PITN: Physics-Informed Temporal Networks
for Cuffless Blood Pressure Estimation

Blood Pressure (BP) is a key cardiovascular parameter commonly used by clinicians to evaluate cardiac and circulatory health [24]. Traditionally, BP is measured using cuff-based (auscultatory or oscillometric) digital BP devices, which compress the user’s arm and often result in an unpleasant user experience [25]. With the increasing use of contactless sensors for health monitoring [26], recent work tends to explore different modal cuffless BP measurement methods (e.g., bioimpedance, millimeter wave, PPG), which use wearable devices to capture pulse activities and infer BP using learning-based methods, like BiLSTM-based HNN [8], and Transformer[9, 4]. However, they require amounts of invasive or cuff-based BP measurement results as ground truth labels to help map pulse activities to BP values. Particularly, beat-to-beat BP estimation and learning algorithms for continuous BP wave function extraction can be viewed as a time series modeling task, which is of significant importance in data mining and has attracted increasing attention in multiple areas [27, 28, 29, 30, 31, 32, 33, 34]. Numerous models have been developed to analyze these beat-to-beat time series signals [35, 36], which includes time domain LSTM-based methods [35], a mixture of time and frequency domain human designed features-based approaches [36]. These methods work well in data-sufficient situations, however, in precision physiological personal modeling, these models often fail due to limited training data and lack of personalized modeling. Unlike previous work, we propose to use a more generalized method for personal modeling which extracts intraperiodic features from the original waveform, and our proposed framework is based on physics constrain and personal modeling which only uses minimal training data while retaining high accuracy.

Numerous studies have provided supporting evidence for the proposition that training with adversarial examples can effectively improve the capabilities of models [37, 38]. For instance, compared to clean examples, adversarial ones contribute to a more effective alignment of network representations with salient data characteristics and human perception [39]. Furthermore, models trained with adversarial examples demonstrate heightened robustness to high-frequency noise [40]. Besides, previous work [41] also demonstrates that by automatically generating adversarial examples, adversarial training will not decrease model performance but benefit both generalization and robustness of models. And He et al. [42] use adversarial training to generate multivariate time-series samples for forecasting tasks. Especially for Physics-Informed Neural Networks (PINNs), of which the robustness can be enhanced by fine-tuning the model with adversarial samples [43]. To the best of our knowledge, we are the first to adopt such an adversarial training method as data augmentation in BP estimation.

Contrastive learning has shown remarkable success in deep learning tasks [44, 45]. Existing techniques like SimCLR [46] and SimCSE [47] have demonstrated superior performance in generating semantically rich image embeddings without the need for labeled data. These methods help pull instances together with the same labels in the latent space, while simultaneously pushing clusters of samples from different classes apart. However, contrastive learning is rarely applied to time series analysis [48]. Several common image augmentations including color changes or rotation may not be as relevant to time series data, especially in medical time series data mining. To address this insufficiency, we introduce contrastive learning to enhance the deep models’ performance by capturing discriminative variations in cardiovascular dynamics.

Given the input time series signal $\mathbf{X}\in\mathbb{R}^{N\times T\times C}$, the goal of cuffless blood pressure is to estimate blood pressure value $y$ (e.g., systolic BP (SBP) or diastolic BP (DBP)). Let $\mathbf{X}=[\mathbf{x}_{1},\mathbf{x}_{2},\ldots,\mathbf{x}_{N}]$ be the continuous bioimpedance (BioZ) signal with $N$ segments, length-$T$ and $C$ channels, for each segment $\mathbf{x}_{i}\in\mathbb{R}^{T\times C}$, we extract physiological feature $\mathbf{u}_{i}\in\mathbb{R}^{M}$ denoted as the $M$ dimensional vector. The neural network can be formulated as $\hat{y}=f(\mathbf{x},\mathbf{u},\theta)$ with input $\mathbf{x}$, $\mathbf{u}$, parameter $\theta$, and the predicted blood pressure $\hat{y}$.

In this part, we review the formal definition of Physics-Informed Neural Networks (PINNs) [11]. PINNs are trained to solve regular supervised tasks while adopting several given laws of physics described by partial differential equations (PDEs) as the network residual. However, the relationships that connect wearable measurements to cardiovascular parameters are not well-defined in the form of generalized PDEs. Following [6], we adopt the idea of building physics constraints with Taylor’s approximation for certain gradually changing cardiovascular phenomena (e.g., establishing the relationship between physiological features extracted from bioimpedance sensor measurements and BP). Here we can define a polynomial with Taylor’s approximation around $i$-th segment as the following:

Following [6], as for physiological feature $\mathbf{u}$, we denote the amplitude change as $u^{1}$, which serves as a proxy for the extent of arterial expansion. The second feature is represented by $u^{2}$, which captures the inverse of the relative time difference between the forward-traveling (i.e., systolic) wave and the reflection wave, providing an estimate of the pulse wave velocity (PWV). The third feature denoted as $u^{3}$, which corresponds to beat-to-beat heart rate. These three physiological features can be formulated as follows:

where $\Delta Z_{\mathrm{max}},\Delta Z_{\mathrm{min}}$ denotes the maximum and minimum variation in the (inverted) bioimpedance signal, $(t_{F},t_{B})$ refer to the trough point in the derivative signal and the zero-crossing point in the derivative signal transitioning to the ascent, and $(t_{J},t_{A})$ denote the peak point in the derivative signal, the zero-crossing point in the derivative signal transitioning to the descent, respectively.

As illustrated in Fig. 2, our framework comprises three key components: 1) Physics-Informed Temporal Network (PITN), 2) adversarial training, and 3) contrastive learning. In the PITN, newly designed temporal blocks are employed to extract personal physiological features, and Projected Gradient Descent (PGD) is tailed to generate adversarial samples for model training augmentation. These adversarial samples, along with clean samples, are disentangled using distinct layer normalization (LN) layers. Additionally, we adopt the contrastive learning loss by comparing clean and adversarial samples, based on their true blood pressure (BP) values.

The whole process is as follows. We first pre-process different modal signals (e.g. bioimpedance signal) to obtain clean input $\mathbf{x}$. Then adversarial sample $\tilde{\mathbf{x}}$ is generated using Project Gradient Descent (PGD). Both clean and adversarial inputs are fed through the temporal block for feature extraction and the output estimated BPs are denoted as $\hat{y}$ and $\tilde{y}$, respectively.

Physics-Informed Neural Networks (PINNs) traditionally use coordinates as input while overlooking potential temporal relationships among these inputs. This omission disregards the crucial inherent temporal dependencies present in practical physical systems, leading to a failure in globally propagating initial condition constraints and capturing accurate solutions across diverse scenarios [15]. In this paper, we introduce a novel temporal modeling block to enhance the capability of PINNs in modeling time-series medical data, by extracting complex temporal variations from transformed 2D tensors using a multi-scale Inception block.

As illustrated in Fig. 3, we organize the temporal block in a residual way [49]. Specifically, for the length-$T$ 1D input signal $\mathbf{x}\in\mathbb{R}^{T\times C}$, we project the raw inputs into the deep features $\mathbf{x}^{0}\in\mathbb{R}^{d_{\mathrm{model}}}$ through the initial embedding layer, represented as $\mathbf{x}^{0}=\mathrm{Embed}(\mathbf{x})$. For the $k$-th layer of temporal block, the input $\mathbf{x}^{k-1}\in\mathbb{R}^{d_{\mathrm{model}}}$ is processed as:

Each temporal block is designed to capture both intraperiod personal characteristics of physiological signals, as shown in Fig. 3. This process includes operations such as $\mathrm{Period}(\cdot)$, two $\mathrm{Reshape}(\cdot)$ functions, a 2D inception convolution $\mathrm{Inception}(\cdot)$ and residual addition. Initially, the 1D signal is extended to a 2D shape by calculating:

For processing extended 2D signal, the temporal block is formulated as follows:

Due to the limited availability of labeled physiological time series data, the results of current methods are still unsatisfactory. To address this issue, we enable blood pressure estimation using adversarial training by generating additional samples, while preserving the performance on clean data [37].

For estimating BP with the function $f(\mathbf{x},\mathbf{u},\theta)$, we formulate the adversarial training as an optimization problem:

In our proposed blood pressure estimation framework, we further regularize the perturbed signal by truncating it within the problem domain $\Omega$:

Although generating additional samples is beneficial for augmenting cuffless blood pressure data, increasing data points can lead to fluctuations in per-subject correlation [6]. Recently, contrastive learning has achieved success in self-supervised learning [53], by introducing additional constraints in the latent space to regularize embedding with the same label. Specifically, in classification tasks, similar data pairs (referred to as positive pairs) are selected for the same class, while all other pairs are treated as negative ones. However, applying contrastive learning to regression tasks remains non-trivial. In this part, we first introduce contrastive learning into the BP regression issue. Here, we define the $i$-th and $j$-th samples as a positive pair based on the following criterion:

We conduct our experiments on Graphene-HGCPT [1], Ring-CPT [5] and Blumio [3] datasets. The first two datasets contain raw time series data obtained from a wearable bioimpedance sensor, with corresponding reference BP values acquired using a medical-grade finger cuff (Finapres NOVA). The Blumio dataset includes data from 115 subjects (aged range 20-67 years), collected using several types of wearable sensors: PPG, applanation tonometry, and the Blumio millimeter-wave radar. Detailed information on each dataset is provided below.

We compare our model against several baseline models and state-of-the-art approaches including methods specifically designed for cuffless blood pressure estimation as Hybrid-LSTM [55], Physics-Informed Neural Networks (PINN) [6] and ResNet1D [4]. Additionally, we evaluate our framework against several general time series models, such as inverted Transformer (iTransformer) [56] and TimesNet [57]. We follow the methodologies provided by each corresponding paper during the experiment. More details refer to the supplementary.

Following the standard practice in blood pressure estimation [9, 6], we adopt common evaluation metrics, including root-mean-square error (RMSE), and Pearson’s correlation coefficient values. RMSE is defined as follows:

where $\hat{y}_{i}$ and $y_{i}$ correspond to $i$-th estimated and true BP value (SBP or DBP), and $N-S$ is the total number of test samples. Pearson’s correlation coefficient $r$ is calculated to measure the linear correlation between the estimated and true BP values, defined as:

where $\bar{Y}$ and $\bar{Y}_{\mathrm{true}}$ denote the mean values of the predicted ($\hat{y}$) and true BP values ($y$), respectively.

Additionally, the mean error (ME) and standard deviation of the error (SDE) are computed following the American National Standards Institute/ Association for the Advancement of Medical Instrumentation/ International Organization for Standardization AAMI standard. This standard requires BP devices to have ME and SDE values less than 5 and 8 mmHg, respectively [58]. Moreover, we utilize a pair-wise t-test [59] to compare the performance of our proposed method and other baseline models, in order to make the experimental results more convincing. More results under the AAMI standard and statistical significance analyses refer to the supplementary.

All implementations are based on the open-source PyTorch framework and trained on a single NVIDIA 3090 GPU. The networks are trained using the Adam optimizer with a learning rate set to 0.001. For generating controllable adversarial samples, we choose a relatively small perturbation parameter $\epsilon$ of 0.2 and set the training steps to 2. During training, we balance the magnitude of different losses by setting hyperparameters based on a parameter sensitivity analysis (see supplementary material). Specifically, we set the weight of physics loss to $\gamma=1$ and the minimal sensitivity of blood pressure to $y_{\mathrm{shift}}=2$.

In this subsection, we conduct ablation studies to examine the effectiveness of the proposed temporal block, adversarial training, and contrastive learning, respectively.