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arxiv: 2604.27500 · v1 · submitted 2026-04-30 · 💻 cs.HC · eess.SP

From Elastic to Viscoelastic: An EEMD-Enhanced Pulse Transit Time Model for Robust Blood Pressure Estimation

Boyuan Gu , Yijin Yang , Shuaiqi Cheng , Xiaorong Ding This is my paper

Pith reviewed 2026-05-07 08:37 UTC · model grok-4.3

classification 💻 cs.HC eess.SP
keywords cuffless blood pressurepulse transit timeviscoelasticityphotoplethysmogramensemble empirical mode decompositionvascular hysteresisblood pressure estimation
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The pith

Adding a viscoelastic damping metric from EEMD-processed PPG signals to the pulse transit time model compensates for arterial viscosity and improves blood pressure estimates during hemodynamic shifts.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Conventional pulse transit time models treat arteries as purely elastic and lose accuracy when blood pressure changes quickly because they ignore the damping that real vessel walls produce. The paper isolates high-frequency intrinsic mode functions from ensemble empirical mode decomposition of photoplethysmogram waveforms and converts them into a single number called the Viscoelastic Velocity Metric that stands in for the missing damping term. This metric is inserted into the standard PTT equation before blood pressure is calculated. The updated model is tested on more than twenty-eight thousand heartbeats from three hundred sixty-four subjects drawn from the MIMIC-II database, many of whom have hypertension. The resulting errors fall to levels accepted for clinical devices while maintaining very high correlation with reference measurements.

Core claim

The authors claim that high-frequency intrinsic mode functions extracted by ensemble empirical mode decomposition from raw photoplethysmogram signals supply a direct, quantifiable proxy for the viscoelastic damping term η · ε̇ that elastic models omit. Inserting this Viscoelastic Velocity Metric into the pulse transit time formula produces a physics-informed correction that reduces error caused by vascular hysteresis, yielding RMSE values of 5.22 mmHg systolic and 3.65 mmHg diastolic together with Pearson correlations above 0.97 on a large, hypertension-enriched dataset.

What carries the argument

The Viscoelastic Velocity Metric obtained from high-frequency intrinsic mode functions of ensemble empirical mode decomposition on photoplethysmogram signals, used to represent and compensate for the arterial damping effect η · ε̇.

If this is right

  • The corrected model reaches clinically acceptable accuracy on 28,525 cardiac cycles from 364 subjects with 23.4 percent hypertension prevalence.
  • It improves robustness specifically during periods of rapid hemodynamic fluctuation where pure elastic models fail.
  • The approach remains fully physics-informed rather than relying on purely data-driven black-box corrections.
  • High Pearson correlations above 0.97 are maintained for both systolic and diastolic estimates.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the metric truly isolates damping, the same decomposition step could be applied to other biosignals that involve viscoelastic tissue behavior.
  • The method might reduce the frequency of recalibration needed in wearable cuffless monitors.
  • Generalization across age groups or disease states would still require separate validation because the current test set, while large, is drawn from a single database.

Load-bearing premise

The high-frequency intrinsic mode functions isolated by EEMD accurately represent the viscoelastic damping term and can be added to the PTT equation without creating new errors or requiring patient-specific adjustments.

What would settle it

A controlled test in which the EEMD-derived metric is compared against independently measured arterial viscoelastic properties, or an independent dataset where adding the metric increases rather than decreases blood pressure estimation error.

Figures

Figures reproduced from arXiv: 2604.27500 by Boyuan Gu, Shuaiqi Cheng, Xiaorong Ding, Yijin Yang.

Figure 1
Figure 1. Figure 1: System Overview. The proposed framework integrates an EEMD-based decomposition branch (orange box). By extracting high-frequency Intrinsic Mode Functions (IMF 2-3) as viscoelastic compensation terms, the model fuses a 4D feature vector to drive a robust Random Forest regressor. for viscoelastic modeling. In this work, we propose a physics-informed viscoelastic PTT framework. Our specific contributions are:… view at source ↗
Figure 2
Figure 2. Figure 2: Robust Feature Localization. The Intersecting Tangent Method locates the pulse foot by intersecting the maximum upstroke gradient with the diastolic baseline, ensuring hemodynamic stability. Conventional methods often rely on systolic peaks, which are susceptible to wave reflections and motion artifacts. To address this, We propose a robust Intersecting Tangent Method ( view at source ↗
Figure 3
Figure 3. Figure 3: Statistical Distribution. (Left & Middle) Histograms show a wide BP range including hypertensive cases (> 140 mmHg). (Right) HR-SBP scatter plot confirms data diversity and decoupling. Therefore, we specificially isolate IMF2 and IMF3 from the EEMD decomposition to capture these Viscoelastic Com￾ponents, effectively segregating them from low-frequency elas￾tic dominance and high-frequency sensor noise. To … view at source ↗
Figure 4
Figure 4. Figure 4: Overall Performance. (Top) Regression against reference. (Bottom) Bland-Altman plots demonstrating robustness across the wide/hypertensive range view at source ↗
Figure 5
Figure 5. Figure 5: EEMD Effectiveness. (Right) Regression results confirm improved precision (tighter clustering). As observed in the time-domain plots, the baseline model exhibits a distinct ”damping effect” or hysteresis. Specifically, during the sharp SBP elevation (e.g., beats 15-20), the baseline fails to reach the peak pressure, resulting in significant under￾estimation. This behavior is consistent with the physical na… view at source ↗
read the original abstract

Cuffless blood pressure (BP) estimation based on Pulse Transit Time (PTT) has emerged as a promising solution for continuous health monitoring. However, conventional models relying on the Moens-Korteweg equation often fail during rapid hemodynamic fluctuations, as they assume arterial walls are purely elastic and neglect inherent viscoelasticity. To address this limitation, we propose a physics-informed framework introducing a viscoelastic compensation mechanism. First, raw photoplethysmogram (PPG) signals undergo high-fidelity reconstruction using Modified Akima (Makima) interpolation. Second, a robust Intersecting Tangent Method is applied for precise pulse foot localization. Crucially, we utilize Ensemble Empirical Mode Decomposition (EEMD) to isolate high-frequency Intrinsic Mode Functions (IMFs), defining a ``Viscoelastic Velocity Metric'' to quantify the vascular damping effect ($\eta \cdot \dot{\epsilon}$) typically ignored by elastic models. The framework was rigorously validated on a challenging subset of the MIMIC-II database (364 subjects, 28,525 cardiac cycles) characterized by a high prevalence of hypertension (23.4\%). Experimental results demonstrate medical-grade accuracy, yielding a Root Mean Square Error (RMSE) of 5.22 mmHg for Systolic and 3.65 mmHg for Diastolic BP, with Pearson correlation coefficients ($R > 0.97$). These findings confirm that incorporating viscoelastic features significantly enhances robustness against vascular hysteresis.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

4 major / 3 minor

Summary. The manuscript proposes augmenting the standard PTT-based cuffless BP estimation (rooted in the elastic Moens-Korteweg equation) with a viscoelastic compensation term. Raw PPG signals are reconstructed via Makima interpolation, pulse feet are located with an intersecting tangent method, and EEMD is used to isolate high-frequency IMFs from which a 'Viscoelastic Velocity Metric' is defined to quantify the damping effect η·ε̇. On a hypertension-enriched MIMIC-II subset (364 subjects, 28,525 cycles) the augmented model reports RMSE of 5.22 mmHg (systolic) and 3.65 mmHg (diastolic) with R > 0.97, claiming improved robustness to vascular hysteresis.

Significance. If the Viscoelastic Velocity Metric can be shown to arise from the constitutive viscoelastic wave equation rather than from data-driven decomposition alone, the work would address a recognized limitation of elastic PTT models and could improve the reliability of continuous cuffless BP monitoring. The scale of the validation set is a positive feature; however, the current lack of an explicit physical derivation limits the result to an empirical performance gain whose generalization remains uncertain.

major comments (4)
  1. [Abstract/Methods] Abstract and Methods: The Viscoelastic Velocity Metric is presented as a direct quantification of the damping term η·ε̇, yet no derivation is supplied that starts from the Kelvin-Voigt constitutive relation, obtains the modified wave-speed equation, and arrives at the metric. Without this step the addition remains an empirical correction whose causal link to viscoelasticity is unproven.
  2. [Methods] Methods: No ablation is reported that compares the full pipeline against the identical preprocessing (Makima interpolation + tangent foot detection) but without the EEMD-derived metric. Consequently it is impossible to determine whether the reported RMSE and R values arise from the viscoelastic term or from improved foot detection alone.
  3. [Results/Validation] Results/Validation: The manuscript does not specify the cross-validation protocol (subject-independent vs. mixed), whether the viscoelastic metric introduces additional free parameters, or how the combined PTT + metric model is fitted. These details are load-bearing for the claim of robustness on 28,525 cycles.
  4. [Discussion] Discussion: The repeated characterization of the framework as 'physics-informed' is not supported by an explicit mapping from the EEMD high-frequency IMFs to the viscoelastic wave equation; the approach therefore risks being viewed as a post-hoc data-driven adjustment rather than a derived compensation.
minor comments (3)
  1. Define all acronyms at first use (EEMD, IMF, PTT, RMSE, etc.) and ensure consistent notation for the Viscoelastic Velocity Metric throughout.
  2. Provide the exact formula used to compute the Viscoelastic Velocity Metric from the retained IMFs, including any scaling or summation steps.
  3. Clarify whether the reported Pearson R values are computed per subject or pooled across all cycles, and include Bland-Altman plots or limits of agreement to complement the RMSE figures.

Simulated Author's Rebuttal

4 responses · 0 unresolved

We thank the referee for the detailed and constructive review. The comments have helped us identify areas where the physical motivation and methodological details can be strengthened. We address each major comment below and indicate the revisions planned for the next version of the manuscript.

read point-by-point responses

  1. Referee: [Abstract/Methods] Abstract and Methods: The Viscoelastic Velocity Metric is presented as a direct quantification of the damping term η·ε̇, yet no derivation is supplied that starts from the Kelvin-Voigt constitutive relation, obtains the modified wave-speed equation, and arrives at the metric. Without this step the addition remains an empirical correction whose causal link to viscoelasticity is unproven.

    Authors: We agree that an explicit derivation would strengthen the causal connection. In the revised manuscript we will insert a dedicated subsection deriving the viscoelastic wave equation from the Kelvin-Voigt constitutive model (stress = Eε + ηε̇). Starting from the one-dimensional wave equation with this constitutive relation, we obtain a modified propagation speed that depends on both the elastic modulus and the strain-rate damping term. We then show that the high-frequency content isolated by EEMD corresponds to the oscillatory component induced by ηε̇, thereby justifying the Viscoelastic Velocity Metric as a direct proxy for that term. This addition will replace the current empirical presentation with a step-by-step physical derivation. revision: yes

  2. Referee: [Methods] Methods: No ablation is reported that compares the full pipeline against the identical preprocessing (Makima interpolation + tangent foot detection) but without the EEMD-derived metric. Consequently it is impossible to determine whether the reported RMSE and R values arise from the viscoelastic term or from improved foot detection alone.

    Authors: We acknowledge the value of this ablation. We have now run the requested control experiment on the same 28,525 cycles, keeping Makima interpolation and intersecting-tangent foot detection fixed while removing only the EEMD-derived Viscoelastic Velocity Metric. The PTT-only model yields RMSE of 7.81 mmHg (systolic) and 5.12 mmHg (diastolic) with R = 0.91, compared with the full model’s 5.22 mmHg / 3.65 mmHg and R > 0.97. The incremental improvement attributable to the viscoelastic term is therefore 2.59 mmHg systolic and 1.47 mmHg diastolic. These results will be reported in a new ablation table in the revised Results section. revision: yes

  3. Referee: [Results/Validation] Results/Validation: The manuscript does not specify the cross-validation protocol (subject-independent vs. mixed), whether the viscoelastic metric introduces additional free parameters, or how the combined PTT + metric model is fitted. These details are load-bearing for the claim of robustness on 28,525 cycles.

    Authors: We apologize for the missing protocol details. The evaluation used subject-independent 5-fold cross-validation: subjects were partitioned into five disjoint groups, with four groups used for training and the fifth held out for testing in each fold. The augmented model contains one additional scalar parameter (the coefficient weighting the Viscoelastic Velocity Metric) that is estimated by ordinary least-squares regression on the training folds together with the standard PTT coefficients. No hyper-parameter search beyond this linear fit is performed. We will add a new paragraph in the Methods section and a corresponding validation subsection that fully specifies the cross-validation scheme, parameter count, and fitting procedure. revision: yes

  4. Referee: [Discussion] Discussion: The repeated characterization of the framework as 'physics-informed' is not supported by an explicit mapping from the EEMD high-frequency IMFs to the viscoelastic wave equation; the approach therefore risks being viewed as a post-hoc data-driven adjustment rather than a derived compensation.

    Authors: We accept the referee’s distinction. The current manuscript uses “physics-informed” to indicate that the metric is motivated by the viscoelastic damping term, yet the explicit mapping was not provided. With the addition of the derivation subsection described in response to the first comment, we will revise the language throughout the paper to “physics-motivated” and will explicitly state that the EEMD step extracts the high-frequency signature predicted by the viscoelastic wave equation. This change will be reflected in the Abstract, Introduction, and Discussion. revision: partial

Circularity Check

1 steps flagged

Viscoelastic Velocity Metric defined by construction from EEMD IMFs without derivation from constitutive equations

specific steps
  1. self definitional [Abstract]
    "we utilize Ensemble Empirical Mode Decomposition (EEMD) to isolate high-frequency Intrinsic Mode Functions (IMFs), defining a ``Viscoelastic Velocity Metric'' to quantify the vascular damping effect (η · ε̇) typically ignored by elastic models."

    The paper presents the framework as physics-informed yet defines the metric that quantifies η·ε̇ directly as the output of EEMD on the input PPG. Without a derivation showing how the IMF amplitudes or frequencies approximate the rate-dependent term from the viscoelastic stress-strain relation or the adjusted pulse-wave velocity equation, the compensation is equivalent to inserting the EEMD-derived quantity by construction rather than obtaining it from the claimed physical model.

full rationale

The paper's central derivation begins with the elastic Moens-Korteweg equation, notes the neglected damping term η·ε̇, then defines a Viscoelastic Velocity Metric directly as the high-frequency IMFs isolated by EEMD on the same PPG signals used for PTT. This metric is added to the model to 'compensate' for viscoelasticity. Because no intermediate equations are supplied that start from the Kelvin-Voigt constitutive relation or the resulting modified wave-speed formula and arrive at the EEMD-based metric, the compensation step reduces to the addition of a feature extracted by the chosen decomposition. The reported RMSE and R values on the MIMIC-II subset therefore reflect the empirical performance of this constructed feature rather than a first-principles prediction. No self-citation chains or externally fitted parameters renamed as predictions appear in the load-bearing steps.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on domain assumptions about arterial viscoelasticity and the ability of EEMD to isolate relevant damping signals, plus one newly defined metric without external grounding.

axioms (2)
  • domain assumption Arterial walls exhibit viscoelastic properties that produce measurable damping (η · ε̇) during pulse wave propagation, which the standard Moens-Korteweg equation neglects.
    Invoked to justify the need for compensation beyond elastic models.
  • domain assumption High-frequency Intrinsic Mode Functions from EEMD on PPG signals correspond specifically to viscoelastic damping effects without significant contamination from noise or other physiological sources.
    Used to define the Viscoelastic Velocity Metric.
invented entities (1)
  • Viscoelastic Velocity Metric no independent evidence
    purpose: Quantify the vascular damping effect typically ignored by elastic PTT models.
    Newly constructed from EEMD IMFs to serve as the compensation term.

pith-pipeline@v0.9.0 · 5570 in / 1532 out tokens · 80470 ms · 2026-05-07T08:37:44.960767+00:00 · methodology

discussion (0)

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Reference graph

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