Non-invasive Pressure-only Aortic Wave Intensity Evaluation Using Hybrid Fourier Decomposition-Machine Learning Approach

Abstract

Objective:

The clinical significance of the wave intensity (WI) analysis for the diagnosis and prognosis of the cardiovascular and cerebrovascular diseases is well-established. However, this method has not been fully translated into clinical practice. From practical point of view, the main limitation of WI method is the need for concurrent measurements of both pressure and flow waveforms. To overcome this limitation, we developed a Fourier-based machine learning (F-ML) approach to evaluate WI using only the pressure waveform measurement.

Methods:

Tonometry recordings of the carotid pressure and ultrasound measurements for the aortic flow waveforms from the Framingham Heart Study (2640 individuals; 55% women) were used for developing the F-ML model and the blind testing.

Results:

Method-derived estimates are significantly correlated for the first and second forward wave peak amplitudes (Wf1, r=0.88, p<0.05; Wf2, r=0.84, p<0.05) and the corresponding peak times (Wf1, r=0.80, p¡0.05; Wf2, r=0.97, p<0.05). For backward components of WI (Wb1), F-ML estimates correlated strongly for the amplitude (r=0.71, p<0.05) and moderately for the peak time (r=0.60, p<0.05). The results show that the pressure-only F-ML model significantly outperforms the analytical pressure-only approach based on the reservoir model. In all cases, the Bland-Altman analysis shows negligible bias in the estimations.

Conclusion:

The proposed pressure-only F-ML approach provides accurate estimates for WI parameters.

Significance:

The pressure only F-ML approach introduced in this work expand the clinical usage of WI into inexpensive and non-invasive settings such as wearable telemedicine.

I. Introduction

There has been an emerging interest in employing wave energy-based approaches for better understanding the cardiovascular function in healthy and diseased conditions. The potential clinical application can range from identifying the therapeutic targets for vascular-related neurodegenerative diseases [1], [2] to hemodynamic analysis of new assist devices for cardiovascular disease patients [3]–[5]. Variety of new medical technologies have been developed in recent years that utilize wave analysis indices to uncover clinical insights affordably and non-invasively [6]–[8]. Wave Intensity (WI) is one such wave energy-based index representing the amount of energy carried by arterial waves generated by the left ventricle [9]. Wave Intensity Analysis (WIA) was introduced by Parker and Jones [10] as a time domain method for hemodynamic wave analysis. In this method, WI is computed as the product of the blood pressure and the velocity changes during short time intervals. The typical pattern of WI consists of a large amplitude forward (positive) peak Wf1 (corresponding to the initial compression caused by a left ventricular contraction), followed by a small amplitude backward (negative) peak Wb1 (corresponding to the reflection of the initial contraction), and finally followed by a moderate amplitude forward decompression wave Wf2 (in protodiastole) [11]. Previous studies have shown the applicability of WIA in the understanding of arterial physiology and pathophysiology in the systemic and pulmonary circulation [12]–[14]. In addition, recent population-based clinical study done by Chiesa et al. [15] has demonstrated that elevated carotid WI, captured in Wf1 amplitudes, predicts faster cognitive decline in long-term follow-ups independently of other cardiovascular risk factors.

While WIA has proved an increasingly valuable approach to understanding the hemodynamics of normal and pathophysiological states, the current requirement of having concurrent measurements of pressure and flow is a limitation for the application of WIA in routine clinical practice [11], [16]. In particular, measuring the aortic flow in the clinical setting can be relatively expensive due to the requirement of aortic flow measurement systems (e.g., echocardiogram) and also time consuming with the need for trained personnel. Recently, Hughes et al. [17] proposed an estimation method for WIA utilizing pressure measurements alone, rather than together with measurements of flow. This method is based on the reservoir model of hemodynamic waves in the cardiovascular system [18], [19]. In this model, the aortic blood pressure is separated into components representing reservoir and excess pressures, where the excess pressure has been shown clinically to be a surrogate of the left ventricular outflow tract flow velocity waveform [20]. The pressure-only estimate demonstrated adequate feasibility and reproducibility in a small dataset (i.e., N=34 healthy participants) [17]. The accuracy of this method is further examined by Aghilinejad et al. [21] on the clincial population-based of Framingham Heart study. They showed that implementing the reservoir-based method on carotid waveform with the assumption of constant peak velocity (1m/s to calibrate excess pressure-based velocity waveform) can outperform the typical use of radial waveform. However, it was shown that the pressure-only WI estimate based on reservoir analysis is not able to capture the backward WI components (amplitudes and timings of Wb1) [21]. In addition, the general agreement of the estimated forward WI components (i.e., Wf1 and Wf2) with the exact one (from pressure and flow) was not strong and there was a bias in the previously proposed estimation method which reduces the generalizability of this technique. Therefore, there is an essential need for introducing a reliable pressure-only estimate of WI to expand the clinical applicability of this method.

The objective of this study is to employ a hybrid Fourier decomposition and machine learning (ML) approach, to estimate WI metrics using only pressure waveform measurement. We used the large heterogeneous population of the Framingham Offspring Cohort to conduct a rigorous analysis of the accuracy of the proposed Fourier-based ML (F-ML) model to estimate pressure-only WI. We also tested the generalizability of this approach using different F-ML algorithms. The systematic work conducted here also lays the groundwork for wider application of F-ML models to estimate energy-based indices such as WI that can provide valuable insights of the cardiovascular system.

II. MATERIALS AND METHODS

A. Participants and Data

We used a subgroup of Framingham Heart Study (FHS) data, a population-based epidemiological cohort analysis in this study. The sample was drawn from the eight-examination cycle of an Offspring cohort which has been previously described [22], [23]. The characteristics of these participants are presented in Table I. The initial population represented a heterogeneous cohort of N=2640 participants including 1201 males and 1439 females with the age range from 40 to 91 years old. This population included 279 patients with cardiovascular-related diseases (defined as myocardial infarction, coronary insufficiency, stroke, heart failure [24]), 177 with valvular diseases and 32 with congestive heart failure (defined based on FHS criteria [25]). All participants provided written informed consent, and the protocols were approved by the Boston University Medical Campus and Boston Medical Center Institutional Review Board. The sample participants all underwent comprehensive and non-invasive assessment of central hemodynamics, providing an extensive collection of tonometry recordings of carotid pressure waveforms [22]. The aortic flow waveforms were obtained via Two-dimensional echocardiography of the left ventricular outflow tract (LVOT) followed by a pulsed Doppler from an apical 5-chamber view to acquire the aortic velocity waveform [2], [22].

TABLE I.

Baseline Characteristics of Patient Data (N = 2640)

Variable	Value
Clinical measures
Age, y	66 ± 9
Women, n (%)	1439 (55)
Height, cm	167 ± 10
Weight, kg	78 ± 17
Body mass index, kg/m2	27.9 ± 5.1
Heart rate, bpm	62 ± 10
Brachial blood pressure, mmHg
Systolic	141 ± 20
Diastolic	69 ± 9
Pulse	72 ± 19
Hypertension, n (%)	978 (37)
Diabetes mellitus, n (%)	229 (9)
Valve Disease, n (%)	177 (5)
Arrythmia, n (%)	182 (5)

All values are (mean ± SD) except as noted.

B. Wave Intensity Analysis

WI is defined as the power per unit cross-sectional area $A$, of an artery due to blood pressure $P=P\left(t\right)$ and average cross-sectional blood flow velocity $U=U\left(t\right)$. The principles and derivation of WI analysis, which enables a decomposition of arterial waves into forward and backward components, have been previously described in detail [9]–[11]. Mathematically speaking, WI is computed as the product of the change in pressure ($dP$) times the change in velocity ($dU$) during a small interval, given by

$$dI=dP\cdot dU.$$ (1)

To remove the dependency of the WI on the sampling time, the increments of pressure and velocity are divided by the time interval, hence the WI in the units of power per unit area per unit time squared (W.s⁻².m⁻²) [11]. WI patterns determine both the direction and intensity of arterial wave propagation at any time instance during a cardiac cycle. For example, if $dI>0$ at a fixed time during the cardiac cycle, forward waves that largely originate from the left ventricle will dominate at that moment. Conversely, if $dI<0$, backward waves (mostly related to wave reflections [11]) will dominate. As can be noted from (1), simultaneous measurements of both pressure and flow (velocity) are needed to determine values of WI which is the typical requirement for the energy-based indices [26]–[28]. Fig. 1 illustrates a typical pattern of WI based on the central blood pressure (top right corner of the figure) where the three major wave peaks (chosen for comparison in this study) are labeled.

Fig. 1.

Schematic of the computation of exact WI and pressure-only estimate of WI based on the proposed Fourier-based ML (F-ML) model. The pressure wave (blue) and the flow wave (red) computed from cross-sectional averaged velocity are required to compute exact WI. In proposed method, only the pressure waveform (blue) is needed to compute WI. Figure also illustrates the three major peaks of WI and the F-ML pipeline.

C. Fourier Representation and Feature Selection

In this study, we utilized Fourier series decomposition for input feature selection (Fig. 1). Originally, the carotid pressure waveforms from tonometry measurements are sampled at the rate of 1000Hz leading to 1000 data points per cycle for single pressure measurement for a typical heart rate of 60 beats per minute. This high dimensionality of the input signal renders ML constructs that are significantly limited for practical data-driven applications [29]. In this work, to transform data from this high-dimensional space to a low-dimensional one, we utilized Fourier-based analysis to retain physical properties of the original data. The Fourier series represents a synthesis of a periodic function by summing harmonically related sinusoids and cosinusoides. An arbitrary periodic pressure function $P=P\left(t\right)$ can be represented as a Fourier series with $N$ oscillatory components. A common form of the Fourier series decomposition, in the Sine-Cosine form is defined as

$$P\left(t\right)=a_0/2+\sum _{\text{n}=1}^{\text{N}}\left(a_n\text{cos}(2\pi nt/T)+b_n\text{sin}(2\pi nt/T)\right),$$ (2)

where $T$ is the period of the pressure function $P=P\left(t\right)$ (i.e., the cardiac cycle or inverse of HR for blood pressure waveform). Here, for conducting the Fourier decomposition, we used the Sine-Cosine form rather than an Amplitude-Phase form to avoid introducing non-linear relationships between ML inputs (i.e., phase inside the Sine or Cosine function). Coefficients $a_n$ and $b_n$ are associated with each individual harmonics (cosine and sine) corresponding to different frequencies $f_n=n/T$, and can be calculated by the Fourier transform given by

$$a_n=2/T{\int }_0^{T}P\left(t\right)\text{cos}(2\pi nt/T)dt,$$ (3)

$$b_n=2/T{\int }_0^{T}P\left(t\right)\text{sin}(2\pi nt/T)dt,n\in (0,\mathrm{\infty }).$$ (4)

Obtaining the coefficients using (3) and (4), the represented pressure waveform $\tilde{\text{P}}$ with finite selected frequency (i.e., $n=0$ to $N$) is achieved by adding up each individual frequency component (2). In this study, the features of the pressure wave were extracted as the first $N$ low frequency components of the waveforms using the Fast Fourier Transform (FFT). Fig. 2 represents the associated error between the clinically-measured pressure waveforms and the reconstructed pressure waveform based on the number of Fourier coefficients. The outliers of the data are identified using the convention of considering four standard deviations from the mean, leading to 263 outliers. Note that after removing the outliers, the mean and the standard deviation is recalculated and then used in the ML computations. As it can be noticed from this figure, the error between the measured pressure and the reconstructed one is marginal after $N=10$. The selected input features in this study (depending on the chosen number of Fourier modes; FN) consists of FN cosine coefficients, (FN - 1) sine coefficients, cardiac time and the notch time, which lead to the input size of (2FN + 1). Fig. S1 also represents the errors between the first derivatives of the reconstructed and the measured pressure waveforms. Sample-reconstructed first derivative of pressure overlaid with measured values is demonstrated in Fig. S2.

Fig. 2.

Associated error between the reconstructed pressure waveform based on a different number of Fourier modes and the measured pressure waveform. The error is reported as the normalized root mean square error (NRMSE) averaged over the whole population.

D. Machine Learning Models

In the present study, we employed six well-established ML algorithms to evaluate the accuracy of the proposed Fourier-based method on WI estimation. These models include Lasso regressor, kernel ridge regressor (KRR), support vector regressor (SVR), random forest regressor, gradient boosted decision-tree, and neural network. Each one of these algorithms is trained on features derived from the Fourier decomposition of the pressure waveform, notch time and cardiac time. Fig. S3 presents the Pearson correlation matrix reporting the interfeature correlations for five Fourier modes (FN=5; input size=11).

The training and testing data split for all F-ML analyses was 70% and 30% respectively. Models are strictly trained on the training population and the test data are only used at the last step for model evaluation. Python’s sklearn and TensorFlow packages are used for data pre-processing (e.g., normalization), training, and testing the algorithms. The pandas and numpy packages were also used for data processing. The hyperparameters in the models were found by a ten-fold cross validation (CV) scheme using the GridSearchCV library. Fig. S4 presents the sample grid search of hyperparameters including the maximum depth, samples required to split, and number of estimators for the random forest regressor. As demonstrated in this figure, the hyperparameters are chosen in a uniform grid with the large range. The comprehensive list of examined hyperparameters for different F-ML models is demonstrated in Table S1 in the supplementary material. The hyperparameters’ values that are not reported in Table S1 were set to their default value. Particularly for neural networks, the “Adam” optimizer was chosen to optimize the weights of neurons in the hidden and output layer, and the number of epochs was set to be equal to 1500. For all models, the combination of hyperparameters attributed to the highest accuracy for training is chosen. All mathematical analysis of the clinical data was performed using custom written codes and algorithms implemented in Python ((Python Software Foundation, Python Language Reference, version 3.9).

E. Statistical Analysis

Baseline characteristics for the study sample were demonstrated in Table I, and continuous variables derived from the sample data were summarized as mean ± standard deviation (SD). Two forward waves (Wf1 and Wf2) and a backward-running wave (Wb1) from the pattern of WI during one cardiac cycle are chosen to evaluate the performance of the proposed F-ML methodology. We investigated both the peak amplitudes with their corresponding timings. Exact WI values for all cases were determined using central flow measurements together with carotid pressure measurements that served as a surrogate for central pressure [30], [31]. In order to assess the error of the estimated values with those of exact WI analysis, we evaluated both Pearson correlation coefficients $r$, and root mean square errors (RMSE) of peak values in the patterns. Statistical significance was defined as $p$-values< 0.05. We also reported the Normalized RMSE (NRMSE) which is computed based on the range of the dependent variable (difference between the maximum and the minimum). The agreement and the bias between the exact WI variables and estimated ones are evaluated by the Bland-Altman analysis presented as mean differences with limits of agreement (mean bias ± 1.96 SD of the differences). Levene’s test for homogeneity of variance is conducted to examine the inter-algorithm differences between F-ML models for evaluating the WI parameters. Lastly, Kruskal-Wallis’ rank-sum test and the Dunn’s test with Bonferroni adjustment are employed for overall and two-by-two comparisons for WI parameters considering non-normal distribution.

III. Results

A. Accuracy of Pressure-only WI Amplitudes

Table II presents the correlation, errors, and agreement between pressure-only estimates of WI with exact WI for the peak amplitudes of Wf1, Wf2, and Wb1. The analysis is conducted for different F-ML models to evaluate the applicability of the Fourier-based decomposition of the pressure waveforms. In this analysis, we utilized the first 20 Fourier modes of the pressure waves, which guarantees the capture of all features of the pressure waveform (see Fig. 2). In all six algorithms, the peak amplitudes are well-correlated, where both forward peaks are significantly correlated, and backward peak is strongly correlated. As demonstrated by the mean difference reported in Table II, the systemic bias between the estimated values and exact WI measurements is negligible.

TABLE II.

Regression statistics and agreement between predicted and exact WI peak amplitudes.

F-ML Algorithm	Lasso	KRR	SVR	Random Forrest	Gradient Boosting	Neural Network
Wf1 Peak Amplitude
Correlation Coefficient (r)	0.83	0.88	0.88	0.83	0.85	0.88
RMSE	36.6	30.8	31.2	38.1	34.5	32.5
Normalized RMSE (%)	8.8	7.4	7.5	9.1	8.3	7.8
Limit of Agreement	143.2	120.7	122.5	150.0	135.6	125.0
Mean Difference	1.6	0.7	−1.1	−0.4	−0.4	1.2
Wf2 Peak Amplitude
Correlation Coefficient (r)	0.75	0.83	0.84	0.77	0.80	0.82
RMSE	13.6	11.3	11.1	13.0	12.2	11.8
Normalized RMSE (%)	10.6	8.8	8.7	10.1	9.5	9.2
Limit of Agreement	53.2	44.2	43.8	51.3	48.6	45.2
Mean Difference	−0.5	−0.2	−0.6	0.4	0.3	0.5
Wb1 Peak Amplitude
Correlation Coefficient (r)	0.57	0.69	0.71	0.63	0.62	0.69
RMSE	3.1	2.8	2.7	3.0	2.9	2.9
Normalized RMSE (%)	13.9	12.4	12.0	13.6	13.3	13.0
Limit of Agreement	12.3	10.8	10.5	11.9	11.6	11.3
Mean Difference	0.1	0.1	0.3	−0.2	−0.3	0.2

^*Correlations have $p$-values< 0.05 unless otherwise indicated. All wave intensity amplitudes are reported in units of $\frac{W}{m^2{s}^2}\times {10}^4$. SVR indicates the support vector regressor. KRR indicates the kernel ridge regressor. Underline values show the highest correlations.

Fig. 3 demonstrates the sample of the Bland-Altman plots indicating the agreement between all intra-individual differences in the three major peak amplitudes (Wf1, Wf2, and Wb1) as well as the corresponding scatter plots between exact and method-derived pressure-only WI using neural network.

Fig. 3.

Scatter and Bland-Altman plots for Wf1, Wf2 and Wb1 amplitudes. The plots are demonstrated for the test data ($N=714$). Correlation (top row) and Agreement (bottom row) of peak amplitude values between exact wave intensities and those estimated by F-ML approach using neural network.

B. Accuracy of Pressure-only WI Timings

Table III presents the corresponding correlations, errors, and agreements of estimated WI peak timings for different F-ML algorithms. For reference, scatter plots for pressure-only estimates of WI peak timing versus the exact ones as well as the Bland-Altman plots of the corresponding differences for the neural network model are presented in Fig. 4.

TABLE III.

Regression statistics and agreement between predicted and exact WI peak times.

F-ML Algorithm	Lasso	KRR	SVR	Random Forrest	Gradient Boosting	Neural Network
Wf1 Peak Time
Correlation Coefficient (r)	0.55	0.75	0.75	0.51	0.68	0.80
RMSE	7.3	5.9	6.1	7.7	6.5	5.4
Normalized RMSE (%)	13.0	10.6	10.8	13.7	11.6	9.8
Limit of Agreement	28.5	23.2	23.8	30.1	25.1	20.5
Mean Difference	−0.3	0.1	0.2	0.0	0.0	0.7
Wf2 Peak Time
Correlation Coefficient (r)	0.92	0.97	0.97	0.84	0.92	0.97
RMSE	11.3	7.1	8.1	16.7	11.7	6.8
Normalized RMSE (%)	5.6	3.5	4.0	8.2	5.7	3.3
Limit of Agreement	44.1	27.9	31.9	65.5	43.9	27.0
Mean Difference	1.1	0.4	0.6	−1.1	−0.8	−2.3
Wb1 Peak Time
Correlation Coefficient (r)	0.49	0.59	0.60	0.45	0.55	0.60
RMSE	24.0	22.0	21.9	24.8	23.0	22.2
Normalized RMSE (%)	14.2	13.0	13.0	14.7	13.6	13.1
Limit of Agreement	94	86.2	85.9	97.4	93.1	88.5
Mean Difference	−0.4	0.3	0.5	−0.3	−0.2	−0.5

^*Correlations have $p$-values< 0.05 unless otherwise indicated. All wave intensity timings are reported in units of ms. SVR indicates the support vector regressor. KRR indicates the kernel ridge regressor. Underline values show the highest correlations.

Fig. 4.

Scatter and Bland-Altman plots for Wf1, Wf2 and Wb1 times. The plots are demonstrated for the test data ($N=714$). Correlation (top row) and Agreement (bottom row) of peak timings between exact wave intensities and those estimated by F-ML approach using neural network

C. Inter-algorithm Comparison for the Proposed Method

Fig. 5 demonstrates the distribution of the testing data among different F-ML algorithms. Levene’s test suggests that for all WI variables including the peak amplitudes and timings, the variances for each group are different. $P$-values from Kruskal-Wallis’ test for overall comparison of the algorithms are also reported in this figure. We also conducted the pairwise comparison using Dunn’s test between the models for Wf1 and Wb1 amplitudes and timings. The results of Dunn’s test suggest that for Wf1 and Wb1 timing there is a significant difference between the neural network model and the other algorithms. The results for the pairwise comparison are shown in Table S2 and Table S3 in the supplementary material. Table S2 represents the comparison for Wf1 time and Table S3 shows the comparison for Wb1 time. For Wf1 and Wb1 amplitudes, there is no significant difference between F-ML models except between neural networks and support vector regressor ($p=0.0236$).

Fig. 5.

Boxplots for WI parameter distributions from different F-ML models. The plots are demonstrated for the test data ($N=714$). All wave intensity amplitudes are reported in units of $\frac{W}{m^2{s}^2}\times {10}^4$ and timings are reported in ms. The reported p-values are from the Kruskal-Wallis’ test for overall comparison of the algorithms.

D. Analytical versus F-ML WI

Table IV presents the correlation and agreement between the pressure-only estimates of WI, the one which is proposed in this study (F-ML) and the previous approach based on the reservoir model of the hemodynamic waves (ODE-based analytical model). The comparison is conducted between the peak amplitudes and times of the WI parameters including Wf1, Wf2, and Wb1. For this comparison, we used the support vector regressor since it did not produce significantly different output compared to other algorithms based on the Dunn’s test (see Table S2 and S3 in the supplementary material). NC in this table indicates no correlation.

TABLE IV.

Comparison between the proposed F-ML and the previous analytical model.

Pressure-only WI Method	Wf1 Amplitude	Wf1 Time	Wf2 Amplitude	Wf2 Time	Wb1 Amplitude	Wb1 Time
F-ML
Correlation Coefficient (r)	0.88	0.75	0.84	0.97	0.71	0.60
Limit of Agreement	125	23.8	45.2	31.9	11.3	85.9
Mean Difference	1.2	0.2	0.5	0.6	0.2	0.5
Analytical Method
Correlation Coefficient (r)	0.85	0.94	0.72	0.98	NC	NC
Limit of Agreement	107	13	54	23	42	250
Mean Difference	−29. 8	2	3.1	6	6.6	52

^*Correlations have $p$-values< 0.05 unless otherwise indicated. All wave intensity amplitudes are reported in units of $\frac{W}{m^2{s}^2}\times {10}^4$ and timings are reported in ms. NC indicates no correlation. The data for ODE-based analytical model are adopted from [21].

E. Sensitivity to input Fourier Modes and Training Size

Fig. 6 presents the Pearson correlation coefficient between the estimated WI and the exact one for three major peak amplitudes and timings as a function of different Fourier modes for the support vector regressor. The number of modes represents the size of the input pressure waveform. Results suggest there is an optimum range for choosing the number of Fourier modes (10 to 20 modes for our database) that can result in the highest correlation.

Fig. 6.

Correlations between the estimated WI and the exact one for three major peak amplitudes and timings as a function of different Fourier modes for the testing data.

Fig. 7 demonstrates the RMSE of the estimated WI peak amplitudes and their corresponding timing as a function of the training size for the support vector regressor. For both timing and amplitude, it can be observed that including more data instances to the training sample after reaching the 40% of the entire dataset (corresponding to 951 instances) only has negligible effects on the accuracy of WI estimation.

Fig. 7.

Sensitivity of precision in terms of normalized root mean square (NRMSE) to the number of the training data. The 100% of the training size corresponds to the whole clinical dataset.

IV. DISCUSSION

In this study, we investigated the accuracy of pressure-only estimates of WI (in terms of peak wave amplitudes and timings) derived from a F-ML approach in a cohort of the Framingham Heart study. Our results show that this approach provides accurate estimations of the main features of WI using only non-invasive pressure measurements. This method significantly outperforms the ODE-based models (reservoir method) [17], [21] for pressure-only WI as demonstrated in Table III. In addition, our results suggest that the Fourier-based representation of the pressure wave can be tuned to reduce the input feature size without any considerable influence on the accuracy of the model. This expands the applicability of the F-ML method for smaller datasets.

A. Principal Findings

In the present study, we have comprehensively investigated, for the first time, the accuracy of wave intensities metrics calculated through use of only single pressure waveform measurements using a novel F-ML approach. One major finding of our study is that aortic WI estimated based on the proposed F-ML approach outperforms estimations from an analytical ODE-based method. Table II suggests that forward contributions of WI (amplitudes of Wf1 and Wf2) are strongly correlated with exact WI (maximum correlation of 0.88 for Wf1 and 0.84 for Wf2). As demonstrated in Table III, there is a strong correlation between the peak timings of predicted Wf1 and Wf2 with the exact one (correlations as high as 0.80 and 0.97 respectively). Regarding the Backward component of WI, in contrast to the previous reservoir-based model, which was not able to capture this feature (as shown in Table IV) [21], the proposed method here can capture the amplitude of Wb1 with strong correlation (0.71 corresponding to SVR in Table II) and its timing with a fairly strong correlation (0.60 corresponding to neural network and SVR in Table III). Table II and Table III reports the agreement between the method-derived estimations of WI (both amplitude and timing) with the exact one and suggest there is a negligible systemic bias between the two. The agreement can be also observed from the sample Bland-Altman plots demonstrated for three peak amplitudes demonstrated in Fig. 3 and three peak times demonstrated in Fig. 4. This is another advantage of the utilized methodology for estimating WI compared to the previous approach as shown in Table IV.

Our results also suggest that choosing the number of Fourier modes to decompose the input pressure waveform to the machine has a substantial impact on the correlation and the error of predicted WI (Fig. 6). Depending on the chosen number of Fourier modes, some information about the pressure waveform may be missed as shown in Fig. 2 (e.g., the exact shape of the notch). However, to capture the WI peaks, these higher frequency features may not be necessarily required and hence it is essential to find the optimum number of Fourier modes. This is particularly important for reduced-order approaches for ML modeling [29]. Results suggest that there is an optimum number of Fourier modes to include as feature dimension for training purposes. Findings show that 10 to 20 modes of Fourier result in the highest correlation based on the utilized population in this study. The presence of an optimum range for choosing the number of Fourier modes is related to the issue of overfitting and underfitting. Employing a very small number of Fourier modes can lead to the inability of input features to sufficiently represent the pressure waveform for the purpose of capturing WI; therefore, the model does not get trained well (underfit). On the other hand, employing too many modes increases the input size and makes the model specific to the training set, ultimately performing less well on never-seen-before data (overfit). Therefore, it is essential to examine the number of Fourier modes as an important parameter in our approach.

In the present study, we also investigated the performance of different F-ML algorithms on Fourier decomposed pressure waveforms with the goal of predicting the WI. Recent advances in AI and ML along with the availability of large clinical datasets bring new research possibilities and approaches to cardiovascular engineering [32]. As examples, Bikia et al. [33]–[35] demonstrated the potential applicability of ML-based methodology for predicting aortic hemodynamics and cardiac contractility for the goal of non-invasive monitoring. They also proposed a methodology that facilitates the clinical use of end-systolic elastance for monitoring the contractile state of the heart in a real-life setting [36]. Tavallali et al. [37] showed the applicability of the regression analysis in estimating carotid-femoral pulse wave velocity which is the gold standard measurement of vascular aging. More recently, Jin et al. [32] proposed a machine learning pipeline to estimate carotid-femoral pulse wave velocity using single peripheral pulse wave in 3082 subjects with ages ranging from 18 to 110 years. Machine learning has been also adopted by Jones. et al. [38] as well as by Wang et al. [39] for early detection of aneurysms. Following the above, this study is in line with the spirit of introducing AI and ML methods into the field of cardiovascular engineering.

The inter-algorithm comparison between different F-ML models, conducted by Kruskal-Wallis’ and pairwise Dunn’s tests, demonstrated that there is no statistical difference in capturing the output of WI peak amplitudes (demonstrated in Fig. 5 and Table S2 in the supplementary material). While non-linear methods such as SVR or neural network results in correlations as high as 0.88, linear models such as Lasso regression predictor still led to strong correlations in capturing WI amplitudes. This is important since such linear models take advantage of better interpretability compared to more complex and computationally expensive ones such as the neural network. On the other hand, for capturing the timings of the amplitude, complex non-linear models (such as SVR) outperform the rest in general. Results from pairwise Dunn’s tests demonstrated that the only model which is statistically different from the rest in capturing WI peak timings is the neural network as shown in Fig. 5. Overall, a more delicate dependence of WI timing with the shape of the waveform necessitates utilizing more complex models for achieving accurate estimations. Lastly, we examined the model’s sensitivity to the relative training size of the utilized dataset in this study. The training size was modified from 90% to 10% of the total number of cases (Fig. 7). Results suggest that NRMSEs decreased gradually with increasing training size. The improvement in the accuracy however is negligible after the training size reaches 40% of the dataset. This sensitivity analysis suggests that the employed training size for this study is sufficient.

B. Study Limitations and Future Work

One limitation in this study can be related to the lack of invasively-measured aortic pressure waveforms for determining exact WI. However, our choice of using the carotid pressure waveform as a surrogate for aortic is well-established [30], [31], [35]. Future studies are needed to independently assess the ability of the proposed pressure-only WI for diagnosis and prognosis of cardiovascular disease.

V. CONCLUSION

This study included a large heterogeneous sample of individuals in order to examine the accuracy of the pressure-only estimation method of WI in the Framingham Heart study population. This study employed a novel F-ML approach to estimate different WI parameters. The strength of our study includes the use of a single-pressure non-invasive waveform measurement for approximating WI. Contrary to conventional WI methodologies which require measurements of both flow and pressure waveforms, the proposed technique requires only the central arterial pressure waveform to perform the analysis. Our results suggest that the pressure-only estimate of this work has straightforward application for large sample sizes and this technique can ultimately increase the clinical usefulness of WI. The statistical analysis (Table II and Table III) comparing the proposed method with conventional WI (which required both pressure and flow wave measurements) suggest that our method is able to capture the WI peak amplitudes and timings. Our results (Table IV) also show that the proposed method outperform the previous approach based on the reservoir model. Hence, it can be used as a reliable model to compute WI based on single pressure measurement. The pressure-only WI estimations of this work provide an important opportunity to further the goal of uncovering clinical insights through wave analysis affordably and non-invasively.