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Physics-Constrained Deep Learning Model for Contactless Blood Pressure Monitoring from Triaxial Bodyseismography

Yuanyuan Zhang, Yida Zhang, Jiahui Li, Yuyan Wu, Fei Dou, Xiao Yin, Zhenlin An, Hae Young Noh, Wenzhan Song

arXiv:2608.23562v1eess.SPcs.AIphysics.bio-ph

TL;DR

The paper addresses robustness problems in contactless BP estimation caused by distorted, axis-incomplete BCG signals and subject-specific hemodynamic variation. It proposes Phy-BP, which combines adaptive quality control, triaxial BSG, and a wave-propagation-based physical constraint. On hospital data, Phy-BP outperformed listed baselines, with reported improvements for triaxial and single-axis comparisons and AAMI-compliant BP errors.

Abstract

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Ballistocardiography (BCG) is promising for unobtrusive long-term blood pressure (BP) monitoring in laboratory settings, but traditional BCG signals are vulnerable to the variations in body-bed interaction with shifted fiducial points in temporal or amplitude axis, and BP varies with personal hemodynamic changes, causing misaligned representations that affect model generalizability and robustness. In this work, we propose a non-invasive BP estimation framework, Phy-BP, based on triaxial bodyseismography (BSG) as an extension of BCG. Firstly, an adaptive quality-control algorithm is designed to select BSG segments enriched with cardiogenic components by jointly considering neighboring beat patterns and universal cardiogenic templates. Furthermore, a physical model is established to describe 3D wave propagation in the body-bed system and is subsequently embedded into the deep learning model to characterize the intrinsic coupling among triaxial BSG signals driven by a single cardiogenic excitation. Thus, multi-axis features are aligned during model training, improving robustness against distortions in real scenarios. Experiments on a 162-hour hospital dataset collected from 21 subjects reveal that the proposed Phy-BP can dynamically filter out low-quality measurements, and the deep learning model training is constrained by physical consistency across different axes to provide faithful BP monitoring, especially when training samples are limited.

I. INTRODUCTION

Contactless BP monitoring needs continuous, reliable measurements, but cuff-based references and 1D BCG signals are limited by intermittent sampling, body-bed variation, and axis-specific signal loss. Phy-BP addresses these issues with invasive ABP supervision, adaptive quality control, triaxial sensing, and physics-constrained feature alignment.

  • Accurate long-term BP monitoring supports nocturnal and longitudinal hemodynamic assessment, but invasive ABP requires catheterization and carries clinical risks.Source: §I · p. 1
  • Cuff-based devices are widely used intermittently, yet their accuracy depends on device algorithms, cuff size, and arm position, while inflation can disrupt sleep and reduce compliance.Source: §I · p. 1
  • Oscillometric ground truth cannot capture beat-to-beat BP dynamics, limiting its suitability as a continuous clinical reference.Source: §I · p. 2
  • 1D BCG may miss vibration energy leaking into X- and Z-axes, while mattress and body-weight changes attenuate or shift fiducial features.Source: §I · p. 2
  • Purely data-driven BCG models leave cardiogenic excitation and wave propagation unmodeled, making representations vulnerable to uncertain BP labels and changing BCG morphology.Source: §I · p. 2
  • Phy-BP uses triaxial BSG, invasive ABP reference measurements, adaptive quality control, and a physical constraint to align features across axes.Source: §I · p. 2
  • 162 hours of hospital data from 21 patients were used for validation, with reported performance outperforming representative BCG- and PPG/ECG-based methods and satisfying AAMI requirements.Source: §I · p. 2

II. BACKGROUND AND CHALLENGES

BP reflects cardiovascular mechanics linked to cardiac output and peripheral resistance, while BSG provides an indirect mechanical view of cardiac activity rather than arterial pressure itself. Triaxial sensing and physics-guided modeling are therefore introduced to capture axis-dependent body-bed responses and align features under changing signal morphology.

III. METHODOLOGY

Phy-BP combines adaptive BSG quality control with a viscoelastic propagation model and physics-constrained triaxial feature alignment for BP estimation.

  • Phy-BP uses hospital-collected, timestamp-synchronized BSG and hemodynamic data for deep-learning BP estimation.Source: A. Overview · pp. 3–4

Quality control

  • Quality control evaluates Y-axis BSG with matched filters to retain segments containing cardiogenic components.The pipeline counts recognized single-cycle BSG patterns to assess segment quality.Source: A. Overview · p. 4, B. Quality Control · p. 4

Physical model

  • The physical model represents heartbeat-driven propagation through the body-bed system as a viscoelastic medium observed by triaxial BSG.It uses a PDE with a point cardiogenic source and models the three-axis measurement as a single-point displacement.Source: A. Overview · p. 4

Physics-constrained deep learning model

  • The physics-constrained layer aligns three-axis features while preserving axis-specific energy leakage and reducing inconsistency from attenuation and temporal distortion.The layer governs feature evolution within an encoder-decoder architecture.Source: A. Overview · p. 4, Figure 3 · p. 4

Matched filter

  • The universal cardiogenic template is a fourth derivative of a Gaussian, but fixed-template filtering can limit adaptability across waveform conditions.Waveform variation across subjects, postures, and mattresses can produce missed or false cycle detections.Source: 1) Matched Filter with Cardiogenic… · pp. 4–5

2) Matched Filter with Dynamic Template Learned from the Current Segment:

The dynamic-template filter learns a representative heartbeat morphology from candidate cycles after correcting their temporal mismatch with linear time warping.

  • A dynamic template is learned from candidate cycles detected by the preceding cardiogenic matched filter.The candidates are expected to contain relatively clean cardiogenic components.Source: 2) Matched Filter with Dynamic… · p. 5
  • Linear time warping rescales each candidate before aggregation to reduce morphology blurring caused by temporal mismatch.Warping stretches or compresses cycles through linear interpolation while preserving overall waveform morphology.Source: 2) Matched Filter with Dynamic… · p. 5
  • The optimization finds scaling lengths and a shared template that make candidate cycles close to one representative morphology.All warped cycles are truncated or padded to the original window length.Source: 2) Matched Filter with Dynamic… · p. 5
  • The scaling ratio Li/Tw is restricted within ±15% to prevent trivial solutions during template learning.Source: 2) Matched Filter with Dynamic… · p. 5

3) Assessment of Signal Quality with Feature Enhancement:

Signal quality assessment identifies high-quality BSG cycles using cardiogenic and dynamic matched filters, then enhances preserved segments with a Hilbert envelope.

2) Finite-Dimensional Dynamic System:

Finite-element discretization converts the body-bed propagation PDE into a high-dimensional dynamical system, which is reduced through modal truncation for efficient physics-constrained learning.

  • Finite-element discretization transforms the propagation PDE into a second-order system with mass, damping, stiffness, and cardiogenic-force terms.q(t) represents nodal displacement states, while M, C, K, and Fc(t) represent system dynamics and excitation.Source: 2) Finite-Dimensional Dynamic System · p. 6
  • Modal truncation is adopted because the finite-element system has too many nodal states for efficient learning and inference.Source: 3) Model Reduction for the Body-bed… · p. 6

Model assumptions

  • The reduction assumes measured BSG is dominated by a limited number of low-frequency vibration modes after removing rigid-body modes.Proportional Rayleigh damping is also assumed.Source: 3) Model Reduction for the Body-bed… · p. 6
  • The reduced model retains the first few dominant modes within the effective BSG frequency band while remaining computationally tractable.An observation matrix maps the latent structural response to sensor channels.Source: 3) Model Reduction for the Body-bed… · p. 6

D. Physics-constrained Deep Learning Model

The model inserts a physics-constrained layer between axis-specific encoders and the decoder to align triaxial BSG representations and regularize their latent dynamics. Training combines BP regression with physics-guided regularization so aligned features retain informative axis-specific components while following a shared dynamical law.

Physics-constrained layer

  • A physics-constrained layer aligns latent representations from three BSG encoders before decoding BP predictions.Its inputs are latent features from the X-, Y-, and Z-axis encoders.Source: 2) Physics-constrained Layer Design · p. 6

Gate-based feature alignment

  • Gate-based alignment normalizes each axis, learns time-varying gates, and selectively preserves informative components while reducing distortion-induced inconsistency.Features are pulled toward a shared prototype when distortion or energy leakage weakens cardiogenic consistency.Source: 2) Physics-constrained Layer Design · pp. 6–7

Physics-constrained latent state regularization

  • Aligned axis features are projected into low-dimensional latent states, while their fused representation provides a shared latent cardiogenic excitation.The individual states need not be identical across axes.Source: 2) Physics-constrained Layer Design · p. 7
  • The regularizer encourages all axis features to follow the same underlying dynamical law by minimizing a dynamics residual.The state-space dynamics use matrices constructed according to a damped second-order dynamical form.Source: 2) Physics-constrained Layer Design · p. 7

Overall loss

  • The overall loss adds a physics-guided regularization term to the BP regression loss, balanced by weighting coefficient λ.LBP measures prediction error against ground-truth BP, while Lphy supplies the physics constraint.Source: 2) Physics-constrained Layer Design · p. 7

IV. DETAILS OF EXPERIMENTS AND DATASET

The study uses geophones mounted under a hospital bed to collect synchronous triaxial BSG and physiological data. Its 162-hour dataset spans 21 people, broad MAP and age ranges, and diverse BP conditions for evaluating robustness.

Dataset collection

  • Three seismic sensors are mounted under the bed frame at different orientations to provide 3-axis sensing in a hospital setting.Each under-bed sensor uses the same geophone and data acquisition board.Source: A. Dataset Collection and Description · p. 7
  • The geophone uses a fixed magnetic core enclosed by wire coils to transduce subtle bed vibrations.This sensor system has been validated for cardiac or sleeping monitoring in prior studies.Source: A. Dataset Collection and Description · pp. 7–8

Dataset description

  • The dataset contains 162 hours of synchronous data from 21 people aged 20 to 84, with ECG, PPG, and PiCCO-derived hemodynamic data collected for comparison or physiological characterization.All vital-sign signals are sampled at 100 Hz.Source: A. Dataset Collection and Description · p. 8
  • The hospital dataset covers a wider MAP distribution than many controlled laboratory datasets and includes a wide age range and diverse BP conditions.These characteristics support more comprehensive evaluation of whether models learn BP-related cardiogenic features rather than population-mean values.Source: A. Dataset Collection and Description · p. 8

B. Implementation Details

Experiments use repeated LOSO cross-validation and compare Phy-BP with BCG, PPG/ECG, and vanilla ResNet-18 baselines using standard BP metrics and AAMI criteria. The evaluation also examines calibration for personal hemodynamic differences and practical robustness.

Training and validation

  • Experiments use leave-one-subject-out cross-validation, with one subject held out for testing while the others provide training and validation data.All experiments are repeated five times and reported as mean values.Source: 1) Model Training Strategy · p. 8

Methods for comparison and metrics

  • Phy-BP is compared with recent BCG and PPG/ECG frameworks, using vanilla ResNet-18 as the baseline.Evaluation metrics include ME, MAE, PCC, and STD.Source: 2) Methods for Comparison and… · p. 8
  • The AAMI criteria require |ME| ≤ 5 mmHg and STD ≤ 8 mmHg for BP prediction studies.The study also calculates averaged relative improvement across SBP, DBP, and MAP tasks and their metrics.Source: 2) Methods for Comparison and… · p. 8

Calibration for personal hemodynamic difference

  • Personal TPR conditions motivate calibration because existing non-invasive measurements do not establish that TPR can be inferred reliably.The dataset includes PiCCO-derived CO, enabling quantitative TPR calculation.Source: 3) Calibration for Personal… · p. 8

Evaluation design

  • The evaluation compares overall performance, module contributions, and robustness across real-world situations.The planned analyses separately assess quality control and the physics-constrained layer, followed by practical robustness and generalizability tests.Source: §V · p. 9

A. Overall Performance

Phy-BP achieved accurate BP estimation across realistic hospital conditions and remained more robust than representative baselines under subject-specific hemodynamic variability. Performance degraded with increasing TPR variability, but Phy-BP generally maintained lower errors and more stable trends.

Overall comparison

  • Phy-BP achieved MAEs of 6.03, 4.23 and 5.07 mmHg for SBP, DBP and MAP, respectively, with a 52.76% improvement over ResNet.The comparison covered representative BP-estimation frameworks under hospital data-collection conditions.Source: 1) Performance Overview · p. 9

Realistic hospital setting

  • The hospital dataset included routine patient movements, varying body-bed interactions and medication-induced hemodynamic changes that challenged compared models even after calibration.These conditions differ from controlled laboratory settings with more stable posture, attachment and morphology.Source: 1) Performance Overview · pp. 9–10

Baseline comparison

  • BCG baselines improved over PPG/ECG methods but remained below Phy-BP, with FSNet and AI-BCG improvements of 24.89% and 29.11%.Single-axis sensing observes only a partial projection of multidirectional cardiogenic vibration propagation.Source: 3) Comparisons to Models using… · p. 10

TPR variability

  • All three models degraded as TPR variability increased, while Phy-BP maintained lower MAE and STD across most subjects with a more stable trend.The analysis treated hemodynamic drift as a common challenge for cuffless and contactless BP estimation.Source: 4) Analysis for Different TPR · p. 10
  • MAP MAE increased from 4.24 mmHg in the low-TPR group to 7.15 mmHg in the high-TPR group, while PCC decreased from 0.88 to 0.67.High-TPR subjects also showed greater variability in MAE and STD.Source: 4) Analysis for Different TPR · p. 11

B. Ablation Studies

Ablation studies show that quality control, multiple BSG axes and physics-constrained learning each contribute to Phy-BP performance. The physical constraint particularly improves error consistency and correlation with BP variation.

Quality control

  • Stricter quality-control cycle requirements generally reduced MAE and STD while improving PCC across all samples.Increasing the threshold progressively removed samples with large prediction errors, although excessively strict thresholds reduce retained data.Source: 1) Effectiveness of Quality Control · p. 11

Input-axis combinations

  • Adding the X-axis increased the overall improvement from 5.76% for Y-only input to 38.22%, demonstrating complementary information across vibration directions.The gain is consistent with cardiogenic energy leaking from the conventional Y-axis into other axes.Source: 2) Necessity of using Triaxial BSG for… · p. 11
  • The Z-axis contained substantial cardiogenic information, while the Y+Z combination reached a 41.33% overall improvement.Z-only performance was close to BCG-based methods such as FSNet and AI-BCG.Source: 2) Necessity of using Triaxial BSG for… · p. 11

Physics constraint

  • Without physical regularization, triaxial inputs achieved a 34.75% overall improvement, whereas the best result reached 52.76% at λ = 1.5.The constrained model reduced STD and increased PCC relative to λ = 0.Source: 3) Effects of Physics Constraint · pp. 11–12

4) Effects of Calibration:

Calibration substantially improves performance when subject-specific hemodynamics are not directly observable, especially for unstable vascular-resistance conditions. Physics-constrained training also preserves more reasonable performance as training data become scarce.

Calibration effects

  • Without calibration, the model showed a −5.24% overall improvement, whereas 1% calibration samples increased the overall score to 42.23%.In this dataset, 1% corresponded to approximately one calibration every 40 hours and reduced MAE/STD for all BP targets.Source: 4) Effects of Calibration · p. 12
  • Subjects with higher TPR variability generally obtained larger MAP MAE improvements from calibration, especially at the 3% and 5% settings.The reported trend links calibration benefit to changes in individual vascular resistance.Source: 4) Effects of Calibration · p. 12

Limited training data

  • As training data were reduced, performance degradation became more pronounced in the low-data regime once diverse body-bed interactions and hemodynamic conditions were insufficiently covered.The degradation was not linear across training-set scales.Source: C. Performance under Limited Training… · p. 13
  • Without the physics constraint, the model became unstable after removing 40% of training data, whereas the constrained Phy-BP maintained reasonable performance using only 40% of the data.The shared dynamical structure among triaxial representations supports learning from limited samples.Source: C. Performance under Limited Training… · p. 13

Overall conclusion

  • Phy-BP achieved MAEs of 6.03, 4.23 and 5.07 mmHg for SBP, DBP and MAP while satisfying the AAMI accuracy standard.The conclusion reports validation on a 162-hour dataset from 21 patients.Source: §VI · p. 13

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