# Computational Analysis of Hemodynamic Indices Based on Personalized Identification of Aortic Pulse Wave Velocity by a Neural Network

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Coronary Circulation Model

^{3}), $A\phantom{\rule{-0.166667em}{0ex}}\left(\right)open="("\; close=")">t,x$ is the vessel cross-section area, p is the blood pressure, $u\phantom{\rule{-0.166667em}{0ex}}\left(\right)open="("\; close=")">t,x$ is the linear velocity averaged over the cross-section and $\psi $ is the friction force.

#### 2.2. Hemodynamic Indices Calculation

#### 2.3. Datasets

#### 2.3.1. Synthetic Database

#### 2.3.2. PWV Dataset

#### 2.3.3. FFR Dataset

#### 2.4. PWV Estimation with Neural Network and Other Machine Learning Methods

#### 2.4.1. Error Estimation

#### 2.4.2. Neural Network

#### 2.4.3. Other Methods

## 3. Results

#### 3.1. AoPWV Estimation

- 1.
- Estimating carotid-femoral PWV from the radial pressure wave using machine learning algorithms [48]. The study population was the Twins UK cohort, containing 3082 subjects aged from 18 to 110 years. The authors used Gaussian process regression and a recurrent neural network to estimate carotid-femoral PWV from the entire radial pressure wave. They report errors of 17–19%.
- 2.
- Estimating carotid-femoral PWV from one carotid waveform measured by tonometry and few clinical variables (age, blood pressure, heart rate, etc.) [48]. The study population included 5050 subjects in the age range of 20 to 69. The authors use the newly developed Intrinsic Frequency algorithm together with neural networks and bootstrap averaging. This algorithm uses an uncalibrated noisy waveform with few additional parameters. The reported error is 14%.
- 3.
- Estimating aortic PWV with ridge regression and a deep neural network from two sets of inputs: a basic set of predictors (age, sex, height, weight, heart rate, systolic and diastolic blood pressure) and an expanded set of predictors (HbA1c, total cholesterol, use of antihypertensive, antidiabetic or cholesterol-lowering medication and smoking status in addition to basic set) [49]. A total of 2254 participants from the Netherlands Epidemiology of Obesity study were included (age 45–65 years). The reported error is 18–22%.

Description | Error |
---|---|

Brachial–ankle PWV with a neural network trained on synthetic data (Table 3) | 16% |

Carotid–femoral PWV with machine learning using peripheral pulse waves [48] | 17–19% |

Carotid–femoral PWV with a neural network using carotid waveform [50] | 14% |

Aortic PWV with a neural network [49] | 18–22% |

The difference between two occasionally different measurements of brachial–ankle PWV by one observer [42] | 10% |

Repeatability of carotid–femoral PWV measurements [41] | 3.4–9.5% |

#### 3.2. FFR Estimation with Predicted AoPWV

#### 3.3. FFR, iFR and CFR Sensitivity to AoPWV

## 4. Discussion

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

PWV | Pulse wave velocity |

RMSE | Root mean square error |

FFR | Fractional flow reserve |

AoPWV | Aortic pulse wave velocity |

CFR | Coronary flow reserve |

iFR | Instantaneous wave-free ratio |

CAD | Coronary artery disease |

LCA | Left coronary artery |

RCA | Right coronary artery |

SBP | Systolic blood pressure |

DBP | Diastolic blood pressure |

MBP | Mean blood pressure |

ECG | Electrocardiogram |

BMI | Body mass index |

CoPWV | Coronary vessels pulse wave velocity |

LAD | Left anterior descending artery |

LADp | Proximal part of the left anterior descending artery |

LADd | Distal part of the left anterior descending artery |

DA | Diagonal artery |

LCX | Circumflex branch of left coronary artery |

## Appendix A

**Figure A1.**Activation functions of neural network (Figure 2). ReLU: $y\left(x\right)=x$ if $x\ge 0$ and $y\left(x\right)=0$ if $x<0$. ELU: $y\left(x\right)=x$ if $x\ge 0$ and $y\left(x\right)={e}^{x}-1$ if $x<0$. Linear: $y\left(x\right)=x$.

Number of Layers | RMSE, m/s | Percentage Error |
---|---|---|

2 layers | 1.55 ± 0.41 | 19% ± 5% |

3 layers | 1.31 ± 0.19 | 16% ± 2% |

4 layers | 1.32 ± 0.22 | 16% ± 3% |

5 layers | 1.37 ± 0.19 | 17% ± 2% |

**Figure A2.**ROC-curves for two approaches that estimate FFR: utilizing fixed AoPWV 7.5 m/s and AoPWV from the neural network. In both cases, stenosis is considered to be significant if FFR < 0.8.

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**Figure 1.**The structure of the computational domain for patient 1. LCA and RCA were extracted from the patient’s CT scans. Each numbered segment corresponds in the model to a one-dimensional tube. Stenosis is a separate segment with decreased diameter. We solve hyperbolic set (1) in the inner points of each segment. We impose mass conservation (5) and continuity of the total pressure (6) on each junction. FFR is measured at segment 6.

**Figure 2.**Structure of neural network: inputs, output, layers, number of neurons in each layer and activation functions (ReLu, ELu and Linear—Figure A1).

**Figure 3.**Bland–Altman plot for brachial–ankle AoPWV: cross-validation on the synthetic database (blue circles) and comparison with brachial–ankle AoPWV calculated for real patients (orange triangles). $\Delta $—difference between brachial–ankle AoPWV predicted by the neural network and subject’s brachial–ankle AoPWV.

**Figure 5.**FFR, iFR and CFR sensitivity to AoPWV for patient 6. Patient 6 had three stenoses located in the proximal part of the left anterior descending artery (LADp), distal part of the left anterior descending artery (LADd) and middle part of the diagonal artery (DA).

**Figure 6.**FFR, iFR and CFR sensitivity to AoPWV for patient 7. Patient 7 had two stenoses located in the left anterior descending artery (LAD) and the left circumflex artery (LCx).

**Figure 7.**FFR, iFR and CFR sensitivity to AoPWV for patients from 8 to 10. Patient 8 had two stenoses located in the left anterior descending artery (LAD) and the left circumflex artery (LCx).

Database | PWV Dataset | Synthetic Database |
---|---|---|

Subjects | 102 | 4374 |

Age, years | 58 ± 15 | 50 ± 17 |

Heart rate, bpm | 68 ± 12 | 76 ± 9 |

Stroke volume, ml | 54.6 ± 19.6 | 60.4 ± 12.3 |

SBP (brachial), mmHg | 104.0 ± 17.1 | 119.1 ± 11.4 |

DBP (brachial), mmHg | 86.7 ± 12.2 | 72.6 ± 7.2 |

MBP (brachial), mmHg | 92.5 ± 12.9 | 93.8 ± 6.75 |

AoPWV (brachial–ankle), m/s | 8.0 ± 1.4 | 9.4 ± 2.1 |

**Table 3.**Comparison of different machine learning methods used for AoPWV estimation. For each case, we present the mean values and standard deviations for seven attempts.

Method | RMSE, m/s | Percentage Error |
---|---|---|

K-nearest neighbors | 1.96 ± 0.09 | 24% ± 1% |

Decision tree | 1.88 ± 0.21 | 23% ± 3% |

Random forest | 1.73 ± 0.14 | 22% ± 2% |

Neural network | 1.31 ± 0.19 | 16% ± 2% |

**Table 4.**FFR values for average and personalized AoPWV. Vessel ID is the notation of the vessel with stenosis, d is the diameter of the non-stenosed part of a vessel, Degree is the percentage diameter in stenosis, $FF{R}_{inv}$ is the invasively measured FFR value, $cFF{R}_{fixed}$ is the calculated FFR with fixed AoPWV for each case ($AoPWV=7.5$ m/s), $PW{V}_{NN}$ is the AoPWV estimated from the neural network and $cFF{R}_{NN}$ is the calculated FFR with AoPWV estimated from the neural network. The bottom of the table contains the mean values ± standard deviations.

Patient | Vessel ID | d, mm | Degree | ${\mathit{FFR}}_{\mathit{inv}}$ | ${\mathit{cFFR}}_{\mathit{fixed}}$ | ${\mathit{PWV}}_{\mathit{NN}},\frac{\mathit{m}}{\mathit{s}}$ | ${\mathit{cFFR}}_{\mathit{NN}}$ |
---|---|---|---|---|---|---|---|

1 | LAD | 1.9 | 46% | 0.89 | 0.90 | 9.7 | 0.89 |

2 | LAD | 3.3 | 61% | 0.86 | 0.87 | 6.4 | 0.87 |

3 | RCA | 3.0 | 61% | 0.88 | 0.91 | 13.5 | 0.89 |

4 | LAD | 2.5 | 48% | 0.82 | 0.83 | 8.8 | 0.83 |

5 | LAD | 1.6 | 55% | 0.82 | 0.81 | 7.3 | 0.81 |

6 | LADp | 2.4 | 38% | 0.90 | 0.92 | 9.1 | 0.91 |

6 | LADd | 2.4 | 28% | 0.82 | 0.86 | 9.1 | 0.85 |

6 | DA | 1.9 | 58% | 0.81 | 0.84 | 9.1 | 0.83 |

7 | LAD | 1.5 | 57% | 0.75 | 0.66 | 7.4 | 0.66 |

7 | LCX | 1.9 | 32% | 0.84 | 0.85 | 7.4 | 0.85 |

8 | LAD | 2.3 | 56% | 0.88 | 0.91 | 7.8 | 0.91 |

8 | LCX | 3.1 | 58% | 0.89 | 0.95 | 7.8 | 0.95 |

9 | LAD | 2.0 | 48% | 0.83 | 0.89 | 6.6 | 0.89 |

10 | LAD | 1.9 | 63% | 0.72 | 0.81 | 14.2 | 0.79 |

2.2 ± 0.5 | 51% ± 11% | 0.84 ± 0.05 | 0.86± 0.07 | 9.1 ± 2.6 | 0.85 ± 0.07 |

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## Share and Cite

**MDPI and ACS Style**

Gamilov, T.; Liang, F.; Kopylov, P.; Kuznetsova, N.; Rogov, A.; Simakov, S.
Computational Analysis of Hemodynamic Indices Based on Personalized Identification of Aortic Pulse Wave Velocity by a Neural Network. *Mathematics* **2023**, *11*, 1358.
https://doi.org/10.3390/math11061358

**AMA Style**

Gamilov T, Liang F, Kopylov P, Kuznetsova N, Rogov A, Simakov S.
Computational Analysis of Hemodynamic Indices Based on Personalized Identification of Aortic Pulse Wave Velocity by a Neural Network. *Mathematics*. 2023; 11(6):1358.
https://doi.org/10.3390/math11061358

**Chicago/Turabian Style**

Gamilov, Timur, Fuyou Liang, Philipp Kopylov, Natalia Kuznetsova, Artem Rogov, and Sergey Simakov.
2023. "Computational Analysis of Hemodynamic Indices Based on Personalized Identification of Aortic Pulse Wave Velocity by a Neural Network" *Mathematics* 11, no. 6: 1358.
https://doi.org/10.3390/math11061358