On analysis of the periodicity attributes of the photo-plethysmograph signal to assess the cardiovascular state
PP Kanjilal*1, S Bandyopadhyay*2, and J Bhattacharya3
*Department of E & ECE, I.I.T., Kharagpur, 721-302, India
1Now at USARIEM, Natick, MA 01760-5007, USA
2Now with Department of Biomedical Engg.,
Johns Hopkins Univ., Baltimore, USA
3Commission for Scientific Visualization, Austrian Academy of Sciences, Sonnenfelsgasse 19/2, A-1010 Vienna, Austria
The photo-plethysmographic signal recorded from the finger, representing the blood pressure excursions, is studied. It is understood that any periodic or nearly periodic series can be considered to be composed of a series of cyclical segments where each segment is characterized by three attributes, namely the periodicity or period length, the periodic pattern and the multiplicative scaling factor. The dynamics of these periodicity (or p-) attributes for the plethysmograph signal are individually as well as collectively analysed to study the cardiovascular state. A new class of surrogate series based on the p-attributes is used for analysis. The signal dynamics is also analysed by mapping the p-attributes in a novel p-attribute space, where each point maps a periodic segment; from the distribution of points, a measure for the assessment of the cardiovascular condition is proposed.
1Introduction
There has been several studies showing the degree of irregularity in the HeartRate Variability (HRV) signal being related to the pathologies of the cardiovascular system [1-5]. The present work attempts to study the cardiovascular dynamics through the analysis of the photo-plethysmographic signal recorded from the finger, which bears similarity to the arterial blood-pressure signal.
Any periodic or nearly periodic signal can be characterized by three specific periodicity attributes or ‘p-attributes’: the periodicity, the repetitive pattern, and the multiplicative scaling factor associated with the successive periodic segments [6,7]; for a plethysmograph signal, all three periodicity attributes may vary with time. While the HRV information, implicit with this signal, is one of the periodicity attributes, the present study tries to assess the cardiovascular status through the analysis of all the three periodicity attributes of the plethysmograph signal.
The study is based on nonlinear dynamical analysis. Surrogate series are generated from the plethysmographic signal and are analysed to assess the determinism associated with the underlying process. To address the question of whether the three periodicity attributes individually influence the nonlinearity, a new class of surrogate series are generated and the degree of determinism is detected in the light of the generated surrogates. A new approach has been proposed for the qualitative assessment of the underlying dynamics of the cardiac process by mapping the plethysmograph series in the orthogonal framework of the three characteristic p-attributes.
2Periodicity detection and decomposition of the signal
2.1 Periodicity detection
{x(k)}is configured into a mxN matrix N, row-wise. Singular value decomposition (SVD) [6,8] of Nis given by N= USVT, where U and V are orthogonal matrices, S is diagonal containing the singular values (s1, s2,..., sr), r = min(m,N), s1 s2... sr. If {x(k)}is strictly periodic with periodicity N, Rank(N)= 1. Again, if {x(k)} is arranged into a matrix iN with row-length iN, i = any positive integer, Rank(iN)= 1. If {x(k)} is nearly periodic with periodicity N, iNcan be full-rank but s1 will be dominant. Thus the presence of a dominant periodic component in {x(k)} will result in repetitive peaks (at multiples of the concerned period length N) in the s1/s2vs. row length spectrum, called the ‘singular value ratio' (SVR)-spectrum or the 'periodicity (or p-)spectrum' [6,7] of {x(k)}. The best rank-1 approximation of the periodic component of periodicity N in {x(k)}is given by the time series represented by u1s1v1T,where u1and v1 are the first columns of U and V respectively.
2.2 Decomposition
The decomposition is performed with a moving data window; the periodicity within the data segment is detected using the p-spectrum and the corresponding periodic pattern is obtained. The successive repetitive segments are linearly stretched or compressed to the detected period length; the corresponding data matrix is singular value decomposed and the pattern and the scaling factor associated with the most dominant component is noted.
Thus the first extracted periodic segment will have the normalised pattern p1, the period length l1, and its scaling factor as (say) a1. The data window is now receded by the length l1 and the process is repeated, finally leading to three sequences (of the extracted regular component) for the successive cyclical segments: {pi}, {li} and {ai}. The nature of the periodicity attributes (for Case-2, Table 1) is shown in Fig. 1.
3 Surrogate analysis
3.1AAFT and IAAFT surrogates
The surrogate data can be used to detect nonlinearity [9]. In the Amplitude Adjusted Fourier Transform (AAFT) surrogate generator [9], the original series {x} is rescaled to conform to Gaussian distribution, Fourier transformed, the phases are randomised, and the rank ordering of the reverse transformed series is used to reorder {x}generating the surrogate. In the improved iterative AAFT (IAAFT) generator [10], {x} is rescaled to {y(i)} conforming to Gaussian distribution, (b) the power spectrum of {y(i)}, is made the same as that of {x} (say, {Xk2}), producing {y*(i)}, and (c) {x} is rank ordered as per {y*(i)} to achieve the similar amplitude distribution producing {y(i+1)}; the steps (b) and (c) are repeated to achieve closeness to the power spectrum and amplitude distribution features of {x}. The proposed surrogate generators follow similar principles, while the three p-attributes are individually shuffled.
3.2 The proposed nonlinear surrogates
The surrogate series with randomised ‘period-length or periodicity’ is generated as follows. The periodicity sequence {li} in randomly shuffled, with the associated scaling and pattern features remaining unchanged; the patterns pi are stretched or compressed as per the new period lengths. The power spectrum of this series is replaced with ({Xk2}), and its successive cyclical segments are rank ordered as per the rank order of the periodicity sequence in the original data. The last two steps are repeated twice to generate the surrogate. The null hypothesis to be tested is that all the information is contained in the pattern and scaling, and not in the periodicity factor.
The surrogates corresponding to the randomisation of the scaling factor and the randomisation of the pattern sequences are generated similarly; in the latter case the rank ordering is defined according to the correlation against the global pattern pg, which is considered to be the average of the local patterns pi. The surrogates are used for the detection of determinism in the plethysmograph signal as follows.
Remark: These surrogates are essentially nonlinear because they all exhibit the noisy limit cycle structure in their phase space (see Fig.2).
4Detection of determinism
Detection of determinism by conventional means can be problematic in case of limited data, lack of stationarity etc. In the present work, a method based on nonlinearly scaled singular value distributions [11] has been used.
{x(k)} and {xsurr(k)} series are configured into different mxn matrices A and Asurr with varying n. For each case the singular values are computed and the total energy in the data is mapped into R (=30, not a limitation) normalised singular values. The averaged values of the R normalized singular values sm(i), (i = 1 to R) over different configurations are computed. The scaled distribution i2sm(i) is plotted against i for both {x(k)} and the three surrogates{xsurr(k)}. Since the singular values are arranged in a non-increasing order, for a purely stochastic series, i2sm(i) will be gradually increasing tending to saturate at a high value, whereas for a deterministic series the singular values si will be having significantly decreasing magnitudes (with increasing i), and hence i2sm(i) will be eventually decreasing tending to saturate at a low value. The distributions of i2sm(i) for {x(k)} and for each of the surrogate series {xsurr(k)} are separately compared using Mann-Whitney (M-W) rank-sum statistic (Z) [12]; if |Z|>1.96, the associated null hypothesis of {x(k)} being generated by a linear stochastic process can be rejected with >95% confidence level [12].
5Proposed p-attribute map and measurement of dynamics
The three periodicity attributes, which are independent of each other, can be used to define a 3-dimensional p-attribute space (or simply p-space), where the individual axes are defined by the periodicity, the scaling factor (each being normalised), and the pattern correlation against the global pattern. A point in this space represents one cyclical segment. Thus the mapping of a signal in the p-space portrays the complete dynamics of the underlying process. The trajectory joining these points sequentially depicts the temporal evolution of the process in the p-space.
The mean value of distances (Cd) moved between successive points in the p-space is proposed to be used as a measure of the dynamics associated with the mapped process.
7 Results
Five subjects were studied and the results are summarised in Table-1.
In case of AAFT surrogates (Fig. 2(e)-(f)), the noisy limit cycle structure associated with the data is destroyed, whereas the same is retained in the proposed nonlinear surrogates (Fig. 2(g)-(j)).
For normal cardiac states (Case-1, Case-3) there is distinct determinism in the signal due to pattern variation; |Zp| drops from 5.2 in Case-1 and 3.39 in Case-3 to lower values for pathological cases (i.e. in Cases 2, 4 and 5) implying lower determinism. Nearly similar behaviour for periodicity variation is observed from the |Zl| values but not for the scaling variations, implying lack of information in the variation of the scaling factors.
The p-map dynamics (Table 1 and Fig.3) providing a collective picture of the p-attribute variations, distinctly follow the cardiovascular stability. Higher dynamics is observed for stable cases (e.g., Cd = 0.3 and 0.42 in Case 1 and 3 respectively), with Cd falling to lower values for pathological cases.
8Conclusions
A generic scheme for analysing the photo-plethysmograph signal in terms of the time-varying periodicity attributes has been presented. First the determinism is detected through scaled singular value distributions of the signal against three specific classes of nonlinear surrogates generated from the periodicity attributes. The effect of the scaling factor is largely stochastic in nature, whereas the degree of determinism in the signal due to the pattern and periodicity variations is seen to be a function of the cardiovascular state.
Further, p-space mapping is shown to be a new way of studying dynamics of an irregular cyclical process. Results show that the cardiovascular status is reflected in the distribution of points in the p-map; decreased dynamics is detected in terms of movements in the p-map in case of pathologies.
TABLE – 1 Summary of the results
Case / History / |Zl| / |Zp| / |Za| / Cd1 / Post operative stable case / 2.90 / 5.20 / 1.57 / 0.30
2 / Post-operative quasi-stable case (cardio thoracic surgery) / 0.65 / 2.96 / 1.39 / 0.14
3 / Patient with no cardiac problem / 3.41 / 3.39 / 1.94 / 0.42
4 / Suffers from pain in heart,
hypertensive. / 2.22 / 2.88 / 1.26 / 0.20
5 / Has a defective valve / 2.44 / 2.72 / 1.76 / 0.24
1
Figure 1 (a) The variations in the periodicity over a segment of data in Case-2, (b) the variations of the scaling factor, (c) the varying pattern profiles of the successive periodic segments with the periodic length defaulted to a fixed value.
Figure 2 (a)-(b) The photo-plethysmograph signal for the post-operative stable patient (Case-1 in Table 1), and its phase-space plot. (c)-(d) The regular component extracted from the signal (a), and the corresponding phase-space plot. (e)-(f) The AAFT surrogate series generated from (c), and the corresponding phase-space plot. (g)-(h) The surrogate generated by shuffling the scaling factors and the corresponding phase-space plot. (i)-(j) The surrogate generated by shuffling the patterns of the periodic segments and the corresponding phase-space plot. Note that in both (h) and (j) the noisy limit cycle structure is retained, whereas in (f) it is destroyed.
Figure 3 (a) The scaled singular value distributions against the same for the surrogates generated through the shuffling of the periodicity, the pattern and the scaling factor sequences for the post-operative stable case (Case-1 in Table-1). (b) The profile of the distances between successive points in the p-map.
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