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

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Modelling Pulse Waves in Baseline Subjects with and without AAAs

#### 2.2. Parameter Sensitivity Analysis

_{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).

_{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).

#### 2.3. Database of In Silico Pulse Waves

#### 2.3.1. Modelling a Database of Pulse Waves in Subjects with and without an AAA

#### 2.3.2. Extracting Pulse Wave Indexes

_{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]):

_{b}to the amplitude of P

_{f}. The time delay (ΔT

_{f-b}) between P

_{b}and P

_{f}was obtained using their zero cross-over as reference points. The AIx was calculated as:

_{fb}the pressure at the time when the P

_{b}adds to the P

_{f}(corresponding to the zero cross-over of P

_{b}).

_{1D}is the inflow to the windkessel segment connecting with the arterial outlet, and Q

_{out}is the outflow (see Figure 1).

#### 2.4. Machine Learning-Based Pulse Wave Analysis

#### 2.4.1. Recurrent Neural Network

_{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].

#### 2.4.2. Training and Testing

## 3. Results

#### 3.1. Parameter Sensitivity Analysis

#### 3.2. Effects of AAA on Pulse Waveforms and Comparison with the Literature

#### 3.3. Comparison of Pulse Wave Indexes Extracted from the Pulse Wave Database

_{b}/P

_{f}) was similar between the IGS and AAA subsets, the wave reflection time (ΔT

_{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).

#### 3.4. AAA Early Detection Using Machine Learning

## 4. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**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.

**Figure 2.**Variations of model parameters used to construct the new database of in silico pulse waves. Panels (

**a**–

**h**) show the variations of heart rate (HR), stroke volume (SV), left ventricular ejection time (LVET), infrarenal abdominal aortic diameter, mean arterial pressure (MAP), peripheral vascular resistance (PVR), AAA maximum diameter, and carotid-femoral pulse wave velocity (PWV), respectively. The AAA maximum diameter and increased PWV values shown in red were added to the original database [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.

**Figure 3.**Schematic of the bidirectional recurrent neural network (BRNN) with long short-term memory (LSTM) used for pulse wave analysis. Simulated PPG pulse waves (PWs) in n = 8748 virtual subjects were used as input data for training a BRNN with LSTM (see Equation (8) for the mathematical expressions of a LSTM unit) to predict the presence (output = 1) or not (output = 0) of an AAA. The BRNN contains input, output and hidden layers, each hidden layer consisting of masking, forward, backward, and dense layers. The forward and backward layers include a series of LSTM units. L and L′ represent the LSTM units in the forward and backward layers, respectively. M and M′ are the corresponding memory cells. X is the input data processed by masking layer for the LSTM units, and H is the output data for the dense layer. PPG PWs are described using discrete values p

_{1}, …, p

_{t}. The subscripts 1, …, t represent each time history point. Details about the symbols in the brown boxes describing LSTM units can be found in Equation (8).

**Figure 4.**Simulated pressure waveforms in the ascending aorta (

**left**), infrarenal abdominal aorta (

**middle**), and femoral artery (

**right**) with individual variations in the AAA-related parameters indicated at the start of each row and illustrated in Figure 1.

**Figure 5.**Simulated flow velocity waveforms in the ascending aorta (

**left**), infrarenal abdominal aorta (

**middle**), and femoral artery (

**right**) with individual variations in the AAA-related parameters indicated at the start of each row and illustrated in Figure 1.

**Figure 6.**Simulated pressure and flow velocity pulse waves at the infrarenal abdominal aorta (labelled in red) and other common measurement sites (labelled in black) for the baseline 65-year-old subject in the IGS subset (without AAA) and AAA subset (D

_{max}= 30 mm, D

_{max}= 40 mm, and D

_{max}= 50 mm). The PPG wave is shown in the digital artery in the time and frequency domains (labelled in blue).

**Figure 7.**Comparison of pulse wave indexes obtained in the baseline and increased global stiffness (IGS) subsets (in black) and the AAA subset with different AAA sizes (in red). SBP: systolic blood pressure (

**a**); DBP: diastolic blood pressure (

**b**); MBP: mean blood pressure (

**c**); P

_{b}/P

_{f}: wave reflection magnitude (

**d**); ΔT

_{f-b}: time delay between P

_{b}and P

_{f}(

**e**); AIx: 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.**Performance of the trained machine learning model for the early detection of AAAs. Test data (

**left**), confusion matrix (

**middle**), and ROC curve (

**right**) when testing in the original (

**top**) and modified (

**bottom**) testing sets. The confusion matrices indicate the number of subjects in each condition together with the proportion of each age and AAA size (if applicable) in brown.

**Table 1.**Carotid-femoral pulse wave velocities measured in control cohorts of healthy subjects and in cohorts of patients with AAA.

Control (Number) | AAA Patient (Number) | Reference |
---|---|---|

10.0 m/s (20) | 14.8 m/s (18) | [43] |

10.03 m/s (42) | 12.99 m/s (108) | [44] |

7.97 m/s (31) | 13.11 m/s (48) | [45] |

9.33 m/s | 13.63 m/s | Mean value |

Source | Value | Reference |
---|---|---|

Growth ratio of elastic modulus from normal to AAA | ||

Measured by magnetic resonance elastography | 96.8% | [42] |

Calculated from the measured pressure and diameter | 49.6% | [46] |

73.2% | Mean value | |

Normal wall thickness | ||

Derived from clinical measurements | 1.4~1.5 mm | [25] |

1.39 mm | [47] | |

1.4 mm | Mean value | |

AAA wall thickness | ||

Derived from clinical measurements | 2 mm | [37] |

1.63 mm | [48] | |

1.48 mm | [49] | |

2.71 mm | [50] | |

2.87 mm | [51] | |

1.64 mm | [52] | |

Used by previous model-based studies | 2 mm | [21,40] |

2 mm | Mean value |

Parameter | Value |
---|---|

Number of LSTM units | 16 |

Batch size | 32 |

Epoch number | 256 |

Optimiser | Adam |

Cost function | Binary cross-entropy |

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**MDPI and ACS Style**

Wang, T.; Jin, W.; Liang, F.; Alastruey, J. Machine Learning-Based Pulse Wave Analysis for Early Detection of Abdominal Aortic Aneurysms Using In Silico Pulse Waves. *Symmetry* **2021**, *13*, 804.
https://doi.org/10.3390/sym13050804

**AMA Style**

Wang T, Jin W, Liang F, Alastruey J. Machine Learning-Based Pulse Wave Analysis for Early Detection of Abdominal Aortic Aneurysms Using In Silico Pulse Waves. *Symmetry*. 2021; 13(5):804.
https://doi.org/10.3390/sym13050804

**Chicago/Turabian Style**

Wang, Tianqi, Weiwei Jin, Fuyou Liang, and Jordi Alastruey. 2021. "Machine Learning-Based Pulse Wave Analysis for Early Detection of Abdominal Aortic Aneurysms Using In Silico Pulse Waves" *Symmetry* 13, no. 5: 804.
https://doi.org/10.3390/sym13050804