An automatic systolic peak detector of blood pressure waveforms using 4th order cumulants

Marcus Schmidt; Andy Schumann; Karl-Jürgen Bär; Georg Rose

doi:10.1515/cdbme-2016-0056

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An automatic systolic peak detector of blood pressure waveforms using 4^th order cumulants

Marcus Schmidt , Andy Schumann , Karl-Jürgen Bär and Georg Rose

Published/Copyright: September 30, 2016

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From the journal Current Directions in Biomedical Engineering Volume 2 Issue 1

Abstract

The arterial blood pressure (ABP) holds a lot of information about the cardiovascular system. To analyse the ABP signal, at first the pulse wave has to be detected correctly. Due to influences, e.g. of noise, an accurate beat detection is a challenging task. In this work, a novel real-time ABP detector based on higher-order statistics is presented. The method uses the 4^th order cumulants for ABP-peak detection. To evaluate this detector, the Fantasia database freely available at physionet was used. In this database, 142,936 systolic peaks were available to check the efficiency of the beat detector. A sensitivity of 99.91% and a positive predictive value of 99.50% were achieved with the novel method.

Keywords: blood pressure; detection; higher-order statistic; 4th order cumulant

1 Introduction

The arterial blood pressure (ABP) is the sum of cardiac pressure and pulse wave reflections and holds a lot of information about the cardiovascular system [1]. The ABP is subdivided in the systole and the diastole which are characterized due to several points. Onset and dicrotic notch mark the beginning and the end of the systole and consequently the beginning and the end of the blood ejection [2]. Hence the beginning and the end of the diastole are represented due to dicrotic notch and onset. To detect the characteristic point of ABP is a difficult task because of the influence of noise to the signal. Before analysing ABP waveforms, at first the systolic peak has to be detected. It represents the highest blood pressure (BP) and its different to the onset is used for diagnostics. Further ABP can used for heart rate variability analysis [3] and reducing false alarms in intensive care units (ICU) [4].

In the past, a lot of ABP detection algorithms were developed. They used the first derivative of the BP-signal in combination with a decision rule [5] and in addition to that a bandpass filter [6]. In another method, also the first derivation was calculated and then summed [7]. This BP-detector is freely available at physionet.org [8]. A BP-detector based on higher order statistics in combination with a decision rule was presented in [9]. Here, the mean value, the variance together with the skewness and the kurtosis were summed. The resulting curve was evaluated concerning systolic peak and dicrotic notch detection.

In this work, an ABP detector based on the 4^th order cumulants with adaptive threshold is developed. The novel method was evaluated on Fantasia database available at physionet.org [8].

2 Theory

Be X a stochastic variable and defined as X = [x₁, x₂, … , x_N], its 4^th order cumulants presented in [10] is given as:

(1)k4=μ4−3⁢μ22.

μ₂ and μ₄ are the 2^nd and 4^th order central moment and can be defined as:

(2)μ2=∑i=1N(xi−μ)2

and

(3)μ4=∑i=1N(xi−μ)4.

The variable μ is the mean value of X. All cumulant can be written as a function of the central moments.

The cumulants have useful properties which are listed in [11], [12]. Important properties which were needed in this work for BP-signal detection are the shift-invariance (see eq. 4 and eq. 5) defined in [11].

(4)k1⁢(X+c)=k1⁢(X)+c,∀c∈ℝ,

(5)kn⁢(X+c)=kn⁢(X),n≥2,

Due to the facts presented in eq. (4) and eq. (5), the cumulants are known as the semiinvariance of X [13].

For two probabilistically independent stochastic variables X and Y a third property of the cumulants is the additivity (eq. 6).

(6)kn⁢(X+Y)=kn⁢(X)+kn⁢(Y).

Combining the property of additivity with the fact that the cumulants are zero for variables with a gaussian distribution shows that white gaussian noise has no impact on the calculation [14].

The 4^th order cumulant is a linear combination of the 2^nd and 4^th order central moment. Both statistical parameter give specific values about the ratio of the mean distribution to its borders. This means that for distributions where most of the values of a stochastic variable are situated on the border the moments exhibit high values. In conclusion the 4^th order cumulants exhibit high values too. This fact can be used for the description of edges. For edges with high slopes the resulted moments and also the cumulants show high values.

3 Material and methods

3.1 Databases

To evaluate the presented methods of the BP-signal morphology the Fantasia database freely available at physionet.org [8] was used. This database consists of 40 datasets recorded from 20 young people (21–34 years old, f1y01–f1y10 and f2y01–f2y10) and 20 elderly people (68–85 years old, f1o01–f1o10 and f2o01–f2o10). The arterial blood pressure was only recorded in half of the datasets (f2y01–f2y10 and f2o01–f2o10) which were used for ABP beat detection. Each of the records had an overall length of 120 min and were digitized with 250 Hz sampling frequency f_s. The Fantasia database was divided into a training database DB_{F_Training} and a test database DB_{F_Test}. The training database consists of three datasets of the young (f2y01–f2y3) and the elderly (f2o01–f2o3) group. In DB_{F_Test} the rest of the datasets was pooled.

For the evaluation of the presented ABP peak detector 144,399 annotated peaks on the basis of the annotated R-waves were available. In some datasets the ABP-signal showed a zero line what makes a correct detection of the ABP peaks impossible. According to this, the number of annotated beats was reduced to 142,936.

3.2 Detection algorithm

The algorithm for the detection of the BP systole (M1) is divided into two main steps. First, a bandpass was applied to reduce high frequently peaks and baseline wander. Here, the cutoff frequency of the lowpass filter was set to 10 Hz. The highpass filter had a cutoff frequency of 0.05 Hz. Both were elliptic filters of 3^rd (lowpass) and 4^th (highpass) order.

After the preprocessing step (see Figure 1A), the 4^th order cumulants (eq. 1) were calculated in a sliding window of 0.02 ⋅ f_s samples. This window was then shoved on sample by sample. The resulting signal of the 4^th order cumulants (Figure 1B, red line) was used for ABP peak detection because of the cumulants which showed high values for the systole in comparison to very low values for the rest of the BP-signal.

Figure 1

Single steps for systolic peak detection: (A) Filtered ABP [mm Hg] (blue) and the resulted 4^th order cumulants [(mm Hg)⁴/1000] (red), (B) 4^th order cumulants (red) and the detected systolic peaks (magenta, green: peak on the top of the 4^th order cumulants maximum) together with the defined threshold (black), (C) Filtered ABP and the detected systolic peaks (magenta, green: peak on the top of the 4^th order cumulants maximum) of the BP-signal.

To detect the ABP peak, a threshold was used and defined as the median of the last 10 maximum peaks of the 4^th order cumulant signal (see the green points in Figure 1B). In a further step the median of the 10 detected cumulants maxima was multiplied with a factor of 0.05 which was experimental determined with the DB_{F_Training}. The resulted threshold of the first 10 maxima is exemplary depicted as a black line in Figure 1B. An ABP peak was detected when the amplitude of the 4^th order cumulants exceeded the threshold. The results of the detected peaks are also shown in Figure 1B as magenta colored points. In Figure 1C, the detected points (magenta) and the maximum of the 4^th order cumulants (green) are depicted together with the filtered BP-signal.

For the reduction of false detection, two decision rules were applied. First, a point on the 4^th order cumulants signal will be detected as true when the difference of the BP-signal of the current sample and the previous sample was positive what indicates a positive slope. It can be assumed that the detected point is part of the ABP peak. A second decision rule, first applied on ECG-signals by Pan and Tompkins [15], prevent further detection for 200 ms after a detected peak.

3.3 Evaluation metric

For each dataset of the DB_Fantasia, a reference annotation was needed to evaluate the presented algorithm. These references were determined using the open source algorithm from Zong et al. [7]. This systolic beat detector is freely available at physionet.org and part of the WFDB toolbox [16] (wabp.m). The results were visually inspected for all datasets by experts and then used as gold standard.

To compare the detection results, the sensitivity (Se), the positive predictive value (+P) and the detection error rate were calculated. By the ANSI/AAMI EC57 standard [17] these three statistical parameters are defined as:

(7)Se=TPTP+FN,

(8)+P=TPTP+FP

and

(9)DER=FP+FNTP+FN.

To calculate Se, +P and DER first the true positive (TP), the false positive (FP) and the false negative (FN) peaks has to be determined. The results of the presented ABP beat detector were compared with two other detectors. The first method was presented in [7] (M2) and based on a windowed slope sum function together with a decision rule. The second method (M3) was presented from Li et al. in [1] and is mainly based on the first derivation evaluation.

4 Results

To find the optimum threshold factor, the DB_{F_Training} was used. The start value was set to 0.01 and increased in steps of 0.01–0.1. As an optimum threshold factor 0.05 was found. The results for the DB_{F_Training} and the DB_{F_Test} are depicted in Table 1.

Table 1

Results of the presented method applied on DB_{F_Training} DB_{F_Test}.

Database	TP	FN	FP	Se (%)	+P (%)	DER (%)
DB_{F_Training}	41,325	31	170	99.93	99.59	0.49
DB_{F_Test}	101,484	96	552	99.91	99.46	0.64

For DB_{F_Training}, a Se of 99.93% and a +P of 99.59% with a DER of 0.49% were reached. Similar values for Se (99.91%) and +P (99.46%) together with a DER of 0.64% are shown.

The detection results of the methods M1, M2 and M3 are shown in Table 2. The results of M2 and M3 were first presented in [1] and taken from their results.

Table 2

Comparison of the results of all methods applied on the presented database.

Method	TP	FN	FP	Se (%)	+ P (%)	DER (%)
M1	142,809	127	722	99.91	99.50	0.59
M2	136,389	1441	2418	98.95	98.26	2.78
M3	1357,48	2082	24	98.29	99.98	1.56

M1 showed the lowest number of FN with 127 and a resulted Se of 99.91%. The lowest number of FP (24) was reached with M3 with a +P of 99.98%. The best DER of 0.59% was achieved using M1.

5 Discussion

A new real-time algorithm was presented based on the 4^th order cumulants. The new detection algorithm was presented and tested on the Fantasia database. The results were compared with two other ABP detectors. The results for M2 and M3 were taken from [1]. Comparing the whole number of beats, a difference appears. In [1], only 137,830 beats were evaluated. In this work, 142,936 systolic beats were used for testing the algorithm. Nevertheless, it can be seen that M1 showed the best results. The reason for that can be found in the sum of FN and FP. M1 shows less FN and FP than in M2 and M3. Adding the missing ABP beats as TP for M2 and M3 it does not change the result for M1’s best performance.

Using the 4^th order cumulants, the novel method shows no influence of a DC-offset. The reason for that is the cumulants property of shift-invariance. With reference to eq. (5) shows that the DC-offset, which represents an simple addition of a constant, disappears.

In Homaeinezhad et al. [9] higher-order statistic based ABP beat detection algorithm was first presented. Here, the mean value, the standard derivation as well as the skewness and the kurtosis were calculated in a sliding window and then summed. The resulted higher-order signal was used for ABP peak detection, but showed high computation complexity because of the number of statistical parameters. With the higher-order statistic signal also ABP analysis like onset or dicrotic notch detection was described in [9]. According to this work, it has to be investigated if equal results for ABP peak detection and a determination of the characteristic points could be achieved using the 4^th order cumulants because of the less computation complexity according to the method presented in [9].

In further work the presented algorithm has to be applied to other databases which include an ABP-signal. Besides, the robustness of noise has to be investigated.

Author’s Statement

Research funding: The author state no funding involved. Conflict of interest: Authors state no conflict of interest. Material and Methods: Informed consent: Informed consent has been obtained from all individuals included in this study. Ethical approval: The research related to human use complies with all the relevant national regulations, institutional policies and was performed in accordance with the tenets of the Helsinki Declaration, and has been approved by the authors’ institutional review board or equivalent committee.

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Published Online: 2016-9-30

Published in Print: 2016-9-1

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Articles in the same Issue

https://doi.org/10.1515/cdbme-2016-0056

Keywords for this article

blood pressure; detection; higher-order statistic; 4th order cumulant

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