# Modeling-based Radial Pressure Waveform Reconstruction Using Photoplethysmography Signals

Jérôme Diaz, François Kimmig, Fabrice Vallée, Arthur Le Gall, Romain

Kirszenblat, Marie Willemet, Philippe Moireau

► To cite this version:

Jérôme Diaz, François Kimmig, Fabrice Vallée, Arthur Le Gall, Romain Kirszenblat, et al. Modeling-based Radial Pressure Waveform Reconstruction Using Photoplethysmography Signals. CinC 2024 - 51st international Computing in Cardiology Conference, Sep 2024, Karlsruhe, Germany. 10.22489/CinC.2024.332 . hal-04870802

HAL Id: hal-04870802

https://inria.hal.science/hal-04870802v1

Submitted on 7 Jan 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
Modeling-based Radial Pressure Waveform Reconstruction using
Photoplethysmography Signals

Jérôme Diaz¹, François Kimmig¹, Fabrice Vallée²,³, Arthur Le Gall², Romain Kirszenblat⁴, Marie Willemet⁴, Philippe Moireau¹

¹ Inria Saclay, Palaiseau, France – École Polytechnique, Palaiseau, France

² Department of Anesthesia and Critical Care Medicine, AP-HP, Hôpital Lariboisière, Paris, France

³ INSERM, Université de Paris, Paris, France

⁴ Withings, Issy-les-Moulineaux, France

Abstract

This study introduces a model linking photoplethysmography (PPG) dynamics to radial pressure waveform (RPW), which could be integrated into digital twins, that enables the reconstruction of RPW from PPG measurements using pulse pressure extrema. Built upon existing literature and supervised symbolic regression on anesthesia data, the model was validated on 581 continuous 10 seconds subsequences from 24 patients. Calibration through an unscented Kalman filter ensured patient-specific accuracy, yielding an averaged Pearson correlation coefficient of 0.955 for the reconstructed signal. The model's ordinary differential equation (ODE) with three parameters showed consistency with existing models. The stable, identifiable parameters underscore the model's robustness. The proposed model gives some insights into the physiology hidden behind the PPG and paves the way for RPW reconstruction using non-invasive measurements.

1. Introduction

PPG is a non-invasive optical technique used to explore vascular functions [1]. It involves emitting light onto the skin surface and measuring the light that either reflects back (reflection PPG) or passes through the tissue (transmission PPG). The tissue's optical properties vary with its composition, and at the heartbeat timescale, the PPG signal reflects the blood volume changes in both macro- and micro-circulation.

The PPG signal is affected by measurement modalities such as sensor contact pressure, sensor geometry – particularly its position with respect to the main arteries – the light wavelength used, and the choice between transmission and reflection PPG. By adjusting these modalities, PPG-based tools can be tailored for a wide range of applications.

In a clinical setting, PPG is utilized for heart rate monitoring, rhythm analysis, blood oxygen saturation measure-

ment, pulse wave velocity measurement, arterial stiffness estimation, breathing monitoring [1,2]. Additionally, over the past decade, PPG has been increasingly integrated into wearable devices for daily cardiovascular health monitoring [2,3], which paves the way for new applications such as atrial fibrillation detection [4], cardiovascular state assessment [5, 6], sleep stage classification [7] or hypertension assessment [8].

With its widespread use, including in commercial applications, research on PPG is very active [2]. Understanding the physical content of the PPG signal continues to be a significant challenge. New sensors are being designed to improve signal quality or target new features. Additionally, innovative analysis methods are being developed to extract more information from the signal, mostly relying on either signal feature extractions or machine learning techniques using the entire PPG waveform as input.

A prominent challenge in PPG analysis is the estimation of blood pressure from the PPG signal. While many attempts have been made to estimate systolic, diastolic or mean arterial pressure [9–17], few methods successfully reconstruct the complete waveform [18].

In this work, we propose a method to dynamically link RPW and PPG, paving the way for the integration of PPG into cardiovascular digital twins. As a first validation, we use it in a standalone fashion to reconstruct the RPW from the PPG signal within the context of vital sign monitoring during anesthesia. PPG signals are collected from the finger using a transmission setting, the calibrated RPW used for the model validation is measured invasively at the wrist and regular cuff blood pressure measurements providing systolic and diastolic pressure values are available.

2. Material and methods

The proposed method relies on an explicit model for-
mulated as a dynamical system that connects the tempo-
ral evolution of pressure with the PPG signal. Following a patient-specific parameter personalization phase conducted using an unscented Kalman filter [19], the model reconstructs the RPW from PPG measurements based on pulse pressure extrema.

## 2.1. Model formulation

The proposed model reads

$$ \dot{\text{PPG}}(t) = \alpha \text{PPG}(t) + \beta \frac{(\text{P}(t) - \text{P}_{\text{min}})}{\Delta \text{P}} + \gamma \frac{\dot{\text{P}}(t)}{\Delta \text{P}}, \quad (1) $$

where $\alpha, \beta$ and $\gamma$ are model parameters that can be calibrated and $\Delta P = P_{\text{max}} - P_{\text{min}}$ corresponds to the pressure range over which the PPG measurements are taken. In this model, the pressure and its time derivative are taken as source terms in order to reconstruct the PPG. It is worth noting that the signs of the parameters are critical to the model's consistency: $\alpha$ is negative in order to avoid exponential growth of PPG while $\beta$ and $\gamma$ are positive such that the dynamics of the pressure source terms drive the evolution of PPG. We also recovered this dynamics through supervised symbolic regression [20], amongst other ODE of varying complexity. The aim of this symbolic regression was to find an ODE able to reproduce the dynamics of the PPG by combining operations on RPW and its time derivative. It minimized the following functional $C_j$ with respect to $f_j$ in a chosen functional space:

$$ C_j(f_j, \mathbf{x}, \mathbf{u}, g) := \int_0^T f_j(\mathbf{x}(t), \mathbf{u}(t))g(t)dt + \int_0^T \mathbf{x}(t)\dot{g}(t)dt, \quad (2) $$

where $\mathbf{x}(t)$ is the target dynamics (PPG in our case), $\mathbf{u}(t)$ the source terms (P and $\dot{P}$) and $g(t) \in C^1$ being a test function.

By rewriting (1) such that $\dot{P}$ appears at the left-hand side, we can get an expression for the dynamics of the pressure as a function of the PPG, as shown hereafter

$$ \dot{\text{P}}(t) = c_1(\text{P}(t) - \text{P}_{\text{min}}) + \Delta P(c_2 \text{PPG}(t) + c_3 \dot{\text{P}}\text{G}(t)). \quad (3) $$

As the PPG's signal intensity cannot be used quantitatively, the subsequence's maximal systolic and minimal diastolic pressures (through $\Delta P$) are needed to rescale the RPW.

## 2.2. Data preprocessing

The data used to perform the symbolic regression procedure and the parameter estimation was taken from a partner hospital anesthesia database of patients. The PPG data was filtered using a band-pass filter with 0.5 Hz and 8 Hz cutoff frequencies in order to get rid of baseline drift as well as high frequency noise [18]. We then segmented the data for RPW and PPG signals using a random forest based algorithm to highlight each heartbeat. Any sequence of data

that did not contain continuous aggregations of heartbeats (which happens when either of the signals flatlines) was removed from the dataset. Furthermore, we looked at duration outliers and discarded any heart rate sequence with a frequency lower than 20 beats per minute (BPM) or higher than 200 BPM. Finally, the signals were resynchronised with one another using cross-correlation [18]. RPW being measured at the wrist while the PPG is taken at the fin-gertip induces a small lag between the signals due to wave travel time, which can be identified by looking at the location of the maximum value for the cross-correlation. The resulting data was then split into 10 seconds of continuous subsequences of RPW and PPG, with no overlap. The PPG data was normalized over each subsequence (not each heartbeat) such that it varies between 0 and 1 while retaining some features of the respiration. At the end of the pre-processing procedure, the data used in our study amounts to 24 patients split into 581 subsequences, which corresponds to 97 minutes of signal in total.

# 3. Results

## 3.1. Model validation

The derived model (1) was first validated in its ability to reconstruct the PPG dynamics using the RPW as its source term. We calibrated the model through an unscented Kalman filter. The PPG reconstruction is done by splitting the 10 seconds signal sequence into two parts: the first 5 seconds are used to calibrate the parameters, and then the PPG is predicted using only the RPW and its time derivative as source terms for the remaining 5 seconds. As a metric to compare the similarities of the two waveforms, we used Pearson's correlation coefficient ($r$) [18], as well as the Mean Absolute Error (MAE). Its averaged value as well as the means and standard deviations of the model's parameters can be found in Table 1a.

## 3.2. RPW reconstruction

We used the same calibration procedure as for the PPG model. An example of the parameter estimation procedure and the resulting predicted RPW can be seen on Fig 1. The model's parameters are identified sequentially during the estimation phase, which can be seen through the narrowing of their covariance. The averaged parameters' value over the different time sequences for (3) and their standard deviations can be seen in Table 1b, as well as the averaged Pearson's correlation coefficient.

# 4. Discussion

The performance of RPW reconstruction achieved with our newly proposed model is comparable to that reported in [18], with an average correlation coefficient of 0.993 (versus 0.955). However, our explicit dynamical system
<table><caption>Table 1. Mean values and standard deviations for the parameters estimated by the Kalman filter, mean Pearson correlation coefficient and MAE over the 581 sub sequences used for the reconstruction of (a) PPG and (b) RPW.</caption><thead><tr><th>(a)</th><th>α (s<sup>-1</sup>)</th><th>β (s<sup>-1</sup>)</th><th>γ</th><th>r</th><th>MAE</th></tr></thead><tbody><tr><td>-7.1 ± 2.5</td><td>7.0 ± 2.6</td><td>1.0 ± 0.15</td><td></td><td>0.961 ± 0.03</td><td>0.07 ± 0.04</td></tr></tbody></table><table><thead><tr><th>(b)</th><th>c<sub>1</sub> (s<sup>-1</sup>)</th><th>c<sub>2</sub> (s<sup>-1</sup>)</th><th>c<sub>3</sub></th><th>r</th><th>MAE (mmHg)</th></tr></thead><tbody><tr><td>-17 ± 17</td><td>17 ± 18</td><td>0.95 ± 0.27</td><td></td><td>0.955 ± 0.03</td><td>5.04 ± 2.47</td></tr></tbody></table>

<figure><img src="image_2.png" /><figcaption>Figure 1. RPW reconstruction and parameter estimation for the model linking pressure and PPG defined in (3). Covariances for the estimation phase are represented in light blue around the estimated trajectories.</figcaption></figure>

formulation offers several advantages over black-box approaches. It can be used, as demonstrated here, as a standalone modality for RPW reconstruction, but more importantly it can also be integrated into larger cardiovascular dynamical models that serve as the core element of a patient's digital twin, enabling the use of PPG measurements to inform and update the digital twin. This newly developed model is consistent with the existing literature [21] and expands on it, serving as a valuable tool to interpret the physiology encoded by the PPG waveform. Indeed, the formulated model in (1) can be linked to a one-stage Windkessel circulation model between the wrist and the fingertip. Assuming that the time derivative of the PPG is proportional to the variations of blood volume measured at the fingertip, $\frac{\beta}{\Delta P}$ plays the role of a resistance and $\frac{\gamma}{\Delta P}$ of a capacitance. The contributions of the pressure term and its time derivative correspond to the blood flux entering the wrist, whereas the PPG term corresponds to the blood flux going to the fingertip with $\alpha$ being a decay

constant. The calibration procedure for both (1) and (3) showed that the parameters were identifiable and had a reasonable overall variability (within 100% of their averaged value), following Gaussian-like distributions. In addition, the reconstructed signals showed high Pearson correlation coefficients, highlighting the potential of the two models for clinical use. However, our method has some shortcomings: the pressure range over which RPW is predicted must be known – typically via a cuff, which is mandatory during anesthesia – and the parameters of the model need a patient specific calibration, which requires a RPW over a few seconds. The requirement for RPW could be mitigated by instead using Doppler ultrasonography at the wrist in order to measure blood flow which could be used to calibrate a local Windkessel model [22]. This personalized circulation model would then be used to generate a RPW suitable for the calibration of (3). The practicability of this calibration procedure will depend on the parameters' intra-patient variability, which remains to be investigated in detail when
the RPW morphology is subject to change.

**5. Conclusion**

Our dynamical PPG model has been established and tested on a small cohort of patients and offers an explicit form, with interpretable parameters and behavior. The results showed that the reconstructed signals were in agreement with the patient data for both models, with high Pearson correlation coefficients. The literature on this model-based approach to PPG signal analysis remains limited [21] despite its advantages over purely signal-based analysis methods. Our model can be used as a bridge between standard cardiovascular models and PPG measurements, thus enhancing digital twin frameworks. Future work may involve enriching the model by incorporating additional terms in order to reconstruct RPW over a wider range of waveforms found in other databases, such as MIMICS-IV. In terms of clinical use, this model may enable the reconstruction of RPW from non-invasive measurements, for instance using a PPG device and a blood pressure cuff during anesthesia where both measures are readily available or with a wearable device embedding both sensors.

**References**

[1] Allen J. Photoplethysmography and its application in clinical physiological measurement. Physiological Measurement February 2007;28(3):R1. ISSN 0967-3334.

[2] Charlton PH, et. al. Wearable Photoplethysmography for Cardiovascular Monitoring. Proceedings of the IEEE 2022; 110(3):355-381. ISSN 1558-2256.

[3] Kirszenblat R, et. al. Validation of the Withings ScanWatch as a Wrist-Worn Reflective Pulse Oximeter: Prospective Interventional Clinical Study. J Med Internet Res 2021; 23(4):e27503. ISSN 1438-8871.

[4] Pereira T, et. al. Photoplethysmography based atrial fibrillation detection: a review. npj Digital Medicine January 2020;3(1):3. ISSN 2398-6352.

[5] Couceiro R, et. al. Assessment of cardiovascular function from multi-Gaussian fitting of a finger photoplethysmogram. Physiological Measurement August 2015; 36(9):1801. ISSN 0967-3334.

[6] Charlton PH, et. al. Assessing hemodynamics from the photoplethysmogram to gain insights into vascular age: a review from VascAgeNet. American Journal of Physiology Heart and Circulatory Physiology April 2022; 322(4):H493-H522. ISSN 0363-6135.

[7] Radha M, et. al. A deep transfer learning approach for wearable sleep stage classification with photoplethysmography. npj Digital Medicine September 2021;4(1):135. ISSN 2398-6352.

[8] Elgendi M, et. al. The use of photoplethysmography for assessing hypertension. npj Digital Medicine June 2019; 2(1):60. ISSN 2398-6352.

[9] Monte-Moreno E. Non-invasive estimate of blood glucose and blood pressure from a photoplethysmograph by means

of machine learning techniques. Artificial Intelligence in Medicine 2011;53(2):127-138. ISSN 0933-3657.

[10] Ruiz-Rodríguez JC, et. al. Innovative continuous non-invasive cuffless blood pressure monitoring based on photoplethysmography technology. Intensive Care Medicine 2013;39(9):1618-1625. ISSN 1432-1238.

[11] Choudhury AD, et. al. Estimating blood pressure using Windkessel model on photoplethysmogram. In 2014 36th Annual International Conference of the IEEE Engineering in Medicine and Biology Society. ISBN 1558-4615, August 2014; 4567-4570.

[12] Radha M, et. al. Estimating blood pressure trends and the nocturnal dip from photoplethysmography. Physiological Measurement 2019;40(2):025006. ISSN 0967-3334.

[13] Slapničar G, et. al. Blood Pressure Estimation from Photoplethysmogram Using a Spectro-Temporal Deep Neural Network. Sensors 2019;19(15). ISSN 1424-8220.

[14] Joachim J, et. al. Real-time estimation of mean arterial blood pressure based on photoplethysmography dicrotic notch and perfusion index. A pilot study. Journal of Clinical Monitoring and Computing 2021;35(2):395-404. ISSN 1573-2614.

[15] Leitner J, et. al. Personalized Blood Pressure Estimation Using Photoplethysmography: A Transfer Learning Approach. IEEE Journal of Biomedical and Health Informatics 2022;26(1):218-228. ISSN 2168-2208.

[16] Wang W, et. al. Cuff-Less Blood Pressure Estimation From Photoplethysmography via Visibility Graph and Transfer Learning. IEEE Journal of Biomedical and Health Informatics 2022;26(5):2075-2085. ISSN 2168-2208.

[17] Chu Y, et. al. Non-invasive arterial blood pressure measurement and SpO2 estimation using PPG signal: a deep learning framework. BMC Medical Informatics and Decision Making 2023;23(1):131. ISSN 1472-6947.

[18] Athaya T, et. al. An Estimation Method of Continuous Non-Invasive Arterial Blood Pressure Waveform Using Photoplethysmography: A U-Net Architecture-Based Approach. Sensors 2021;21(5). ISSN 1424-8220.

[19] Chapelle D, et. al. Fundamental principles of data assimilation underlying the Verdandi library: applications to biophysical model personalization within euHeart. Medical Biological Engineering Computing November 2012; 51(11):1221-1233.

[20] Qian Z, et. al. D-CODE: Discovering closed-form ODEs from observed trajectories. In International Conference on Learning Representations. 2022;.

[21] Cook LB. Extracting arterial flow waveforms from pulse oximeter waveforms: apparatus. Anaesthesia 2001; 56(6):551-555.

[22] Jana B, et. al. Windkessel model-based cuffless blood pressure estimation using continuous wave doppler ultrasound system. IEEE Sensors Journal 2020;20(17):9989-9999.

Address for correspondence:

Jérôme Diaz

Inria, 1 Rue Honoré d'Estienne d'Orves, 91120 Palaiseau

jerome.diaz@inria.fr