# Time-Frequency Machine Learning Transfer Function for Central Pressure Waveforms

Soha Niroumandi, Heng Wei, Faisal Amlani, M Hossein Gorji, Rashid Alavi,

Julio Chirinos, Niema Pahlevan

► To cite this version:

Soha Niroumandi, Heng Wei, Faisal Amlani, M Hossein Gorji, Rashid Alavi, et al. Time-Frequency Machine Learning Transfer Function for Central Pressure Waveforms. European Heart Journal Open, 2025, 10.1093/ehjopen/oeaf082. hal-05136152

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# Time-frequency machine learning transfer function for central pressure waveforms

Soha Niroumandi ID<fnref n="1" />, Heng Wei ID<fnref n="1" />, Faisal Amlani ID², Hossein Gorji ID³, Rashid Alavi ID<fnref n="1" />, Julio A. Chirinos ID4,5, and Niema M. Pahlevan ID1,6,*

Received 17 March 2025; revised 14 May 2025; accepted 10 June 2025; online publish-ahead-of-print 23 June 2025

Handling Editor: Anders Gottsäter

## Aims

Clinical studies show that pulsatile haemodynamics and pressure waveform analysis are valuable for the diagnosis and prognosis of hypertension and heart failure (HF). While generalized transfer functions (GTFs) have shown clinical significance, some studies report limitations with GTF in capturing central pulsatile haemodynamics. This study introduces a hybrid time-frequency, machine learning-based transfer function that reconstructs central pressure waveforms from peripheral measurements, accurately capturing central pulsatile haemodynamics and arterial wave-based information.

## Methods and results

Our method uses Fourier harmonics for approximating the pressure waveform. The model is trained on these harmonics using a feed-forward neural network (FNN) with a custom time-domain cost function that captures the full temporal dynamics of physiological events during a cardiac cycle. The final hybridized-FNN transfer function model is trained, tested, and validated on data from the Framingham Heart Study (6698 participants). Our method produces carotid waveforms with median normalized mean squared error (%NMSE) values of 0.09 and 0.10 for brachial and radial inputs, compared to 0.42 and 0.26 for GTF, with similar accuracy improvements in other metrics. Correlation coefficients for the first and second forward wave times and amplitudes are 0.97, 0.93, 0.82, and 0.79 with brachial input, and 0.97, 0.92, 0.87, and 0.80 with radial input, vs. as low as 0.22 and 0.31 for GTF. Overall, our method significantly improved correlations across similarity, morphology, and wave-based parameters.

## Conclusion

Our hybridized FNN transfer function approach enables robust calculation of the central arterial pressure waveform from a single measured peripheral waveform, preserving key physiological sequences in a cardiac cycle.

<footnote n="1">¹Department of Aerospace and Mechanical Engineering, University of Southern California, 3650 McClintock Ave. Room 400, Los Angeles, CA 90089, USA; ²Université Paris-Saclay, CentraleSupélec, ENS Paris-Saclay, CNRS, LMPS—Laboratoire de Mécanique Paris-Saclay, 4 Av. des Sciences, 91190 Gif-sur-Yvette, France; ³Swiss Federal Laboratories for Materials Science and Technology (EMPA), Ueberlandstrasse 129, 8600 Dübendorf, Switzerland; ⁴Division of Cardiovascular Medicine, Hospital of the University of Pennsylvania, 3400 Civic Center Blvd., Smilow TRC 11th Floor, Philadelphia, PA 19104, USA; ⁵Perelman School of Medicine, University of Pennsylvania, 3400 Civic Center Blvd, Philadelphia, PA 19104, USA; and ⁶Division of Cardiovascular Medicine, Keck School of Medicine, University of Southern California, 1975 Zonal Ave., Los Angeles, CA 90033, USA</footnote>

<footnote>* Corresponding author. Tel: +1 213 740 7182, Email: pahlevan@usc.edu.</footnote>

<footnote>© The Author(s) 2025. Published by Oxford University Press on behalf of the European Society of Cardiology.</footnote>

<footnote>This is an Open Access article distributed under the terms of the Creative Commons Attribution-NonCommercial License (https://creativecommons.org/licenses/by-nc/4.0/), which permits non-commercial re-use, distribution, and reproduction in any medium, provided the original work is properly cited. For commercial re-use, please contact reprints@oup.com for reprints and translation rights for reprints. All other permissions can be obtained through our RightsLink service via the Permissions link on the article page on our site—for further information please contact journals permissions@oup.com.</footnote>
**Graphical abstract**

<figure><img src="image_2.png" /></figure>

**Keywords**

Cardiovascular transfer function • Central blood pressure • Arterial haemodynamics • Time-frequency machine learning

**Introduction**

Aortic pressure waveforms can be directly used to derive informative haemodynamic indices such as central pulse pressure, augmentation index, and ejection duration. In addition, aortic pressure waveform data can be combined with corresponding information on aortic flow or ventricular volume to derive key metrics of left ventricular afterload, cardiac performance, as well as ventricular-arterial interactions.¹⁻⁵ Aortic pressures can also provide incremental information for cardiovascular risk prediction compared to peripheral pressures.⁶,⁷

The pressure wave measured at the common carotid artery can be used as a surrogate for aortic pressure waveforms.⁸ However, measuring reproducible pressure waves is simpler at the radial and brachial arteries than at the common carotid artery. On the other hand, the morphological characteristics of the carotid (central) waveform correlate more strongly to left ventricular (LV) dynamics and LV-aortic coupling than radial or brachial (peripheral) waveforms.⁹⁻¹³ A variety of transfer function techniques (often frequency-dependent) have been proposed to obtain central pressure waveforms from radial or brachial waveforms.¹⁴⁻¹⁹ Such transfer functions relate the frequency components of the input (i.e. peripheral) with the output (i.e. central), facilitated by the use of Fourier transforms (obtained over 8–10 s of steady-state data).¹⁴⁻¹⁹ Although generalized transfer functions (GFTs) are well established, individual or adaptive transfer functions may yield superior performance since they are derived from the specific physical parameters of each individual.¹⁴⁻¹⁶,²⁰ Researchers have also proposed physics-based adaptive transfer functions by employing principles of fluid dynamics.¹⁷⁻²⁰ However, these models either require multiple measurements at different sites, some of which may be challenging to acquire, or they have not demonstrated improvements in the timing features of the central waveforms compared to the GFT.¹⁷⁻²⁰

Recently, machine learning (ML) has demonstrated great potential in deriving transfer functions between central and peripheral waveforms

given its capabilities in treating complex data relationships, non-linear patterns, and inverse problems.¹⁷⁻¹⁹,²¹,²² In particular, a number of ML-based models have been proposed to directly approximate central waveforms from peripheral measurements using limited patient or animal data.¹⁵,¹⁹,²¹⁻²³ In this work, we aim to exploit ML to develop more practical and robust transfer functions for accurate reconstructions of central waveforms from measured data at the periphery. Here, we take a hybrid time-frequency ML approach that leverages the high-order interpolating capability of a Fourier series to reconstruct the complete central (carotid) waveform from the peripheral (brachial, radial) waveform. Our method employs a reduced-order approach to generate an accurate approximating basis (such as an interpolating function) to precisely approximate the pressure waveform. The model is then trained on this reduced-order data using a feed-forward neural network (FNN) with a customized cost function. This cost function is defined in terms of the time-domain reconstruction of the full-field vector waveform, capturing the full-field temporal dynamics that account for physiological sequences during a cardiac cycle. The final hybridized feed-forward neural network (NN) transfer function model (hereafter referred to as 'Hybridized FNN TF') is trained, tested, and validated using data from the Framingham Heart Study (FHS) (6698 participants). We hypothesize that such an approach can outperform a conventional GTF in capturing significant time-domain waveform features (due to the choice of loss function) and can be applied to any arbitrary-sized input signal (due to the choice use of data dimensionality reduction).

**Methods**

**Participants**

We utilized data from three different cohorts of the FHS (the Original, Offspring, and Third Generation groups), in which non-invasive examinations of central haemodynamics were performed, yielding successful results
from 293 individuals from the original cohort, 2484 from the Offspring cohort, and 3921 from the third-generation cohort (for a total of 6698 participants). Participant ages ranged from 19 to 90 years, and the cohorts were comprised of community-dwelling individuals with and without clinically apparent cardiovascular disease.¹,¹³ Carotid, brachial, and radial pressure waveforms were acquired, calibrated, and processed as previously described.¹³,²⁴

## Dimensionality-reduction for machine learning

The tonometry-based carotid pressure waveforms in the Framingham dataset were sampled at a rate of 1000 Hz. Although all sampling rates are identical, the length of the cardiac cycle of each measurement j (hereinafter denoted $T = T_j$ in seconds) varies as a function of heart rate (HR), leading to a heterogeneously sized training database (the total sample size of each measured pressure waveform varies from participant to participant). That is, each discrete waveform $\mathbf{P} \in \{\mathbf{P}_{\text{data}}\}_{j=\{1,...,N_{\text{data}}}\}$ in a database of size $N_{\text{data}}$ has a discrete-time vector of length $N_{tj}$. Such varying input (and output) data lengths are not suitable for the FNN architecture considered in this work (the input size is fixed by definition). This, together with the large number of samples of a waveform for a typical HR (e.g. $N_{tj} = 1000$ for $T = 1$ s), suggests the use of a reduced-order representation (of fixed size) to both accurately compress the data (dimensionality reduction) as well as to homogenize the input and output parameter sizes to a fixed number across the entirety of a database (enabling application of the final model to any arbitrarily sized/sampled input brachial/radial waveform measurement). Such a reduced-order approach is efficient in dealing with variable input/output size vs. 'full-field' representations, with more competitive offline training and subsequent online inference speeds.²⁵

We develop our reduced-order approach for the NN model as follows: given the continuous counterpart $p(t)=[0, T] \to \mathbb{R}$ of a uniformly sampled discrete pressure waveform $\mathbf{P} = [p(t_1), ..., p(t_{N_t})]^{\text{T}} \in \mathbb{R}^{N_t}$ (e.g. measurements at $t = t_k$, $k = 1, ..., N_t$), we seek an accurate approximating (e.g. interpolating) basis $\{\phi_k(t)\}_{k=0,...,K}$ such that $p(t)$ can be approximated with high accuracy by an expression given by

$$p(t) \approx p_K(t) = \sum_{k=0}^{K} \hat{p}_k \phi_k(t) \quad (1)$$

for constant (possibly complex-valued) coefficients $\hat{p}_k$. A number of interpolating bases (e.g. algebraic polynomials) are possible, with very different accuracies and convergence rates. Due to the smooth periodic nature of the pressure signal, an ideal candidate is to use a trigonometric polynomial (e.g. a truncated or band-limited Fourier series) of degree $K$, i.e.

$$p_K(t) \approx \sum_{k=0}^{2K} \hat{p}_k z^{k-K}, \quad z = e^{2\pi i t} \quad (2)$$

Such a choice of reduced-order basis is known to be rapidly convergent (highly accurate) and well-behaved, particularly for periodic signals containing mostly a handful of lower frequencies such as pressure waveforms.²⁶,²⁷ The problem for constructing the corresponding interpolant is hence given by finding the coefficients such that the interpolant is as close as possible to the raw waveform $p(t)$, i.e. the $\hat{p}_k$, $k = 0, ..., 2K$ corresponding to the solution of the least-squares minimization problem given by

$$\min_{\{p_k\}} \| p(t) - p_K(t) \|_2^2. \quad (3)$$

The solution to this problem yields a classical spectral approximation given by the Fourier coefficients for the corresponding mode at wavenumber $k$, i.e.

$$\hat{p}_k = \begin{cases} 1 & p(t) e^{-2\pi i (k-K)t/T} dt. \\ 0 & \end{cases} \quad (4)$$

For the discrete sample vector of measurements $\mathbf{P} = [p(t_1), ..., p(t_{N_t})]^{\text{T}}$ of length $N_t$ samples, the corresponding Fourier coefficients can be calculated by the discrete Fourier transform (DFT):

$$\hat{p}_k = \frac{1}{N_t} \sum_{i=0}^{N_t-1} p(t_i) e^{-2\pi i (k-K) t_i/T} \quad (5)$$

The calculation of the above expression is rapidly facilitated by an application of the Fast Fourier Transform (FFT). Since pressure is real-valued, the coefficients are conjugate symmetric, i.e.

$$\hat{p}_k = \overline{\hat{p}_{2K-k}} \quad (6)$$

and hence only $\hat{p}_k$, $k = 0, ..., K$ are needed to represent the reduced-order interpolant formulation (for a total of $K+1$ complex-valued parameters). Additionally, by subtracting out the mean of the signal before performing a DFT (so that the waveform carries a mean value of zero), the $k=0$ can also be eliminated. In other words, the trigonometric polynomial interpolant reduces each sampled pressure vector $\mathbf{P}_j \in \{\mathbf{P}_{\text{data}}\}_{j=\{1,...,N_{\text{data}}}\}$ of arbitrary size $N_{tj}$ to the representation of just $K$ complex values (or $2K$ values when storing the real and imaginary parts of each coefficient separately for machine learning), where $K \ll N_{tj}$ is the same fixed value for all pressure measurements.

Hence for each element (measurement) $\mathbf{P}_{\text{data}}$ of the set of arbitrary-length input pressure signals $\{\mathbf{P}_{\text{data}}\}_{j=\{1,...,N_{\text{data}}}\}$ (where $\cdot_{\text{data}}$ denotes the labelled data for training), the associated vector of reduced-order parameters via trigonometric polynomial interpolation is given by $\hat{\mathbf{P}}_{(K)}^{\text{data}} = (\hat{p}_0, ..., \hat{p}_K)^{\text{T}}$, and hence the final database constitutes the set given by

$$\{\mathbf{P}_{(K)}^{\text{data}}\}_{j=\{1,...,N_{\text{data}}}\} \quad (7)$$

## Feed-forward neural network-based transfer function model

The proposed ML architecture for the final NN-based transfer function model is an FNN. Figure 1 illustrates the database generation (reduced-order processing of the raw waveforms) as well as the FNN-based model design for the proposed transfer function. The goal of the latter is to predict the coefficients $\{\hat{\mathbf{p}}_{\text{transf}}\}$ that describe the corresponding carotid pressure waveform $\mathbf{P}^{\text{transf}}$ from an input brachial or radial waveform $\mathbf{P}^{\text{meas}}$ measured at $N_t$ uniformly spaced sample points in time. The full predicted carotid waveform can then be reconstructed via Equation 2.

The model is hence trained on the reduced-order database $\{\mathbf{P}_{(K)}^{\text{data}}\}_{j=\{1,...,N_{\text{data}}}\}$ introduced in the previous subsection. A corresponding customized loss function is proposed as follows: supervised learning of the FNN model is governed by a mean square error (MSE) loss defined in terms of the time-domain reconstructed full-field vector waveform of length $N_j$ samples, i.e. the Euclidean error distance between the prediction and ground truth (data) temporal waveforms. Defining

$$\mathbf{P}_{(K)} = \left( \sum_{k=0}^{2K} \hat{p}_k(z(t_1))^{k-K}, \dots, \sum_{k=0}^{2K} \hat{p}_k(z(t_{N_t}))^{k-K} \right)^{\text{T}}, z(t) \\ = \exp(2\pi i t/T) \quad (8)$$

to be the time domain reconstruction corresponding to Fourier coefficients $\hat{\mathbf{P}}_{(K)} = (\hat{p}_0, ..., \hat{p}_K)^{\text{T}}$, the proposed loss function is hence given by

$$\mathcal{L} = \frac{1}{N_j} \| \mathbf{P}_{(K)}^{\text{pred}} - \mathbf{P}_{(K)}^{\text{data}} \|_2^2 \\ = \frac{1}{N_j} \sum_{n=1}^{N_j} \left( \sum_{k=0}^{2K} \hat{p}_k^{\text{pred}}(z(t_n))^{k-K} - \sum_{k=0}^{2K} \hat{p}_k^{\text{data}}(z(t_n))^{k-K} \right)^2 \quad (9)$$
<figure><img src="image_3.png" /><figcaption>**Figure 1** Data preprocessing and model description for the wave morphology-based hybridized feed-forward neural network transfer function.</figcaption></figure>

As opposed to defining loss as the MSE directly between the Fourier coefficients (i.e. between $\hat{\mathbf{P}}$ and $\hat{\mathbf{P}}^{\text{data}}$), the definition above incorporates the full-field temporal dynamics which accounts for physiological sequences during a cardiac cycle. Optimizing with respect to the time-domain reconstructed waveform can better preserve overall waveform morphology including corresponding time-dependent features (e.g. reflected arrival times, notches, etc.). Indeed, minimizing with respect to the coefficients alone does not perform as well as that of Equation (9) than the loss function proposed above (see Supplementary material online for tests of alternative loss functions).

The final overall model utilizes a degree $K = 14$ trigonometric polynomial (for a total of $2K = 28$ real and imaginary components constituting the input/output parameter sizes) for the reduced-order waveform representations. The complete FNN structure is comprised of four layers: an input layer with 28 nodes (corresponding to the input parameters of Fourier coefficients), two hidden layers with 64 and 128 nodes, respectively, and an output layer with 28 nodes (corresponding to the output parameters). The number of hidden layers is chosen in order to keep the architecture relatively simple and efficient in terms of required training data size and subsequent computational complexity. ReLU activation functions are used in the hidden layers to ensure faster convergence and avoid the vanishing gradient problem.²⁸ Optimization for training with respect to the loss function of Equation (9) is facilitated by a stochastic gradient descent algorithm (ADAM optimizer with $\ell^2$ regularization to minimize overfitting, using a weight decay coefficient of $\lambda = 0.00001$ and a learning rate of $\alpha = 0.001$). Early stopping is also employed to terminate training when the validation performance ceases to improve (up to 1500 epochs). All the aforementioned hyperparameter values have been determined by identifying, via a coarse grid search across predefined ranges, suitable combinations that optimize model performance in terms of convergence rate and maximum allowable loss. Training and validation of the FNN are conducted on 60% and 20% of the 6698 patient data, respectively, with blind testing on the remaining 20%. The test dataset is not used during any stage of model development and is reserved solely for evaluating the model's performance (all results of this paper are on the test dataset). Data preprocessing and model development have been carried out using Python's `skLearn`, `PyTorch`, `pandas`, and `NumPy` packages.

## Summary of the overall model

An algorithmic summary of the proposed hybridized FNN TF model based on trigonometric polynomials is provided in Algorithm 1.

## Analysis methods

We assess the accuracy and fidelity of the hybridized FNN TF-modelled waveforms using the metrics presented in Table 1 and briefly described in what follows.

The considered physiological metrics are based on morphology and offer information about specific waveform characteristics: end-systolic time, end-systolic pressure,²⁹ Systolic area ratio, subendocardial viability ratio [Buckberg index,³⁰ as well as the first and second cardiovascular intrinsic frequencies (IFs: $\omega_1, \omega_2$)].¹⁰,¹¹,¹³,³¹⁻³⁹ Aortic characteristic impedance⁴⁰ is additionally considered by calculating its value through the measured aortic flow waveform from FHS⁴¹ and the corresponding measured or transferred central pressure waveform (carotid). Conventional wave-based metrics are also employed for studying the efficacy of the proposed approach, particularly those provided by wave intensity (WI) analysis⁴² of the predicted carotid waveform (which has been shown to be physiologically relevant:⁴²) the first/second forward and backward wave times as well as the first/second forward and backward wave amplitudes. WI analysis is performed using the relevant pressure waveforms together with FHS-measured velocity (flow) waveforms (further details are provided in Supplementary material online).

The statistical methods incorporate both the use of error-based and correlation-based metrics including percentage normalized mean squared error (%NMSE), percentage dynamic time warping distance, normalized Euclidean distance (NE-dist), and interclass correlation (ICC) to quantitatively assess the quality of the fit between the calculated and original (measured) central waveforms. The former most is given by

$$ \text{NMSE} := \frac{\frac{1}{N_j} \sum_{n=1}^{N_j} \left( \sum_{k=0}^{2K} \hat{p}_k^{\text{pred}}(z(t_n))^{k-K} - \sum_{k=0}^{2K} \hat{p}_k^{\text{data}}(z(t_n))^{k-K} \right)^2}{\left( \frac{1}{N_j} \sum_{n=1}^{N_j} \left( \sum_{k=0}^{2K} \hat{p}_k^{\text{data}}(z(t_n))^{k-K} \right) \right)^2} \quad (10) $$

Percentage dynamic time warping distance is calculated using the fastdtw library in Python and normalized by the sum of the lengths of the sequences.⁴³ Euclidean distance between the waveforms is calculated using the Euclidean distance function from the SciPy library and normalized by the length of the waveforms.⁴⁴ Second and third ICCs are calculated using the SciPy signal sub-library correlation functions. In the results that follow,
<table><caption>**Table 1 Analysis metrics and their abbreviations**</caption><thead><tr><th>Metric</th><th>Abbreviation</th><th>Metric</th><th>Abbreviation</th></tr></thead><tbody><tr><td>Normal mean squared error</td><td>NMSE</td><td>First main intrinsic frequency</td><td>ω<sub>1</sub></td></tr><tr><td>Interclass correlation</td><td>ICC</td><td>Second main intrinsic frequency</td><td>ω<sub>2</sub></td></tr><tr><td>Normalized dynamic time warping distance</td><td>NDT-dist</td><td>First forward wave peak time</td><td>Wf1-T</td></tr><tr><td>Normalized Euclidean distance</td><td>NE-dist</td><td>Second forward wave peak time</td><td>Wf2-T</td></tr><tr><td>End-systolic time</td><td>NT</td><td>First forward wave peak amplitude</td><td>Wf1-A</td></tr><tr><td>End-systolic pressure</td><td>NP</td><td>Second forward wave peak amplitude</td><td>Wf2-A</td></tr><tr><td>Early systolic inflection pressure</td><td>P1</td><td>First backward wave peak time</td><td>Wb1-T</td></tr><tr><td>Systolic area ratio</td><td>SA</td><td>First backward wave peak amplitude</td><td>Wb1-A</td></tr><tr><td>Buckberg index</td><td>BBinx</td><td>Aortic characteristic impedance</td><td>Zao</td></tr></tbody></table>

<table><caption>**Table 2 Error and similarity metrics**</caption><thead><tr><th>%NMSE<br>[median,<br>range]</th><th>NDT-dist<br>[median,<br>range]</th><th>NE-dist<br>[median,<br>range]</th><th>ICC<br>[ICC2,<br>ICC3]</th><th>%NMSE<br>[median,<br>range]</th><th>NDT-dist<br>[median,<br>range]</th><th>NE-dist<br>[median,<br>range]</th><th>ICC<br>[ICC2,<br>ICC3]</th></tr></thead><tbody><tr><td>Carotid waveforms calculated from <i>brachial</i> waveforms using GTF<br>[0.42, 9.78]</td><td>[61.46,555.31]</td><td>[0.064,0.257]</td><td>[0.83, 0.72]</td><td>Carotid waveforms calculated from <i>brachial</i> waveforms using hybridized FNN TF<br>[0.090, 3.55]</td><td>[13.2, 221.16]</td><td>[0.029, 0.158]</td><td>[0.96, 0.93]</td></tr><tr><td>Carotid waveforms calculated from <i>radial</i> waveforms using GTF<br>[0.26, 12.44]</td><td>[34.34, 371.15]</td><td>[0.050, 0.319]</td><td>[0.90, 0.84]</td><td>Carotid waveforms calculated from <i>radial</i> waveforms using hybridized FNN TF<br>[0.099, 6.33]</td><td>[12.9, 136.57]</td><td>[0.031, 0.216]</td><td>[0.96, 0.93]</td></tr></tbody></table>

FNN, feed-forward neural network; GTF, generalized transfer function; NMSE, normalized mean squared error; TF, transfer function.

(i.e. higher accuracy and clinical relevancy); and (iii) it can be applied to signals of any arbitrary sampling rate.

## Clinical importance of transfer functions and current needs

The clinical significance of the transfer function approach has been demonstrated in several studies.⁴⁵⁻⁴⁸ A notable recent study involving participants from the IDCARS (International Database of Central Arterial Properties for Risk Stratification) cohorts showed that wave analysis metrics derived from central waveforms, synthesized using the GTF from radial tonometry pressure waveforms, were predictive of composite cardiovascular outcomes.⁴⁸ However, negative findings have also been reported regarding the usefulness of the GTF method.⁴⁹⁻⁵¹ These negative results may be attributed to the GTF's inability to capture the dynamic features of the central pressure waveform. Overall, the application of transfer functions has largely remained within the domain of clinical research. In recent years, in addition to the GTF, various other more specialized approaches—whether frequency-based, physics-based, or ML-based—have been developed and validated.¹⁷,¹⁹,²¹,²²,⁵² These new transfer functions can all provide central pressure waveforms from measured peripheral waveforms such as radial, brachial, femoral, or photoplethysmography waveforms.¹⁷,¹⁹,²¹,²²,⁵² While each of these methods has its strengths and advantages (contributing significantly to the transfer function concept), they also have their limitations.¹⁷,¹⁹,²¹,²²,⁵²

Clinical studies have established that pulsatile haemodynamics and pressure waveform analysis hold prognostic value for cardiovascular

diseases such as HF, and access to this information can significantly improve patient treatment.¹³,⁵³⁻⁵⁶ More importantly, recent clinical studies have demonstrated that improvements in HF patient treatment cannot be fully captured using brachial (cuff) pressures alone.⁵⁶ Instead, central pressure waveform analysis offers more patient-specific insights, aiding in therapeutic optimization.⁵⁶ Furthermore, recent clinical data on patients with HF with preserved ejection fraction suggest that non-invasive assessment of pulsatile left ventricular workload, which can be accurately assessed through central pressure waveforms, can further improve therapeutic interventions.⁵³ Our proposed hybridized FNN transfer function addresses these needs by synthesizing central pressure waveforms that accurately capture important wave-based characteristics and pulsatile haemodynamic metrics. Our method has been specifically designed to preserve this critical information, as demonstrated by our results. To the best of our knowledge, wave-based assessments (e.g. WI, IF) across a large cohort, including both healthy individuals and patients with cardiovascular disease (as conducted in this work), have not been performed for other recently developed transfer function methods.

## Hybridized feed-forward neural network as a novel dynamics-preserving transfer function

As demonstrated in Table 2, the proposed hybridized FNN TF significantly improves quantitative similarity metrics, including the NMSE, normalized dynamic time warping distance (NDT-dist), NE-dist, and
<table><caption>**Table 3** Morphology-based parameters (R is the Pearson correlation coefficient and R² is R² score)</caption><thead><tr><th>NT<br>[R, R2, bias]</th><th>NP<br>[R, R2, bias]</th><th>SA<br>[R, R2, bias]</th><th>BBindx<br>[R, R2, bias]</th><th>P1<br>[R, R2, bias]</th><th>SP<br>[R, R2, bias]</th><th>PP<br>[R, R2, bias]</th><th>NP/PP<br>[R, R2, bias]</th><th>Zao<br>[R, R2, bias]</th></tr></thead><tbody><tr><td>Carotid waveforms calculated from brachial waveforms using GTF<br>[0.657, -0.012,<br>-0.007]</td><td>[0.973, 0.928, +<br>2.057]</td><td>[0.759, 0.177, +<br>0.265]</td><td>[0.650, -1.448, +<br>0.265]</td><td>[0.908, 0.443,<br>10.945]</td><td>[0.967, 0.799,<br>-7.993]</td><td>[0.962, 0.742,<br>-7.993]</td><td>[0.914, 0.179, +<br>0.319]</td><td>[0.860, 0.555, 0.015]</td></tr><tr><td>Carotid waveforms calculated from brachial waveforms using hybridized FNN TF<br>[0.820, 0.432, 0.004]</td><td>[0.975, 0.922, +<br>2.463]</td><td>[0.858, 0.722,<br>-0.031]</td><td>[0.859, 0.723,<br>-0.030]</td><td>[0.946, 0.881,<br>-2.014]</td><td>[0.984, 0.966,<br>-0.390]</td><td>[0.979, 0.957, +<br>0.387]</td><td>[0.945, 0.868,<br>-0.056]</td><td>[0.922, 0.810,<br>-0.012]</td></tr><tr><td>Carotid waveforms calculated from radial waveforms using GTF<br>[0.640, -0.022,<br>-0.008]</td><td>[0.964, 0.914, +<br>4.063]</td><td>[0.721, 0.147,<br>-0.066]</td><td>[0.737, -0.273, +<br>0.102]</td><td>[0.879, 0.499, 9.285]</td><td>[0.963, 0.847,<br>-6.108]</td><td>[0.957, 0.812,<br>-6.108]</td><td>[0.918, 0.495, +<br>0.231]</td><td>[0.802, 0.040, 0.026]</td></tr><tr><td>Carotid waveforms calculated from radial waveforms using hybridized FNN TF<br>[0.778, 0.365, +<br>0.092]</td><td>[0.971, 0.896, +<br>3.205]</td><td>[0.861, 0.730,<br>-0.017]</td><td>[0.862, 0.731,<br>-0.017]</td><td>[0.945, 0.888,<br>-1.208]</td><td>[0.982, 0.962,<br>-0.684]</td><td>[0.977, 0.954, +<br>0.680]</td><td>[0.948, 0.861,<br>-0.074]</td><td>[0.915, 0.781,<br>-0.013]</td></tr></tbody></table>

FNN, feed-forward neural network; GTF, generalized transfer function; TF, transfer function.

ICC. The improvements are consistent for both radial-based and brachial-based input waveforms, though slightly better for the brachial-based transferred waveforms. Figure 3 illustrates these improvements qualitatively for sample waveforms of classical Types A, B, and C. Additionally, as shown in Figure 2 (A and B), the most common absolute per cent error between the calculated carotid pressure waveforms using the hybridized FNN TF and the measured carotid waveforms (for both radial and brachial source waveforms) is close to 2%. Figure 2 (C and D) further indicates a strong correspondence between each Fourier mode of these hybridized FNN TF carotid waveforms and the tonometry-measured carotid waveforms.

The improvements in physiologically relevant morphological metrics, such as systolic period (NT), early systolic inflection pressure (P1), area under the systolic portion of the pressure curve (SA), and the subendocardial viability ratio (Buckberg ratio; BBindx), are significant for both brachial and radial waveforms (Table 3). These improvements range from 13% to 32% in correlation coefficient (from R = 0.65 to R = 0.86 for the largest improvement). Similar improvements are also observed for aortic characteristic impedance in Table 3, a well-established pressure-flow-based marker of pulsatile load, dependent on aortic size and wall stiffness. Although the GTF performs reasonably well in capturing systolic pressure, pulse pressure, and the ratio of end-systolic pressure to pulse pressure (NP/PP) (all with R > 0.91), our proposed hybridized FNN TF still demonstrates superior improvement (all with R > 0.94). The improved performance of our proposed method vs. GTF is highlighted further when comparing R² scores, where the former demonstrates a strong correlation as opposed to the latter. As demonstrated in Table 4, the carotid waveforms generated by our proposed methodology preserve the energy transmitted through the arterial system more effectively than those calculated using GTF. This is evident from the improvement in the first four WI parameters: the first and second forward wave amplitudes and their respective times (Wf1-A, Wf2-A, Wf1-T, and Wf2-T). Notably, a significant improvement is seen in the reflection times (Wf1-T and Wf2-T). Additionally, the significant improvement observed in intrinsic frequency parameters such as ω₁ and ω₂ (calculated from the transferred carotid pressure waveform) strongly indicates that the proposed hybridized FNN TF approach provides more detailed insights into the coupled dynamics of the heart and the arterial system when compared with a conventional GTF.

## Strengths and limitations

Our proposed transfer function is based on a reduced-order representation of the waveforms, allowing it to handle various data sizes and sampling rates. It requires only one peripheral pressure measurement to perform the analysis/transfer. Although our study has several strengths it also has some limitations. All carotid pressure waveforms were measured using a single type of platform: piezoelectric arterial application tonometry (Cardiovascular Engineering, Inc.). However, in principle, our model should be applicable to any fidelity pressure waveform measurement, regardless of source or sampling rate. Furthermore, our data predominantly consist of White participants of Western European descent from the FHS, which may limit the generalizability of our models to other racial or ethnic groups. An additional limitation of the hybridized FNN TF itself relates to its increased complexity when compared to traditional GTF approaches (which can be simpler to build and apply). However, this complexity may be acceptable since it allows for improved accuracy (as demonstrated by the results in this work). Finally, direct predictions of outcomes have not been investigated using the new transfer function methodology. Studies towards comprehensively characterizing the diagnostic and prognostic performance of metrics computed from the hybridized FNN TF-derived central pressure waveforms for specific individual and composite endpoints are a subject of future work.
<table><caption>**Table 4** Correlation of wave intensity and IF parameters (see Table 1 for metric definitions)</caption><thead><tr><th>Wf1-T</th><th>Wf1-A</th><th>Wf2-T</th><th>Wf2-A</th><th>Wb1-T</th><th>Wb1-A</th><th>ω<sub>1</sub></th><th>ω<sub>2</sub></th></tr></thead><tbody><tr><td colspan="8">Carotid waveforms calculated from <i>brachial</i> waveforms using GTF</td></tr><tr><td>0.865</td><td>0.840</td><td>0.343</td><td>0.695</td><td>0.215</td><td>0.278</td><td>0.878</td><td>0.797</td></tr><tr><td colspan="8">Carotid waveforms calculated from <i>brachial</i> waveforms using hybridized FNN TF</td></tr><tr><td>0.967</td><td>0.933</td><td>0.824</td><td>0.792</td><td>0.500</td><td>0.742</td><td>0.946</td><td>0.899</td></tr><tr><td colspan="8">Carotid waveforms calculated from <i>radial</i> waveforms using GTF</td></tr><tr><td>0.946</td><td>0.861</td><td>0.591</td><td>0.751</td><td>0.343</td><td>0.310</td><td>0.893</td><td>0.803</td></tr><tr><td colspan="8">Carotid waveforms calculated from <i>radial</i> waveforms using hybridized FNN TF</td></tr><tr><td>0.971</td><td>0.922</td><td>0.874</td><td>0.795</td><td>0.493</td><td>0.743</td><td>0.946</td><td>0.880</td></tr></tbody></table>

FNN, feed-forward neural network; GTF, generalized transfer function; TF, transfer function.

<figure><img src="image_4.png" /><figcaption>**Figure 2** Histogram of mean per cent absolute error between the ground truth (tonometry-measured) and hybridized feed-forward neural network transfer function model-calculated carotid waveforms from brachial (A) and radial (C) input waveforms. Corresponding relative errors as a function of harmonic between the calculated and target waveforms for the brachial (B) and radial (D) cases. Error bars depict the 95% confidence interval for the mean of the relative differences between the frequency components of the normalized measured and the hybridized feed-forward neural network transfer function calculated carotid waveforms.</figcaption></figure>

## Perspectives

We have introduced a novel hybridized FNN transfer function with a customized time-domain cost function (for ML) that uses as input the harmonics from the frequency domain. Consequently, the approach enables a robust calculation of the central arterial pressure waveform

from a single measured peripheral pressure waveform that preserves important physiological sequences within a cardiac cycle while requiring orders-of-magnitude fewer degrees-of-freedom. This innovative approach proposed here enables the non-invasive estimation of the central blood pressure waveform from a single non-invasively measured peripheral waveform and offers a significant improvement over existing
<figure><img src="image_5.png" /><figcaption>**Figure 3** Fully reconstructed waveforms calculated by our hybridized feed-forward neural network transfer function and by a conventional GTF (overlaid with real measurements in solid line) for three classes of target carotid waveform types based on peak systolic pressure and augmentation index (Alx). (A) and (B) illustrate Type A carotid waveforms, where peak systolic pressure occurs after the shoulder, transferred from brachial and radial waveforms, respectively. (C) and (D) illustrate Type B carotid waveforms, where peak systolic pressure occurs after the shoulder, transferred from brachial and radial waveforms, respectively. (E) and (F) illustrate Type C carotid waveforms, characterized by a dominant initial pressure peak and an inflection on the downstroke of the pressure waveform, transferred from brachial and radial waveforms, respectively. The waveforms produced by the hybridized feed-forward neural network transfer function (dashed line) clearly and strongly resemble the tonometry-measured waveforms, unlike the generalized transfer function (dotted line).</figcaption></figure>

methods. Such an accurate reconstruction can facilitate arterial wave-based analysis and pulsatile haemodynamic quantification that can potentially offer more patient-specific insight towards therapeutic optimization. Future studies are needed to test this methodology on other large cohorts, as well as to directly investigate the predictive value in isolation of the transferred waveforms (i.e. without consideration of the ground truth values).

The model introduced in this work can be integrated into various testing and diagnostic platforms, which are much more widely available for peripheral waveforms than for carotid waveforms. For example, non-invasive peripheral waveform measurements can be acquired not

only by computer-controlled tonometry devices but also by wearable devices (e.g. smartwatches with mounted pressure sensors), which can help facilitate broader ambulatory applications. Moreover, invasive radial waveform measurements for continuous pressure monitoring over relatively long time intervals of hospitalized patients may benefit by the application of the proposed approach to facilitate the monitoring of central parameters in addition to the measured peripheral parameters. Finally, the methodology proposed herein may help facilitate the prediction of incident cardiovascular risk through confident reconstructions of central waveforms. Studies towards comprehensively characterizing the diagnostic and prognostic performance of predicted
central pressure waveforms for specific individual and composite end-points is the subject of future work.

## Lead author biography

<figure><img src="image_6.png" /></figure>

Soha Niroumandi, MSc, is a PhD candidate in mechanical and cardiovascular engineering at the Michelson Center for Convergent Bioscience at the University of Southern California (USC). Her research combines medical engineering and computer science to develop non-invasive and inexpensive smartphone-based methods for cardiovascular and cerebrovascular monitoring using signal processing and machine learning. She is a recipient of the American Heart Association Predoctoral Fellowship and the Phi Kappa Phi National Honor Society's 2025 Student Recognition Award for her contributions to advancing the early detection of dementia.

## Supplementary material

Supplementary material is available at *European Heart Journal Open* online.

## Author contributions

Soha Niroumandi (Conceptualization [equal], Data curation [lead], Formal analysis [lead], Methodology [equal], Software [equal], Visualization [equal], Writing—original draft [equal], Validation [equal], Writing—review & editing [equal]), Heng Wei (Conceptualization [equal], Data curation [supporting], Formal analysis [supporting], Methodology [equal], Software [equal], Visualization [equal], Validation [equal], Writing—review & editing [equal]), Faisal Amlani (Conceptualization [equal], Data curation [supporting], Formal analysis [supporting], Methodology [equal], Software [equal], Visualization [equal], Writing—review & editing [equal]), Hossein Gorji (Methodology [equal], Software [equal], Visualization [equal], Validation [equal], Writing—review & editing [equal]), Rashid Alavi (Conceptualization [equal], Data curation [supporting], Formal analysis [supporting], Methodology [equal], Software [equal], Visualization [equal], Validation [equal], Writing—review & editing [equal]), Julio A. Chirinos (Conceptualization [equal], Supervision [supporting], Writing—review & editing [equal]), Niema M. Pahlevan (Conceptualization [equal], Methodology [equal], Software [equal], Visualization [equal], Supervision [lead], Validation [equal], Writing—review & editing [equal]).

## Acknowledgements

The Framingham Heart Study is supported by Contract No. HHSN2682015000011 from the National Heart, Lung, and Blood Institute (NHLBI) with additional support from other sources. This manuscript was not prepared in collaboration with investigators of the Framingham Heart Study and does not necessarily reflect the opinions or conclusions of the Framingham Heart Study or the NHLBI. The authors would like to thank Sohrab P. Mazandarani for his valuable insight and for productive discussions.

## Funding

None.

**Conflict of interest:** None declared.

## Data availability

The data used in this study can be requested directly from the Framingham Heart Study. It is publicly available to qualified investigators with approved research proposals. The data underlying this article will be shared on reasonable request to the corresponding author.

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