# Blood Pressure Estimation by Photoplethysmogram Decomposition into Hyperbolic Secant Waves

^{1}

^{2}

^{3}

^{4}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

_{1}, the renal reflection pulse as P

_{2}, and the iliac reflection pulse as P

_{3}. The amplitude ratios P

_{2}to P

_{1}, the T

_{13}interval, and the time delay between systolic (P

_{1}) and iliac peak (P

_{3}) are used as parameters to estimate the blood pressure. Their results show statistically significant correlations between P

_{2}, P

_{1}, T

_{13}, and blood pressure as measured by central line catheters (systole: R

^{2}: 0.92, p < 0.0001, diastole: R

^{2}: 0.78, p < 0.0001). Several PDA parameters, such as systolic upstroke time, diastolic time, 2/3, and 1/2 pulse amplitude, are possible candidates for estimating blood pressure from pulse waves using PPG [8,9,10,11]. Despite the success of PDA in monitoring changes in BP using a finger PPG, the degree of uncertainty in the fit parameters determined from literature PDA models is high (up to 30%) [12,13,14], and a potentially large number of heartbeats are required to obtain acceptable signal-to-noise blood pressure estimates. As an alternative Gaussian PDA, the use of secant hyperbolic waves (sech) for the PDA of PPG signals was considered in [14]. Sech may provide a better PDA for blood pressure estimation because it represents a possible solution to the Moens–Korteweg equation describing a pressure pulse in an elastic tube. Indeed, in [14], their results suggest that the hyperbolic secant wave function performs better for three-wave PDA than previous functions in the PDA literature, as sech reduces pulse-to-pulse parameter noises in the reflected waves P

_{2}and P

_{3}.

## 2. Methods

#### 2.1. Pulse Wave Model for Blood Pressure Estimation

#### 2.1.1. Mathematical Models for Pulse Decomposition Analysis

#### 2.1.2. Features Extraction from Decomposed Waves Based on Pulse Wave

#### 2.1.3. Multiple Regression for the Relationship between the Features and Blood Pressure

#### 2.2. Dataset Construction for the Verification of Our Proposed Method

#### 2.2.1. Vital Signs Acquisition

#### 2.2.2. Data Pre-Processing

#### 2.3. The Verification of Our Proposed Method

## 3. Results

^{2}= 0.77, p = 0.0000) underperformed compared with the prediction power of the sech PDA (R

^{2}= 0.81, p = 0.0000).

^{2}= 0.58, p = 0.0000) underperformed compared with the prediction power of the sech-PDA (R

^{2}= 0.76, p = 0.0000).

## 4. Discussion

^{2}correlations with central line catheter SBP and DBP to be 0.92 and 0.78, respectively. By comparison, our study shows the R

^{2}was 0.58 for SBP based on the Gaussian PDA (Figure 8a), and 0.76 for SBP based on the sech PDA (Figure 8b), with ground truth reference BP peripherally measured in the finger. While [7] used data from 38 male and 25 female patients of 62.7 ± 11.5 years old, our study used data of healthy volunteers 22.4 ± 1.7 years old (8 males and 2 females) whose blood pressures varied by up to 60% during data collection, which made the prediction in our case more challenging. More detailed tests on clinical datasets will be the subject of future work.

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

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**Figure 5.**Box plots with 60 systolic blood pressure (sbp) for each subject. Each sbp point is the average of an 8-s window.

**Figure 6.**Multiple correlation coefficients for each subject data as the training data and the correlation coefficients calculated by Leave One Out for each subject dataset. (

**a**) Results for each subject independently. (

**b**) Mean value of results from (

**a**). For the group, the differences between the Gaussian and sech results are not statistically significant.

**Figure 7.**Scatter plots for each subject data as training data in multiple regression. (

**a**) Result based on Gaussian PDA. (

**b**) The result based on sech PDA multiple regression was performed with 60 simultaneous 8-s segments of continuous systolic blood pressure vs. PPG pulse decomposition features for each subject data. Each scatterplot contains 600 data points. For both plots, p = 0.0000.

**Figure 8.**Scatter plots for each subject data as test data by multiple regression using Leave One Out. (

**a**) Result based on Gaussian PDA. (

**b**) Result based on sech PDA. In Leave One Out, we estimated systolic blood pressure with an 8-s segment based on the multiple regression model learned with the other segments of the subject data, and we repeated the process to test all 60 segments data comprehensively. Each scatterplot contains 600 data points. For both plots, p = 0.0000.

**Figure 9.**The multiple correlation coefficients for all the subjects’ data and the correlation coefficients for all the subjects’ data with Leave One Out analysis. The reduction in the coefficients compared to Figure 7 implies that PDA for each individual must be calibrated by an independent regression.

**Figure 10.**Mean squared error between the original pulse wave and the reproduced wave based on Gaussian PDA or sech PDA in each subject.

Combination of Features | ||
---|---|---|

Subject Number | Sech-PDA | Gaussian-PDA |

1 | Δt_{0,1}, FWHM_{2} | Δt_{0,2}, a_{1}, a_{2}, FWHM_{1}, FWHM_{2}, FWHM_{3} |

2 | a_{3}, FWHM_{1}, FWHM_{2}, FWHM_{3} | Δt_{0,2}, FWHM_{1} |

3 | Δt_{0,1}, a_{1}, a_{2}, FWHM_{2} | Δt_{0,1}, a_{2} |

4 | Δt_{0,1}, a_{1}, a_{2}, FWHM_{1} | Δt_{0,1}, FWHM_{1} |

5 | FWHM_{2}, FWHM_{3} | Δt_{0,1}, FWHM_{1}, FWHM_{2}, FWHM_{3} |

6 | Δt_{0,1}, a_{2}, a_{3}, FWHM_{3} | Δt_{0,1}, Δt_{0,2}, a_{1}, a_{2}, FWHM_{1} |

7 | a_{1}, a_{3}, FWHM_{2}, FWHM_{3} | Δt_{0,1}, FWHM_{3} |

8 | Δt_{0,1}, a_{3} | Δt_{0,1}, a_{1}, a_{2}, a_{3}, FWHM_{1}, FWHM_{2}, FWHM_{3} |

9 | Δt_{0,1}, a_{2}, FWHM_{1}, FWHM_{2} | Δt_{0,1}, a_{2}, FWHM_{1}, FWHM_{2} |

10 | Δt_{0,1}, Δt_{0,2}, a_{2}, FWHM_{1}, FWHM_{2} | Δt_{0,1}, a_{1}, a_{2}, FWHM_{2}, FWHM_{3} |

all | a_{2}, FWHM_{1}, FWHM_{2} | Δt_{0,1}, Δt_{0,2}, a_{2}, FWHM_{2}, FWHM_{3} |

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**MDPI and ACS Style**

Nagasawa, T.; Iuchi, K.; Takahashi, R.; Tsunomura, M.; de Souza, R.P.; Ogawa-Ochiai, K.; Tsumura, N.; Cardoso, G.C. Blood Pressure Estimation by Photoplethysmogram Decomposition into Hyperbolic Secant Waves. *Appl. Sci.* **2022**, *12*, 1798.
https://doi.org/10.3390/app12041798

**AMA Style**

Nagasawa T, Iuchi K, Takahashi R, Tsunomura M, de Souza RP, Ogawa-Ochiai K, Tsumura N, Cardoso GC. Blood Pressure Estimation by Photoplethysmogram Decomposition into Hyperbolic Secant Waves. *Applied Sciences*. 2022; 12(4):1798.
https://doi.org/10.3390/app12041798

**Chicago/Turabian Style**

Nagasawa, Takumi, Kaito Iuchi, Ryo Takahashi, Mari Tsunomura, Raquel Pantojo de Souza, Keiko Ogawa-Ochiai, Norimichi Tsumura, and George C. Cardoso. 2022. "Blood Pressure Estimation by Photoplethysmogram Decomposition into Hyperbolic Secant Waves" *Applied Sciences* 12, no. 4: 1798.
https://doi.org/10.3390/app12041798