Stochastic Modeling of the PPG Signal: A Synthesis-by ... - IEEE Xplore

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IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 60, NO. 9, SEPTEMBER 2013

Stochastic Modeling of the PPG Signal: A Synthesis-by-Analysis Approach With Applications Diego Mart´ın-Mart´ınez∗ , Student Member, IEEE, Pablo Casaseca-de-la-Higuera, Marcos Mart´ın-Fern´andez, and Carlos Alberola-L´opez, Senior Member, IEEE

Abstract—In this paper, we propose a stochastic model of photoplethysmographic signals that is able to synthesize an arbitrary number of other statistically equivalent signals to the one under analysis. To that end, we first preprocess the pulse signal to normalize and time-align pulses. In a second stage, we design a single-pulse model, which consists of ten parameters. In the third stage, the time evolution of this ten-parameter vector is approximated by means of two autoregressive moving average models, one for the trend and one for the residue; this model is applied after a decorrelation step which let us to process each vector component in parallel. The experiments carried out show that the model we here propose is able to maintain the main features of the original signal; this is accomplished by means of both a linear spectral analysis and also by comparing two measures obtained from a nonlinear analysis. Finally, we explore the capability of the model to: 1) track physical activity; 2) obtain statistics of clinical parameters by model sampling; and 3) recover corrupted or missing signal epochs by synthesis. Index Terms—ARMA, benchmarking, modeling, PCA, photoplethysmography, signal synthesis, statistical validation, subject simulation.

I. INTRODUCTION CCORDING to the World Health Organization [1], cardiovascular diseases are the main cause of death in the world. Their prevention and prompt diagnosis constitute major challenges in medical and scientific fields. Nowadays, diagnosis and follow up of cardiovascular pathologies is rarely performed without resorting to functional information that physiological signals convey [3]. In order to extract meaningful interpretations from these signals, this information has to be “decoded” by either simple processes such as visual inspection or more

A

Manuscript received November 12, 2012; revised February 28, 2013; accepted March 27, 2013. Date of publication April 12, 2013; date of current version August 16, 2013. This work was supported in part by the University of Valladolid and Banco Santander under the FPI-UVa Fellowship, the Spanish Ministerio de Ciencia e Innovaci´on and the Fondo Europeo de Desarrollo Regional (FEDER) under Research Grant TEC2010–17982, the Spanish Instituto de Salud Carlos III (ISCIII) under Research Grants PI11–01492, PI11-02203, and by the Spanish Centro para el Desarrollo Tecnol´ogico Industrial (CDTI) under the cvREMOD (CEN–20091044) project. Asterisk indicates corresponding author. ∗ D. Mart´ın-Mart´ınez is with the Laboratorio de Procesado de Imagen (LPI), Universidad de Valladolid, 47002 Valladolid, Spain (e-mail: dmarmar@ lpi.tel.uva.es). P. Casaseca-de-la-Higuera, M. Mart´ın-Fern´andez and C. Alberola-L´opez are with the Laboratorio de Procesado de Imagen (LPI), Universidad de Valladolid, 47002 Valladolid, Spain (e-mail: [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TBME.2013.2257770

involved procedures that require a higher degree of processing; signal analysis is occasionally complemented with signal synthesis. In the literature, attempts have been described to synthesize signals, with a special focus on ECGs [4]–[6]. Signal synthesis have then been used for a number of applications, such as model-based filtering [6], model-based compression [7], classification with clinical applications [8], or even person authentication in biometric systems [9]. Such models may be trained using well-known public databases, such as PhysioBank from the PhysioNet project [10]. However, other types of signals, such as photoplethysmographic (PPG), have received, to the best of our knowledge, considerable less attention. Specifically, in [11], a periodic component of the PPG signal is sought and dynamics are studied by means of nonlinear methods. Evidence is gathered about the differences between controls and cases in terms of the correlation dimensionality. Another attempt is [12], in which the spectral content of the PPG signal is found different between cases and controls. These two papers, however, concentrate on the analysis part but do not seem to provide a methodology for signal synthesis; such an attempt is described in [13], in which different parametric models are compared and the authors conclude that an autoregressive model with exogenous input provides the better fit among those studied. These models are only appropriate for short PPG records since they do not consider the nonstationary behavior of the signal. The synthesis-by-analysis paradigm is a powerful tool which can provide: 1) specific signal models that summarize patient properties in a reduced set of parameters; 2) model-based rules for signal synthesis that can be used to simulate either pathologies or patient evolution by simply changing the model parameters; 3) completion of missing epochs in failed acquisitions; 4) a common reference framework to compare different patients and construct automatic diagnostic aid systems; 5) benchmarking for testing numerous biomedical signal processing techniques at controlled distorsion or noise levels with different sampling frequencies. All these tasks can be carried out with the added value that both density functions as well as confidence intervals of any parameter of interest can be easily approximated by sampling the model. This provides an objective information that can be used, for instance, in personalized health systems, to trigger alarms in real time when parameters go beyond the safety intervals derived from the model variability. This constitutes a statistical atlas of a patient, in which not only the main features of the signal, but also

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MART´IN-MART´INEZ et al.: STOCHASTIC MODELING OF THE PPG SIGNAL: A SYNTHESIS-BY-ANALYSIS APPROACH WITH APPLICATIONS

Fig. 1.

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Pipeline of the analysis methodology for the PPG signal modeling.

their statistical characterization, can be summarized in a number of parameters. This approach, to the best of our knowledge, has not been sufficiently explored in the literature and this paper is a contribution in this direction. Statistical atlases, however, are common practice in the medical imaging community [14]–[16]. In this paper, we propose a modeling methodology aimed at the construction of patient-specific atlases of PPG signals. The proposed method is based on a shape parameterization of the PPG wave and a nonstationary model of its time evolution. The analysis methodology leads to a set of parameters that summarize patient information and allows for generating statistically equivalent signals at will; the equivalence is measured both in terms of spectral content as well as in terms of two nonlinear features commonly used in the literature. Three applications are described in the paper: first, the model is able to track the physical activity, by gradually shifting from a basal condition to different intensities of sport activity. Second, statistics of a PPGderived clinical parameter [the inflection point area (IPA1 )] are obtained from model sampling and compared at the mentioned intensities of physical activity. Finally, corrupted or missing signal epochs are recovered by synthesizing equivalent signals out of the model. II. MATERIALS We have used 76 PPG signals; 26 of them have been acquired with the Omicron FT Surveyor device (RGB Medical Devices LTD.), the acquisition interval lasted 5 min with a sampling period of Ts = 15 ms (fs = 66.6 Hz), the sensor was located in the right index finger and subjects were under relaxed conditions. The 50 remaining signals have been obtained from the public database multiparameter intelligent monitoring in intensive care II (MIMIC-II) included in the PhysioNet project [10], [18]. Since signals from MIMIC-II are longer than 5 min, they have been cropped for the sake of homogeneity. The employed signals have been classified as controls (10 signals), cases (16 signals), and MIMIC-II (50 signals). The second set consists of patients that arrived at the cardiology service of any of the two main public hospitals in Valladolid, Spain. Patients included in the study are affected of, at least, one of these pathologies: supra-ventricular tachycardia (ST), ventricular tachycardia (VT), atrial fibrillation (AF), atrio-ventricular blockade (AVB), extra-systole supra-ventricular (ESV), previous myocardial infarction (MI), high blood pressure (HBP), and cardiac failure (CF). III. METHODS Fig. 1 shows an overview of the modeling method we here describe; as indicated in the figure, the procedure consists of three stages, namely, preprocessing, shape modeling (also called parameterization), and temporal modeling. The first stage splits the 1 This parameter is commonly used to estimate the Cardiac Output using Windkessel models or the inflection harmonic area ratio (IHAR) [17].

Fig. 2. Results at the preprocessing stage. (a) Pulses corresponding to each beat are identified; (b) original pulses superimposed; (c) warped pulses. x(t) is the original PPG signal with continuous (DC) component removed.

signal x(t) into a number N of normalized pulses, {˜ xn (t)}N n =1 . One pulse is considered per heartbeat; the number N denotes the total number of beats in x(t). The second stage finds the shape parameters of each pulse and gives rise to a time series: W[n], n = 1, . . . , N , the statistics of which are dealt with at the third stage. We now describe the details of each stage. A. Preprocessing The preprocessing stage consists of two subprocedures: pulse delineation and temporal normalization. 1) Pulse Delineation: By means of the pulse delineation, the signal x(t) is split into a number of pulses xn (t), n = 1, . . . , N , one per beat [see Fig. 2(a)]. To this end, some methods, such as [19], are described in the literature; however, if signals are only moderately corrupted, simpler methods —based on lowpass filtering and valley (local minimum) detection— can be used, instead. Since we only pursue the position of the valleys corresponding to the onset/end of pulses, we have used a lowpass filtered version of the signal (with cut-off frequency close to 0.15 Hz) to locate these two points. Using this methodology, we achieved a 99% of correctly delineated pulses. 2) Temporal Normalization: The need of temporal normalization stems from the fact that pulses are incoherent, i.e., 1) the maxima are not located at the same time instant; and 2) the pulse lengths are not necessarily equal. This is obvious from Fig. 2(b), in which pulses have been overlapped for comparison. As for modeling, it is far more convenient that pulses show more coherence. Therefore, we have applied a temporal normalization procedure, i.e., a time warping, bearing in mind the following two criteria for the normalized pulses: C1 : Maxima should be located at the average position of the original maxima. C2 : Lengths should be equal to the average length of the original pulses. Fig. 2(c) illustrates the results of the transformation applied. Specifically, assuming that all the pulses start at t = 0, the warping transformation is defined as follows: ⎧ t ⎪ ⎪ · pηT 1 if 0 ≤ t ≤ tp eak ⎨t p eak T (t) = t − tp eak ⎪ ⎪ · pηT 2 + ηT 1 if tp eak < t ≤ tend ⎩ tend − tp eak (1) where tp eak and tend are the time instants at which the maximum and the end of the pulse take place, respectively, and ηT 1 , ηT 2

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Fig. 3. Box-plots of the amplitudes of the pulse set: (a) prior to and (b) after warping. The length is measured in samples. L m a x denotes the length of the largest pulse and L av the average length.

are the mean values of T1 , T2 , respectively, which are the random variables that model tp eak and tend − tp eak over the pulses within the registry [see Fig. 4(a)]. Therefore, temporally normalized pulses are given by   (2) x ˜n (t) = xn T −1 (t) . Fig. 3 shows the box-plots of the instant amplitudes of the pulse set before (a) and after (b) the warping. Clearly, the original pulses show more spread (interquartile —Iq— ranges are larger and the amount of outliers is considerably greater). Additionally, we have also observed two effects after applying the warping, namely, 1) differences between distributions of contiguous points are smoother and 2) amplitude distributions are closer to a Gaussian distribution law.

Fig. 4. (a) Scheme for the parameterization of PPG pulses. (b) Real example of the shape model; continuous line is given by the model, while dashed line represents the original signal.

Fig. 5. Pipeline of the estimation of the parameter vector of a pulse. Parameters of the SoG 2 model are estimated by means of the nonlinear least squares method using the Levenberg–Marquardt algorithm for function minimization procedures. The estimation of the residue —˜ r (t)— is also indicated, however, it will not be mentioned from now on due to the reduced interest provided by this signal. r˜(t) can be modeled as white noise zero-mean Gaussian with variance σ r2˜ .

conditions [20]. That is, for y(t) (t ∈ R), the conditions lim y(t) = lim y(t) = 0, and

t→−∞

(5)

t→∞

∂y(t) ∂y(t) = lim =0 (6) t→∞ ∂t ∂t hold. These conditions, however, are not observed in the original pulses, so this effect should be accounted for to get a better fit. For this purpose, the model incorporates a linear local trend, defined as the line that passes through both the onset and the end of the pulse (see Fig. 4):

 (t − b1 )2 y(t) = a1 exp − c21

 (t − b2 )2 + a2 exp − c22 lim

t→−∞

B. Pulse Shape Modeling (Parameterization) Each normalized pulse will be modeled according to  ˜ x ˜n (t) = Γ W[n], t + r˜n (t), t ∈ T

(3)

˜ ˜ t) is a curve defined by the parameter vector W, where Γ(W, r˜(t) is the residue, and T ≡ [0, ηT 1 + ηT 2 ] is the normalized time domain in which pulses are defined. n = 1, . . . , N is the beat counter.2 Since the pulses in the original signal are due to the superposition of a forward and a backward wave, we have used a model consisting of the summation of two Gaussian curves (SoG2 ):

 (t − b1 )2 y(t) = a1 exp − c21

 (t − b2 )2 + a2 exp − (4) c22 where the parameters ai model the amplitude of the curves, bi the location of their maxima, and ci their width. The SoG2 model, in addition, satisfies Dirichlet and Neumann boundary

+ m0 + m1 · t.

(7)

Once this trend is removed, pulses are closer to satisfy the boundary conditions of the SoG2 model. Therefore, the estimation of the model parameters is to be sequentially done, as indicated in the pipeline depicted in Fig. 5. Recalling equations (1)–(7), each pulse is represented by the parameter vector [t1 , t2 , m0 , m1 , a1 , b1 , c1 , a2 , b2 , c2 ]

(8)

which will be considered as a realization of the random variable W = [T1 , T2 , M0 , M1 , A1 , B1 , C1 , A2 , B2 , C2 ] .

2 We

have used a specific notation in order to differentiate between random variables and the values they actually take. Thus, we have used bold capital letters to denote scalar random variables (Z), and have established differences for vector random variables (Z). As for the specific values, we have used regular (nonbold) font: Z (scalars), and Z (vectors). Since no random matrices variables are used, matrices are referred to as Σ.

(9)

However, both T1 and T2 are not necessary for the characterization of x ˜n (t). Hence, normalized pulses are parameterized by means of the variable ˜ = [M , M , A , B , C , A , B , C ] W (10) 0

1

1

1

1

2

2

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MART´IN-MART´INEZ et al.: STOCHASTIC MODELING OF THE PPG SIGNAL: A SYNTHESIS-BY-ANALYSIS APPROACH WITH APPLICATIONS

Fig. 6. Pipeline of the methodology proposed to model the temporal evolution. Blocks named “ARMA(2,2)” are devoted to the estimation of the ARMA(2,2) model parameters, which compose the “ModelX , Y ”.

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Fig. 7. (a) Temporal series of Y[n]; abscissae represent beats. Note that the variance of Y j [n] is higher than the one of Y k [n] for j < k. (b) Normalized 0 ; abscissae denote the beat lag. autocorrelation function for {Y j }1j = 1

such that ˜ t). x ˜sn (t) ← y(t) = Γ(W,

(11)

We have observed that the elements of W are correlated and follow a Gaussian distribution law,3 so the vector is modeled as a multivariate Gaussian random variable  W ∼ N10 η W , ΣW , (12) where η W ∈ R10 and ΣW ∈ M10×10 are the mean vector and the covariance matrix, respectively. The statistics of this vector change with time so the series W[n] is a random process, the temporal dependence of which is to be modeled as we now indicate.

IV. EXPERIMENTS AND DISCUSSION

C. Temporal Evolution Modeling Fig. 6 shows the two stages in which the temporal evolution is modeled: S1 : Principal component analysis of the elements of W[n]. S2 : Modeling of the power spectrum of each channel. It should be noted that index n denotes “beats” instead of time. This should be borne in mind to deal with the base variable of the forthcoming power spectra. 1) Decorrelation: In order to avoid the complexity that stems from multichannel approaches, we have resorted to the principal component analysis —PCA— transformation [22] to split the problem into [see equation (9)] ten uncorrelated channels. Specifically, we will process the series Y[n] defined as Y[n] = H · pX[n]

(13)

where X[n] = W[n] − η W and H is the matrix of eigenvectors of the covariance matrix of X, denoted as ΣX ∈ M10×10 (ΣX = ΣW ). The estimation of H has been done using all available data. We have observed that, even though W[n] is nonstationary, the eigenvectors of a locally estimated version of H remain fairly unchanged in time. Fig. 7(a) shows the signals in the ten channels obtained after applying equation (13), sorted naturally by the relative weight in terms of the variance; Fig. 7(b) shows the autocorrelation function of each channel; it can be appraised from the figure that channels are increasingly closer to white noise as its variance decreases. 3p

> 0.05 for the χ 2 -GOF (Goodness of Fit) test [21].

2) Temporal Modeling: In the light of literature on temporal modeling, an autoregressive moving average (ARMA)-based proposal has been used. As indicated in Fig. 6, each component of Y[n] is modeled by means of two ARMA models: one for the trend —YT ,k [n]— and one for the residue —YR ,k [n]. Specifically, the trend is a low-pass signal derived by filtering Y[n] with a FIR 30-coefficient-long filter, while the residue is the difference between the original series and the trend. These signals are orthogonal and we have observed that they are closer to stationarity than the original. We have resorted to two ARMA(p = 2, q = 2) models to approximate each spectrum.

This section describes several experiments aimed at showing the adequacy of the model here proposed; the first two experiments demonstrate the capability of the model to synthesize PPG signals with high statistical similarity to the original series from a twofold perspective, i.e., both in terms of a spectral analysis as well as in terms of a nonlinear analysis. The capability of the model to track physical activity is evaluated in experiment 3. The fourth experiment shows the capability of the proposed model to reconstruct corrupted or missing signal epochs. A. Experiment 1: Spectral Similarity Assessment To assess the similarity between the spectra of the corresponding channels of the original and the synthetic series W[n], we have resorted to both a randomized test [23] on the periodogram as well as a significance test on the spectral coherence [24]. Randomized Test: Due to the fact that the series are nonstationary, we divide each series into 20-point windows and the FFT is calculated in each (square modulus); comparisons are carried out at frequencies 0, fs /4, and fs /2 to check similarity at low, middle, and high frequency content while keeping the spectral points far apart so as to facilitate uncorrelation between every pair. Let M denote the number of 20-point windows each series is split into. For each series and each frequency, the M FFT values are stacked and the 25, 50, and 75 percentiles of the amplitude distribution are calculated; the value used for test is the difference between every two corresponding percentiles in each frequency of the original and the synthetic series. The acceptance/rejection region is determined by randomizing the set of 2M points (for each frequency), finding the differences in percentiles, and finding the region in which the λ% (for acceptance)

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TABLE I MEAN ACCEPTANCE RATE OF THE H 0 HYPOTHESIS FOR THE SPECTRUM COMPARISON BETWEEN THE SERIES FORMING W[n] BY MEANS OF THE RANDOMIZED TEST

original–synthetic pairs. The lower row shows sample values of spectral coherence as a function of the normalized frequencies for (left) synthetic–synthetic pairs and (right) original–synthetic pairs. Clearly, the low-pass character of the PPG signal reveals that coherence is fairly high for low-frequency spectral components; for higher frequencies results lose meaning since the spectral moduli in these regions are arbitrarily small. B. Experiment 2: Similarity Assessment by Nonlinear Analysis

of these differences (for each percentile and frequency) fall into. The value λ is obtained after applying the Bonferroni correction [21], so that the overall level of the test is 5%. We have adopted the criterion that for two spectra to be similar, the three percentile differences for the three frequencies should pass the test. Results of this test are presented in Table I. This table shows the mean ± the standard deviation of the acceptance rate of the hypothesis H0 (i.e., similar spectra) on each channel, where the acceptance rate is defined as the number of synthetic series that pass the test divided by the total amount of simulations. This table shows higher results for the MIMIC-II group, which implies a better model fit for this group. This stems from the number of registries composing each group; since the MIMIC-II is the largest group, outliers are less critical. Regarding the control and case groups, results are worse for the last group as a consequence of the more variable shape of pulses from pathological signals. Spectral Coherence: In addition, spectral coherence between the original signal and the set of synthetic signals has been assessed; we have now resorted to a different procedure in which one of the capabilities of the model, i.e., the possibility of finding statistics of the parameters involved by sampling the model, is exploited. Specifically, a number of pairs of original and synthetic independent signals have been generated (processing length of K = 20 signal points as in the randomized tests have been used) and spectral coherence has been calculated for (K/2) + 1 frequencies. We have considered that two signals are spectrally coherent if all its frequencies are coherent. The overall level of the significance test has been set 5%; Bonferroni correction has once again been applied to determine the test level at each frequency; cut-off regions in the distribution of spectral coherence in each frequency have been determined by sampling the model with as many pair samples so as to have an expected number of one hundred rejected cases in each frequency. Once the acceptance intervals are estimated, the test is applied to a number of original and synthetic signal pairs to find out the probability of acceptance. Results of this test are shown in Table II for each component of vector W[n]. Under first block, we show the results of a 20-point window located at the beginning of the record; last block denotes results from the last 20-point window. For illustrative purposes, Fig. 8 shows (upper row) the histograms of the coherence for two frequencies from (left) synthetic–synthetic pairs and (right)

We have also assessed the similarity of the original and the synthetic series by means of two nonlinear techniques, namely, symbolic dynamics (SD), where the features are the probability of appearance of each word and approximate entropy (ApEn). Both magnitudes are usually used to get a measurement of nonlinear variability/regularity of time series; we will analyze these magnitudes within the series W[n]. Symbolic Dynamics (SD): A broad description of this technique can be found in [25]–[27]. Mainly, the series is encoded according to an alphabet and a quantification law in order to achieve a string of symbols. Then, symbols (coming from the alphabet A with a number of symbols (A)) are combined to create words (M symbols long), the probabilities of the appearance of which over the aforementioned string are to be computed. Both (A) and M are the parameters of this method: SD((A), M ). The aforementioned probabilities are, actually, the features representative of regularity/variability: regular signals would have flat words (consecutive repetitions of one symbol) with higher probabilities whereas variable signals would have fickle words (with alternate symbols) with higher probabilities. Approximate Entropy (ApEn): Entropy can be defined as the rate of information production. ApEn was introduced by Pincus [28] as a measurement of the system complexity closely related to entropy. Richman and Moorman provide an accurate definition of the magnitude and its parameters in [29]: “ApEn(m, r, N ) is approximately equal to the negative average natural logarithm of the conditional probability that two sequences that are similar for m points remain similar, that is, within a tolerance r, at the next point.” According to this definition, lower values are achieved by regular signals, while variable signals yield higher values. As for the parameters involved, the choice has been carried out as follows: with respect to SD, we have set it to (4, 4) as a tradeoff between exhaustive description of the signal and capability of estimation of the underlying probabilities of occurrence; as for ApEn, we have resorted to commonly used values [30]–[32]: m = 2 and r = 0.15 × σ, where σ denotes the standard deviation of the series. As for the methodology of analysis, we have applied a similar procedure to that used for testing the spectral coherence, i.e., the parameters resulting from the nonlinear analysis of the original series have been compared to the distribution of parameters from the synthetic series created by the model, fed with the parameters extracted from the original signal in the analysis stage; cutoff points of each distribution are estimated by sampling the model to get 1000 realizations. Specifically, Table III shows the

MART´IN-MART´INEZ et al.: STOCHASTIC MODELING OF THE PPG SIGNAL: A SYNTHESIS-BY-ANALYSIS APPROACH WITH APPLICATIONS

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TABLE II MEAN ACCEPTANCE RATE OF THE H 0 HYPOTHESIS FOR THE SPECTRUM COMPARISON BETWEEN THE SERIES FORMING W[n] BY MEANS OF THE SPECTRAL COHERENCE BASED TEST

Fig. 8. Upper row: Histogram of the spectral coherence of the second component of vector W[n] at normalized frequency 0.182 for (left) synthetic-synthetic signal pairs and (right) original-synthetic signal pairs. Lower row: sample spectral coherence functions for (left) a synthetic-synthetic signal pair and (right) an original-synthetic signal pair (both for the second component of vector W[n]. TABLE III PERCENTAGE OF TIMES THAT PARAMETERS OF THE ORIGINAL SERIES ARE INSIDE THE REGION OF NORMALITY OF THE DISTRIBUTION CREATED BY THE VALUES OF THE SYNTHETIC SERIES

For both the control and case groups, relative frequencies have been computed using all the signals within each group, that is, 10, 16, and 50 signals, respectively. In the light of these results, we can maintain that even though the model has been conceived to emulate the spectral behavior (by means of an ARMA model), it turns out that the model accounts for nonlinear variability/regularity issues as well, as indicated by this nonlinear analysis. In addition, as it was also the case in the experiments carried out in Section IV-A, better results are achieved for the control and MIMIC-II groups. We understand that the cause is the same as that stated in the foregoing section (i.e., more variability in the case group and more weight of the outliers). It is worth-noting that this experiment is a nice complement of that described in Section IV-A since nonlinear features not considered there, but tightly related to the time behavior of the signals, are here accounted for. Consider the following example: let q(t) denote the level of battery of any system; the modulus of the spectrum of q(t), t ∈ [0, T ] is the same as that of q(T − t); however, if q(t) shows short times of charging and long times of discharging, q(T − t) shows exactly the opposite behavior. This sort of nuances, however, are actually considered by employing nonlinear methods for variability assessment as we do with the analysis in this section.

C. Experiment 3: Performance Assessment on an Exercise Scenario

relative frequency that each component of W[n] from the original series falls inside the region of normality4 of the distribution created by the corresponding index from the synthetic series. 4 Region

comprised between the 2.5 and 97.5 percentiles.

As the literature reveals, the shape of PPG pulses varies with the stress level induced by physical activity [33]; expected changes are a decrease in both the pulse amplitude and the pulse length with the intensity level of the exercise; this adaption should be more noticeable in T2 than in T1 . This experiment evaluates the capability of the model to cope with the aforementioned effect. To accomplish this objective, two controls have been monitored under the following protocol: 3 min under resting conditions (hereafter S1 ), 9 min cycling at low intensity (S2 ), 6 min cycling at medium intensity (S3 ), 4 min cycling at high intensity (S4 ), 1 min cycling at low intensity (S5 ), and 4 min at rest (S6 ). If the model is able to capture the exercise influence, the model parameters should grab all the information needed so as to be able to synthesize pulses that resemble those of the

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in each of the six stages in the exercise. Bearing in mind these two figures, the following conclusions can be extracted. 1) The model is robust to nonstationarities, since synthetic signals of parameters can follow the nonstationarities induced by the multistage experiment. 2) Deviations in Fig. 9 of the average synthetic cardiac pulse with respect to the original give rise to deviations in Fig. 10 between the average synthetic pulses with respect to the original. Hence, the largest differences in shape are obtained for the first and the fifth stages. This difference is more noticeable during the stage S1 for subject 1, while for subject 2 highest differences occur during the stage S6 (see Fig. 10). 3) Point-wise discrepancies are expected to be lower if more samples are obtained from the model or function β (cpo [n], cps [n]) in equation (see Section IV-C) is redesigned. D. Experiment 4: Density Functions of Clinical Parameters Fig. 9. Top line: temporal evolution of cardiac period (T 1 + T 2 ) during the experiment. Black lines are from the original registries, while red ones are synthesized according to the model previously estimated from the original signals. Bold line represents moving averaged values, solid thin lines represent the acceptance band, and dashed line stands for raw values. S j denotes the jth stage of the experiment. Bottom line: original —ppgo (t)— and synthetic —ppgs (t)— PPG signals.

original signal for every stage of the experiment; the purpose of this experiment is to check whether this will be the case. Similarity assessment between the synthetic and the original signals has been carried out in previous sections by comparing ensemble parameters; however point-wise correspondences cannot be guaranteed with a single realization due to the stochastic nature of the model. Nevertheless, such correspondences can be approximated by means of model sampling. Specifically, we have chosen a signal derived from the original, namely, the cardiac period defined as cp = T1 + T2 , which is the parameter that best reflects the onset and the end of every stage constituting the experiment. Thus, we pursue to build the synthesized cardiac period signal, which is the signal that minimizes ε=

NB

β (cpo [n], cps [n]) · cpo [n] − cps [n]

(14)

n =1

where NB is the number of beats that the experiment consists of and β(·, ·) is an indication function the value of which is unity if the beat falls inside the acceptance band —BA [n] ≡ [ηcp o [n] − σcp o [n], ηcp o [n] + σcp o [n]]5 — and 0 otherwise, i.e.,

0, if cps [n] ∈ BA [n], ∀n ∈ [1, NB ] β (cpo [n], cps [n]) = 1, otherwise. In Fig. 9, the cardiac period signals, as well as the original and synthetic PPG signals, are presented for both subjects. In addition, Fig. 10 shows the mean pulse shape as well as its deviation 5 Statistical moments have been estimated using a sliding window of length 60 beats.

One of the main advantages of using the synthesis-analysis paradigm is its capability to provide density functions and confidence intervals for meaningful clinical parameters. This feature allows to check if a parameter estimated from the acquired signal ranges between reasonable limits (those obtained from the confidence intervals) for a specific patient. If this is not true an alarm can be raised as diagnostic support. Since common acquisitions are usually too short to obtain reliable estimations of the parameter distributions, sampling the model constitutes an interesting alternative. The parameters of the model are first estimated from the acquired signal and a synthesis procedure is then carried out to get a representative sample of the clinical parameter from a longer time series. We have carried out the following experiment to illustrate this attribute of the model. We use the IPA parameter [17], which is defined for each pulse as depicted in Fig. 11, where ton , tinf , and tend denote the onset, the inflection, and the end time of the pulse, respectively. The IPA has been computed for every stage of the exercise described in Experiment 3 (see Section IVC) both from the original signal and from a longer synthesized signal (104 pulses long) created with the model estimated from the corresponding stage under study. In Fig. 12, the normalized histograms of IPA estimation are plotted for both subjects. Smoother histograms (less spikes, if any) have been obtained for synthetic data as a consequence of the larger size of data used for the estimation. This more reliable density estimation leads to a better characterization of the statistics of IPA, which is desirable for its use as a diagnostic index. E. Experiment 5: Signal Reconstruction The synthesis capability of the model described in this paper can also be applied to the reconstruction of missing or corrupted epochs in PPG signals. To that end, the model parameters are estimated out of the largest window of uncorrupted signal values; then, we synthesize a large number of candidate

MART´IN-MART´INEZ et al.: STOCHASTIC MODELING OF THE PPG SIGNAL: A SYNTHESIS-BY-ANALYSIS APPROACH WITH APPLICATIONS

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Fig. 10. Shape of pulses for every stage of the experiment. S 1 : 3 min under resting conditions before exercise; S 2 : 9 min cycling at low intensity; S 3 : 6 min cycling at medium intensity; S 4 : 4 min cycling at high intensity; S 5 : 1 min cycling at low intensity; S 6 : 4 min resting. Black lines represent the average pulse (bold), the average ± the standard deviation (medium) and the average ± two times the standard deviation (thin) of the original signal; red line represents the average pulse of the synthetic signal. Figures of the top row are from a healthy male patient 35 years old, whereas figures from the bottom row are from a healthy male patient 26 years old.

Fig. 13. Reconstructed piece with the optimal simulation (cyan line) obtained from a PPG acquired signal with a 23-s missing epoch using the methodology in [35]. The missing piece is delimited by the red lines.

Fig. 11. Schematic representation of the nth PPG pulse for the IPA computation.

Fig. 12. Normalized histograms for the IPA parameter estimations in every stage. Black areas are referred to estimations from the original signal, while red areas are for estimations from synthetic signals.

epochs (in our experiment, 10 000 candidates) from which one is selected. Our model is designed to simulate series of parameters (W[n]) that last for a specific number of beats; however the number of beats to simulate (Nb ) is unknown in this experiment and it is estimated as follows:  Nb = 2 ·

ηT 1

τ + ηT 2

 (15)

where τ is the length of the damaged piece. As a consequence of this estimation, synthetic pieces are usually larger than the artifacted epoch so the leftover beats must be removed. Simulation of the model creates series Nb beats long using as initial conditions the last two beats before the beginning of the corrupted piece. To evaluate the adequacy of the candidates which replace the damaged epoch, they are evaluated according to the following three criteria: 1) Temporal Error (εt ):which measures the error between the time of maxima from original and synthetic signals during the first four beats after the temporal position of the corrupted piece. 2) Amplitude Error (εx ): defined as the mean absolute error between the original and synthetic signals during the first four beats after the temporal position of the corrupted piece. 3) Statistical Similarity Through the Williams’ Index (W I): It gives a measurement of how the covariance matrix of the synthetic epoch agrees with those matrices estimated from previous and subsequent signal values as compared with how they agree with each other [34]. Needless to say, the best candidate is chosen as that with lower temporal and amplitude errors and closer-to-one Williams’ index. In order to test how the model here exposed is able to reconstruct corrupted or missing epochs of the acquired signal, we have applied this methodology to an acquired PPG signal with a missing epoch lasting for 23 s. Results of the reconstruction can be observed in Fig. 13 and Table IV. Further details on this experiment can be found in [35].

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TABLE IV EVALUATION SUMMARY OF THE TEN BEST SIMULATIONS SYNTHESIZED FOR THE RECONSTRUCTION

V. CONCLUSION In this paper, we have proposed a model of PPG signals that summarizes a patient record in a number of parameters, which can then be used for further purposes; throughout the paper, we have in mind the construction of statistical atlases, which contain not only the main features of the signal but also their statistical characterization. Additionally, due to the fact that the model can be reversed, i.e., synthesis-by-analysis is possible, the model is able to synthesize signal intervals; this is of special importance in ambulatory measurements with signals acquired through wearable devices, which are prone to artifacts and missing data. The generation of synthetic intervals also allows for statistical characterization of clinical parameters. This constitutes a useful tool for diagnostic support, since it can be used to assess safety intervals according to the model variability. These safety intervals can be used to trigger alarms when specific parameter measures do not range between their limits. The model, as described in the paper, has three stages: (a) pulses are first delineated and time-normalized; then (b) each pulse is modeled in isolation; and (c) the time evolution of the parameter vector obtained in (b) is also tracked by means of PCA and two ARMA models. The experiments described show the validity of the model as well as its capability to track sports activity, to derive probabilistic distributions of clinical parameters, and to reconstruct corrupt or missing epochs of the signal. Further research on this topic will include the construction of population atlases as well letting these ideas carry over to ECG signals. ACKNOWLEDGMENT The authors would like to thank the support of the Cardiology Services at both the Hospital Cl´ınico and Hospital R´ıo Hortega, Valladolid, Spain, as well as the company RGB Medical Devices, for kindly sharing us the measurement devices REFERENCES [1] World Health Statistics. World Health Organization, Geneva, Switzerland, 2011. [2] L. S¨ornmo and P. Laguna, Bioelectrical Signal Processing in Cardiac and Neurological Applications, 1st ed. Burlington, MA, USA: Elsevier Academic Press, 2005.

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[27] M. Baumert, V. Baier, S. Truebner, A. Schirdewan, and A. Voss, “Shortand long-term joint symbolic dynamics of heart rate and blood pressure in dilated cardiomyopathy,” IEEE Trans. Biomed. Eng., vol. 52, no. 12, pp. 2112–2115, Dec. 2005. [28] S. Pincus, “Approximate entropy as a measure of system complexity,” in Proc. Natl. Acad. Sci., USA, vol. 88, 1991, pp. 2297–2301. [29] J. Richman and J. Moorman, “Physiological time-series analysis using approximate entropy and sample entropy,” Amer. J. Physiol. Heart Circ. Physiol., vol. 278, pp. H2039–H2049, 2000. [30] S. Pincus, I. Gladstone, and R. Ehrenkranz, “Approximate entropy: A regularity statistic for medical data analysis,” J. Clin. Monit., vol. 7, pp. 335– 345, 1991. [31] S. Pincus and R. Viscarello, “Approximate entropy: A regularity measure for fetal heart rate analysis,” Obstet. Gynecol., vol. 79, pp. 249–255, 1992. [32] S. Pincus, T. Cummings, and G. Haddad, “Heart rate control in normal and aborted-sids infants,” Amer. J. Physiol., vol. 264, pp. R638–R646, 1993. [33] K. Chellappan, E. Zahedi, and M. Mohd Ali, “Effects of physical exercise on the photoplethysmogram waveform,” in Proc. 5th Student Conf. Res. Dev., Selangor, Malaysia, Dec. 2007, pp. 1–5. [34] G. Williams, “Comparing the joint agreement of several raters with another rater,” Biometrics, vol. 32, pp. 619–627, 1976. [35] D. Mart´ın-Mart´ınez, P. Casaseca-de-la-Higuera, M. Mart´ın-Fern´andez, and C. Alberola-L´opez, “Cardiovascular signal reconstruction based on parameterization –shape modeling– and non-stationary temporal modeling,” in Proc. 20th Eur. Signal Proc. Conf., Bucharest, Romania, Aug. 2012, pp. 1826–1830.

Diego Mart´ın-Mart´ınez (S’12) received the Ingeniero de Telecomunicaci´on and M. Eng. degrees both from the University of Valladolid, Valladolid, Spain, in 2009 and 2011, respectively. He is currently working toward the Ph.D. degree within the Laboratory of Image Processing (LPI) at the same University. In October 2009, he joined the LPI as a Researcher and has since contributed to several research projects. He is currently working as a Researcher at the Signal Theory, Communications and Telematic Engineering Department, E.T.S. de Ingenieros de Telecomunicaci´on, University of Valladolid, with a research fellowship (FPI-UVa). His research interests are stochastic modeling and statistical and nonlinear methods for signal processing, mainly with clinical application. He is a Reviewer of several international conferences and journals.

Pablo Casaseca-de-la-Higuera received the Ingeniero de Telecomunicaci´on and the Ph.D. degrees from the University of Valladolid, Valladolid, Spain, in 2000 and 2008, respectively. From December 2000 to November 2003, he worked as a Design Engineer for Alcatel Espacio S.A. (currently Thales Alenia Spazio), where he contributed to several space programs including the European satellite navigation project Galileo. His activities there were all related to digital signal processing and radio frequency design for the telemetry, tracking and command subsystem. After this period, he joined the Laboratory of Image Processing (LPI), University of Valladolid, with a research fellowship, which finished in October 2005, when his academic activities started. He is currently an Assistant Professor (tenured) at the ETSI Telecomunicaci´on, University of Valladolid, where he performs his research within the LPI. His research interests include statistical modeling and nonlinear methods for biomedical signal and image processing, and network traffic analysis.

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Marcos Mart´ın-Fern´andez received the Ingeniero de Telecomunicaci´on and the Ph.D. degrees from the University of Valladolid, Valladolid, Spain, in 1995 and 2002, respectively. He is an Associate Professor at the ETSI Telecomunicaci´on, University of Valladolid, where he is currently teaching and supervising several Master and Ph.D. students. From March 2004 to March 2005, he was a Visiting Assistant Professor of radiology at the Laboratory of Mathematics in Imaging (Surgical Planning Laboratory, Harvard Medical School, Boston, MA, USA) thanks to a Fulbright fellowship grant. His research interests include statistical methods for signal and image segmentation and filtering in multidimensional signal processing. He also works on the application of mathematical methods to solve image processing problems. He is also investigating the fields of magnetic resonance imaging, ultrasonic imaging, and electrophysiological signals analysis and synthesis. He is within the Laboratory of Image Processing, University of Valladolid, where he is currently performing his research. He is a Reviewer of several international scientific journals and Member of the scientific committee of some conferences (ICIP, MICCAI, EUSIPCO, BIBE, ICASSP and ISBI). He has 120 published papers in scientific Journals and Conferences. His Hirsch index is 10, with more than 250 received cites (SCOPUS).

Carlos Alberola-L´opez (S’94–M’96–SM’06) received the Ingeniero de Telecomunicaci´on and Ph.D. degrees both from the Politechnical University of Madrid, Madrid, Spain, in 1992 and 1996, respectively. In 1997, he was a Visiting Scientist at the Thayer School of Engineering, Dartmouth College, NH, USA. He is currently a Professor at E.T.S. Ingenieros de Telecomunicaci´on, University of Valladolid, Spain, where he is also the Head of the Laboratory of Image Processing. He is a Coeditor and/or Reviewer of several scientific journals and main conferences and he is a Consultant of the Spanish Government for the evaluation of research initiatives. He has coauthored several book chapters and more than one hundred journal and conference papers. His research interests include statistical signal and image processing applications specially focused on, but not limited to, the field of bioengineering.