Machine Learning-Based Pulse Wave Analysis for Early Detection of Abdominal Aortic Aneurysms Using In Silico Pulse Waves

An abdominal aortic aneurysm (AAA) is usually asymptomatic until rupture, which is associated with extremely high mortality. Consequently, the early detection of AAAs is of paramount importance in reducing mortality; however, most AAAs are detected by medical imaging only incidentally. The aim of this study was to investigate the feasibility of machine learning-based pulse wave (PW) analysis for the early detection of AAAs using a database of in silico PWs. PWs in the large systemic arteries were simulated using one-dimensional blood flow modelling. A database of in silico PWs representative of subjects (aged 55, 65 and 75 years) with different AAA sizes was created by varying the AAA-related parameters with major impacts on PWs—identified by parameter sensitivity analysis—in an existing database of in silico PWs representative of subjects without AAAs. Then, a machine learning architecture for AAA detection was trained and tested using the new in silico PW database. The parameter sensitivity analysis revealed that the AAA maximum diameter and stiffness of the large systemic arteries were the dominant AAA-related biophysical properties considerably influencing the PWs. However, AAA detection by PW indexes was compromised by other non-AAA related cardiovascular parameters. The proposed machine learning model produced a sensitivity of 86.8 % and a specificity of 86.3 % in early detection of AAA from the photoplethysmogram PW signal measured in the digital artery with added random noise. The number of false positive and negative results increased with increasing age and decreasing AAA size, respectively. These findings suggest that machine learning-based PW analysis is a promising approach for AAA screening using PW signals acquired by wearable devices.


Introduction
An abdominal aortic aneurysm (AAA) is usually defined as the irreversible localized dilatation of the infrarenal abdominal aorta, which is usually asymptomatic until rupture [1]. The morbidity of AAA is significantly higher in men than in women (1.3% to 8.9% vs. 1.0% to 2.2%) and increases as a result of various factors such as tobacco smoking, ageing, and a family history of AAAs [1][2][3]. Given that the rupture of an AAA is often lethal with mortality reaching about 90% [4,5], timely diagnosis and appropriate treatment are crucial for patients with an AAA. Large AAAs in underweight subjects can often be detected by physical examination, but accuracy depends on the examiner's skills and is considerably reduced for obese body habitus and small AAA size [6]. In current clinical practice, AAAs are most often detected as incidental findings of ultrasonography, abdominal computed tomography, or magnetic resonance imaging performed for flow rate and PPG. The baseline subject without an AAA was adapted from the baseline 65-year-old subject derived from the database of in silico PWs developed by Charlton [14], given that it is usually elderly people who suffer from AAA [2,22]. Herein, two biophysical properties of the original model were adapted to make the subjects with and without AAA comparable: (i) the tapered infrarenal abdominal aorta was replaced by a straight tube with a diameter equal to the average value of the original proximal and distal diameters, and (ii) wall viscosity in this segment was removed to improve numerical stability. The baseline subject with an AAA was then obtained by modifying the baseline subject without AAA ( Figure 1). First, the shape of the infrarenal abdominal aorta was transformed from a straight line (i.e., constant diameter) into a cosine curve with the same length and the maximum diameter being set at 30 mm. Second, the stiffness of this segment, which was quantified by the product of elastic modulus and wall thickness (Eh), was decoupled from the diameter and thereby maintained constant.

Modelling Pulse Waves in Baseline Subjects with and without AAAs
One-dimensional blood flow modelling in the lager systemic arteries (see Figure 1) was used to simulate several PW signals: blood pressure, blood flow velocity, blood flow rate and PPG. The baseline subject without an AAA was adapted from the baseline 65year-old subject derived from the database of in silico PWs developed by Charlton [14], given that it is usually elderly people who suffer from AAA [2,22]. Herein, two biophysical properties of the original model were adapted to make the subjects with and without AAA comparable: (i) the tapered infrarenal abdominal aorta was replaced by a straight tube with a diameter equal to the average value of the original proximal and distal diameters, and (ii) wall viscosity in this segment was removed to improve numerical stability. The baseline subject with an AAA was then obtained by modifying the baseline subject without AAA (Figure 1). First, the shape of the infrarenal abdominal aorta was transformed from a straight line (i.e., constant diameter) into a cosine curve with the same length and the maximum diameter being set at 30 mm. Second, the stiffness of this segment, which was quantified by the product of elastic modulus and wall thickness (Eh), was decoupled from the diameter and thereby maintained constant.

Parameter Sensitivity Analysis
Based on the reference model of a subject aged 65 years old with an AAA introduced in Section 2.1, several AAA-related biophysical properties-the type of AAA shape, AAA Figure 1. The one-dimensional model of pulse wave propagation with a normal infrarenal abdominal aorta (left) and the parameter variations considered to simulate an AAA (right). These include (1) type of AAA shape, (2) AAA maximum diameter (D max ), (3) AAA length (L), (4) AAA local stiffness (Eh; i.e., elastic modulus multiplied by wall thickness), and (5) global stiffness of the larger systemic arteries. The model contains the arterial segments making up the larger systemic arteries, an aortic inflow waveform prescribed at the aortic root, and lumped outflow boundary conditions at each terminal segment representing vascular beds.

Parameter Sensitivity Analysis
Based on the reference model of a subject aged 65 years old with an AAA introduced in Section 2.1, several AAA-related biophysical properties-the type of AAA shape, AAA maximum diameter, AAA length, AAA local stiffness and global stiffness of the larger systemic arteries-were varied individually to investigate their influence on the simulated PWs (see Figure 1).
Although AAAs in real patients are usually fusiform and asymmetric [1,21,37], previous model-based studies have simulated AAAs as being axisymmetrical with a geometry that can be described by several types of mathematical functions [33,35]. In the present Symmetry 2021, 13, 804 4 of 18 study, three common types of AAA shapes derived from previous studies were considered: multiple tangential arcs [33] (Shape 1), a cosine curve [35,38] (Shape 2), and a parabolicexponential curve [39] (Shape 3) (see Figure 1). The mathematical functions of the radius (y) against the distance from the vessel inlet (x) for the three types of shapes were respectively expressed by: where d is the radius of the straight infrarenal abdominal aorta without AAA, L is the vessel length and h is the difference between the AAA maximum diameter and d (see Figure 1). The AAA size is usually evaluated by the maximum diameter (D max ) in clinical practice, with D max larger than 30 mm being diagnosed as AAA and D max larger than 50 mm being stratified as a serious AAA [2,40]. Since this study focused on the early detection of AAAs, herein three levels of D max were considered in the parameter sensitivity analysis: 30, 40 and 50 mm. Besides the maximum diameter, the length of an AAA also shows inter-patient difference [18,41], which was considered in the present study by increasing the length of the reference model from 106 to 120 mm (see Figure 1).
Besides these morphological features, the infrarenal abdominal aorta in subjects with an AAA usually has a higher elastic modulus and a thicker vascular wall compared with normal subjects [25,42]. In the model this was simulated by coupling the stiffness Eh of the AAA with the diameter (D, varying along the infrarenal abdominal aorta) using: The values of the empirical constants k 1 , k 2 , and k 3 were taken from a previous study [14]. Variations in AAA local stiffness were described by multiplying the above expression by a cosine function that made the maximum Eh value decrease or increase by 20% (see Figure 1). Moreover, clinical studies have found larger carotid-femoral pulse wave velocities (cf-PWVs) in patients with AAA than in age-matched healthy subjects [43][44][45]. The values of cf-PWV for control and AAA patients reported by three clinical studies are listed in Table 1. From these data a cf-PWV increase of 4.3 m/s was calculated in patients suffering from AAA. This change was considered in the parameter sensitivity analysis by increasing the prescribed desired cf-PWV as described by Charlton et al. [14]. The entire database of in silico PWs created in this study contains three subsets: (i) baseline subset, (ii) increased global stiffness (IGS) subset, and (iii) AAA subset. The first two subsets consist of subjects without AAAs, while the third subset contains subjects with different AAA sizes.
The baseline subset was modelled based on the database developed by Charlton et al. [14], which contains six age groups with 729 subjects in each group. In the present study, only the three groups with the older subjects (55, 65 and 75 years old) were used since AAA usually occurs in the elderly rather than the young [2,22]. The baseline subset was obtained by modifying each of the original 2187 (729 × 3) subjects as described in Section 2.1. The IGS subset (also 2187 subjects) was created from the baseline subset, by increasing the desired cf-PWV by 4.3 m/s as described above (see Table 1).
According to the parameter sensitivity analysis, the AAA-related parameters that strongly influence the simulated PWs were AAA maximum diameter and global stiffness of the systemic arteries (see Section 3.1). Variations in global stiffness, which was quantified by cf-PWV, were already considered in the IGS subset. Accordingly, the AAA subset was created by introducing an AAA with maximum diameters of 30, 40, or 50 mm to each subject of the IGS subset. The resulting AAA subset thereby included 6561 (2187 × 3) subjects with three different AAA sizes. The methodology used to simulate an AAA for each subject followed the steps described in Section 2.1 to obtain the baseline subject with an AAA, except for the calculation of AAA local stiffness. Herein, higher Eh values with cosine distributions were assigned instead of the constant Eh values used in the baseline and IGS subsets.
A literature review provided the elastic modulus, E, and wall thickness, h, of the normal infrarenal abdominal aorta and AAA, as summarized in Table 2. The mean increase in E from normal to AAA was 73.2%. The average h increased by 42.9%, from 1.4 mm (normal condition) to 2 mm (AAA) and was found to be independent of AAA size [42]. Accordingly, the Eh value of the infrarenal abdominal aorta for the baseline 65-year-old subject in the AAA subset was assigned a cosine distribution with the mean value increasing by 147.43% with respect to the baseline subset. At the two extremes in the vessel, Eh was assigned the same values as for the corresponding subject in the IGS subset. All other 6560 subjects in the AAA subset were prescribed the same distribution of Eh in the infrarenal abdominal aorta as the baseline 65-year-old subject. This distribution was linearly scaled to maintain the value at the two vessel extremes used in the IGS subset.   Figure 2 shows the variations in the model input parameters with age for the entire database. These include the cardiovascular parameters prescribed by Charlton et al. [14] and the AAA-related parameter changes introduce in this study; i.e., the AAA maximum diameter and global stiffness of the systemic arteries (shown in red dots and lines).
Measured by magnetic resonance elastography 96.8% [42] Calculated from the measured pressure and diameter 49.6% [46] 73  Figure 2 shows the variations in the model input parameters with age for the entire database. These include the cardiovascular parameters prescribed by Charlton et al. [14] and the AAA-related parameter changes introduce in this study; i.e., the AAA maximum diameter and global stiffness of the systemic arteries (shown in red dots and lines).  [14] (with parameters shown in black) to simulate increased stiffness and AAAs. Panel (i) shows the PWV as a function of the arterial diameter for the baseline model of each age, in the baseline (black) and IGS/AAA (red) subsets.

Extracting Pulse Wave Indexes
The cf-PWV and the augmentation index (AIx) have been shown to be significantly higher in patients with AAA compared to normal subjects [43][44][45]. Accordingly, for each subject in this study, cf-PWV was calculated from the carotid-femoral pulse transit time (cf-PTT) and corresponding arterial path length, with the cf-PTT measured from the pressure PWs in the carotid and femoral arteries using the foot-to-foot method [14,53]. Wave separation analysis [54] was used to calculate the AIx for each virtual subject, as well as the timing and magnitude of wave reflection. The forward (P f ) and backward (P b ) components of the pressure PW in the ascending aorta were calculated from the pressure (P), flow (Q), and local characteristic impedance (Z c , herein the average value of the modulus of the 3rd to 10th harmonics of the input impedance [55]): The wave reflection magnitude was calculated as the ratio of the amplitude of P b to the amplitude of P f . The time delay (∆T f-b ) between P b and P f was obtained using their zero cross-over as reference points. The AIx was calculated as: with SBP the systolic blood pressure, PP the pulse pressure, and P fb the pressure at the time when the P b adds to the P f (corresponding to the zero cross-over of P b ). The PPG PW in the digital artery of each virtual subject was extracted for further analysis using the mathematical expression [14]: where Q 1D is the inflow to the windkessel segment connecting with the arterial outlet, and Q out is the outflow (see Figure 1).

Recurrent Neural Network
RNN is a machine learning architecture that is widely applied to handle time-series data in which the inputs at each time step include not only the input from the current time step, but also the output from the previous step [56]. For the purpose of improving the accuracy of prediction, bidirectional RNN (BRNN)-an advanced architecture-may be chosen when all time steps of the sequential input data are available [57]. Among the different types of nodes for RNN, the long short-term memory (LSTM) unit is one of the most suitable units for data with long duration, which is able to stop vanishing gradients by keeping track of dependencies between different time steps [58]. Accordingly, BRNN with LSTM was employed to perform the machine learning-based PW analysis in this study. Figure 3 shows a schematic of this machine learning architecture together with the workflow followed to train the machine learning model using the in silico PWs from the new database.  In the present machine learning architecture, first a batch of PWs were imported into the input layer, where the waves with short durations were extended to the duration of the longest wave by filling dummy values (i.e., the largest floating-point number that the system can support) at the end. Subsequently, the masking layer reacted to exclude the dummy values from being considered when the data were processed in the following layers. After the forward and backward layers processed the time-series data, a dense layer with an activation function (herein the sigmoid function) was used to estimate the output result. The forward and backward layers consisted of a series of LSTM units (see Figure  3), with the structure of each unit described as where Hk j − 1 and Hk j are the input and output, respectively, of the kth LSTM unit in the jth layer, and Mk j is the memory cell. U and W are the weights, and b is the bias. The subscripts f, i, and o represent the forget gate, input gate, and output gate, respectively. σ and tanh represent the sigmoid function and tanh function, respectively. The parameters of the present machine learning architecture are summarized in Table 3. This architecture was constructed using the open-source library TensorFlow 2.1 together with the high-level application programming interface Keras. The development of this architecture referred to a previous study as well as the corresponding online repository [59]. In the present machine learning architecture, first a batch of PWs were imported into the input layer, where the waves with short durations were extended to the duration of the longest wave by filling dummy values (i.e., the largest floating-point number that the system can support) at the end. Subsequently, the masking layer reacted to exclude the dummy values from being considered when the data were processed in the following layers. After the forward and backward layers processed the time-series data, a dense layer with an activation function (herein the sigmoid function) was used to estimate the output result. The forward and backward layers consisted of a series of LSTM units (see Figure 3), with the structure of each unit described as where H k j−1 and H k j are the input and output, respectively, of the kth LSTM unit in the jth layer, and M k j is the memory cell. U and W are the weights, and b is the bias. The subscripts f, i, and o represent the forget gate, input gate, and output gate, respectively. σ and tanh represent the sigmoid function and tanh function, respectively. The parameters of the present machine learning architecture are summarized in Table 3. This architecture was constructed using the open-source library TensorFlow 2.1 together with the high-level application programming interface Keras. The development of this architecture referred to a previous study as well as the corresponding online repository [59]. PPG PWs in the digital arteries were extracted from the new database of in silico PWs to train and test the machine learning architecture (see Figure 3). The entire database included 10,935 subjects with (6561) and without (4374) an AAA, randomly split into a training (80%, 8748) and testing (20%, 2187) set. The infrarenal abdominal aorta was assigned a label condition of '0' if an AAA was not present and '1' if an AAA was present. The PPG PWs in the training set together with their corresponding conditions were employed in the training process. Subsequently, the trained machine learning model was tested on the testing set to evaluate its performance in the early detection of AAAs (i.e., classification of the normal (0) and AAA (1) conditions). Moreover, the trained machine learning model was further tested on a modified testing set which consisted of randomly varied PPG PWs obtained by superposing random variations (uniform distribution between −50% and 50%) to the first twenty harmonics of each PW. These variations reduced the similarity of the testing set to the training set and made the PWs more realistic by simulating the presence of measurement errors. Figure 4 shows the simulated pressure PWs in the infrarenal abdominal aorta, and an upstream (ascending aorta) and downstream (femoral artery) vessel with individual variations in the AAA-related parameters described in Section 2.2. The maximum AAA diameter was the dominant morphological feature affecting the PW shape in these three arteries, introducing oscillations to the baseline PW with their amplitudes increasing with the diameter size. In contrast, the type of AAA shape and AAA length only had a small influence on the simulated pulse waveforms (see the first three rows of Figure 4). Arterial stiffness also influenced pulse waveforms (see the last two rows of Figure 4). Although the AAA local stiffness only had a moderate influence, the global stiffness of the larger systemic arteries considerably affected PW morphology. The pulse pressure increased with the stiffening of the vascular wall in all one-dimensional model arteries (global stiffening) under both normal and AAA conditions (see the last row of Figure 4). Under AAA conditions, the increased pulse pressure combined with the oscillations introduced by the localized change in diameter in the infrarenal abdominal aorta. The simulated flow velocity PWs at the same three arterial sites were also mainly affected by changes in maximum AAA diameter and overall stiffness (see Figure 5). However, only the infrarenal abdominal aorta and femoral artery were affected by these changes, since the velocity PW in the ascending aorta was determined primarily by the prescribed inflow waveform at the aortic root and the ascending aorta diameter, and neither was varied to simulate arterial haemodynamics with the presence of an AAA. Furthermore, the amplitude of the velocity PW in the infrarenal abdominal aorta was reduced by more than 50% for all AAA-related parameters, since the AAA increased the luminal cross-sectional area in that vessel without changing the flow rate. The simulated flow velocity PWs at the same three arterial sites were also mainly affected by changes in maximum AAA diameter and overall stiffness (see Figure 5). However, only the infrarenal abdominal aorta and femoral artery were affected by these changes, since the velocity PW in the ascending aorta was determined primarily by the prescribed inflow waveform at the aortic root and the ascending aorta diameter, and neither was varied to simulate arterial haemodynamics with the presence of an AAA. Furthermore, the amplitude of the velocity PW in the infrarenal abdominal aorta was reduced by more than 50% for all AAA-related parameters, since the AAA increased the luminal cross-sectional area in that vessel without changing the flow rate. y 2021, 13, x FOR PEER REVIEW 11 of 18  Figure 6 shows pressure, flow velocity and PPG PWs for the baseline 65-year-old subjects in the IGS and AAA subsets at various measurement sites. The presence of an AAA introduced oscillations to the PWs in the infrarenal abdominal aorta (where the AAA is located), as well as in other upstream and downstream arteries. The amplitudes of these oscillations increased with increasing maximum AAA diameter. Further comparison of the PPG PWs in the digital artery in the frequency domain revealed increasing discrepancies between the magnitudes of the 5th and the 6th harmonics with the increasing maximum AAA diameter. These findings suggest that PWs measured in peripheral sites, such as the digital artery in the finger, could be used to detect the presence of an AAA in clinical practice.

Effects of AAA on Pulse Waveforms and Comparison with the Literature
According to in vivo measurements reported by previous studies, the flow velocity PW in a normal infrarenal abdominal aorta has a triphasic waveform due to the highly resistive blood flow, while the presence of an AAA leads to a much slower and less resistive waveform [8]. These results are qualitatively comparable with the simulated velocity PWs in the middle of the infrarenal abdominal aorta shown in Figure 6. Moreover, in vitro measurements in a hydraulic bench model showed similar trends to the simulated results of the present study, such as the oscillations in the pressure PW in the ascending aorta in  Figure 6 shows pressure, flow velocity and PPG PWs for the baseline 65-year-old subjects in the IGS and AAA subsets at various measurement sites. The presence of an AAA introduced oscillations to the PWs in the infrarenal abdominal aorta (where the AAA is located), as well as in other upstream and downstream arteries. The amplitudes of these oscillations increased with increasing maximum AAA diameter. Further comparison of the PPG PWs in the digital artery in the frequency domain revealed increasing discrepancies between the magnitudes of the 5th and the 6th harmonics with the increasing maximum AAA diameter. These findings suggest that PWs measured in peripheral sites, such as the digital artery in the finger, could be used to detect the presence of an AAA in clinical practice.

Effects of AAA on Pulse Waveforms and Comparison with the Literature
According to in vivo measurements reported by previous studies, the flow velocity PW in a normal infrarenal abdominal aorta has a triphasic waveform due to the highly resistive blood flow, while the presence of an AAA leads to a much slower and less resistive waveform [8]. These results are qualitatively comparable with the simulated velocity PWs in the middle of the infrarenal abdominal aorta shown in Figure 6. Moreover, in vitro measurements in a hydraulic bench model showed similar trends to the simulated results of the present study, such as the oscillations in the pressure PW in the ascending aorta in the presence of an AAA, and the increased magnitude and oscillations in the velocity PW at the inlet of the AAA [36].   Figure 7 compares PW indexes obtained from pressure measurements in the ascending aorta (except for cf-PWV and digital PPG indexes) under normal conditions (baseline and IGS subsets) and with an AAA (AAA subset). These indexes were selected because they have been shown to differ in subjects with and without AAA in clinical studies [43][44][45]. The mean blood pressure (MBP) in the ascending aorta was similar between the two conditions, but the systolic blood pressure (SBP) increased in the IGS and AAA subsets and the diastolic blood pressure (DBP) decreased due to arterial wall stiffening increasing pulse pressure (Figure 7a-c), as well as carotid-femoral pulse wave velocity (Figure 7g). Moreover, although the wave reflection magnitude (Pb/Pf) was similar between the IGS and AAA subsets, the wave reflection time (ΔTf-b) increased significantly in the AAA condition (Figure 7d,e). The AIx did not show a clear trend, though it was considerably higher in the AAA subset with the largest AAA diameters than in the baseline subset (Figure 7f).

Comparison of Pulse Wave Indexes Extracted from the Pulse Wave Database
Indexes calculated from the peripheral PW were also considered in this study. The differences between the magnitudes of the 5th and the 6th harmonics of the PPG PWs in the digital artery, which showed a significant increase with the presence of an AAA in the baseline 65-year-old subject (see Figure 6, last plot), were compared among different subsets. As shown in Figure 7h,i, the mean value of the differences in magnitude were similar among different subsets, but the standard deviations of the AAA subset were relatively larger, indicating a wide range of values in the AAA subset. This result suggests that the difference between the magnitudes of the 5th and the 6th harmonics of the digital PPG PW (as well as the PW itself) was strongly influenced by various cardiovascular parameters in addition to the AAA size. Therefore, the detection of an AAA by individual analysis of PPG PW indexes is a challenging task.  Figure 7 compares PW indexes obtained from pressure measurements in the ascending aorta (except for cf-PWV and digital PPG indexes) under normal conditions (baseline and IGS subsets) and with an AAA (AAA subset). These indexes were selected because they have been shown to differ in subjects with and without AAA in clinical studies [43][44][45]. The mean blood pressure (MBP) in the ascending aorta was similar between the two conditions, but the systolic blood pressure (SBP) increased in the IGS and AAA subsets and the diastolic blood pressure (DBP) decreased due to arterial wall stiffening increasing pulse pressure (Figure 7a-c), as well as carotid-femoral pulse wave velocity (Figure 7g). Moreover, although the wave reflection magnitude (P b /P f ) was similar between the IGS and AAA subsets, the wave reflection time (∆T f-b ) increased significantly in the AAA condition (Figure 7d,e). The AIx did not show a clear trend, though it was considerably higher in the AAA subset with the largest AAA diameters than in the baseline subset (Figure 7f).

Comparison of Pulse Wave Indexes Extracted from the Pulse Wave Database
Indexes calculated from the peripheral PW were also considered in this study. The differences between the magnitudes of the 5th and the 6th harmonics of the PPG PWs in the digital artery, which showed a significant increase with the presence of an AAA in the baseline 65-year-old subject (see Figure 6, last plot), were compared among different subsets. As shown in Figure 7h,i, the mean value of the differences in magnitude were similar among different subsets, but the standard deviations of the AAA subset were relatively larger, indicating a wide range of values in the AAA subset. This result suggests that the difference between the magnitudes of the 5th and the 6th harmonics of the digital PPG PW (as well as the PW itself) was strongly influenced by various cardiovascular parameters in addition to the AAA size. Therefore, the detection of an AAA by individual analysis of PPG PW indexes is a challenging task. y 2021, 13, x FOR PEER REVIEW 13 of 18  Figure 8 shows the confusion matrices and receiver operator characteristic (ROC) curves of the AAA prediction using both the original testing set (top) and the modified testing set with randomly varied PPG PWs (bottom). The trained machine learning model showed near to perfect performance when using the original set. The area under the curve (AUC) was approximately equal to 1, and almost all subjects were classified correctly (see top panels in Figure 8). When the modified testing set was employed (Figure 8, bottom), the resulting AUC could reach 0.928, suggesting that the trained machine learning model was competent in classifying normal and AAA conditions using digital PPG PWs that had less resemblance to the PWs of the training set.

AAA Early Detection Using Machine Learning
Further analysis of the type of subjects in each condition of the confusion matrix revealed that misdiagnosis (i.e., false positive) usually occurred in older subjects (age 75: 89%, when using the original testing set; 53%, when using the modified testing set), while younger subjects with relatively smaller AAA were more likely to suffer from missed diagnosis (i.e., false negative) (see central column in Figure 8). Nonetheless, the ratio of misdiagnosis and missed diagnosis were overall low in the predicted results using both the original and modified testing sets (misdiagnosis: 9/853 (original), 117/853 (modified); missed diagnosis: 13/1334 (original), 176/1334 (modified)). These findings suggest that the machine learning model trained with a database of in silico PWs could be used for the early detection of AAAs in real subjects. augmentation index (f); cf-PWV: carotid-femoral pulse wave velocity (g); difference between the magnitudes of the 5th and the 6th harmonics of the digital PPG pulse wave (h), and the same difference normalized by the magnitude of the 1st harmonic (i). Dots indicate mean values and error bars represent standard deviations. All indexes were calculated in the ascending aorta except for cf-PWV and digital PPG indexes. Figure 8 shows the confusion matrices and receiver operator characteristic (ROC) curves of the AAA prediction using both the original testing set (top) and the modified testing set with randomly varied PPG PWs (bottom). The trained machine learning model showed near to perfect performance when using the original set. The area under the curve (AUC) was approximately equal to 1, and almost all subjects were classified correctly (see top panels in Figure 8). When the modified testing set was employed (Figure 8, bottom), the resulting AUC could reach 0.928, suggesting that the trained machine learning model was competent in classifying normal and AAA conditions using digital PPG PWs that had less resemblance to the PWs of the training set.

AAA Early Detection Using Machine Learning
Further analysis of the type of subjects in each condition of the confusion matrix revealed that misdiagnosis (i.e., false positive) usually occurred in older subjects (age 75: 89%, when using the original testing set; 53%, when using the modified testing set), while younger subjects with relatively smaller AAA were more likely to suffer from missed diagnosis (i.e., false negative) (see central column in Figure 8). Nonetheless, the ratio of misdiagnosis and missed diagnosis were overall low in the predicted results using both the original and modified testing sets (misdiagnosis: 9/853 (original), 117/853 (modified); missed diagnosis: 13/1334 (original), 176/1334 (modified)). These findings suggest that the machine learning model trained with a database of in silico PWs could be used for the early detection of AAAs in real subjects. y 2021, 13, x FOR PEER REVIEW 14 of 18

Discussion
We have provided a proof of concept for the feasibility of early detection of AAAs by machine learning-based pulse wave (PW) analysis. Given the high mortality of AAA rupture, early detection of AAAs is crucial for an effective treatment to reduce the risk of rupture. AAAs are often detected as incidental findings of medical imaging assessments which are costly and can only be performed at specialised centres. Analysis of PPG PWs, on the other hand, offers the possibility to screen the wider population, since these signals can be acquired by a wide range of ubiquitous devices such as smartphones, tablets, and fitness devices. Using a newly created database of in silico PWs in subjects with and without AAAs, we have demonstrated that AAA detection using individual PW indexes is a challenging task, but machine learning-based PW analysis is a more promising approach. By using a recurrent neural network, we have obtained a sensitivity of 86.8% and a specificity of 86.3% in AAA detection when tested in a modified dataset representative of a sample of subjects with and without AAAs with random noise added to their PPG PWs.
Our results also suggest that PW indexes may not be sufficient to detect the presence of an AAA. However, we have not conducted a comprehensive study involving all available PW indexes and the different methods for calculating them. For example, the AIx was only calculated using wave separation analysis from pressure and flow waveforms in the ascending aorta, but the calculation using the second or fourth derivative of the pressure waveform may be more convenient and improve the results of AAA detection [54]. In addition, the combination of multiple PW indexes may improve the ability to detect the presence of an AAA, which needs further research.
The database of in silico PWs was created using a physics-based computational blood flow model, which allowed us to study the interpretability of the machine learning results. The simulated AAAs in the virtual subjects of the database covered a range of AAAfrom small to moderate-which enabled evaluation of the ability of the machine learning model to predict the presence of an AAA in its early stage of development. The size of an AAA is usually quantified by the maximum diameter which, according to the results of our parameter sensitivity analysis, is the major morphological factor influencing PWs.

Discussion
We have provided a proof of concept for the feasibility of early detection of AAAs by machine learning-based pulse wave (PW) analysis. Given the high mortality of AAA rupture, early detection of AAAs is crucial for an effective treatment to reduce the risk of rupture. AAAs are often detected as incidental findings of medical imaging assessments which are costly and can only be performed at specialised centres. Analysis of PPG PWs, on the other hand, offers the possibility to screen the wider population, since these signals can be acquired by a wide range of ubiquitous devices such as smartphones, tablets, and fitness devices. Using a newly created database of in silico PWs in subjects with and without AAAs, we have demonstrated that AAA detection using individual PW indexes is a challenging task, but machine learning-based PW analysis is a more promising approach. By using a recurrent neural network, we have obtained a sensitivity of 86.8% and a specificity of 86.3% in AAA detection when tested in a modified dataset representative of a sample of subjects with and without AAAs with random noise added to their PPG PWs.
Our results also suggest that PW indexes may not be sufficient to detect the presence of an AAA. However, we have not conducted a comprehensive study involving all available PW indexes and the different methods for calculating them. For example, the AIx was only calculated using wave separation analysis from pressure and flow waveforms in the ascending aorta, but the calculation using the second or fourth derivative of the pressure waveform may be more convenient and improve the results of AAA detection [54]. In addition, the combination of multiple PW indexes may improve the ability to detect the presence of an AAA, which needs further research.
The database of in silico PWs was created using a physics-based computational blood flow model, which allowed us to study the interpretability of the machine learning results. The simulated AAAs in the virtual subjects of the database covered a range of AAA-from small to moderate-which enabled evaluation of the ability of the machine learning model to predict the presence of an AAA in its early stage of development. The size of an AAA is usually quantified by the maximum diameter which, according to the results of our parameter sensitivity analysis, is the major morphological factor influencing PWs. This finding is consistent with the fact that many computational model-based studies simulating PWs in the presence of AAAs have varied the AAA maximum diameter [10,33,36]. In addition, the virtual subjects in our new database had their global stiffness increased to make the simulated PWs more realistic. Many clinical studies have observed an increase in the overall stiffness of patients with an AAA [43][44][45], although this has not been considered in previous computational model-based studies investigating the influence of AAAs on PW propagation [10,[33][34][35][36]. Our parameter sensitivity analysis has suggested that the global stiffness of large systemic arteries has a considerable effect on the discrepancies of the PWs under normal and AAA conditions, highlighting the importance of simulating the effect of increased global stiffness with AAAs. Moreover, the false positive and false negative results of the machine learning prediction were closely related to the age of the subject and AAA size, suggesting that these two factors should be considered to optimise the predictive power of machine learning algorithms. Lastly, although the PPG PW was used in the training and testing of the machine learning architecture used in this study, PWs measured along the arteries in the arm were similarly affected by the presence of an AAA. As a result, it is expected that any of these PWs could be used for AAA detection by machine learning-based PW analysis, although the PPG PW at the digital artery or radial artery may be easier to acquire in real subjects.
The findings of this study must be considered in the context of certain limitations. Firstly, the computational model used in the present study is a one-dimensional model rather than a three-dimensional model that should be more accurate in simulating blood flow in the presence of an AAA. Nonetheless, a recent study has demonstrated that onedimensional modelling of the large arteries is accurate compared to three-dimensional modelling to simulate PW propagation across AAA [60]. Secondly, in patients with AAAs, calcification and intraluminal thrombus usually occur within the AAA, which change the stiffness (both the elastic modulus and wall thickness) of the AAA and significantly influence the distribution of AAA wall stresses [61][62][63]. However, one-dimensional modelling is not able to directly simulate calcification and intraluminal thrombus. In this study, the effects of calcification and intraluminal thrombus were described by increasing the local stiffness of AAAs, which was also the approach adopted by previous studies using one-dimensional modelling [10,35,36]. Lastly, although the variations of many influential cardiovascular parameters were considered to generate the present in silico PW database, the conditions of real subjects were not fully described since they are more complicated and not so systematic. The testing set was derived from the in silico PW database and in vivo measurements were not used, making our study a proof-of-concept. Further research is warranted to test the machine learning model trained in this study using PPG PWs measured in real subjects.
A well-trained and tested machine learning model would be a very useful mathematical tool for the early detection of AAAs by pulse wave analysis. The proliferation of commercial wearable devices that can accurately measure PPG signals on the wrist or finger offers an opportunity to screen the wider population for AAAs in daily life. A diagnosis of AAA should ultimately be confirmed by a medical imaging examination. However, AAA detection by PPG PW analysis may help improve AAA diagnosis by identifying those subjects with a potential AAA, thus paving the way for more effective treatment that reduces the current mortality rates.

Data Availability Statement:
The data presented in this study are available in haemod.uk/aaa.