA against every SI.

Fig. 9.

(Color online) Waveforms of separated A2 and P2 using ICA.

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Fig. 10. (Color online) Waveforms of separated A2 and P2 using the proposed algorithm.

Fig. 11. (Color online) Box plots of RMSE as separation performance of ICA and the proposed algorithm.

because both methods obtained good RMSE when the SI is 27 ms or more. The mean value ofthe proposed algorithm is 0.032, which is 0.017 smaller than the ICA value of 0.049, indicatinghigh accuracy. The interquartile range of the proposed algorithm is 0.011, which is 0.024 smaller
than that of ICA (0.035), indicating stable results with small variation.
4.2

Estimation of SI between A2 and P2

The SI was estimated from the A2 and P2 waveforms separated in the previous section using
the time-centroid method. We discuss only the results when the SI is less than 20 ms, in which it
is difficult to detect the A2–P2 splitting visually. Figure 12(a) shows the relationship between thetrue SI and the SI estimated by ICA. The slope of the regression line denoted by the dashed line
is 0.659, and the correlation coefficient is 0.844. The closer these values are to 1, the morecorrectly the SI is estimated. Figure 12(a) shows that the SI is not estimated well, especially inthe region where the SI is close to 0 ms. On the other hand, Fig. 12(b) shows the result of the
proposed algorithm. The slope of the regression line and the correlation coefficient are 0.959 and
0.982, respectively, indicating that the proposed algorithm can estimate the SI with higher
accuracy than ICA.

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(a)

(b)

Fig. 12. (Color online) Comparison of estimated SI with true SI using (a) ICA and (b) the proposed algorithm. The
dashed line is the linear regression line.

Figure 13 shows the Bland–Altman analysis(17) of these results, which plots the difference
between the estimated SI and true SI values against the mean of the estimated and true SI values.
Figure 13(a) shows the result of ICA. In particular, the estimation accuracy is very low andunstable when the mean of the estimated and true SI is between 0 ms and about 8 ms. The 95%
limit of agreement (LOA), denoted by the dashed line, is −0.649 ± 6.85 ms. On the other hand,Fig. 13(b) shows the result of the proposed algorithm: the estimation accuracy is stable regardlessof SI, and LOA is −0.305 ± 2.15 ms, which is about 9.40 ms smaller than that of ICA. These
indicate that the proposed algorithm can estimate SI with less error than ICA.
Although there is no general required accuracy about the SI estimation yet, it can beconsidered as follows. The only method that can directly measure PAP is pulmonary artery
catheterization, and it is invasive. The only practical method of non-invasive but restrictive
estimation of PAP is Doppler echocardiography, and its standard error of the estimate (SEE)against the catheterization is 8 mmHg.(18) On the other hand, our proposed method by SI
estimation can calculate PAP (as mean of systole and diastole) using Eqs. (5) and (6).(2,19)
PPAP =−13.7 + 19.2 I NSII NSI =ISI RHR
600

(5)

(6)

Here, P PAP is PAP (mmHg), INSI is normalized SI (ms), ISI is SI (ms), and RHR is heart rate(bpm). From these equations, 8 mmHg as SEE can be converted to LOA. The converted LOA ismean ±3.48 ms (mean is not important here) assuming a heart rate of 60. The LOA of our
proposed method is ± 2.15 ms, which is lower than that of Doppler echocardiography (±3.48 ms).

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(a)

(b)

Fig. 13. (Color online) Bland–Altman analysis of the estimated SI using (a) ICA and (b) the proposed algorithm.
The solid line indicates the mean of the difference between the estimated and true SIs. The dashed lines indicate
LOA.

It indicates that the proposed method is practical. However, it is important to note that theseresults are based on simulations, and it is necessary to continue experiments on actual heartsounds and improvements on the algorithm.
4.3

Limitations

Figure 11 shows the separation performance of the proposed algorithm in terms of RMSE.
The overall RMSE is low, indicating that the proposed algorithm has stable and high separation
performance compared with ICA. On the other hand, the RMSE of some samples exceeded 0.08.
The reason for this is that the proposed algorithm mistakenly selected a model that has a similar
waveform, but the generation timing is off by one wavelength. However, Fig. 13(b) shows that
the SI estimation result of the proposed algorithm is very stable. The reason for this is that even
if a wrong waveform is mistakenly selected, the proposed algorithm does not distort the A2 andP2 waveforms significantly. For that reason, the time-centroid method is performed well withsome wrong selection. As a result, the proposed algorithm can estimate the SI stably with LOA
of −0.305 ± 2.15 ms, which is 9.40 ms smaller than the ICA.
The proposed algorithm needs to select the two models that are most likely to be A2 and P2,
as shown by two arrows in Fig. 4. This time, this selection process worked well. However, if, forexample, A2 and P2 have the same parameters ( f1 and f2) and the SI is 0 ms, i.e., A2 and P2 arenot split, the minima to be selected may overlap. Then the overlapping minima will be selected
as the A2 model, and then the model with the second lowest RMSE will be selected as the P2
model, even though it is not the correct model. In other words, the model selection process of the
proposed algorithm does not work in this situation. Therefore, we are trying to solve this problem
by determining that the SI is close to 0 ms by considering the respiratory variability of the SI.

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5.

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Conclusions

A novel algorithm to separate A2 and P2 from S2 with optimizing demixing vectors using
RMSE between outputs and A2/P2 signal models as cost function is successfully demonstrated.
This proposed algorithm does not require an assumption that A2 and P2 are statisticallyindependent. In this algorithm, the nonlinear transient chirp signal model is introduced as amodel of A2 and P2. First, this algorithm seeks two models that are most close to multiplicationof the observed sound and the optimized demixing vector. Then, by optimizing the separated
sound to be close to the sought models, the proposed algorithm can separate S2 into the mostA2/P2-like sounds. To evaluate the proposed algorithm, 120 types of S2 with different SIs are
simulated, and the separation of A2/P2 and the estimation of SI are performed using the
proposed algorithm. As a result, in the waveform separation, the mean value of RMSE, whichindicates the magnitude of waveform distortion, was reduced by about 35% and the interquartile
range by about 69% using the proposed algorithm compared with the conventional ICA
algorithm. In the estimation of SI, the correlation coefficient between the SI estimated using the
proposed algorithm and the true SI is 0.982. Furthermore, the proposed algorithm can be used toestimate the SI with an error of −0.305 ± 2.15 ms as LOA. This error range is about 69% smaller
than that obtained by the conventional method (−0.649 ± 6.85 ms). These mean that the proposed
algorithm can be used to estimate SI stably regardless of the independence of A2 and P2. By
estimating PAP from SI calculated by this method, we can establish an accurate and safe noninvasive evaluation method for pulmonary arterial hypertension.

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