# Central Arterial Dynamic Evaluation from Peripheral Blood Pressure Waveforms Using CycleGAN: An In Silico Approach

^{1}

^{2}

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## Abstract

**:**

^{2}, respectively. Regarding the pressure–strain elastic modulus, it is achieved a mean absolute percentage error of 6.5 ± 5.1%. GAN-based deep learning models can recover the pressure–strain loop of central arteries while observing pressure signals from peripheral arteries.

## 1. Introduction

#### 1.1. Arterial Stiffness

#### 1.2. Arterial Stiffness and Machine Learning

#### 1.3. Virtual Databases in Research

## 2. Methodology

#### 2.1. Dataset

^{2}) and flow velocity (m/s), corresponding to different sites of the arterial tree. Furthermore, because the arterial wall viscosity property is considered in the database [9], the hysteresis phenomenon between pressure and strain can be evaluated, as shown in Figure 1.

_{Bra-Rad}) signals correspond to the X domain (the blue and green lines in Figure 2) and the aortic abdominal pressure and area (P-A

_{Abdo}) signals correspond to the Y domain (the red lines in Figure 2). Although PWs are noiseless, a pre-processing stage is performed to prepare the model’s input. First, the area magnitude is converted to cm

^{2}. Second, because PWs have different durations, the time-window is fixed to the maximum PW duration across the database; shorter pulses are repeated until time-windows are fulfilled, considering a stationary situation. Third, a MinMax [−1, 1] normalization is performed for both pressure and area magnitude. Finally, PWs are resampled from 500 Hz to 256 Hz, which is done for two reasons: (1) to reduce computational cost, and (2) because 256 Hz is a common sampling frequency used in commercial devices (SphygmoCor®XCEL, AtCor Medical, Sydney, Australia).

#### 2.2. Pressure–Strain Elastic Modulus

_{P-$\epsilon $}[16], is considered as the measurement of arterial stiffness:

_{P-$\epsilon $}is observed as an evaluation parameter.

#### 2.3. CycleGAN Model

_{Bra-Rad}and P-A

_{Abdo}signals, respectively.

#### 2.3.1. General Architecture

#### 2.3.2. Architecture of Generators and Discriminators

#### 2.3.3. Loss Functions

- where LSGAN and WGAN-GP measure the Pearson ${\chi}^{2}$ divergence and the Wasserstein distance, respectively. For the sake of simplicity, only the case where ${G}_{xy}:X\to Y$ (the second term in Equation (5)) is written, because ${G}_{yx}:Y\to X$ is defined analogously by replacing the X with the Y domain and vice versa. The objective for ${\mathcal{L}}_{GAN}({G}_{xy},{D}_{y},X,Y)$ considering LSGAN is defined as follows:

#### 2.3.4. Hyperparameters and Experimental Settings

#### 2.4. Evaluation

_{P-$\epsilon $}was calculated for both the ground truth and estimated pulses. The strain $\epsilon $ was derived from the area PW, then E

_{P-$\epsilon $}is computed as the slope $\beta $ of a simple linear regression:

_{P-$\epsilon $}as follows:

## 3. Results

^{2}and 0.2 ± 0.2 cm

^{2}, respectively. Figure 5 compares the true and predicted samples.

_{P-$\epsilon $}, the ME for the best LSGAN and WGAN-GP results were 13.1 ± 56.5 mmHg/% and 70.6 ± 216.0 mmHg/%, respectively. Finally, according to MAPE, the LSGAN and WGAN-GP results were 6.5 ± 5.1% and 28.6 ± 19.3%, respectively. Figure 6 shows a hysteresis comparison for the same signals in Figure 5, where the black dotted line represents the $\beta $ and $\widehat{\beta}$ parameters from Equation (10).

## 4. Discussion

_{P-$\epsilon $}. It is worth noting that the obtained P-D loops correspond to an aortic site where the real measurement would, while highly invasive, be very useful regarding the potential information that could extracted [23,24].

_{Bra-Rad}and P

_{Abdo}reduce the learning difficulty and offer bounds on the estimated PW. For example, systolic P

_{Bra-Rad}values are always greater than P

_{Abdo}, and the model can easily learn this, while the shape of the PW remains as a challenging task. On the contrary, the calibration of the area PW values has no reference points from the pressure values, meaning that the learning task increases in complexity. In particular, for the pressure PW, in comparison with previous works [25,26,27] it is apparent that the error as presented in Table 2 is lower. It should be noted, however, that these previous works were obtained from real subjects, which is a more challenging task.

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

BP | blood pressure |

CNN | convolutional neural network |

CV | cardiovascular |

CVD | cardiovascular disease |

DBP | diastolic blood pressure |

E | elastic module |

E_{P-$\epsilon $} | pressure–strain elastic modulus |

GAN | generative adversarial network |

GRU | gated recurrent unit |

LOA | limit of agreement |

LR | learning rate |

LSGAN | least-square GAN |

MAPE | mean absolute percentage error |

ME | mean error |

ML | machine learning |

NN | neural network |

P-D | pressure–diameter |

PPG | photoplethysmography |

PW | pulse wave |

PWV | pulse wave velocity |

RMSE | root mean squared error |

SBP | systolic blood pressure |

SVM | support vector machine |

WGAN-GP | Wasserstein GAN with gradient penalty |

## Appendix A

${\mathcal{L}}_{\mathit{GAN}}$ | ${\mathit{G}}_{\mathit{GRU}}$ | ${\mathit{D}}_{\mathit{l}=1}$ | ${\mathit{D}}_{\mathit{Iters}}$ | ${\mathit{\lambda}}_{\mathit{cyc}}$ | Experiment |
---|---|---|---|---|---|

LSGAN | 128 | 8 | 1 | 5 | A |

WGAN-GP | 64 | 6 | 15 | 25 | B |

LSGAN | 64 | 6 | 1 | 15 | C |

64 | 6 | 1 | 25 | D | |

64 | 6 | 1 | 5 | E | |

WGAN-GP | 128 | 8 | 15 | 5 | F |

64 | 6 | 10 | 5 | G | |

64 | 6 | 15 | 15 | H | |

64 | 6 | 15 | 5 | I | |

64 | 6 | 50 | 15 | J | |

64 | 6 | 50 | 25 | K |

${\mathcal{L}}_{\mathit{GAN}}$ | Experiment | Pressure [mmHg] | Area [cm^{2}] | ${\mathit{E}}_{\mathit{P-\epsilon}}$ [mmHg/%] | |
---|---|---|---|---|---|

RMSE | RMSE | ME | MAPE | ||

LSGAN | A | 0.8 ± 0.4 | 0.1 ± 0.1 | 13.1 ± 56.5 | 6.5 ± 5.1 |

WGAN-GP | B | 1.7 ± 0.8 | 0.2 ± 0.2 | 70.6 ± 216.0 | 28.6 ± 19.3 |

LSGAN | C | 4.4 ± 1.2 | 0.2 ± 0.1 | 229.3 ± 304.6 | 45.0 ± 25.7 |

D | 27.1 ± 6.3 | 0.2 ± 0.1 | 654.2 ± 265.4 | 137.5 ± 14.6 | |

E | 5.8 ± 3.3 | 0.3 ± 0.2 | 18.0 ± 137.7 | 17.3 ± 13.1 | |

WGAN-GP | F | 2.7 ± 1.7 | 0.4 ± 0.2 | 3.1 ± 349.1 | 59.7 ± 54.4 |

G | 3.8 ± 1.8 | 0.2 ± 0.1 | 84.3 ± 201.0 | 28.7 ± 19.5 | |

H | 1.5 ± 0.7 | 0.4 ± 0.3 | 104.8 ± 222.3 | 28.7 ± 22.7 | |

I | 3.4 ± 2.1 | 0.3 ± 0.2 | 10.9 ± 273.3 | 39.3 ± 31.9 | |

J | 1.6 ± 0.7 | 0.4 ± 0.2 | 12.7 ± 300.1 | 48.2 ± 47.0 | |

K | 2.0 ± 0.8 | 0.2 ± 0.2 | 68.4 ± 242.7 | 32.9 ± 23.6 |

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**Figure 1.**Abdominal aorta pressure–strain hysteresis loop; the color of the line refers to the time.

**Figure 2.**

**Left**: Brachial, radial and abdominal aorta pressure signals (blue, green, and red lines, respectively).

**Right**: Abdominal aorta area signal.

**Figure 5.**Comparison between true and predicted signals for pressure (

**left**) and area (

**right**); blue and red correspond to the true and predicted signals, respectively.

**Figure 6.**Comparison between true (blue) and predicted (red) hysteresis cycles. Line brightness refers to time; the black dashed lines refer to $\beta $ and $\widehat{\beta}$ from Equation (10).

**Figure 7.**Bland–Altman plots showing the difference between the maximum and minimum values of pressure (

**top**) and diameter (

**bottom**) for true and estimated signals; the plots were obtained for the best LSGAN model.

**Table 1.**Grid search of hyperparameters. ${\mathcal{L}}_{GAN}$ refers to the adversarial loss function. ${G}_{GRU}$ and ${D}_{l=1}$ refer to the feature size of the generator’s GRU units and the first layer output’s discriminators. ${D}_{Iters}$ refers to the number of discriminators per generator updates (only for WGAN-GP; for LSGAN it is always 1).

${\mathcal{L}}_{\mathit{GAN}}$ | ${\mathit{G}}_{\mathit{GRU}}$ | ${\mathit{D}}_{\mathit{l}=1}$ | ${\mathit{D}}_{\mathit{Iters}}$ | ${\mathit{\lambda}}_{\mathit{cyc}}$ |
---|---|---|---|---|

[LSGAN, WGAN-GP] | [64, 128] | [6, 8] | [5, 15, 25] | [5, 15, 25] |

**Table 2.**Mean ± standard deviation of error for Brachial–Radial to Aortic Abdominal case on the test set.

${\mathcal{L}}_{\mathit{GAN}}$ | Experiment | Pressure [mmHg] | Area [cm^{2}] | ${\mathit{E}}_{\mathit{P-\epsilon}}$ [mmHg/%] | |
---|---|---|---|---|---|

RMSE | RMSE | ME | MAPE | ||

LSGAN | A | 0.8 ± 0.4 | 0.1 ± 0.1 | 13.1 ± 56.5 | 6.5 ± 5.1 |

WGAN-GP | B | 1.7 ± 0.8 | 0.2 ± 0.2 | 70.6 ± 216.0 | 28.6 ± 19.3 |

_{GRU}= 128, D

_{t=1}= 8, D

_{Iters}= 1, λ

_{cyc}= 5. B→ G

_{GRU}= 64, D

_{t=1}= 6, D

_{Iters}= 15, λ

_{cyc}= 25.

**Table 3.**Mean ± standard deviation of error for Brachial–Radial to Aortic Abdominal case on training set.

${\mathcal{L}}_{\mathbf{GAN}}$ | Experiment | Pressure [mmHg] | Area [cm^{2}] | ${\mathit{E}}_{\mathit{P-\epsilon}}$ [mmHg/%] | |
---|---|---|---|---|---|

RMSE | RMSE | ME | MAPE | ||

LSGAN | A | 0.8 ± 0.4 | 0.1 ± 0.1 | 13.4 ± 51.5 | 6.2 ± 4.9 |

WGAN-GP | B | 1.8 ± 0.9 | 0.2 ± 0.2 | 71.1 ± 209.3 | 28.3 ± 20.8 |

_{GRU}= 128, D

_{t=1}= 8, D

_{Iters}= 1, λ

_{cyc}= 5. B→ G

_{GRU}= 64, D

_{t=1}= 6, D

_{Iters}= 15, λ

_{cyc}= 25.

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**MDPI and ACS Style**

Aguirre, N.; Cymberknop, L.J.; Grall-Maës, E.; Ipar, E.; Armentano, R.L.
Central Arterial Dynamic Evaluation from Peripheral Blood Pressure Waveforms Using CycleGAN: An In Silico Approach. *Sensors* **2023**, *23*, 1559.
https://doi.org/10.3390/s23031559

**AMA Style**

Aguirre N, Cymberknop LJ, Grall-Maës E, Ipar E, Armentano RL.
Central Arterial Dynamic Evaluation from Peripheral Blood Pressure Waveforms Using CycleGAN: An In Silico Approach. *Sensors*. 2023; 23(3):1559.
https://doi.org/10.3390/s23031559

**Chicago/Turabian Style**

Aguirre, Nicolas, Leandro J. Cymberknop, Edith Grall-Maës, Eugenia Ipar, and Ricardo L. Armentano.
2023. "Central Arterial Dynamic Evaluation from Peripheral Blood Pressure Waveforms Using CycleGAN: An In Silico Approach" *Sensors* 23, no. 3: 1559.
https://doi.org/10.3390/s23031559