Assessing pressure wave components for aortic stiffness monitoring through spectral regression learning

Abstract Aims The ageing process notably induces structural changes in the arterial system, primarily manifesting as increased aortic stiffness, a precursor to cardiovascular events. While wave separation analysis is a robust tool for decomposing the components of blood pressure waveform, its relationship with cardiovascular events, such as aortic stiffening, is incompletely understood. Furthermore, its applicability has been limited due to the need for concurrent measurements of pressure and flow. Our aim in this study addresses this gap by introducing a spectral regression learning method for pressure-only wave separation analysis. Methods and results Leveraging data from the Framingham Heart Study (2640 individuals, 55% women), we evaluate the accuracy of pressure-only estimates, their interchangeability with a reference method based on ultrasound-derived flow waves, and their association with carotid-femoral pulse wave velocity (PWV). Method-derived estimates are strongly correlated with the reference ones for forward wave amplitude (R2=0.91), backward wave amplitude (R2=0.88), and reflection index (R2=0.87) and moderately correlated with a time delay between forward and backward waves (R2=0.38). The proposed pressure-only method shows interchangeability with the reference method through covariate analysis. Adjusting for age, sex, body size, mean blood pressure, and heart rate, the results suggest that both pressure-only and pressure-flow evaluations of wave separation parameters yield similar model performances for predicting carotid-femoral PWV, with forward wave amplitude being the only significant factor (P < 0.001; 95% confidence interval, 0.056–0.097). Conclusion We propose an interchangeable pressure-only wave separation analysis method and demonstrate its clinical applicability in capturing aortic stiffening. The proposed method provides a valuable non-invasive tool for assessing cardiovascular health.


Spectral Regression Learning Method
The first step to implement the spectral regression learning method is to transform data from a high-dimensional input signal space to a low-dimensional space.To do so, we employed Fourierbased decomposition.The Fourier series is a method of combining harmonically related sinusoids and cosines to synthesize a periodic function.Any periodic pressure function P(t) can be expressed as a Fourier series with N oscillatory components.A typical representation of the Fourier series decomposition is as follows: where T is the period of the pressure function P(t) (i.e., the cardiac cycle or inverse of HR for blood pressure waveform).In conducting the Fourier decomposition here, we used the sinusoidcosinusoide form rather than an Amplitude-Phase form to make the inputs independent from each other.This is because the system of all sine and cosine functions at different frequencies builds a complete orthonormal set.Coefficients  # and  # are associated with each individual harmonics (cosine and sine) corresponding to different frequencies  # = # % , and can be calculated by the Fourier transform given by: , (0, ∞). (3) Obtaining the coefficients using Eq. ( 2) and (3), the represented pressure waveform  : () with finite selected frequency (i.e., n = 0 to N) is achieved by adding up each individual frequency component (Eq.( 1)).In this study, the features of the pressure wave were extracted as the first  *+ low frequency components of the waveforms using the Fast Fourier Transform (FFT).Previous studies suggested that to fully recover the shape of the pressure waveform using Fourier mode decomposition, the first 20 modes are enough and the error between the measured pressure and the reconstructed after 20 modes is marginal.The selected input features in this study (depending on the chosen number of Fourier modes; FN) consists of FN cosine coefficients and (FN -1) sine coefficients (sine  ) is always equal to zero).
In the next step, we use appropriate regression models to build the spectral regression learning model for estimating the Fourier modes of the flow waveform.In this study, the support vector regressor was selected to train and test the regression model due to its proven applicability and performance in previous studies.As a brief summary of this model, a Support Vector Regressor is a machine learning tool utilized for predicting numerical values, such as the amplitude of the sinusoidal-cosinusoidal components of the flow waveform in this study.It operates by identifying a "support vector," essentially a subset of data points crucial for accurate predictions.The Support Vector Regressor aims to create a model with a margin of tolerance on either side of the predicted values by establishing a hyperplane that maximizes the margin between the predicted values and the line.Some points are considered support vectors, indicating their significance in determining the position of this line.SVR is particularly useful when dealing with data that doesn't follow a simple linear pattern, requiring a more flexible approach to capture underlying relationships.This training and testing is 70% and 30%, respectively.The pandas and numpy packages are also utilized for data processing.The hyperparameters in the models are determined using a ten-fold cross-validation (CV) scheme with the GridSearchCV library.

Forward and Backward Wave Separation Components
In the arterial system, the pressure waveform comprises forward (Pf) and backward (Pb) components.The forward component (Pf) signifies the contribution of forward waves to measured pressure, while the backward component (Pb) represents the contribution of backward waves, resulting in the equation P = Pf + Pb.Pressure wave separation, initially described by Westerhof et al., can be applied to identify the forward and backward components of the measured pressure waveform: Here, Q is the measured flow profile, and  , is the characteristic impedance, representing the intrinsic relationship between pressure and flow when waves travel in one direction only.It can be estimated from measured pressure and flow signals historically achieved in the frequency domain (via Fourier transform).This involves calculating the average ratio of pressure and flow harmonics within a specific frequency range, such as high frequency with the result appearing relatively insensitive to the chosen range (Dujardin and Stone, 1981).Following the computation of forward and backward waves, the maximum amplitude of these waves can define other wave separation parameters.The reflection index (RI) is a significant parameter, defined as: To compute the time delay between the forward and backward waves, half the time of the maximum cross-correlation between Pf and Pb was utilized, following the works of Qasem and Avolio.

Supplemental Figures
Figure S1 demonstrates scatter and Bland-Altman plots indicating the agreement between the measured and estimated uncalibrated averages of the mean flow profile, revealing the method's ability to capture the shape of the flow profile.

Figure S1
Scatter and Bland-Altman plots indicating the agreement between the measured and estimated uncalibrated averages of the mean flow profile using spectral regression learning method.
2 = 0.39 Figure S2 demonstrates the distributions of the residuals between reference and estimated variables with respect to age.

Figure S2
The distributions of the residuals between reference and estimated variables with respect to age for all four variables derived from wave separation analysis including forward wave amplitude (Pf), backward wave amplitude (Pb), reflection index (RI), and time delay (TD) between forward and backward waves.
model is trained on features derived from the Fourier decomposition of the pressure waveform, as illustrated in the graphical abstract.Initially, the clinical dataset is divided into training and testing data for all regression learning analyses.The models are exclusively trained on the training population based on the Fourier decomposition of the central pressure waveform and central flow waveform.The models are trained with the Fourier modes of the pressure waveforms as input and the Fourier modes of the flow waveforms as output.After training, the testing dataset is used only once to assess the model's accuracy.The pressure waveform in the testing dataset is first decomposed into the Fourier modes and fed into the predictive model.The output of the predictive models, representing the estimated modes of the corresponding flow waveform, is then inverse Fourier transformed into the time domain using the computed modes and the length of the signal.The combination of the predictive model and the inverse Fourier transform is referred to as the spectral regression learning model.To implement this model, Python's scikit-learn (sklearn) and TensorFlow packages are employed for training and testing the algorithms.The data split for