Cardiovascular digital twins using a Windkessel physics informed neural network

Abstract

Cardiovascular digital twins (CDTs) have the potential to transform precision medicine by enabling tailored insights, continuous monitoring, and personalized simulations of cardiovascular dynamics through virtual representations of the cardiovascular system. Accurately building these representations requires precise estimation of personalized parameters such as arterial compliance and peripheral resistance. However, current methods are often burdensome, rely on invasive procedures, or require large datasets. To address these limitations, we present Windkessel physics-informed neural networks (WPINNs), a framework combining Windkessel models with physics-informed neural networks (PINNs) to estimate personalized cardiovascular parameters and predict blood pressure (BP) waveforms from noninvasive bioimpedance (Bio-Z) wearables. WPINNs embed the governing differential equations of Windkessel models into the training process, enabling interpretable and accurate BP predictions with minimal ground truth data. We validate WPINNs using Bio-Z datasets from healthy and hypertensive individuals, achieving a 12%–25% reduction in error compared to traditional data-driven deep learning models. Additionally, WPINNs estimate arterial compliance and peripheral resistance with high accuracy, resulting in errors from 0.77% to 6.07% utilizing a synthetic cardiovascular waveform dataset. This work establishes WPINNs as a strong foundation for noninvasive and interpretable CDT frameworks.

Introduction

Digital twins for cardiovascular health offer a promising avenue for precision medicine by enabling real-time simulations, continuous monitoring, and prediction of individual health trajectories¹. Cardiovascular digital twins (CDTs) are personalized on an individual basis through the continuous bi-directional flow of information between the individual and a virtual representation of their cardiovascular system². As the global burden of cardiovascular disease (CVD) continues to rise, leading to conditions such as hypertension, CDTs provide an opportunity to enhance patient-specific care through early detection, personalized monitoring, and targeted treatment interventions^3–5. At the core of the CDTs, the virtual representation aims to replicate the underlying complex dynamic behavior of an individual’s cardiovascular system under various physiological states, including physiological fluctuations, responses to treatments, and disease progression, validated by a human-in-the-loop expert (e.g., cardiologist), illustrated in Fig. 1^2,6. However, constructing CDTs with accurate virtual representations requires estimating numerous personalized parameters, a process that inherently depends on large amounts of ground truth data or hard-to-access data such as clinical images⁷. Existing methods rely on invasive or bulky procedures often fed to complex computational models or population-level assumptions, which limit both personalization and real-time updates to the virtual representation^1,2. To address these limitations, we propose a CDT framework that leverages noninvasive data from wearables to estimate personalized cardiovascular parameters and physiological waveforms, with minimal dependency on the ground truth data.

Fig. 1. Cardiovascular digital twin framework cycle.

Illustration of cardiovascular digital twin framework highlighting the bi-directional flow of data between the physical counterpart and the virtual representation.

In healthcare settings, individualized physiological data is often scarce and predominantly limited to baseline states, which may not accurately reflect pathophysiological conditions such as heart failure; moreover, data capturing disease-specific manifestations at the personalized level remains limited⁸. In settings with limited data, scientific machine learning (SciML) offers a promising paradigm by integrating machine learning with physical laws to ensure scientifically grounded predictions⁹. Particularly, a subset of SciML models called physics-informed neural networks (PINNs) is gaining traction within precision health applications for their ability to embed physiological physics, in the form of differential equations, directly into neural network training as an additional component in the neural network loss function^10,11. PINNs have been shown to reduce the dependence on ground truth data; furthermore, they support personalization and extraction of interpretable physiological parameters^12,13. Nevertheless, conventional PINNs using one-dimensional (1-D) or three-dimensional (3-D) physics for modeling cardiovascular dynamics rely on geometrical information from invasive or imaging-based procedures, severely limiting their scalability and applicability to ambulatory settings^14,15. A promising alternative involves the zero-dimensional (0-D) Windkessel models, which represent the cardiovascular system using lumped parameters without the need for anatomical data¹⁶. The model is defined by ordinary differential equations (ODEs), relates blood pressure (BP) and blood flow through clinically relevant parameters such as arterial compliance and peripheral resistance¹⁷. These interpretable parameters are highly personalized and key to diagnosing vascular stiffness and understanding BP regulation^18,19. Dynamic fluctuations in these parameters are triggered by stimuli resulting in vascular vasodilation or vasoconstriction, offering insights into cardiovascular function and treatment efficacy^20,21. Although validation methods for the Windkessel model, such as diastolic time decay analysis and mock circulatory systems, can effectively assess its ability to capture underlying hemodynamics, validating the model with real patient data often requires invasive procedures to collect data from aortic or central arteries near the heart^22,23. Moreover, traditional Windkessel modeling approaches treat parameters as static values, limiting their real-world representation of dynamic physiology. Research supports dynamic pressure-dependent formulations and beat-to-beat estimates to more accurately represent cardiovascular physiology, though this comes at the cost of increased computational demands due to nonlinear ODEs^24,25 along with the need for more data for personalizing the complex parameters. A gap remains in developing personalized virtual representations of the cardiovascular system using PINNs that integrate Windkessel models with dynamically defined personalized parameters, to be used in CDT applications.

In this work, we present the Windkessel PINNs (WPINNs) for establishing a dynamic virtual representation of an individual’s cardiovascular system, with reduced dependency on ground truth data. By incorporating the Windkessel model’s ODEs into the PINN training process, we estimate personalized cardiovascular parameters and accurately predict key waveforms such as BP. Our framework extracts blood flow waveforms from calibrated wearable bioimpedance data collected from more easily accessible peripheral arteries, eliminating the need for invasive or burdensome imaging data from aortic or central arteries. We test these models using peripheral arteries, including brachial, radial, and digital, due to the ground truth extracted from the Finapres NOVA, a finger cuff-based system for collecting BP waveforms, and these arteries still show a compliant behavior. These peripheral arteries retain a compliant behavior due to the presence of a smooth muscle layer in the arterial wall; as a result, the pressure-area relationships remain physiologically meaningful in these vessels, appropriate to employ the Windkessel models to represent the hemodynamic behavior²⁶. Figure 2a illustrates our end-to-end framework. WPINNs lay the foundation for accurate, clinically interpretable, and convenient CDTs.

Fig. 2. Windkessel physics-informed neural network (WPINN).

a WPINN framework flows from wearable data inputs (blue) to outputs (green) of cardiovascular metrics and uncertainty quantification (dark blue). b 3-element (3E) Windkessel illustration c Data collection protocol diagram for Bio-Z datasets. d Illustration of WPINN deep learning architecture using convolutional neural network (CNN) downsampling layers (white and green), transformer encoder (yellow), and CNN upsampling layers (white and light blue).

We evaluated the effectiveness of WPINN CDTs through a case study on cuffless BP estimation using wearable bioimpedance sensors. Bioimpedance (Bio-Z) is a noninvasive sensing technique that delivers a high-frequency, low-amplitude electrical current into the body, where the resulting impedance changes reflect tissue and fluid composition²⁷. Bio-Z sensors placed over arteries capture impedance variations caused by arterial volume fluctuations as pulse waves travel through the vessels²⁸. These oscillating Bio-Z signals contain rich hemodynamic information that can be used to predict cardiovascular metrics such as BP, supporting noninvasive applications like cuffless BP monitoring^29,30.

The contributions of this framework includes: (i) predicting critical cardiovascular variables such as continuous BP waveform from extensive wearable Bio-Z datasets collected from a cohort of healthy and hypertensive individuals, trained on minimal ground truth data, (ii) assessing optimal Windkessel model complexities (based on number of modeled elements) and defined parameter complexities (whether parameters are constant, vary beat-to-beat, or vary dynamically with pressure) for predicting BP, and associated computational tradeoffs, (iii) validating the personalized parameter estimation dynamics of WPINN using an extensive dataset of simulated subjects with realistic blood flow and BP waveforms, (iv) including clinically relevant interpretability metrics, such as pressure-compliance curves and arterial resistance dynamics extracted from WPINN at personalized levels, and (v) utilizing uncertainty quantification through the calculated physics error from WPINN.

The wearables used in this study include Bio-Z rings and wristbands. Separate datasets were utilized from previous studies for each modality, with N = 6 presumed healthy participants using the Bio-Z ring and N = 23 hypertensive participants using the Bio-Z wristbands, representing the largest known Bio-Z dataset to date. Participants underwent external stimuli, including hand gripper exercises, Valsalva maneuvers, and cold pressor tests, to induce fluctuations in BP from baseline. These interventions produced a diastolic BP (DBP) range of 57 mmHg to 134 mmHg and a systolic BP (SBP) range of 92 mmHg to 203 mmHg, captured beat-to-beat using the Finapres NOVA system, highlighting the wide range of captured BP and diverse cardiovascular properties. Our WPINN model demonstrated strong predictive BP performance, achieving 12%–25% improvement over conventional data-driven models, achieving a root mean squared error (RMSE) as low as 6.49 mmHg and 3.99 mmHg for SBP and DBP, respectively. Additionally, the model produced highly accurate estimates of personalized, dynamic cardiovascular parameters, with percentage errors ranging from 5.78% to 6.07% for arterial compliance and 0.77% to 1.01% for peripheral resistance, when tested on an extensive synthetic cardiovascular waveform dataset. These results highlight the robustness of WPINN as a foundational component for CDT systems.

Results

Windkessel physics informed neural network

The BP waveform is driven by a physiological mechanism that is affected by factors including stroke volume (SV), heart rate (HR), arterial compliance, and vascular resistance. SV and HR are cardiac parameters: SV represents the volume of blood ejected into the aorta from the left ventricle, while HR refers to the number of times the heart beats within a given time period, typically per minute. The remaining factors, arterial compliance and vascular resistance, depend on an individual’s vascular system properties, which can vary based on demographics³¹. Healthy vascular function is characterized by higher compliance and lower resistance, whereas reduced compliance and increased resistance indicate potential cardiovascular health issues^32,33.

Unhealthy vascular function is typically clinically assessed using pulse wave velocity (PWV), a metric that measures the speed at which a pulse wave travels between two arterial sites. An increased PWV indicates stiffer arteries, which are consequently less compliant. More details regarding the derivation of PWV can be found in the “Methods” section³⁴.

Similarly, metrics of vascular resistance named total peripheral resistance (TPR), also known as systemic vascular resistance, can also be expressed using mean arterial pressure (MAP) and cardiac output (CO), mathematically expressed under the “Methods” section^35,36. These physiological factors can vary not only according to individual demographics, but also as a result of external stimuli such as physical or mental stressors³⁶. These stress stimuli excite the sympathetic nervous system, fight or flight response, leading to increases in cardiac output, vasoconstriction, which is the narrowing of the blood vessels, and subsequently increases BP³⁷. Once the stimuli are mitigated or diminished, the body will return to baseline³⁸.

The Windkessel models, describing the relationship between blood flow waveforms, expressed as the change in blood volume through time, to BP waveforms, incorporate the physiologically relevant compliance and resistance parameters. The simplest Windkessel model is mathematically expressed using the 2-element (2E) Windkessel model, an ordinary differential equation (ODE) detailed under the “Methods” section. As an extension of the 2E model, the 3-element (3E) model, illustrated in Fig. 2b, includes an additional impedance term called characteristic impedance ($Z_c$).

The parameters of the Windkessel models are specific for each individual and can be expressed with varying degrees of complexity. (i) Assuming the personalized parameters (C, $R_p$, $Z_c$) are constant across all beats in time, (ii) parameters fluctuate on a beat-to-beat basis, and (iii) C and $Z_c$ are expressed as pressure-dependent functions, hence $C\left(P\right)$ and $Z_c\left(P\right)$ parameter functions are defined¹⁶. Further details regarding these parameter functions are found under the “Methods” section.

In PINN architecture, two losses are conventionally defined, which include ${\mathcal{L}}_{\mathrm{data}}$, the supervised mean squared error loss between predicted and ground truth data, and ${\mathcal{L}}_{\mathrm{residual}}$, which is the mean squared error loss of the related function to inform the predictions of network^10,12. In our Windkessel PINN (WPINN), the residual physics loss ${\mathcal{L}}_{\mathrm{WK}}$ is either the 2E or 3E ODEs, which integrates personalized parameters with varying degrees of complexity. Two identical network architectures are defined, where one is our baseline only driven by the supervised data loss and the other is our proposed WPINN models integrating the total loss function defined under the “Methods” section. The input for each model includes time, Bio-Z waveform, and calibrated blood flow waveform of fixed-length arrays; in addition, some models require calculated SV inputs to calculate beat-to-beat parameter values. Detailed information regarding the specific network architectures, calibrated blood flow waveforms, and calculated SV can be found in the “Methods” section.

Study design

Two previously collected datasets containing Bio-Z waveforms, BP waveforms, and SV values are utilized in this study. Reference ground truth BP waveforms and SV values are collected using the Finapres NOVA, an oscillometric finger cuff device for noninvasive cardiovascular monitoring³⁹. The first dataset, titled ring-CPT, consists of Bio-Z measurements whose extracted impedance waveforms are utilized as inputs to our models, and their temporal derivatives are employed as proxies for arterial blood flow, collected from a Bio-Z ring wearable, placed over the digital artery of the finger. Additionally, information regarding the Bio-Z data collection and flow calibration process can be found under the “Methods” section of the manuscript. Participants (N = 6, Beat-Samples = 8012, Age (years) = 21 ± 3, Gender = 1: Female & 5: Male, Height (cm) = 173 ± 6, Weight (kg) = 89 ± 48) performed the cold pressor test maneuver to induce a sympathetic response, leading to BP fluctuations. The diastolic and systolic BP ranges in this dataset are 57 mmHg to 134 mmHg and 92 mmHg to 203 mmHg, respectively³⁰. The second dataset, titled hypertensive-HGVCPT, contains Bio-Z measurements collected from a Bio-Z watch wearable, positioned over the radial artery of the wrist. Participants (N = 23, Beat-Samples = 46,096, Age (years) = 56 ± 14, Gender = 11: Female & 12: Male, Height (cm) = 168 ± 10, Weight (kg) = 93 ± 25) were diagnosed hypertensive patients who performed three maneuvers to induce BP fluctuations: (i) hand gripper exercise, (ii) the Valsalva maneuver, (iii) and the cold pressor test⁴⁰. The diastolic and systolic BP ranges in the hypertensive-HGVCPT dataset are 59–119 mmHg and 111–188 mmHg, respectively. Utilization of both datasets provides a diversified evaluation of the proposed model’s ability to generalize across different physiological conditions, including both presumably healthy and hypertensive individuals. Figure 2c displays this data collection protocol.

WPINN model validation and uncertainty quantification

Ensuring robust validation and uncertainty quantification is critical for ensuring the accuracy of predicted BP and underlying hemodynamics. Our proposed WPINN not only predicts cuffless BP from wearables but also provides interpretable insights into the underlying hemodynamic parameters that drive fluctuations in BP. Validating the WPINN to conventional data-driven models across multiple datasets, including data-scarce scenarios and synthetic ground truth simulations, demonstrates generalizability and consistency to adhere to the defined physiology. Additionally, integrating the Windkessel models as a residual physics loss allows for physics-driven uncertainty quantification to identify potentially erroneous predictions based upon the deviation from the defined physics. The analysis performed to ensure robust validation and uncertainty quantification includes:

• Validation of forward BP predictions on real-world healthy and hypertensive cohorts, under limited data scenarios.

• Validate inverse parameter inference utilizing a simulated dataset, recovering Windkessel parameters, and confirming the relationship with known waveforms.

• Propose physics-driven uncertainty quantification using the Windkessel residual error and clustering techniques to identify potentially erroneous predictions.

Cuffless blood pressure validation

The evaluation of validating our proposed Windkessel models, in comparison to a baseline purely data-driven deep neural network, emphasizes enhanced performance in two key scenarios: (i) when limited personal data is available, reflecting real-world constraints, and (ii) the ability to interpolate unseen BP distributions. To meet these criteria, personalized models are trained for each participant, with their dataset split according to the bottom and top percentiles of either SBP or DBP for training and validation. Additionally, we calculate the total waveform BP (WBP) error for waveforms split according to DBP values. Specifically, we examined the performance of selecting the bottom and top 2.5th and 5th percentiles of BP data, corresponding to a minimal 5% and 10% of the participants’ data being used for model training and validation. The remaining data, representing a larger range, is withheld for testing, as illustrated in Fig. 3. The root mean squared error (RMSE) performance metric is reported for the total number of participants in each dataset for both SBP and DBP. More information regarding the pre- and post-processing procedures of the data can be found under the “Methods” section.

Fig. 3. Training, validation, and test dataset splits.

Dataset split for diastolic (DBP) (Left) and systolic (SBP) (Right), where top and bottom 2.5th percentiles are used for training and validation (green), with a large range and number of BP samples set aside for testing (blue).

A comprehensive analysis of our WPINN models, illustrated architecture in Fig. 2d, integrating the various complexities of the defined residual Windkessel ODE and associated parameter definitions. Specifically, each of the WPINN models can be grouped into the following categories: 2E or 3E WPINNs, integrating one of the two Windkessel equations provided in the “Methods” section, and defined parameter complexities, which include constant parameters (CP), beat-to-beat parameters (BBP), or pressure-dependent parameters (PDP). Figure 4a illustrates the circuit models and parameter fluctuations through time for each defined parameter complexity. A more in-depth description and mathematical formulation of these parameters are provided in the “Methods” section. Furthermore, these models are compared with baseline models that share the same inputs and network architecture trained only with the conventional loss term, to properly evaluate the improvement that our best-performing WPINN offers over the conventional approach. A baseline conventional (CONV) data-driven model is created with identical network architecture and inputs of Bio-Z, calibrated blood flow, and time arrays, with the same targeted output, the BP waveform.

Fig. 4. Data modeling and results.

a Illustration of parameter definitions for compliance (dashed green) and peripheral resistance (dashed blue), with circuit illustrations and parameter changes through time. b Bland-Altman and c Pearson correlation plots for predicted and reference blood pressure (BP) waveforms. d Histogram plots for parameters, compliance at diastolic BP (DBP) in dark green, systolic BP (SBP) in light green, peripheral resistance in teal, and BP values, DBP in purple, and SBP in blue; in addition to an illustration of simulated BP waveforms distributions. e Example loss curves, including training loss (solid dots), validation loss (hollow dots), physics loss (dashed lines), and test loss (triangle points) for conventional (CONV) models, highlighting overfitting. f Example loss curves for Windkessel physics-informed neural network (WPINN) model without severe overfitting.

The results of this comprehensive model analysis are illustrated in Table 1, for each respective dataset, Windkessel ODE, and parameter complexity. Across both datasets, the WPINN integrating the 2E Windkessel ODE residual loss and PDP parameters yielded the best results in comparison to a baseline purely data-driven model. An explanation for the 2E Windkessel model outperforming the 3E model, with regard to SBP and DBP prediction, is that the reduced model mitigates over-parameterization in scarce data problem settings, such as in this work, leading to better generalizing parameter estimates than the 3E formulation. In addition, it is important to note that the difference in RMSEs between 2E and 3E WPINNs is small, almost negligible at around 0.2 mmHg. For the ring-CPT dataset, the average RMSE percent improvements from the baseline model across both SBP, DBP, and minimal training data sizes equated to upwards of 25%. Likewise, for the larger hypertensive-HGVCPT dataset, an average RMSE improvement of 12% was observed. These significant improvements are illustrated in Fig. 4b, c using Bland-Altman and Pearson correlation plots for the best-performing model complexity, which is 2E-PDP Windkessel models.

Table 1.

Bio-Z dataset results

Dataset	Percentiles	Model	Parameter complexity	SBP RMSE (mmHg)	DBP RMSE (mmHg)	WBP RMSE (mmHg)	Training time (s)	Cost per epoch
ring-CPT	5th	CONV	-	8.49	6.37	8.00	25.5 ± 8.3	$\mathcal{O}\mathcal{(}N\cdot L)$
		WPINN (2E)	CP	7.35	5.24	7.27	31.5 ± 10.9	$\mathcal{O}\mathcal{(}N\cdot L\cdot D)$
			BBP	8.11	4.79	7.54	32.0 ± 10.7	$\mathcal{O}\mathcal{(}N\cdot L\cdot D\cdot B)$
			PDP	7.03	4.13	7.54	35.5 ± 12.8	$\mathcal{O}\mathcal{(}N\cdot L\cdot D\cdot B\cdot P)$
		WPINN (3E)	CP	7.85	6.06	7.84	30.9 ± 10.1	$\mathcal{O}\mathcal{(}N\cdot L\cdot D)$
			BBP	8.90	11.44	9.03	33.5 ± 12.0	$\mathcal{O}\mathcal{(}N\cdot L\cdot D\cdot B)$
			PDP	6.49	5.05	7.25	34.2 ± 10.8	$\mathcal{O}\mathcal{(}N\cdot L\cdot D\cdot B\cdot P)$
	2.5th	CONV	-	9.72	6.85	9.02	25.9 ± 8.8	$\mathcal{O}\mathcal{(}N\cdot L)$
		WPINN (2E)	CP	9.29	5.31	8.32	31.9 ± 11.1	$\mathcal{O}\mathcal{(}N\cdot L\cdot D)$
			BBP	8.93	6.82	9.48	31.9 ± 10.9	$\mathcal{O}\mathcal{(}N\cdot L\cdot D\cdot B)$
			PDP	8.33	4.65	8.04	34.6 ± 12.1	$\mathcal{O}\mathcal{(}N\cdot L\cdot D\cdot B\cdot P)$
		WPINN (3E)	CP	8.92	6.37	8.70	30.8 ± 9.3	$\mathcal{O}\mathcal{(}N\cdot L\cdot D)$
			BBP	8.98	12.02	9.81	31.8 ± 10.4	$\mathcal{O}\mathcal{(}N\cdot L\cdot D\cdot B)$
			PDP	7.84	4.50	8.75	34.2 ± 12.7	$\mathcal{O}\mathcal{(}N\cdot L\cdot D\cdot B\cdot P)$
hypertensive-HGVCPT	5th	CONV	-	8.43	4.15	7.15	35.7 ± 14.3	$\mathcal{O}\mathcal{(}N\cdot L)$
		WPINN (2E)	CP	7.44	4.09	7.21	42.4 ± 17.3	$\mathcal{O}\mathcal{(}N\cdot L\cdot D)$
			BBP	7.35	4.04	7.59	45.8 ± 18.6	$\mathcal{O}\mathcal{(}N\cdot L\cdot D\cdot B)$
			PDP	7.00	3.99	7.12	48.2 ± 18.9	$\mathcal{O}\mathcal{(}N\cdot L\cdot D\cdot B\cdot P)$
		WPINN (3E)	CP	7.78	5.13	6.69	42.5 ± 16.9	$\mathcal{O}\mathcal{(}N\cdot L\cdot D)$
			BBP	8.34	7.08	8.42	42.5 ± 17.8	$\mathcal{O}\mathcal{(}N\cdot L\cdot D\cdot B)$
			PDP	7.60	4.03	6.66	47.2 ± 19.4	$\mathcal{O}\mathcal{(}N\cdot L\cdot D\cdot B\cdot P)$
	2.5th	CONV	-	8.76	5.24	7.78	35.1 ± 14.1	$\mathcal{O}\mathcal{(}N\cdot L)$
		WPINN (2E)	CP	8.20	4.60	7.42	41.9 ± 17.0	$\mathcal{O}\mathcal{(}N\cdot L\cdot D)$
			BBP	8.22	4.46	8.15	43.6 ± 18.4	$\mathcal{O}\mathcal{(}N\cdot L\cdot D\cdot B)$
			PDP	7.77	4.40	7.73	46.2 ± 18.4	$\mathcal{O}\mathcal{(}N\cdot L\cdot D\cdot B\cdot P)$
		WPINN (3E)	CP	8.53	5.50	7.25	42.0 ± 17.2	$\mathcal{O}\mathcal{(}N\cdot L\cdot D)$
			BBP	8.90	8.21	9.02	41.8 ± 17.5	$\mathcal{O}\mathcal{(}N\cdot L\cdot D\cdot B)$
			PDP	8.83	4.44	7.08	46.4 ± 18.4	$\mathcal{O}\mathcal{(}N\cdot L\cdot D\cdot B\cdot P)$

Blood pressure (BP) prediction and training time comparisons for conventional (CONV), Windkessel physics-informed neural network (WPINN) 2-element (2E), and WPINN 3-element (3E) models. Bolded values indicate the lowest root mean square error (RMSE) for respective BP type, dataset, and percentile split. Additionally, WPINN models are specified with parameter complexities. Where $N$ is the number of subject sets, $L$ is the number of layers in the network, $D$ is the number of derivatives taken in the physics loss, $B$ is the number of beats, and $P$ is the number of pressure points per beat.

Further analysis was conducted by solving the Windkessel ODE directly using traditional numerical methods and the inferred parameter values of peripheral resistance and compliance obtained from the best-performing WPINN model. Reference-calibrated blood flow and parameters were utilized as model inputs to solve the ODE and reconstruct the same BP waveforms in the test datasets, under the 5th percentile training scenario. Nevertheless, the resulting BP waveform RMSEs were considerably higher, from 20.78 mmHg to 23.96 mmHg. This highlights a critical insight: while WPINNs effectively regularize the learning process and improve predictive performance, the direct translation to BP via the Windkessel ODE is a complex and nontrivial task. The elevated errors from numerical ODE evaluation suggest that real-world BP dynamics involve additional physiological complexities beyond what the simplified Windkessel model can capture. These findings underscore the value of hybrid modeling approaches that combine data-driven flexibility with physiologically grounded constraints to enhance both generalizability and interpretability in personalized cardiovascular modeling.

Validation of arterial parameters using the AorticFlowWave dataset

The superior performance of our model in accurately predicting cuffless BP, particularly in data-scarce scenarios, was highlighted in the previous section. Additionally, our framework can model the dynamic parameters associated with the underlying Windkessel model. However, the ground truth values for these parameters are often unknown in real-world scenarios, especially on a beat-to-beat or pressure-dependent basis.

To validate our model’s ability to estimate these parameters accurately, we created a synthetic dataset comprising N = 50 subjects. Aortic blood flow waveforms were generated using the AorticFlowWave script developed by the Hae Mod research group⁴¹. Each subject was assigned a varied number of samples, ranging from 288 to 1011. Unique compliance and peripheral resistance values were assigned per subject, with pressure-dependent compliance values spanning from 0.13 $\mathrm{mL\; mmH}{\mathrm{g}}^{-1}$ to 2.68 $\mathrm{mL\; mmH}{\mathrm{g}}^{-1}$ and resistance values from 0.44 $\mathrm{mmHg\; s\; m}{\mathrm{L}}^{-1}$ to 2.20 $\mathrm{mmHg\; s\; m}{\mathrm{L}}^{-1}$ across the entire dataset. The resulting simulated data produced SBP values from 94 mmHg to 223 mmHg and DBP values from 53 mmHg to 85 mmHg. Parameter and BP distributions are illustrated in Fig. 4d. We executed the same testing scheme that was performed with the real-world datasets. The prediction results for BP and parameter values are summarized in Table 2. Low percentage errors were reported for estimated parameter values, with a range from 5.68% to 6.07% for the more dynamic pressure-dependent compliance values and 0.77% to 1.01% for peripheral resistance, validating our WPINN’s ability to accurately capture the underlying cardiovascular parameters.

Table 2.

AorticFlowWave dataset results

Simulated dataset	Percentiles	SBP RMSE (mmHg)	DBP RMSE (mmHg)	WBP RMSE (mmHg)	$\mathrm{C}$ % error	${\mathrm{R}}_{\mathrm{p}}$% error
WPINN (2E) PDP	5th	3.27	1.50	2.00	5.73–6.07	0.77–0.78
	2.5th	3.69	1.75	2.38	5.68–5.72	0.87–1.01

Simulated dataset results included blood pressure (BP) root mean squared errors (RMSEs) and percentage (%) error ranges for Windkessel model parameter values.

Further analysis that investigated the impact of noise added to the input waveforms at various signal-to-noise ratios (SNRs) from 3 dB to 30 dB, detailed in Table 3, revealed errors trends that affected each prediction differently. SBP RMSE was impacted the most and nearly tripled as the SNR was reduced from 30 dB to 3 dB; however, DBP and WBP RMSE only increased by 2–3 mmHg. With regards to the predicted parameters, arterial compliance error increased the least, where at SNRs above 15 dB the percentage error remained between 5 and 6%, and below 15 dB increased to between 6 and 7%. Nevertheless, peripheral resistance percentage errors significantly increased with each reduction in SNR, where the largest percentage error of around 20% was observed at 3 dB. An explanation for these differences between the prediction errors can be attributed to differences in variation within the dataset, especially when comparing SBP and DBP, and how the parameters are mathematically related to the inputs, where peripheral resistance relies on the accurate determination of the mean flow from the model input, which was affected by added noise in this analysis.

Table 3.

AorticFlowWave dataset results with Gaussian noise

Simulated dataset	Percentiles	SNR (dB)	SBP RMSE (mmHg)	DBP RMSE (mmHg)	WBP RMSE (mmHg)	C % error	R % error
WPINN (2E) PDP	5th	30	3.62	1.63	2.03	5.53–5.69	1.12–1.14
		15	4.43	1.73	2.40	5.46–5.52	4.51
		6	6.85	2.26	3.46	5.95–6.03	12.84–13.19
		3	9.40	2.93	4.41	6.80–6.81	20.60–20.66
	2.5th	30	3.73	1.85	2.47	5.49–5.52	1.20–1.23
		15	4.97	1.90	2.73	5.30–5.41	4.50–4.54
		6	8.02	2.59	4.04	6.24–6.90	12.97–12.98
		3	9.84	3.65	5.55	6.59–6.90	20.62–20.93

Simulated dataset results with added input noise included (BP) root mean squared errors (RMSEs) and percentage (%) error ranges for Windkessel model parameter values.

Model convergence

We present an additional analysis of the convergence behavior of the best-performing WPINN architecture compared to its baseline data-driven model. The training, physics, validation, and test losses are plotted for an exemplary subject and compared against the baseline model’s losses. The loss curves reveal that the baseline model exhibits overfitting, whereas the WPINN model achieves stable generalization, with the test loss plateauing at the optimal validation loss point in Fig. 4e, f. The complexity of the ODE formulation and model parameters can significantly increase training time, as reported in Table 1. Additionally, in extreme data-scarce instances, we observe WPINN SBP and DBP RMSE performance as a function of a limited number of training samples, shown in Fig. 5 for an exemplary individual. This analysis revealed DBP RMSE to plateau after around 5 training sample instances at an RMSE of 5 mmHg, whereas SBP RMSE plateaued at around 15 training samples at an RMSE of 10 mmHg. This difference is likely due to the lower variability of DBP compared to SBP, which explains both the lower RMSE and the fewer samples needed to reach a performance plateau.

Fig. 5. Training sample size and error trend.

Systolic blood pressure (SBP) and diastolic blood pressure (DBP) root mean squared errors (RMSE) over varying amounts of labeled instances.

Interpretation of physiological parameters

As a result of integrating the cardiovascular parameters of arterial compliance and resistance, it is possible to interpret how these parameters influence fluctuations in BP. The pressure-compliance curve is utilized to illustrate differences in compliance for respective individuals. Compliance typically decreases with increasing age and increasing pressure. We illustrate differences in pressure-compliance curves for selected individuals with distinctive ages to highlight this physiological trend in Fig. 6a. In addition, the relationship between peripheral resistance and mean arterial pressure can be exemplified as a generally linear relationship. Increases in peripheral resistance are subsequently related to an increase in mean arterial pressure, which is demonstrated in Fig. 6b for an exemplary participant. Combining these visualizations allows for a more comprehensive understanding of BP fluctuations with clinical relevance.

Fig. 6. Parameter fluctuations.

a Pressure-compliance curves illustrating differences in compliance for selected individuals varying in ages: 37-year-old (yro) (diamond), 55 yro (square), and 75 yro (triangle). b Peripheral resistance v. mean arterial pressure scatterplot illustrating linear relationship for an exemplar individual. c Percent change in diastolic blood pressure (DBP), d Percent change in compliance, e Percent change in resistance with regard to DBP percentile ranges for ring-CPT (solid dot) and hypertensive-HGVCPT (hollow dot) datasets.

To further illustrate the fact that our framework captures dynamic fluctuations not only with regard to BP but also the underlying physiological parameters, Fig. 6c–e visualizes the percentage change in DBP and parameter values with regard to percentile ranges of DBP. In accordance with physiology, increases from baseline BP come as a result of arterial stiffening and vasoconstriction, which is exemplified through decreasing arterial compliance and increasing peripheral resistance. These expected physiological trends are demonstrated across both datasets, highlighting the robustness of our framework to uncover the root cardiovascular influences for fluctuating BP.

Furthermore, to highlight our best-performing WPINN model successfully captures the underlying parameters unique for each individual, we plot the distinctive distributions of parameter estimates for three exemplary participants. Figure 7a illustrates the individual distributions of peripheral resistance ($R_p$), mean flow (${\mu }_{I\left(t\right)}$), and mean arterial pressure (${\mu }_{P\left(t\right)}$), where the relationships between these variables are described by $R_p={\mu }_{P\left(t\right)}/{\mu }_{I\left(t\right)}$. Figure 7b illustrates the individual distributions of beat-to-beat arterial compliance ($C$), pulse pressure ($\mathit{PP}$), and stroke volume ($\mathit{SV}$), where the relationships between these variables are described by $C=\mathit{SV}/\mathit{PP}$. These scatter plots highlight the mathematical relationships between parameters and emphasize the individuality of parameter distributions for each individual. To further validate that each WPINN is successfully trained for each respective individual, we analyze the model’s performance to predict BP for other participants, from the ring-CPT dataset, which it was not trained on. Figure 7c, d validates that each model is individually trained for a respective participant, with SBP and DBP RMSEs significantly increasing when the model is used to predict BP for unintended participants.

Fig. 7. Subject-specific parameter distributions and personalized modeling trends.

a Peripheral resistance ($R_p)$, mean flow (${\mu }_{I\left(t\right)})$, and mean arterial pressure (${\mu }_{P\left(t\right)})$, scatterplot relationships described by $R_p={\mu }_{P\left(t\right)}/{\mu }_{I\left(t\right)}$ for three exemplary participants. b Beat-to-beat arterial compliance ($C)$, pulse pressure ($\mathit{PP})$, and stroke volume ($\mathit{SV})$, scatterplot relationships described by $C=\mathit{SV}/\mathit{PP}$ for three exemplary participants. c Heatmaps of systolic blood pressure (SBP) root mean squared errors (RMSEs) and d diastolic blood pressure (DBP) RMSEs for each respective participant using a respective trained model, highlighting that the models are only personalized for the defined individual. e Density plots for predicted Windkessel parameters, illustrating distributions that are statistically different from one another. f Histogram illustrating percent errors for beat-to-beat values of Windkessel parameters, compliance (blue), and peripheral resistance (green) for real participants.

To statistically quantify the differences in the individual distributions of arterial compliance and peripheral resistance across all participants, the ANOVA test (N = 3241, α = 0.05) was employed, revealing statistically significant differences (p < 0.001) for parameter values across all participants. Figure 7e illustrates the density plot distributions for the Windkessel parameters for a subset of participants, highlighting the individuality of the predicted parameters. Furthermore, we can further validate these predicted parameters using the previously mentioned relationships: $R_p={\mu }_{P\left(t\right)}/{\mu }_{I\left(t\right)}$ and $C=\mathit{SV}/\mathit{PP}$, using ground truth Finapres NOVA BP and calibrated flow waveforms to calculate reference beat-to-beat resistance and compliance values for collected participants. Results yielded low mean absolute percentage errors of 3.67% for peripheral resistance and 8.61% for arterial compliance, with error distributions illustrated in Fig.7f.

Physics-driven uncertainty quantification

We hypothesized that the Windkessel-informed physics loss computed by the trained WPINNs can serve as an indicator of prediction reliability. By evaluating this loss across the entire dataset, we aim to investigate whether higher physics errors are associated with increased prediction errors on test set samples. Specifically, we propose that predicted samples with larger physics errors reflect deviation from the assumptions of the Windkessel model; therefore, they are more likely to exhibit greater prediction inaccuracies.

For the best-performing WPINN model, the waveform physics RMSE and prediction RMSE are grouped into three clusters using the K-means clustering algorithm, based on the physics RMSE values. The optimal number of clusters is established using the elbow method demonstrated in Fig. 8a. Furthermore, the distributions of the clusters according to Windkessel/physics RMSE are highlighted in Fig. 8b, c. An observed correlation between increased physics error and higher test set error is shown in Fig. 8d. Furthermore, boxplots comparing prediction error distributions for samples with physics errors for respective physics RMSE ranges reveal a statistically significant difference using the ANOVA test (N = 22,729, α = 0.05) with p < 0.001, in Fig. 8e. Additionally, Fig. 8f demonstrates the identification of an erroneous prediction for an exemplary period of data. These erroneous predictions could be influenced by unwanted noises such as body movement or loss of electrode contact with skin¹². These findings suggest that the physics error from WPINN models can be used as an uncertainty metric to identify predictions prone to higher errors.

Fig. 8. Uncertainty analysis.

a Plot for identifying the optimal number of clusters using the elbow method; within-cluster sum of squares metric (solid green) and optimal number of clusters (dashed red). b Density and c histogram plots illustrating distribution of clusters; cluster 1 (green), cluster 2 (orange), and cluster 3 (red), according to Windkessel/Physics root mean squared error (RMSE). d 2-D density plots illustrating the distribution of clusters according to Windkessel/Physics and predicted blood pressure (BP) waveform RMSEs. e Boxplot for each cluster where differences in distribution means are statistically significant. f Time plot for an exemplar period highlighting the successful identification of an erroneously predicted beat (red dots), among low error predictions (green dots) and ground truth (dashed black).

WPINN update analysis

For an exemplary participant, an initial WPINN model was trained using baseline data with low BP variability, as shown in Fig. 9a. A separate region corresponding to elevated BP during a maneuver was reserved to assess the model’s adaptability to changing physiological states before and after updating. Additionally, Fig. 9a illustrates BP predictions from the model before updating, where the results have poor generalization and low variability, representative of the baseline data it was initially trained on. Figure 9b highlights the data points from the maneuver region with the highest Windkessel/Physics-based errors, which were selected to update the model. In addition, BP predictions from the updated model using these selected data points for training are highlighted with improved generalization to the trends in BP. The statistical impact, using a paired t-test (N = 90, α = 0.05), of this update is shown in Fig. 9c, d, where significant reductions in waveform Windkessel/Physics and WBP are observed along with beat-to-beat measurements of SBP and DBP in Fig. 9e, f. Additionally, consistent with physiological expectations, the updated model exhibits a decrease in arterial compliance at DBP and an increase in peripheral resistance during the maneuver phase, illustrated in Fig. 9g, h, demonstrating both improved model accuracy and parameter responsiveness.

Fig. 9. Before and after model update trends.

a Plot comparing reference and predicted systolic blood pressure (SBP) & diastolic blood pressure (DBP) values (Reference SBP dashed blue, reference DBP dashed purple, predicted SBP green, predicted DBP teal, and update sample red dot), showing consistently low variability relative to baseline training data throughout the maneuver period prior to model update. b Same plot as in (a), now including additional sampled data from the maneuver period, used to update the model and improve SBP & DBP predictions. Boxplots showing the reduction in (c) Windkessel/Physics & (d) waveform blood pressure (WBP) errors before (navy) and after (green) updating the model with maneuver data. Boxplots showing the reduction in beat-to-beat (e) SBP & (f) DBP prediction errors after updating the model with maneuver data. Boxplots showing changes in estimated (g) arterial compliance & (h) peripheral resistance after incorporating maneuver data into the model update.

Discussions

In this work, we introduce the WPINN framework, an adaptation of conventional PINNs that integrates the Windkessel ODE into the physics-informed loss. This study provides a comprehensive evaluation of the predictive and interpretative capabilities of WPINNs. Specifically, we assess the BP prediction performance of WPINN models with varying complexities, considering both the defined ODE formulations, namely, the 2E and 3E Windkessel models, and the complexity of the defined parameters, which include CP, BBP, and PDP definitions. These models are compared against baseline approaches that do not incorporate the Windkessel-informed loss. Furthermore, we investigate the consistency of prediction improvements when training data is limited by comparing scenarios where only 10% and 5% of participant data are available. In addition, we analyze the total training time across different model complexities and provide exemplary loss function curves to illustrate how purely data-driven baseline models tend to overfit, even with a held-out validation set, whereas WPINN models exhibit improved generalization. Furthermore, we validated an accurate estimation of the underlying cardiovascular parameters using a synthetic dataset of subjects with various cardiovascular properties. The interpretability of the WPINN framework is also examined by extracting clinically relevant parameters such as arterial compliance and peripheral resistance, which are then analyzed in relation to BP. The relationship between these parameters, age demographics, and changes in BP is further explored to assess their variability. Furthermore, we leverage the physics error from the physics-informed Windkessel loss as an uncertainty quantification metric to identify beat-samples that are more prone to higher evaluation errors. This analysis demonstrates the advantages of WPINNs in integrating physiological principles into deep learning models, leading to improved BP prediction accuracy, enhanced interpretability, and robust uncertainty estimation.

Further analysis of SBP, DBP, and WBP prediction results highlights that the simpler 2E ODE model consistently outperforms the more complex 3E model in terms of RMSE. Conversely, the best-performing parameter definitions were obtained using the most complex pressure-dependent equation for arterial compliance and defining peripheral resistance as a beat-to-beat value, a function of mean flow and mean pressure. These findings underscore the delicate balance between model complexity and performance. While simpler models may struggle to fully capture complex hemodynamics, more complex models might require additional data to achieve comparable accuracy, particularly in data-scarce scenarios like those presented in this study.

With enhanced predictive performance, WPINN models that incorporate PDP parameters require the longest training time. As shown in Table 1, the total execution time for these models is significantly higher compared to baseline non-physics-informed models. This increase can be attributed to the use of complex nonlinear equations that model the relationship between pressure and compliance. Additionally, WPINN models solve a multi-objective optimization problem by integrating two weighted loss functions, further contributing to the extended training duration. Despite the increased computational demands, the WPINN framework offers two key advantages: improved predictive accuracy and clinically relevant interpretability.

Estimating arterial compliance and peripheral resistance within a robust, noninvasive framework can offer clinicians a more specialized approach to understanding evolving cardiovascular dynamics. In this study, we validate the accurate estimation of these parameters with percentage errors from 0.77% to 6.07%. Additionally, we present examples of pressure-compliance curves for three individuals across a wide age range. Consistent with clinical literature, the compliance trend decreases with age, which is illustrated by curves where increases in pressure lead to greater decreases in compliance, indicative of a less compliant arterial system. Additionally, an exemplar linear relationship between peripheral resistance and mean arterial pressure is demonstrated, aligning with the traditional definition of peripheral resistance as the ratio of mean pressure to mean flow. Understanding the trends between these parameters and BP not only enhances clinical interpretability, providing deeper insight into the causes of hypertension and age-related vascular changes, but also supports the extrapolation of unseen BP distributions.

In predicting unseen BP distributions, this work demonstrates that the Windkessel/physics residual error can serve as an effective measure of uncertainty quantification. Specifically, data samples with higher residual RMSE values are more likely to result in greater prediction errors. This relationship is illustrated in Fig. 8f, which shows a consecutive waveform example highlighting the correlation between residual RMSE and BP prediction RMSE. Elevated residual errors may stem from increased noise in the input flow waveform or heightened model uncertainty in BP predictions, both contributing to larger Windkessel errors. Ongoing exploration of this approach, evaluating the residual component of the physics loss on unseen samples, could lead to the development of more sophisticated uncertainty modeling strategies to further investigate and leverage this trend.

Limitations of this work that can be addressed in future research include the use of more sophisticated transformations of peripheral blood flow data, acquired in this study using Bio-Z wearables, to better represent central arterial dynamics. In this study, the reference ground truth BP waveforms obtained from the Finapres NOVA are representative of brachial artery dynamics at the heart level, contrary to aortic or central artery waveforms, which are more proximal to the heart. Such transformations could incorporate physiological characteristics such as arterial aging and wave transmission effects and could be validated through the collection of central arterial waveforms using advanced noninvasive ultrasound techniques.

Furthermore, we acknowledge the dependence on Finapres-derived SV as a limitation. Finapres SV is noninvasively estimated through the volume-clamp method and its proprietary pulse contour analysis software, which includes formulations derived from the Langewouters model. Elements of the Langewouters model are utilized in this study within our CDT framework, introducing a potential methodological coupling between the SV estimates used during calibration and the physiological model embedded in our WPINN approach. As a result, a portion of the observed agreement may arise from shared underlying assumptions rather than fully independent measurements. Although the SV values are used only in the training set for calibration, these SV values are not introduced into the validation or test sets. We acknowledge that this methodological overlap using the Langewouters model may influence the learned flow scaling. Future research should therefore investigate calibration strategies using independently measured noninvasive SV methods as alternatives to invasive gold-standard techniques or develop approaches that reduce the dependence on calibration.

This research successfully developed an optimized WPINN CDT framework for enhanced BP waveform prediction, offering both clinical interpretability and a means of uncertainty quantification. It provides novel insights into the interplay between Windkessel model complexity, computational tradeoffs, clinically validated trends in cardiovascular parameters, and the use of physics-based loss for uncertainty estimation. Moving forward, future work could focus on the development of more advanced uncertainty quantification techniques, such as Bayesian modeling, to better assess prediction confidence. This study lays a strong foundation for CDTs, which can be further refined through the integration of new network architectures or alternative computational strategies.

Methods

Physiological formulations

Pulse wave velocity (PWV), the clinical metric for assessing arterial health, can be expressed as a function of arterial elasticity modulus (E), also known as Young’s modulus,

$$\mathrm{PWV}=\sqrt{\mathit{Eh}/\rho D}$$ 1

for a given arterial wall thickness ($h$), arterial diameter (D), and constant blood density ($\rho$). Furthermore, compliance (C), which is defined by change of volume (V), or proportional area (A), over the change in pressure (P), is proportional to the inverse square of PWV in Eq. (2), which can be used to derive the inverse relationship between compliance and elasticity in Eq. (3)^42,43.

$$C=\frac{\mathit{dV}}{\mathit{dP}}\propto \frac{\mathit{dA}}{\mathit{dP}}=\frac{A}{\rho {\mathit{PWV}}^2}$$ 2

$$C\propto \frac{1}{E}$$ 3

Furthermore, cardiac output (CO) is the simple multiplication of stroke volume (SV) and heart rate (HR)⁴⁴. The expressions for CO and total peripheral resistance (TPR), the ratio between mean arterial pressure (MAP) and CO, are illustrated in the following Eqs. (4, 5).

$$\mathit{CO}=\mathit{SV}\times \mathit{HR}$$ 4

$$\mathit{TPR}=\mathit{MAP}/\mathit{CO}$$ 5

As previously mentioned, the Windkessel models are mathematically expressed as ODEs, where the 2E Windkessel model is written as,

$$\frac{P\left(t\right)}{R_p}+C\frac{\mathit{dP}\left(t\right)}{\mathit{dt}}-I\left(t\right)=0$$ 6

where $I\left(t\right)$ represents blood flow, and $R_p$ represents peripheral resistance. The extension of the 2E model, integrating an additional parameter called characteristic impedance ($Z_c$) is the 3E Windkessel model and is written in Eq. (7).

$$\frac{P\left(t\right)}{R_p}+C\frac{\mathit{dP}\left(t\right)}{\mathit{dt}}-\left(1+\frac{Z_c}{R_p}\right)I\left(t\right)-C{Z}_c\frac{\mathit{dI}\left(t\right)}{\mathit{dt}}=0$$ 7

WPINN loss functions

Our WPINN incorporates two loss functions. The first is the conventional data-driven mean squared error (MSE) between ground truth and predicted BP waveforms for N samples in the training set.

$${\mathcal{L}}_{\mathit{data}}=\frac{1}{N}{\sum }_{i=1}^N{\left|{\hat{P}}_i-P_i\right|}^2$$ 8

Second is the Windkessel residual loss, which is evaluated over M number of samples across training, testing, and validation datasets. The MSE of the Windkessel ODEs is evaluated as a function of predicted pressure, estimated parameters, input flow, and time, which are detailed below in Eq. (9) and Eq. (10).

$${\mathcal{L}}_{2\mathrm{E\; residual}}=\frac{1}{M}{\sum }_{j=1}^M{\left|\frac{{\hat{P}}_j}{{R_p}_j}+C_j\frac{d{\hat{P}}_j}{d{t}_j}-I_j\right|}^2$$ 9

$$\mathcal{L}=\frac{1}{M}\sum _{j=1}^M{∣\frac{{\widehat{P}}_j}{{R_p}_j}+C_j\frac{d{\widehat{P}}_j}{d{t}_j}-\left(1+\frac{{Z_c}_j}{{R_p}_j}\right)I_j-C_j{Z_c}_j\frac{d{I}_j}{d{t}_j}∣}^2$$ 10

The total loss for our WPINN is written as the weighted sum of the conventional data and Windkessel-informed residual losses. Specifically, in this work, we included a more sophisticated adaptive weighting scheme using a self-adaptive loss method that dynamically updates the weights of each loss term in every epoch, based on Gaussian probabilistic models of uncertainty. Using this self-adaptive weighting scheme, the total loss of the WPNN models based on the maximum likelihood estimation can be written as,

$${\mathcal{L}}_{\mathit{total}}=\frac{1}{{\epsilon }_{\mathrm{data}}^2}{\mathcal{L}}_{\mathrm{data}}+\frac{1}{{\epsilon }_{\mathrm{WK}}^2}{\mathcal{L}}_{\mathrm{WK}}+\log \left({\epsilon }_{\mathrm{data}}{\epsilon }_{\mathrm{WK}}\right)$$ 11

where the ε terms are the adaptive weight variables that represent the uncertainty of the specific function⁴⁵. Both ε uncertainty terms are initialized to 2, as recommended by the researchers of the work.

Parameter equations

The cardiovascular parameters utilized in the Windkessel equation can be estimated as a function of pressure, flow, or a set of trainable variables within the network. The simplest parameter definitions are constant parameters (CP), which are participant-specific singular values that do not fluctuate. Models incorporating the CP parameters are estimated using trainable variables within the network to minimize the residual loss. The next level of parameter definitions, named beat-to-beat parameters (BBP), fluctuate on a per-beat basis and are mathematically calculated as follows:

$$C=\frac{\mathit{SV}}{\mathit{PP}}$$ 12

$$R_p=\frac{{\mu }_{P\left(t\right)}}{{\mu }_{I\left(t\right)}}$$ 13

$$Z_c=\frac{\max \left(\mathit{dP}\left(t\right)\right)}{\max \left(\mathit{dI}\left(t\right)\right)}$$ 14

where PP represents pulse pressure, the difference between SBP and DBP, and $\mu$ represents the mean for respective pressure and flow waveforms⁴⁶. Furthermore, the most complex parameter definitions utilized in this work describe $C$ and $Z_c$ as pressure-dependent functions, which we call pressure-dependent parameter (PDP), defined using the Langewouters model. These nonlinear functions rely on the initial relationship of aortic cross-sectional area ($A\left(P\right)$), and compliance per unit length ($C^{\prime }\left(P\right)$) definitions written in Eq. (15) and Eq. (16).

$$A\left(P\right)=A_{\max }\left(0.5+\frac{1}{\pi }{\tan }^{-1}\left(\frac{P-P_0}{P_1}\right)\right)$$ 15

$$C^{\prime }\left(P\right)=\frac{\frac{A_{max}}{\pi P_1}}{1+{\left(\frac{P-P_0}{P_1}\right)}^2}$$ 16

Where $A_{max}$ represents the maximum cross-sectional area of the aorta, and $P_{\mathrm{0,1}}$ are pressure-related parameters. With these functions defined, $C$(P) and $Z_c\left(P\right)$ can be written as follows:

$$C\left(P\right)={\mathit{lC}}^{\mathit{\prime }}\left(P\right)$$ 17

$$Z_c\left(P\right)=\sqrt{\frac{\rho }{A\left(P\right)C^{\prime }\left(P\right)}}$$ 18

with $l$ representing the effective aortic length and $\rho$ the density of blood^25,47.

Network architecture

Given that this research focuses on translating time-series waveforms collected from wearables into BP waveforms, we propose a deep learning architecture that combines 1-D convolutional layers with a transformer encoder. The initial part of the network uses downsampling convolutional layers to compress the input waveform into a lower-dimensional feature space. This is followed by a transformer encoder, which captures global temporal dependencies within the latent representation. Finally, the network concludes with upsampling 1-D convolutional layers that reconstruct the predicted BP waveform output. All the activations within the hidden layers of the network utilize the “swish” activation function. Specifically, the hyperparameters for each layer of the network are as follows: first layer number of filters: 32, kernel size: 5, second layer max-pooling pool size: 2, third layer number of filters: 64, kernel size: 3, fourth layer max-pooling pool size: 2, fifth layer incorporating positional encoding to the latent feature space, sixth layer is the transformer encoder layer with number of heads: 4, hidden dimension size: 16, fully connected layer size: 16, and dropout of 10%. The seventh layer number of filters: 64, kernel size: 3, the eighth layer upsampling layer of size: 2, ninth layer number of filters: 32, kernel size: 3, tenth layer upsampling layer of size: 2, and the final output convolutional layer with number of filters: 1, kernel size: 5, and a linear activation function.

Data collection protocol

Participants from the ring-CPT dataset were enrolled under IRB2017–0086D approved by Texas A&M University, while those in the hypertensive-HGVCPT dataset participated under IRB2000032876, approved by the Yale School of Medicine. All participants provided written informed consent to take part in the experiments. All participants performed the following protocols in a controlled lab environment while seated with an upright posture. For the ring-CPT study, Bio-Z rings, custom-fabricated using silicone and silver electrodes, were individually fitted to each participant. The ring electrodes were placed on top of the digital artery of the finger, precisely located using a handheld ultrasound device, for optimal Bio-Z sensing of arterial blood flow. Participants performed up to four iterations of the cold pressor test (CPT), each followed by a 2-min recovery period. Both the CPT and the recovery phases lasted two minutes each. In contrast, the hypertensive-HGVCPT protocol employed a Bio-Z wristband wearable, placed on top of the radial artery along the wrist, which was also precisely located using the same handheld ultrasound device, instead of the ring, and included additional maneuvers: the handgrip (HG) exercise and the Valsalva maneuver (V). The full protocol began with a 3-min baseline recording, followed by a 1-min Valsalva maneuver and a 3-min recovery. This was followed by 2-min of handgrip exercise, another 3-min recovery, and then a 2-min cold pressor test, concluding with a final 3-min recovery period. Simultaneous noninvasive arterial BP waveforms and derived measurements of SV were collected using the Finapres NOVA device, an oscillometric finger cuff device allowing for the continuous noninvasive measurements of these critical cardiovascular metrics.

Preprocessing and flow calibration

Collected Bio-Z and BP waveforms are aligned according to the correlation in HR, where trials of data with HR correlations lower than 0.7 after a moving average filter of window size 10 are omitted. Once aligned, the process of calibrating the blood flow waveform begins with a reference Bio-Z waveform and Finapres-derived SV value. Bio-Z waveforms are a proxy for the volumetric changes in blood flow over an area of a targeted artery. By taking the derivative of this waveform with respect to time, we gain a proxy to blood flow expressed as the change in volume over time. However, this proxy to blood flow needs to be calibrated to the correct amplitude to properly represent blood flow at the same height as the heart. The calibration process involves a simple linear transformation of the Bio-Z derivative, which can be written as

$$I_{\mathit{BioZ}}\left(t\right)=\frac{\mathit{SV}}{X_{\mathrm{systolic}}}\frac{\mathit{dX}}{\mathit{dt}}$$ 19

where $\mathrm{X}$ is the Bio-Z waveform, and ${\mathrm{X}}_{\mathrm{systolic}}$ is the systolic amplitude of the Bio-Z waveform, commonly denoted as the maximum peak of the inverse waveform. The coefficients to linearly transform the derivative of Bio-Z to blood flow in mL s⁻¹ can be averaged across multiple waveforms provided in calibration. We specifically used the reference Finapres SV and Bio-Z waveforms only from the training set to perform this calibration process. Additionally, to acquire estimated ${\mathrm{SV}}_{\mathrm{BioZ}}$ from the $I_{\mathrm{BioZ}}\left(t\right)$, a simple integral is defined,

$${\mathit{SV}}_{\mathrm{BioZ}}={\int }_0^{t_{\mathrm{systolic}}}{I}_{\mathrm{BioZ}}\left(t\right)\mathit{dt}$$ 20

allowing for the calculation of arterial compliance defined in Eq. (12) using the calibrated flow waveform from Bio-Z data and pulse pressure.

Train/Test/Validation Scheme

To highlight the robustness of our WPINN framework to make more accurate predictions when minimal data is available, on unseen BP distributions, we split the training, validation, and test sets according to lower and upper percentiles according to SBP and DBP. When evaluating the total waveform BP (WBP) error, we utilize the split percentile splitting scheme according to DBP. For the training and validation datasets, data below the lower defined percentile and above the upper percentile are extracted, where 80% of the identified data is used in training, and the other 20% is used in validation. The middle region between the lower and upper limits is set aside for testing. Specifically, the lower and upper 5th and 2.5th percentiles of blood pressure are defined for splitting the sets, which equates to a minimal 10% and 5% of the overall participant data used for training and validation.

All models are trained for a total of 64 epochs, where the best model is saved according to the lowest RMSE on the validation set BP waveforms. Additionally, for test set predictions, we evaluated the RMSEs not only for the entire waveform (WBP) but also for critical SBP and DBP points.

Synthetic dataset generation

Aortic blood flow waveforms were acquired from the AorticFlowWave script developed by the Hae Mod research group⁴¹. A total of N = 50 synthetic subjects were simulated, with each assigned between 256 and 1024 samples. Peripheral resistance for each subject and beat was defined as the ratio of a constant, randomly selected from a uniform distribution between 80 and 105, divided by the mean aortic flow waveform. Arterial compliance was defined using the nonlinear pressure-dependent relationship defined in Eqs. (15–18), where for each subject the scalar parameters $A_{\max }$, $P_0$, $P_1$, and $l$ were sampled from uniform distributions ranging from baseline values of 5, 40, 40, and 65, respectively, up to a 10% increase. The resulting pressure waveforms were generated using a defined 2E Windkessel ODE and the Runge–Kutta numerical method.

To further investigate the robustness of the proposed WPINN models to input noise, Gaussian noise was introduced to the input waveforms at signal-to-noise ratios (SNRs) of 30, 15, 6, and 3 dB. The same training, validation, and testing split strategy was maintained across all noise levels, using the same BP percentiles splits. Model performance was evaluated using the same SBP, DBP, and WBP RMSE, along with percentage errors in estimated compliance and peripheral resistance for each respective SNR.

Parameter interpretability processing

The interpretability of cardiovascular parameters is evaluated for the best-performing WPINN model complexity. Differences in the pressure-compliance curves are analyzed for individuals selected by significant age gaps. As indicated in the literature, as individuals age, their arteries become stiffer and less compliant, resulting in a decreasing shift in overall compliance. Additionally, the relationship between peripheral resistance and mean arterial pressure is examined for a representative individual. According to the literature, this relationship can be exhibited as linear. To quantify this linearity, the Pearson correlation value is reported, providing a measure of the strength of the relationship between mean arterial pressure and peripheral resistance.

To validate that the best-performing WPINN models are robustly personalized, the ANOVA test was employed to determine whether the predicted Windkessel parameters of arterial compliance and peripheral resistance differ significantly across participants. Further analysis was conducted to evaluate each model’s performance in predicting BP for individuals it was not trained on, highlighting the individuality and subject-specific nature of the models. Additionally, using Eqs. (12, 13), beat-to-beat reference parameter estimates were calculated from ground truth Finapres NOVA BP and calibrated flow waveforms. These reference values are then compared to the parameters derived using predicted blood pressure, further validating the model’s ability to adhere to the defined Windkessel physics relationship.

Uncertainty quantification processing

To approximate uncertainty in the predicted results, we evaluate the physics RMSE across all data samples in the test set. We utilize the K-means unsupervised learning technique to cluster the data into distinctive clusters according to the physics error. The elbow method is utilized to calculate the optimal number of cluster groups into which our data is divided. For each cluster, the distribution of BP waveform prediction RMSE is analyzed to assess the relationship between physics error and prediction accuracy. Additionally, representative waveforms from a selected period are visualized to illustrate the corresponding levels of Windkessel-based uncertainty.

Furthermore, the same physics-driven uncertainty metric is utilized for identifying data to best sample from to update the WPINN model and CDT framework during a change in physiological state. Statistical analysis using a paired t-test is performed comparing distributions of errors and parameter values before and after updating the model for an exemplary participant, which is hypothesized significantly to reduce model errors and more accurately represent the underlying physiology.

Author contributions

D.O., K.S., and R.J. conceived the idea of the construction of the WPINN model design. D.O. designed and optimized machine learning models. D.O. compiled the models and analyzed the data. E.S. contributed to the design of the experiments and provided clinical guidance on study methodology and interpretation. R.J. supervised the complete process. The manuscript was written with the contributions of all authors. All authors have approved the final version of the manuscript.

Data availability

The ring-CPT and hypertensive-HGVCPT datasets are currently not publicly available due to participant privacy considerations, but are available from the corresponding author on reasonable request. Synthetically generated data and code to reproduce the analyses are available at https://github.com/TAMU-ESP/WPINN.

Code availability

The neural network models are available on GitHub at https://github.com/TAMU-ESP/WPINN. The custom code used for data visualization is available from the corresponding author upon request.

Competing interests

The authors declare no competing interests.