A Physiological-Model-Based Neural Network Framework for Blood Pressure Estimation from Photoplethysmography Signals

A flow diagram of the procedure, starting from data collection, is shown in Figure 1. The study was approved by the Ethics Committee Computer and Information Science (EC-CIS) of the University of Twente (No. 240831) and conducted in Roessingh Research and Development (RRD). 10 healthy participants aged between 20 and 29 were enrolled in the study. Informed consent was received from all the participants and the study was performed in accordance with the declaration of Helsinki.

Participants were asked to perform three exercise activities in 35 minutes. They started with sitting 5 minutes on a chair and standing 5 minutes. Then they were asked to cycle on an Bremshey® exercise bike at a fixed speed of 45 rpm, power of 50($\pm$10) watts for 10 minutes. After 1 minute’s resting, they started cycling again on the same bike at a fixed speed of 45 rpm, power of 100($\pm$10) watts for 10 minutes. Exercise intensities were set to assure patients trained within the aerobic threshold, and 10 minutes period length was chosen to guarantee the blood pressure, total peripheral resistance, and arterial compliance reached a steady-state, free from transient fluctuations at the start of exercise. During the whole study, subjects kept their left hand in a sling to prevent hydrostatic pressure artifacts. PPG signal was collected on the left index finger (Shimmer 3 GSR+ Unit, Shimmer Wearable Sensory Technology) and BP was measured continuously using volume-clamp PPG on the left middle finger (Finometer® Model-2, Finapres Medical Systems, Enschede, the Netherlands). During the whole study, the left hand was placed in a sling suspended around the neck where the left hand’s index and middle fingers were kept at heart level to prevent hydrostatic pressure artifacts for blood pressure and PPG’s reliable measurement. Each participant has conducted the whole experiment twice on different days.

Beat-to-beat systolic and diastolic blood pressure (SBP, DBP) signals, cardiac output (CO)(in mL/min) and total peripheral resistance (TPR)(mmHg·s/cm³) were exported from BeatScope® Easy software with Modelflow algorithm embeded, along with the estimated stroke volume (SV) on the beat to beat arterial waveform analysis. A second order Savitzky-Golay filter with the window size of 5 cycles was applied to smoothen the irregular turbulence of the above variables (SBP, DBP, CO, TPR).

The finger PPG signal from Shimmer GSR+ unit was exported from Consensys® software with a sampling rate of 64Hz. The preprocessing pipeline for PPG signals consisted of four key steps to enhance data quality and minimize artifacts. First, baseline drift was removed using a linear detrending method, ensuring the signal remained centered around its baseline. Second, motion artifacts were mitigated through the application of a median filter with a window size of 0.5 seconds, to reduce transient spikes caused by sudden movements. Third, a fourth-order Butterworth bandpass filter with cutoff frequencies of 0.5 Hz and 5.0 Hz was applied to isolate the physiological frequency range of PPG signals while suppressing noise outside this range. The cutoff frequencies were selected based on the Nyquist frequency, computed based on the sampling rate. Lastly, the denoised signal underwent a moving average filter with a 0.1-second window to further smoothen residual noise.

The systolic upstroke time ($T_{s}$) and diastolic time ($T_{d}$) from PPG were set as model inputs, while systolic pressure ($P_{s}$) and diastolic pressure ($P_{d}$) were set as outputs. Figure 2 illustrated these variables in one cycle’s PPG and BP signal. For signal synchronization, both BP and PPG preprocessed signals were aligned according to the time stamps of each activity and each device’s system time. Since PPG and BP were collected at the middle knuckle of index and middle finger on left hand, respectively, the physiological blood-flow time difference measured by the two signals was assumed as negligible.

The physiological model was defined by a set of validated equations (Eq.1 and Eq.2) [11] based on the two-element Windkessel model proposed by Otto Frank [12], that describes the mechanical properties of the arterial vessels in terms of total peripheral resistance and arterial compliance.

$$\begin{split}P_{s}&=P(t|t=T_{s})\\ &=P_{ts}e^{-T_{s}/RC}+\frac{I_{0}T_{s}C\pi R^{2}}{T_{s}^{2}+C^{2}\pi^{2}R^{2}}\left(1+e^{-T_{s}/RC}\right)\end{split}$$

(1)

$$P_{d}=P(t|t=T_{d})=P_{td}e^{-T_{d}/RC}$$

(2)

where the units for $T_{s}$ and $T_{d}$ are in seconds and for $P_{s}$ and $P_{d}$ are in mmHg. $P_{ts}$ and $P_{td}$ are the initial values of $P_{s}$ and $P_{d}$, respectively. Every cardiac cycle is assumed to start with a systolic period, and $P_{ts}$ is set to an initial value as the condition of first cardiac cycle. $R$ and $C$, in all the following text, refer to TPR and AC in units of mmHg·s/cm³ and cm³/mmHg, respectively. The blood flows from the ventricle to the aorta was assumed as a sine wave where $I_{0}$ is the peak amplitude, calculated as Eq.3.

$$\ I_{0}=\frac{CO\times T_{c}}{60\times\int_{0}^{T_{s}}\sin\left(\frac{\pi t}{T_{s}}\right)dt}\$$

(3)

where $CO$ is cardiac output (in mL/min) and $T_{c}$ (= $T_{s}$ + $T_{d}$) is the duration of one heart cycle.

Eq.1 and 2, as our physiological model, representing a continuous simulated correlation from $T_{s}$/$T_{d}$ to $P_{s}$/$P_{d}$. It is applied to add physiological constraint in neural network structure.

Figure 3 reveals the structure of our PMB-NN model. It consists of two parts, neural network and composed loss. We constructed a fully connected neural network (FCNN) with single input T and single output P, and 1 input layer, 3 hidden layers, and1 output layer (1-128-128-128-1) with ReLU activation functions mapping the input $T$ (which comprises $T_{s}$ and $T_{d}$) to the output $\hat{P}$. The input vector $T$ is merged of the two feature arrays ($T_{s}$ and $T_{d}$) alternatively and the target vector $P$ is merged in the same way, either.

The composed loss, consists of one data fitting term and two physiological constraint terms derived from Eq.1 and Eq.2, was used to optimize the hyperparameters (weights and biases) in the FCNN training. The data fitting term, $L_{data}$ is calculated as a $L_{2}$ loss function in Eq.4.

$$L_{data}=\frac{1}{N}\sum_{i=1}^{N}(\hat{P_{i}}-P_{i})^{2}$$

(4)

where $N$ is the length of $P$. $\hat{P}_{i}$ and $P_{i}$ are the FCNN output and ground truth of blood pressure, respectively. The physiological constraint terms, $L_{E1}$ and $L_{E2}$, are defined as another two $L_{2}$ loss functions in Eq.5 and 6 according to the parametric model in Eq.1 and 2. Initial values for $R$ and $C$ were both set as 1.

$$L_{E1}=\frac{1}{n}\sum_{i=1}^{n}(\hat{P}_{s,i}-f_{1}(P_{d,i-1},T_{s},R,C))^{2}$$

(5)

$$L_{E2}=\frac{1}{n}\sum_{i=1}^{n}(\hat{P}_{d,i}-f_{2}(P_{s,i},T_{d},R,C))^{2}$$

(6)

where $n$ equals to $0.5N$, is the length of $P_{s}$/$P_{d}$. $f_{1}$ and $f_{2}$ are the functions at the right side of Eq.4 and 5. $\hat{P}_{s,i}$ and $\hat{P}_{d,i}$ represent the predicted systolic and diastolic pressure in the $i^{th}$ cycle. $P_{d,i-1}$ and $P_{s,i}$ are real diastolic pressure in the ${i-1}^{th}$ cycle and systolic pressure in the $i^{th}$ cycle.

The optimization adjusts the model’s parameters, including neural network weights, biases, and physiological parameters $R$ and $C$, to minimize the total loss, $L_{tot}$. The Adam optimizer, combined with a learning rate scheduler with initial rate as 0.01, was used to iteratively refine these parameters and ensure stable convergence. The model was iteratively trained until $L_{tot}$ drops below a predefined threshold of 10 mmHg² within 1000 iterations, ensuring the model achieves a balance between accuracy and physical plausibility.

Training and testing data sets were the same length (10 minutes each activity for different types) collected on the same participant on different dates, respectively. We tested the performance of PMB-NN by comparing $P_{esti}$ and estimated and measured pressure values counted on systolic pressure and diastolic pressure, separately. Both performances were revealed in terms of root mean squared error (RMSE) (in mmHg) and mean absolute percentage error (MAPE) (in %). The estimated $R$ was compared with the golden standard $R$ which was calculated by the Beatscope software, while we only reported the estimated $C$ given the lack of measured $C$ using Finometer Model-2 devices.