- Email: [email protected]
Chaotic dynamics of respiratory sounds
Chaos, Solitons and Fractals 29 (2006) 1054–1062 www.elsevier.com/locate/chaos
Chaotic dynamics of respiratory sounds C. Ahlstrom a
a,b,*
, A. Johansson a, P. Hult
a,b
, P. Ask
a,b
Department of Biomedical Engineering, Linko¨pings Universitet, IMT/LIU, Universitetssjukhuset, S-58185 Linko¨ping, Sweden b ¨ rebro University Hospital, S-70185 O ¨ rebro, Sweden Biomedical Engineering, O Accepted 24 August 2005
Abstract There is a growing interest in nonlinear analysis of respiratory sounds (RS), but little has been done to justify the use of nonlinear tools on such data. The aim of this paper is to investigate the stationarity, linearity and chaotic dynamics of recorded RS. Two independent data sets from 8 + 8 healthy subjects were recorded and investigated. The first set consisted of lung sounds (LS) recorded with an electronic stethoscope and the other of tracheal sounds (TS) recorded with a contact accelerometer. Recurrence plot analysis revealed that both LS and TS are quasistationary, with the parts corresponding to inspiratory and expiratory flow plateaus being stationary. Surrogate data tests could not provide statistically sufficient evidence regarding the nonlinearity of the data. The null hypothesis could not be rejected in 4 out of 32 LS cases and in 15 out of 32 TS cases. However, the Lyapunov spectra, the correlation dimension (D2) and the Kaplan–Yorke dimension (DKY) all indicate chaotic behavior. The Lyapunov analysis showed that the sum of the exponents was negative in all cases and that the largest exponent was found to be positive. The results are partly ambiguous, but provide some evidence of chaotic dynamics of RS, both concerning LS and TS. The results motivate continuous use of nonlinear tools for analysing RS data. 2005 Elsevier Ltd. All rights reserved.
1. Introduction The origin of normal respiratory sounds (RS) is not completely clear, and it is likely that many different mechanisms are involved. Lung sounds (LS) are probably induced by turbulence at the level of lobar or segmental bronchi [1]. The branching patterns of the airways form a fractal structure where each branch repeats itself over multiple length scales [2]. It is known that dynamic processes, such as acoustic waves, propagating over fractal structures exhibit fluctuations in time that follow power law distributions [3]. Since power laws are closely related to fractals, it is reasonable to believe that LS are also fractal. Tracheal sounds (TS) differ from LS. Although both are driven by airflow, the two sounds are generated from different sources [4]. Possible chaotic behavior of TS can be motivated by cavitation noise (which is known to be chaotic) in the vocal tract. Nonlinear analyses have previously been used for detection of adventitious LS such as crackles [5] and wheezes [6]. Various algorithms for calculation of LS fractal dimension have been investigated in [7], the Hurst exponent of TS was * Corresponding author. Address: Department of Biomedical Engineering, Linko¨pings Universitet, IMT/LIU, Universitetssjukhuset, S-58185 Linko¨ping, Sweden. Tel.: +46 13 222484; fax: +46 13 101902. E-mail address: [email protected] (C. Ahlstrom).
0960-0779/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.08.197
C. Ahlstrom et al. / Chaos, Solitons and Fractals 29 (2006) 1054–1062
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scrutinized in [8] and TS dynamics was analyzed in [9]. Despite the growing interest in nonlinear methods for RS analysis, little has been done to justify the use of these techniques. Specifically, many authors assume that RS are stationary, high dimensional and nonlinear. The aim of this paper is to verify these assumptions and to demonstrate the chaotic dynamics of RS. Tests of stationarity, nonlinearity as well as an analysis of chaotic behavior are performed on recorded RS data. One LS set and one TS set, acquired with different measurement equipment to assure independency, are investigated.
2. The experimental data sets This study was approved by the ethical committee at Linko¨ping University Hospital and was performed on sixteen healthy subjects (age 20–30 years). All test subjects gave their informed consent. A first data set (LS) was acquired in an ordinary, not soundproof, environment from eight subjects (6 male, 2 female). The recordings were taken at the left lung apex with an electronic stethoscope (M30, Meditron ASA, Norway) and digitized at 44.1 kHz with 16 bits per sample (Analyzer, Meditron ASA, Norway). An airflow reference signal was recorded with a Fleisch tube (Hugo Sachs Elektronik, Germany) and digitized at 1 kHz with 12-bits per sample (DAQCard-700, National Instruments, USA). A second data set (TS) was acquired in a soundproof room from the other eight subjects (2 male, 6 female). These recordings were taken at the throat over the trachea, with a contact accelerometer (EMT25C, Siemens, Sweden) and digitized together with the flow reference signal at 5 kHz (DAQCard-500, National Instruments, USA). All subjects were asked to breathe normally with a medium flow rate (about 15 mL/s/kg) for 1 min in the sitting position (LS set) or in the supine position (TS set). No further instructions were given. The recorded signals in the LS set were downsampled to 5 kHz using an anti-aliasing lowpass filter during the resampling process. Inspirations and expirations were extracted from all signals, inspiratory data was defined as sections corresponding to flow rates above 60% of the measurement maximum flow, and correspondingly for expiratory sounds. Inspiratory and expiratory data for both sets was tested for stationarity (Section 3.1), nonlinearity (Section 3.2) and chaotic behavior (Section 3.3). Processing and analysis were conducted in Matlab (The MathWorks Inc., USA), TSTool [10] and TISEAN [11].
3. Methodology The dynamics of a time discrete system is determined by its possible states in a multivariate vector space (called state space or phase space). The transitions between the states are described by vectors, and these vectors form a trajectory describing the time evolution of the system. An observed signal S is a projection from this multivariate state space onto a one-dimensional time-series. S can be considered as a set of n scalar measurements S ¼ fsk g k ¼ 1; 2; . . . ; n
ð1Þ
from which a sequence of N d-dimensional vectors ai can be constructed using Takens delay embedding theorem ai ¼ fsi ; siþs ; siþ2s ; . . . siþðd1Þs g
i ¼ 1; 2; . . . ; N
ð2Þ
where s is a delay parameter and d is the embedding dimension [12]. The purpose of the embedding is to unfold the projection back into a reconstructed state space that is dynamically and topologically equivalent to the state space that generated the process S [13]. In this study, s and d were chosen based on the techniques of average mutual information [12] and false nearest neighbors (Caos method) [14]. 3.1. Test of stationarity Nonstationarity introduces a tendency that points close in phase space are also close in time. To investigate this property, recurrence plots (RP) were used. RPs graphically displays the recurrence of states of a system (i.e. how often a small region in state space is visited) and can be used on rather short time series. An RP is a nearly symmetric N · N matrix where a point (i, j) represents the Euclidian distance (or any other norm) between ai and aj: RPði;jÞ ¼ Hðei kai aj kÞ
ð3Þ
where i, j = 1, . . . , N, ei is a cut-off distance, k Æ k is the Euclidian norm and H(Æ) is the Heaviside function. A significant density change near the diagonal in an RP is an indication of drift and hidden periodicities, and hence nonstationarity.
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RPs were calculated, using embedding parameters according to Table 2 and ei = 0.1, for each breathing cycle of LS and TS. The results were examined manually. 3.2. Tests of nonlinearity An analysis commonly used for determining if data is consistent with various linear systems (or if it is consistent with something else, i.e. a nonlinear system) is the method of surrogate data [15]. Surrogate data, generated to represent a null hypothesis, is compared to the original data in order to reject or approve the null hypothesis. In this study two different null hypotheses were used; that the signal was a realization of a linear Gaussian stochastic process or a realization of a linear Gaussian stochastic process followed by a static (possibly) nonlinear observation function. The surrogate data was created using phase randomization (FT) or iterative Amplitude Adapted Fourier Transform (iAAFT) [15]. Fifty surrogate time series were created for each investigated signal and for each null hypothesis, giving a two-sided test at the 95% level of significance. A third order quantity, asymmetry under time reversal, was used as a measure of nonlinearity [15]: /ðdÞ ¼
N X 1 ðsk skd Þ3 N d k¼dþ1
ð4Þ
where d is a time lag. A simple rank-testing procedure was used for statistically testing whether the original data was drawn from the same distribution as the surrogate data [15]. Basically, /(d) is sorted in increasing order and the rank index for the investigated (original) data is returned. If this rank is unity or 51, this means that the original data lies in the tail of the distribution, and the null can be rejected. The tests for nonlinearity were performed on inspiration as well as expiration for both LS and TS. 3.3. Extraction of chaotic invariants There are no single criteria to decide whether a given time series is the result of a chaotic process. Instead a set of criteria, established in [15], is used to investigate the chaotic dynamics of inspiratory and expiratory LS and TS: 1. The process should be nonlinear. 2. The correlation dimension (D2) of the process should be fractal. The value of D2 should also converge to a constant value for increasing embedding dimension. 3. At least one of the Lyapunov exponents (ki) should be positive (to ensure that the system is sensitive to its initial conditions). 4. The sum of all Lyapunov exponents should be negative (as the dynamics have to be physically realizable). 5. The Kaplan–Yorke dimension (DKY) should be close in numerical value to D2. The investigation of criteria 2–4 for the experimental data is briefly described in the following paragraphs. D2 reflects the way in which the delay vectors are distributed in state space. D2 is hence a measure of the underlying dynamical systems complexity. To minimize the influence of noise, the maximum likelihood estimation procedure outlined in [16] was used to estimate D2. One of the defining attributes of a strange attractor is that it displays exponential sensitivity to initial conditions. The Lyapunov exponent is a measure of the separation rate of close trajectories in state space. There are as many Lyapunov exponents as there are state space dimensions, each indicating the divergence (positive exponents) or convergence (negative exponents) of trajectories in different directions. The sum of Lyapunov exponents, i.e. the rate of state space contraction or expansion, should be negative for a physical process. However, the largest exponent is always positive in a chaotic system. In this study, the method of local linear fits was employed to calculate the spectrum of Lyapunov exponents [11,17]. The maximum exponent and the sum of all exponents were calculated to check criteria 3 and 4. Finally, DKY gives another measure of the dimension of the strange attractor (defined in terms of the attractors Lyapunov exponents [18]).
4. Results Table 1 summarizes the amount of data used in the study and a typical example of LS can be seen in Fig 1. Fig. 2a shows an example of the average mutual information for inspiratory LS data from test subject one. The optimal s,
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Table 1 Number of breaths and total signal length of inspiration data and expiration data for LS and TS LS set (n = 8)
Number of breaths Total signal length (s)
TS set (n = 8)
Inspiration
Expiration
Inspiration
Expiration
5.7 ± 3.2 7.9 ± 2.5
5.9 ± 3.2 9.1 ± 3.1
2.0 ± 0.0 1.6 ± 0.7
2.0 ± 0.0 2.1 ± 0.7
Amplitude (V)
0.6
Inspiration
Expiration
0.4 0.2 0 –0.2 –0.4 17
18
19
20
21
22
23
Time (s) Fig. 1. Example of typical LS data from test subject 1 showing one respiratory cycles.
(a)
(b)
1 0.8
3
Cao’s method
Mutual Information
4
2
1
0.6 0.4 0.2
0 0
50
100
Time Delay (sample)
5
10 15 Dimension (d)
20
Fig. 2. Determination of the time delay s (using mutual information) and the embedding dimension d (using Caos method) from a typical inspiratory LS from test subject 1 (resulting in s = 40 and d = 7).
where the mutual information reaches a minimum for the first time, was 40. Note that the corresponding minimum value of the mutual information is small, indicating that the signal samples are essentially independent when s = 40, and yet it is nonzero, indicating that they are correlated with each other. The optimal embedding dimension for the same signal example was found to be d = 7, see Fig. 2b. Results from the estimation of embedding parameters for all subjects are given in Table 2. The dimensions for all signals were around d = 7, while the time delay differed between the groups (LS: s 40 and TS: s 3). The embedding parameters found for inspiration and expiration within LS and TS were similar. The recurrence plots indicated that both LS and TS data were nonstationary. However, when looking at segments of inspiratory and expiratory data separately, fairly uniform recurrence patterns were found. Fig. 3 gives an example where a more dense recurrence plot can be seen in expiration compared to inspiration. An example of the distribution of the test statistic (Eq. (4)) is illustrated in Fig. 4 and results from the rank order procedure for both tests and all signals can be seen in Table 3. From Table 3, it can be seen that most LS data is classified as nonlinear with both FT and iAAFT (in 28 out of the 32 cases). For the TS set, the null hypotheses could not be rejected in 15 out of the 32 cases.
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Table 2 Embedding parameters for the state space reconstruction of RS during inspiration and expiration Subject
Time delay (s)
Embedding dim (d)
Inspiration
Expiration
Inspiration
Expiration
LS set 1 2 3 4 5 6 7 8 Mean Std.
39 41 45 36 38 28 41 41 38.63 5.04
40 41 44 38 41 31 40 41 39.50 3.82
7 6 8 7 6 6 6 6 6.63 1.06
7 7 6 6 6 6 7 6 6.50 0.53
TS set 9 10 11 12 13 14 15 16 Mean Std.
2 3 5 3 4 3 2 2 3.00 1.07
3 3 3 3 4 3 3 3 3.13 0.35
6 8 8 6 6 6 7 8 7.13 1.25
7 7 7 6 7 7 8 6 6.88 0.64
Fig. 3. Examples of recurrence plots from subject 1 (LS data) during inspiration and expiration. The data within the inspiratory segment and the expiratory segment are fairly uniform, while there is a clear difference in density between the two segments.
D2 and DKY are shown side-by-side for all test data in Table 4. Both D2 and DKY are fractal and even though their values are not equal, they are in the same order of magnitude. The numbers of calculated Lyapunov exponents varied between 7, 8 and 9 corresponding to the embedding dimension. Table 5 summarizes the Lyapunov spectra for inspiratory and expiratory data for the LS set while results from the TS set are given in Table 6. The sum of the Lyapunov exponents was found to be negative in all cases and the largest Lyapunov exponent was found to be positive.
5. Discussion Recorded RS signals have been tested for stationarity, nonlinearity and chaotic behavior. The results indicate that the data is stationary within the rather stable flow plateaus of inspiration and expiration. Surrogate tests for nonlinearity indicated that LS are nonlinear, but the null hypothesis could neither be confirmed nor rejected for TS. This could
C. Ahlstrom et al. / Chaos, Solitons and Fractals 29 (2006) 1054–1062
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10
8
6
4
2
0 –15
–10
–5
φ
rev
0
(τ)
5 –7
x 10
Fig. 4. Example of a histogram (subject 1) showing the distribution of the test statistic under the assumption of the null hypothesis (iAAFT). The vertical line denotes the value of the test statistic for the original lung sound data during inspiration. Clearly, the original data is not drawn from this distribution and the null hypothesis can be rejected.
Table 3 Results from the rank order test, using surrogate data created with FT and iAAFT Surrogate
Inspiration
Expiration
LS set FT iAAFT
0 0
0 4
TS set FT iAAFT
3 5
3 4
The table shows the number of cases (out of 8) where the null hypothesis could not be rejected.
either mean that TS data is linear or that the recorded signals, at this particular sampling, only cover a fraction of the rich dynamics and hence appear linear stochastic in some cases. The analysis of chaotic properties indicated that D2 is fractal and rather close to DKY. From the Lyapunov analysis it was observed that the Lyapunov exponents were essentially independent of test subject, that the largest exponents were positive and that the sums of the exponents were negative. Even though s, D2, DKY, and the Lyapunov exponents are fairly consistent within the two data sets, they differ considerably between LS and TS. A likely explanation is the low pass filtering effect of the chest wall. Sounds on the chest surface have a lower amplitude than TS, are louder at low frequencies and decrease in amplitude with increasing frequency [19]. This means that LS are obscured or damped compared to TS, and that some of the dynamics are probably filtered out. Another difference between TS and LS are that they are generated from different geometrical structures and a third reason could be differences in the measurement equipment, particularly pre-filters in the microphone amplifiers. Dynamical analysis of experimental time series requires the data to be drawn from a stationary process. For example, when using surrogate data, a significant result of the test does not necessarily indicate a nonlinear or even chaotic process underlying the data. It might simply be caused by a nonstationarity of the process. If RS are stationary in some way, it is in a quasistationary way. By using data from the rather stable flow plateaus of inspiration and expiration, we intend to reduce the problem of nonstationarity as much as possible. The results demonstrate the chaotic dynamics of RS. However, there are several reasons why the outcome should be interpreted with care. The data set consists of 16 healthy individuals, and the results are thus based on a rather small population. Furthermore, the results indicate that LS and TS should be handled separately, not as two independent sources of the same phenomena. Calculations of the Lyapunov spectrum assume that there exist well defined Jacobians. In particular, when attractors are thin in the embedding space, some (or all) of the local Jacobians might be estimated poorly. This also affects the estimation of DKY, and it is not surprising to see that DKY somewhat differs from D2. This
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Table 4 Correlation dimension (D2) and Kaplan–Yorke dimension (DKY) of all test data Subject
Inspiration
Expiration
D2
DKY
D2
DKY
LS set 1 2 3 4 5 6 7 8 Mean Std.
5.900 5.661 5.957 5.993 6.605 4.750 6.269 5.572 5.838 0.548
6.308 5.842 7.466 6.622 5.842 5.775 5.716 5.603 6.145 0.632
5.069 5.146 5.400 5.837 6.414 4.650 6.428 4.770 5.461 0.695
6.429 6.326 5.598 5.922 5.763 6.209 6.415 5.418 6.010 0.391
TS set 9 10 11 12 13 14 15 16 Mean Std.
4.083 4.469 5.740 4.225 4.621 4.432 4.185 3.982 4.467 0.556
3.099 5.142 5.163 3.974 4.130 3.608 3.913 4.470 4.187 0.716
4.395 4.359 4.777 4.102 4.750 4.185 4.250 4.478 4.412 0.247
4.488 4.384 4.738 4.011 4.615 4.311 5.059 3.536 4.393 0.464
Table 5 Lyapunov exponents of inspiratory data Subject
k1
LS set 1 2 3 4 5 6 7 8
9.84 13.26 9.00 13.07 14.53 18.17 12.53 12.16
5.23 6.70 5.18 6.93 6.89 8.96 6.12 5.98
2.05 2.09 2.79 2.96 2.38 3.26 1.99 1.91
0.44 1.41 0.88 0.03 1.44 1.84 1.61 1.55
3.25 6.21 1.06 3.24 6.63 8.72 5.94 6.24
7.28 17.31 3.07 7.77 18.80 25.25 17.71 17.51
TS set 9 10 11 12 13 14 15 16
41.78 32.17 22.55 40.93 33.84 32.44 38.28 35.28
6.24 20.61 11.33 16.91 13.39 10.31 14.03 18.31
38.46 3.36 0.04 11.77 7.13 13.77 12.44 4.38
96.79 13.88 8.53 44.98 33.41 50.87 44.44 23.69
193.54 35.89 19.46 108.09 75.00 106.88 88.22 54.31
387.30 62.32 36.44 246.96 177.67 264.45 145.48 91.77
k2
k3
k4
k5
k6
k7
k8
19.44 6.46 20.59
16.11
116.06 65.79
235.29 144.01
344.77 167.79
362.09
P
i ki
13.29 2.88 8.86 8.67 3.06 5.43 4.63 5.26 668.08 407.30 240.32 353.96 245.97 393.22 583.05 650.42
3
All numbers have been multiplied with 10 .
could also be due to the unavoidable presence of noise or differences between the algorithms used for estimating D2 and DKY. One of the main driving forces behind RS research is to find useful measurements to interpret differences between RS in health and disease. The results of this study indicate that healthy RS are chaotic, but whether this holds true for pathologic situations remains to be investigated.
C. Ahlstrom et al. / Chaos, Solitons and Fractals 29 (2006) 1054–1062
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Table 6 Lyapunov exponents of expiratory data Subject
k1
LS set 1 2 3 4 5 6 7 8
10.13 9.15 10.52 15.67 12.88 12.19 10.18 10.76
4.78 5.22 3.26 6.31 5.33 6.52 5.38 4.73
2.94 2.93 2.58 3.52 3.01 3.36 2.81 2.50
0.20 0.44 1.79 1.52 1.52 4.76 0.22 2.26
3.08 3.03 6.13 6.69 6.18 10.89 2.95 6.81
7.07 6.84 16.97 19.65 17.78 27.01 7.02 18.45
TS set 9 10 11 12 13 14 15 16
39.89 32.87 42.10 39.89 31.28 32.83 33.00 30.56
18.95 13.89 20.28 13.77 16.44 12.57 19.35 12.09
0.28 4.51 1.42 11.91 0.71 6.09 3.87 16.76
25.75 26.18 26.69 47.47 18.79 28.46 15.28 48.36
59.21 57.51 56.48 107.52 46.01 59.47 35.54 108.95
106.26 110.13 105.98 250.48 79.87 114.61 62.08 267.82
k2
k3
k4
k5
k6
k7
k8
17.99 17.92
i ki
10.49 10.93 8.52 2.35 4.26 20.59 9.86 9.53
18.04
250.58 252.30 240.13 186.91 253.56 111.11
P
241.71
383.24 403.88 368.31 363.73 284.57 416.79 409.50 399.23
All numbers have been multiplied with 103.
Acknowledgements This study was supported by the Swedish Agency for Innovation Systems and the Swedish Research Council.
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[18] Chlouverakis KE, Sprott JC. A comparison of correlation and Lyapunov dimensions. Physica D—Nonlin Phenom 2005;200:156–64. [19] Pasterkamp H, Kraman SS, Wodicka GR. Respiratory sounds. Advances beyond the stethoscope. Am J Respir Crit Care Med 1997;156:974–87.
Chaotic dynamics of respiratory sounds C. Ahlstrom a
a,b,*
, A. Johansson a, P. Hult
a,b
, P. Ask
a,b
Department of Biomedical Engineering, Linko¨pings Universitet, IMT/LIU, Universitetssjukhuset, S-58185 Linko¨ping, Sweden b ¨ rebro University Hospital, S-70185 O ¨ rebro, Sweden Biomedical Engineering, O Accepted 24 August 2005
Abstract There is a growing interest in nonlinear analysis of respiratory sounds (RS), but little has been done to justify the use of nonlinear tools on such data. The aim of this paper is to investigate the stationarity, linearity and chaotic dynamics of recorded RS. Two independent data sets from 8 + 8 healthy subjects were recorded and investigated. The first set consisted of lung sounds (LS) recorded with an electronic stethoscope and the other of tracheal sounds (TS) recorded with a contact accelerometer. Recurrence plot analysis revealed that both LS and TS are quasistationary, with the parts corresponding to inspiratory and expiratory flow plateaus being stationary. Surrogate data tests could not provide statistically sufficient evidence regarding the nonlinearity of the data. The null hypothesis could not be rejected in 4 out of 32 LS cases and in 15 out of 32 TS cases. However, the Lyapunov spectra, the correlation dimension (D2) and the Kaplan–Yorke dimension (DKY) all indicate chaotic behavior. The Lyapunov analysis showed that the sum of the exponents was negative in all cases and that the largest exponent was found to be positive. The results are partly ambiguous, but provide some evidence of chaotic dynamics of RS, both concerning LS and TS. The results motivate continuous use of nonlinear tools for analysing RS data. 2005 Elsevier Ltd. All rights reserved.
1. Introduction The origin of normal respiratory sounds (RS) is not completely clear, and it is likely that many different mechanisms are involved. Lung sounds (LS) are probably induced by turbulence at the level of lobar or segmental bronchi [1]. The branching patterns of the airways form a fractal structure where each branch repeats itself over multiple length scales [2]. It is known that dynamic processes, such as acoustic waves, propagating over fractal structures exhibit fluctuations in time that follow power law distributions [3]. Since power laws are closely related to fractals, it is reasonable to believe that LS are also fractal. Tracheal sounds (TS) differ from LS. Although both are driven by airflow, the two sounds are generated from different sources [4]. Possible chaotic behavior of TS can be motivated by cavitation noise (which is known to be chaotic) in the vocal tract. Nonlinear analyses have previously been used for detection of adventitious LS such as crackles [5] and wheezes [6]. Various algorithms for calculation of LS fractal dimension have been investigated in [7], the Hurst exponent of TS was * Corresponding author. Address: Department of Biomedical Engineering, Linko¨pings Universitet, IMT/LIU, Universitetssjukhuset, S-58185 Linko¨ping, Sweden. Tel.: +46 13 222484; fax: +46 13 101902. E-mail address: [email protected] (C. Ahlstrom).
0960-0779/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.08.197
C. Ahlstrom et al. / Chaos, Solitons and Fractals 29 (2006) 1054–1062
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scrutinized in [8] and TS dynamics was analyzed in [9]. Despite the growing interest in nonlinear methods for RS analysis, little has been done to justify the use of these techniques. Specifically, many authors assume that RS are stationary, high dimensional and nonlinear. The aim of this paper is to verify these assumptions and to demonstrate the chaotic dynamics of RS. Tests of stationarity, nonlinearity as well as an analysis of chaotic behavior are performed on recorded RS data. One LS set and one TS set, acquired with different measurement equipment to assure independency, are investigated.
2. The experimental data sets This study was approved by the ethical committee at Linko¨ping University Hospital and was performed on sixteen healthy subjects (age 20–30 years). All test subjects gave their informed consent. A first data set (LS) was acquired in an ordinary, not soundproof, environment from eight subjects (6 male, 2 female). The recordings were taken at the left lung apex with an electronic stethoscope (M30, Meditron ASA, Norway) and digitized at 44.1 kHz with 16 bits per sample (Analyzer, Meditron ASA, Norway). An airflow reference signal was recorded with a Fleisch tube (Hugo Sachs Elektronik, Germany) and digitized at 1 kHz with 12-bits per sample (DAQCard-700, National Instruments, USA). A second data set (TS) was acquired in a soundproof room from the other eight subjects (2 male, 6 female). These recordings were taken at the throat over the trachea, with a contact accelerometer (EMT25C, Siemens, Sweden) and digitized together with the flow reference signal at 5 kHz (DAQCard-500, National Instruments, USA). All subjects were asked to breathe normally with a medium flow rate (about 15 mL/s/kg) for 1 min in the sitting position (LS set) or in the supine position (TS set). No further instructions were given. The recorded signals in the LS set were downsampled to 5 kHz using an anti-aliasing lowpass filter during the resampling process. Inspirations and expirations were extracted from all signals, inspiratory data was defined as sections corresponding to flow rates above 60% of the measurement maximum flow, and correspondingly for expiratory sounds. Inspiratory and expiratory data for both sets was tested for stationarity (Section 3.1), nonlinearity (Section 3.2) and chaotic behavior (Section 3.3). Processing and analysis were conducted in Matlab (The MathWorks Inc., USA), TSTool [10] and TISEAN [11].
3. Methodology The dynamics of a time discrete system is determined by its possible states in a multivariate vector space (called state space or phase space). The transitions between the states are described by vectors, and these vectors form a trajectory describing the time evolution of the system. An observed signal S is a projection from this multivariate state space onto a one-dimensional time-series. S can be considered as a set of n scalar measurements S ¼ fsk g k ¼ 1; 2; . . . ; n
ð1Þ
from which a sequence of N d-dimensional vectors ai can be constructed using Takens delay embedding theorem ai ¼ fsi ; siþs ; siþ2s ; . . . siþðd1Þs g
i ¼ 1; 2; . . . ; N
ð2Þ
where s is a delay parameter and d is the embedding dimension [12]. The purpose of the embedding is to unfold the projection back into a reconstructed state space that is dynamically and topologically equivalent to the state space that generated the process S [13]. In this study, s and d were chosen based on the techniques of average mutual information [12] and false nearest neighbors (Caos method) [14]. 3.1. Test of stationarity Nonstationarity introduces a tendency that points close in phase space are also close in time. To investigate this property, recurrence plots (RP) were used. RPs graphically displays the recurrence of states of a system (i.e. how often a small region in state space is visited) and can be used on rather short time series. An RP is a nearly symmetric N · N matrix where a point (i, j) represents the Euclidian distance (or any other norm) between ai and aj: RPði;jÞ ¼ Hðei kai aj kÞ
ð3Þ
where i, j = 1, . . . , N, ei is a cut-off distance, k Æ k is the Euclidian norm and H(Æ) is the Heaviside function. A significant density change near the diagonal in an RP is an indication of drift and hidden periodicities, and hence nonstationarity.
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RPs were calculated, using embedding parameters according to Table 2 and ei = 0.1, for each breathing cycle of LS and TS. The results were examined manually. 3.2. Tests of nonlinearity An analysis commonly used for determining if data is consistent with various linear systems (or if it is consistent with something else, i.e. a nonlinear system) is the method of surrogate data [15]. Surrogate data, generated to represent a null hypothesis, is compared to the original data in order to reject or approve the null hypothesis. In this study two different null hypotheses were used; that the signal was a realization of a linear Gaussian stochastic process or a realization of a linear Gaussian stochastic process followed by a static (possibly) nonlinear observation function. The surrogate data was created using phase randomization (FT) or iterative Amplitude Adapted Fourier Transform (iAAFT) [15]. Fifty surrogate time series were created for each investigated signal and for each null hypothesis, giving a two-sided test at the 95% level of significance. A third order quantity, asymmetry under time reversal, was used as a measure of nonlinearity [15]: /ðdÞ ¼
N X 1 ðsk skd Þ3 N d k¼dþ1
ð4Þ
where d is a time lag. A simple rank-testing procedure was used for statistically testing whether the original data was drawn from the same distribution as the surrogate data [15]. Basically, /(d) is sorted in increasing order and the rank index for the investigated (original) data is returned. If this rank is unity or 51, this means that the original data lies in the tail of the distribution, and the null can be rejected. The tests for nonlinearity were performed on inspiration as well as expiration for both LS and TS. 3.3. Extraction of chaotic invariants There are no single criteria to decide whether a given time series is the result of a chaotic process. Instead a set of criteria, established in [15], is used to investigate the chaotic dynamics of inspiratory and expiratory LS and TS: 1. The process should be nonlinear. 2. The correlation dimension (D2) of the process should be fractal. The value of D2 should also converge to a constant value for increasing embedding dimension. 3. At least one of the Lyapunov exponents (ki) should be positive (to ensure that the system is sensitive to its initial conditions). 4. The sum of all Lyapunov exponents should be negative (as the dynamics have to be physically realizable). 5. The Kaplan–Yorke dimension (DKY) should be close in numerical value to D2. The investigation of criteria 2–4 for the experimental data is briefly described in the following paragraphs. D2 reflects the way in which the delay vectors are distributed in state space. D2 is hence a measure of the underlying dynamical systems complexity. To minimize the influence of noise, the maximum likelihood estimation procedure outlined in [16] was used to estimate D2. One of the defining attributes of a strange attractor is that it displays exponential sensitivity to initial conditions. The Lyapunov exponent is a measure of the separation rate of close trajectories in state space. There are as many Lyapunov exponents as there are state space dimensions, each indicating the divergence (positive exponents) or convergence (negative exponents) of trajectories in different directions. The sum of Lyapunov exponents, i.e. the rate of state space contraction or expansion, should be negative for a physical process. However, the largest exponent is always positive in a chaotic system. In this study, the method of local linear fits was employed to calculate the spectrum of Lyapunov exponents [11,17]. The maximum exponent and the sum of all exponents were calculated to check criteria 3 and 4. Finally, DKY gives another measure of the dimension of the strange attractor (defined in terms of the attractors Lyapunov exponents [18]).
4. Results Table 1 summarizes the amount of data used in the study and a typical example of LS can be seen in Fig 1. Fig. 2a shows an example of the average mutual information for inspiratory LS data from test subject one. The optimal s,
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Table 1 Number of breaths and total signal length of inspiration data and expiration data for LS and TS LS set (n = 8)
Number of breaths Total signal length (s)
TS set (n = 8)
Inspiration
Expiration
Inspiration
Expiration
5.7 ± 3.2 7.9 ± 2.5
5.9 ± 3.2 9.1 ± 3.1
2.0 ± 0.0 1.6 ± 0.7
2.0 ± 0.0 2.1 ± 0.7
Amplitude (V)
0.6
Inspiration
Expiration
0.4 0.2 0 –0.2 –0.4 17
18
19
20
21
22
23
Time (s) Fig. 1. Example of typical LS data from test subject 1 showing one respiratory cycles.
(a)
(b)
1 0.8
3
Cao’s method
Mutual Information
4
2
1
0.6 0.4 0.2
0 0
50
100
Time Delay (sample)
5
10 15 Dimension (d)
20
Fig. 2. Determination of the time delay s (using mutual information) and the embedding dimension d (using Caos method) from a typical inspiratory LS from test subject 1 (resulting in s = 40 and d = 7).
where the mutual information reaches a minimum for the first time, was 40. Note that the corresponding minimum value of the mutual information is small, indicating that the signal samples are essentially independent when s = 40, and yet it is nonzero, indicating that they are correlated with each other. The optimal embedding dimension for the same signal example was found to be d = 7, see Fig. 2b. Results from the estimation of embedding parameters for all subjects are given in Table 2. The dimensions for all signals were around d = 7, while the time delay differed between the groups (LS: s 40 and TS: s 3). The embedding parameters found for inspiration and expiration within LS and TS were similar. The recurrence plots indicated that both LS and TS data were nonstationary. However, when looking at segments of inspiratory and expiratory data separately, fairly uniform recurrence patterns were found. Fig. 3 gives an example where a more dense recurrence plot can be seen in expiration compared to inspiration. An example of the distribution of the test statistic (Eq. (4)) is illustrated in Fig. 4 and results from the rank order procedure for both tests and all signals can be seen in Table 3. From Table 3, it can be seen that most LS data is classified as nonlinear with both FT and iAAFT (in 28 out of the 32 cases). For the TS set, the null hypotheses could not be rejected in 15 out of the 32 cases.
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Table 2 Embedding parameters for the state space reconstruction of RS during inspiration and expiration Subject
Time delay (s)
Embedding dim (d)
Inspiration
Expiration
Inspiration
Expiration
LS set 1 2 3 4 5 6 7 8 Mean Std.
39 41 45 36 38 28 41 41 38.63 5.04
40 41 44 38 41 31 40 41 39.50 3.82
7 6 8 7 6 6 6 6 6.63 1.06
7 7 6 6 6 6 7 6 6.50 0.53
TS set 9 10 11 12 13 14 15 16 Mean Std.
2 3 5 3 4 3 2 2 3.00 1.07
3 3 3 3 4 3 3 3 3.13 0.35
6 8 8 6 6 6 7 8 7.13 1.25
7 7 7 6 7 7 8 6 6.88 0.64
Fig. 3. Examples of recurrence plots from subject 1 (LS data) during inspiration and expiration. The data within the inspiratory segment and the expiratory segment are fairly uniform, while there is a clear difference in density between the two segments.
D2 and DKY are shown side-by-side for all test data in Table 4. Both D2 and DKY are fractal and even though their values are not equal, they are in the same order of magnitude. The numbers of calculated Lyapunov exponents varied between 7, 8 and 9 corresponding to the embedding dimension. Table 5 summarizes the Lyapunov spectra for inspiratory and expiratory data for the LS set while results from the TS set are given in Table 6. The sum of the Lyapunov exponents was found to be negative in all cases and the largest Lyapunov exponent was found to be positive.
5. Discussion Recorded RS signals have been tested for stationarity, nonlinearity and chaotic behavior. The results indicate that the data is stationary within the rather stable flow plateaus of inspiration and expiration. Surrogate tests for nonlinearity indicated that LS are nonlinear, but the null hypothesis could neither be confirmed nor rejected for TS. This could
C. Ahlstrom et al. / Chaos, Solitons and Fractals 29 (2006) 1054–1062
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10
8
6
4
2
0 –15
–10
–5
φ
rev
0
(τ)
5 –7
x 10
Fig. 4. Example of a histogram (subject 1) showing the distribution of the test statistic under the assumption of the null hypothesis (iAAFT). The vertical line denotes the value of the test statistic for the original lung sound data during inspiration. Clearly, the original data is not drawn from this distribution and the null hypothesis can be rejected.
Table 3 Results from the rank order test, using surrogate data created with FT and iAAFT Surrogate
Inspiration
Expiration
LS set FT iAAFT
0 0
0 4
TS set FT iAAFT
3 5
3 4
The table shows the number of cases (out of 8) where the null hypothesis could not be rejected.
either mean that TS data is linear or that the recorded signals, at this particular sampling, only cover a fraction of the rich dynamics and hence appear linear stochastic in some cases. The analysis of chaotic properties indicated that D2 is fractal and rather close to DKY. From the Lyapunov analysis it was observed that the Lyapunov exponents were essentially independent of test subject, that the largest exponents were positive and that the sums of the exponents were negative. Even though s, D2, DKY, and the Lyapunov exponents are fairly consistent within the two data sets, they differ considerably between LS and TS. A likely explanation is the low pass filtering effect of the chest wall. Sounds on the chest surface have a lower amplitude than TS, are louder at low frequencies and decrease in amplitude with increasing frequency [19]. This means that LS are obscured or damped compared to TS, and that some of the dynamics are probably filtered out. Another difference between TS and LS are that they are generated from different geometrical structures and a third reason could be differences in the measurement equipment, particularly pre-filters in the microphone amplifiers. Dynamical analysis of experimental time series requires the data to be drawn from a stationary process. For example, when using surrogate data, a significant result of the test does not necessarily indicate a nonlinear or even chaotic process underlying the data. It might simply be caused by a nonstationarity of the process. If RS are stationary in some way, it is in a quasistationary way. By using data from the rather stable flow plateaus of inspiration and expiration, we intend to reduce the problem of nonstationarity as much as possible. The results demonstrate the chaotic dynamics of RS. However, there are several reasons why the outcome should be interpreted with care. The data set consists of 16 healthy individuals, and the results are thus based on a rather small population. Furthermore, the results indicate that LS and TS should be handled separately, not as two independent sources of the same phenomena. Calculations of the Lyapunov spectrum assume that there exist well defined Jacobians. In particular, when attractors are thin in the embedding space, some (or all) of the local Jacobians might be estimated poorly. This also affects the estimation of DKY, and it is not surprising to see that DKY somewhat differs from D2. This
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Table 4 Correlation dimension (D2) and Kaplan–Yorke dimension (DKY) of all test data Subject
Inspiration
Expiration
D2
DKY
D2
DKY
LS set 1 2 3 4 5 6 7 8 Mean Std.
5.900 5.661 5.957 5.993 6.605 4.750 6.269 5.572 5.838 0.548
6.308 5.842 7.466 6.622 5.842 5.775 5.716 5.603 6.145 0.632
5.069 5.146 5.400 5.837 6.414 4.650 6.428 4.770 5.461 0.695
6.429 6.326 5.598 5.922 5.763 6.209 6.415 5.418 6.010 0.391
TS set 9 10 11 12 13 14 15 16 Mean Std.
4.083 4.469 5.740 4.225 4.621 4.432 4.185 3.982 4.467 0.556
3.099 5.142 5.163 3.974 4.130 3.608 3.913 4.470 4.187 0.716
4.395 4.359 4.777 4.102 4.750 4.185 4.250 4.478 4.412 0.247
4.488 4.384 4.738 4.011 4.615 4.311 5.059 3.536 4.393 0.464
Table 5 Lyapunov exponents of inspiratory data Subject
k1
LS set 1 2 3 4 5 6 7 8
9.84 13.26 9.00 13.07 14.53 18.17 12.53 12.16
5.23 6.70 5.18 6.93 6.89 8.96 6.12 5.98
2.05 2.09 2.79 2.96 2.38 3.26 1.99 1.91
0.44 1.41 0.88 0.03 1.44 1.84 1.61 1.55
3.25 6.21 1.06 3.24 6.63 8.72 5.94 6.24
7.28 17.31 3.07 7.77 18.80 25.25 17.71 17.51
TS set 9 10 11 12 13 14 15 16
41.78 32.17 22.55 40.93 33.84 32.44 38.28 35.28
6.24 20.61 11.33 16.91 13.39 10.31 14.03 18.31
38.46 3.36 0.04 11.77 7.13 13.77 12.44 4.38
96.79 13.88 8.53 44.98 33.41 50.87 44.44 23.69
193.54 35.89 19.46 108.09 75.00 106.88 88.22 54.31
387.30 62.32 36.44 246.96 177.67 264.45 145.48 91.77
k2
k3
k4
k5
k6
k7
k8
19.44 6.46 20.59
16.11
116.06 65.79
235.29 144.01
344.77 167.79
362.09
P
i ki
13.29 2.88 8.86 8.67 3.06 5.43 4.63 5.26 668.08 407.30 240.32 353.96 245.97 393.22 583.05 650.42
3
All numbers have been multiplied with 10 .
could also be due to the unavoidable presence of noise or differences between the algorithms used for estimating D2 and DKY. One of the main driving forces behind RS research is to find useful measurements to interpret differences between RS in health and disease. The results of this study indicate that healthy RS are chaotic, but whether this holds true for pathologic situations remains to be investigated.
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Table 6 Lyapunov exponents of expiratory data Subject
k1
LS set 1 2 3 4 5 6 7 8
10.13 9.15 10.52 15.67 12.88 12.19 10.18 10.76
4.78 5.22 3.26 6.31 5.33 6.52 5.38 4.73
2.94 2.93 2.58 3.52 3.01 3.36 2.81 2.50
0.20 0.44 1.79 1.52 1.52 4.76 0.22 2.26
3.08 3.03 6.13 6.69 6.18 10.89 2.95 6.81
7.07 6.84 16.97 19.65 17.78 27.01 7.02 18.45
TS set 9 10 11 12 13 14 15 16
39.89 32.87 42.10 39.89 31.28 32.83 33.00 30.56
18.95 13.89 20.28 13.77 16.44 12.57 19.35 12.09
0.28 4.51 1.42 11.91 0.71 6.09 3.87 16.76
25.75 26.18 26.69 47.47 18.79 28.46 15.28 48.36
59.21 57.51 56.48 107.52 46.01 59.47 35.54 108.95
106.26 110.13 105.98 250.48 79.87 114.61 62.08 267.82
k2
k3
k4
k5
k6
k7
k8
17.99 17.92
i ki
10.49 10.93 8.52 2.35 4.26 20.59 9.86 9.53
18.04
250.58 252.30 240.13 186.91 253.56 111.11
P
241.71
383.24 403.88 368.31 363.73 284.57 416.79 409.50 399.23
All numbers have been multiplied with 103.
Acknowledgements This study was supported by the Swedish Agency for Innovation Systems and the Swedish Research Council.
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