# Estimation of Aortic Stiffness with Bramwell–Hill Equation: A Comparative Analysis with Carotid–Femoral Pulse Wave Velocity

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## Abstract

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## 1. Introduction

## 2. Methods

#### 2.1. Carotid–Femoral Pulse Wave Velocity

#### 2.2. Ultrasound

#### 2.3. Pulse Pressure

- Using the PulsePen, the central pressure was derived from carotid pressure waves, which have been shown to be similar to those recorded in the ascending aorta [20].

#### 2.4. Bramwell–Hill Equation

- $\rho $ represents the density of the blood;
- A is the mean area of the blood vessel;
- $\Delta A$ is the difference between the maximum and minimum area of the blood vessel during a cardiac cycle;
- $\Delta P$ is the difference between the central systolic and diastolic pressures;
- $PW{V}_{est}$ is the estimated PWV obtained from the measurement of pulsatility ($\Delta A/A$) and pressure variation ($\Delta P$).

#### 2.5. Experimental Data

#### 2.6. Statistical Analysis

## 3. Results

- 1.
- carotid–femoral pulse wave velocity (cf-PWV);
- 2.
- pulsatility ($\Delta A/A$), indicated as the ratio between the range (i.e., maximum minus minimum value) and the mean value of the area of a portion of the abdominal aorta measured by automated edge tracking from ultrasound scans;
- 3.
- pressure variation ($\Delta P$), measured using either PulsePen (first seven subjects) or SphygmoCor (remaining subjects).

## 4. Discussion

#### 4.1. Outcomes

#### 4.2. Significance

#### 4.3. Limitations

#### 4.4. Future Perspectives

## 5. Conclusions and Further Work

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

cf-PWV | carotid–femoral Pulse Wave Velocity |

PWV | Pulse Wave Velocity |

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**Figure 1.**Experimental data from two subjects with different carotid–femoral pulse wave velocities (cf-PWV): (

**A**,

**B**) subject with medium PWV; (

**C**,

**D**) subject with high PWV. Ultrasound scans are considered in (

**A**,

**C**), showing two frames corresponding to local minimum and maximum average diameter (left) and the diameter over time (right). The indication of average pulsatility $\Delta A/A$ and pressure variation ($\Delta P$, obtained using PulsePen in these cases) are provided. Tonometry is shown in (

**B**,

**D**), for both the carotid and femoral artery, with the measured cf-PWV indication.

**Figure 2.**Experimental measurements, including repetitions for specific subjects: carotid–femoral pulse wave velocity (cf-PWV); pulsatility ($\Delta A/A$), indicated as the ratio between the range and the mean value of the area of the aorta measured by automated edge tracking from ultrasound scans; and pressure variation using either PulsePen or SphygmoCor ($\Delta P$).

**Figure 3.**Comparison of PWV measured estimated by the Bramwell–Hill equation using the measurements of pulsatility and pressure variation (correlation coefficient 17.8%). Scatter plot (

**left**) and Bland–Altman representation (

**right**).

**Figure 4.**Comparison of measured and estimatedpulsatility by inverting Bramwell–Hill equation and using the measurements of PWV and pressure variation (correlation coefficient 56.1%). Scatter plot (

**left**) and Bland–Altman representation (

**right**).

**Figure 5.**Comparison of pressure variation either measured or estimated by inverting Bramwell-Hill equation and using the measurements of PWV and pulsatility (correlation coefficient 54.5%). Scatter plot (

**left**) and Bland-Altman representation (

**right**).

**Figure 6.**Sensitivity of PWV estimated by Bramwell–Hill equation to certain variables: blood density, error in measuring pulsatility, error in the estimation of pressure variation.

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**MDPI and ACS Style**

Mesin, L.; Floris, L.; Policastro, P.; Albani, S.; Scacciatella, P.; Pugliese, N.; Masi, S.; Grillo, A.; Fabris, B.; Antonini-Canterin, F.
Estimation of Aortic Stiffness with Bramwell–Hill Equation: A Comparative Analysis with Carotid–Femoral Pulse Wave Velocity. *Bioengineering* **2022**, *9*, 265.
https://doi.org/10.3390/bioengineering9070265

**AMA Style**

Mesin L, Floris L, Policastro P, Albani S, Scacciatella P, Pugliese N, Masi S, Grillo A, Fabris B, Antonini-Canterin F.
Estimation of Aortic Stiffness with Bramwell–Hill Equation: A Comparative Analysis with Carotid–Femoral Pulse Wave Velocity. *Bioengineering*. 2022; 9(7):265.
https://doi.org/10.3390/bioengineering9070265

**Chicago/Turabian Style**

Mesin, Luca, Luca Floris, Piero Policastro, Stefano Albani, Paolo Scacciatella, Nicola Pugliese, Stefano Masi, Andrea Grillo, Bruno Fabris, and Francesco Antonini-Canterin.
2022. "Estimation of Aortic Stiffness with Bramwell–Hill Equation: A Comparative Analysis with Carotid–Femoral Pulse Wave Velocity" *Bioengineering* 9, no. 7: 265.
https://doi.org/10.3390/bioengineering9070265