# Uncertainty Quantification in the In Vivo Image-Based Estimation of Local Elastic Properties of Vascular Walls

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Imaging Analysis

#### 2.2. Evaluation of the Elastic Module from PCMRI

#### 2.3. Uncertainty Quantification

#### 2.4. In Silico Modeling

#### 2.5. Post-Processing

## 3. Results

## 4. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Mesh Sensitivity

**Figure A1.**Mesh sensitivity analyses of five meshes (3, 2, 1, 0.5, and 0.25 mm) conducted on relative area change (

**a**) and maximum flow rate (

**b**) measured at a specific model’s cross-section.

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**Figure 1.**CT imaging processing: sagittal (

**a**), axial (

**b**), and coronal (

**c**) planes of CT dataset, raw segmentation result (

**d**), and refined model of the aorta (

**e**).

**Figure 2.**Evenly spaced subsample of PCMRI frames, with aortic lumen countered with ROI (magenta) in both magnitude (

**a**) and phase images (

**b**); (

**c**) flow velocity (magenta boxes) obtained from PCMRI segmentation of the lumen of the aorta.

**Figure 3.**Fluid domain mesh (

**a**) and plane cut showing internal elements (

**b**) with grid details at wall surface (

**c**) and at plane cut (

**d**).

**Figure 4.**Location of the cross-sections extracted from the FSI simulations, perpendicular to the model centerline: $C{S}_{1}$ (green) proximal to inlet valve; $C{S}_{2}$ (blue) at the ascending aorta; $C{S}_{3}$ (yellow) after supra-aortic branches; and $C{S}_{4}$ (pink) and $C{S}_{5}$ (orange) at the descending aorta, with this latter closer to the outlet section.

**Figure 5.**(

**a**) Flow and area variations along the cardiac cycle obtained from the segmentation of the patient’s PCMRI data; (

**b**) QA loop (white circles) with indication of the points belonging to the early systole period (black circles) used for the linear interpolation (black line) to compute the $PWV$.

**Figure 6.**Localizer image (

**a**) used to define the acquisition plane ${\Sigma}_{PCMRI}$ of the PCMRI at the descending aorta (

**b**) and equivalent plane ${\Sigma}_{FSI}$ in the FSI domain (

**c**); (

**d**) comparison between flow curves as extracted from PCMRI data of the patient (black line with diamond markers) and FSI simulation (blue line with circle markers) at the same plane of the descending aorta.

**Figure 7.**Mean and standard deviation computed along the selected cross-sections of the coefficient ratio ${a}_{r}/{a}_{0}$ of the gPC polynomial expansion for the performed stochastic analyses.

**Figure 8.**PDF (color scale), mean (straight line), and standard deviation (dashed line) plots of flow variations along the cardiac cycle at cross-sections $C{S}_{1}$ (

**a**), $C{S}_{2}$ (

**b**), $C{S}_{3}$ (

**c**), $C{S}_{4}$ (

**d**), and $C{S}_{5}$ (

**e**), respectively identified by the green, blue, yellow, pink, and orange slices in the aortic model (

**f**).

**Figure 9.**PDF (color scale), mean (straight line), and standard deviation (dashed line) plots of area variations along the cardiac cycle at cross-sections $C{S}_{1}$ (

**a**), $C{S}_{2}$ (

**b**), $C{S}_{3}$ (

**c**), $C{S}_{4}$ (

**d**), and $C{S}_{5}$ (

**e**), respectively identified by the green, blue, yellow, pink, and orange slices in the aortic model (

**f**).

**Figure 10.**PDFs of area variation at the five cross-sections, i.e., $C{S}_{1}$ (

**a**,

**b**), $C{S}_{2}$ (

**c**,

**d**), $C{S}_{3}$ (

**e**,

**f**), $C{S}_{4}$ (

**g**,

**h**), and $C{S}_{5}$ (

**i**,

**j**) along the entire cardiac cycle (

**a**,

**c**,

**e**,

**g**,

**i**) and at the time instances ${t}_{1}$, ${t}_{2}$, and ${t}_{3}$ of the cardiac cycle (

**b**,

**d**,

**f**,

**h**,

**j**).

**Table 1.**List of the four quadrature points ${x}_{i}$, calculated using the Gauss–Legendre integration rule.

Quadrature Points ${\mathit{x}}_{\mathit{i}}$ | |||
---|---|---|---|

${x}_{1}$ | ${x}_{2}$ | ${x}_{3}$ | ${x}_{4}$ |

$1.2\widehat{E}$ | $1.08\widehat{E}$ | $0.92\widehat{E}$ | $0.8\widehat{E}$ |

**Table 2.**$RCR$ values of the Windkessel models for the boundary conditions assigned to the aorta outlets: the three supra-aortic vessels, i.e., the brachiocephalic artery, the left common carotid artery, and the left subclavian artery, and the descending aorta.

${\mathit{R}}_{\mathit{p}}$ (Kg s${}^{-1}$ m${}^{-4}$) | C (m${}^{3}$ Pa ${}^{-1}$) | ${\mathit{R}}_{\mathit{d}}$ (Kg s${}^{-1}$ m${}^{-4}$) | |
---|---|---|---|

Brachiocephalic artery | $1.3\times {10}^{7}$ | $1.5\times {10}^{-9}$ | $1.3\times {10}^{9}$ |

Left common carotid artery | $5.1\times {10}^{7}$ | $3.8\times {10}^{-10}$ | $5.0\times {10}^{9}$ |

Left subclavian artery | $1.1\times {10}^{7}$ | $1.7\times {10}^{-9}$ | $1.1\times {10}^{9}$ |

Descending aorta | $2.5\times {10}^{6}$ | $7.7\times {10}^{-9}$ | $2.4\times {10}^{8}$ |

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**MDPI and ACS Style**

Fanni, B.M.; Antonuccio, M.N.; Pizzuto, A.; Berti, S.; Santoro, G.; Celi, S.
Uncertainty Quantification in the In Vivo Image-Based Estimation of Local Elastic Properties of Vascular Walls. *J. Cardiovasc. Dev. Dis.* **2023**, *10*, 109.
https://doi.org/10.3390/jcdd10030109

**AMA Style**

Fanni BM, Antonuccio MN, Pizzuto A, Berti S, Santoro G, Celi S.
Uncertainty Quantification in the In Vivo Image-Based Estimation of Local Elastic Properties of Vascular Walls. *Journal of Cardiovascular Development and Disease*. 2023; 10(3):109.
https://doi.org/10.3390/jcdd10030109

**Chicago/Turabian Style**

Fanni, Benigno Marco, Maria Nicole Antonuccio, Alessandra Pizzuto, Sergio Berti, Giuseppe Santoro, and Simona Celi.
2023. "Uncertainty Quantification in the In Vivo Image-Based Estimation of Local Elastic Properties of Vascular Walls" *Journal of Cardiovascular Development and Disease* 10, no. 3: 109.
https://doi.org/10.3390/jcdd10030109