# Biomechanical Characterisation of Thoracic Ascending Aorta with Preserved Pre-Stresses

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Vessel Geometries

#### 2.1.1. Cylindrical Geometry

#### 2.1.2. Image-Based Aorta Geometry

#### 2.2. Finite Element Analysis

#### 2.3. Mechanical Parameter Estimation

#### 2.3.1. Experimental Data Used for Fitting

#### 2.3.2. Algorithm Description

^{2}—the norm of the nodal differences of the experimental diastolic FE geometry (${\mathit{x}}_{\mathit{\alpha},\mathit{d}\mathit{i}\mathit{a}\mathit{s}}^{\mathit{e}\mathit{x}\mathit{p}})$ and the diastolic geometry $\left({\mathit{x}}_{\mathit{\alpha},\mathit{d}\mathit{i}\mathit{a}\mathit{s}}^{\mathit{e}\mathit{s}\mathit{t}\mathrm{*}}\right)$ produced using the new mechanical parameter ${c}^{\mathrm{*}}.$ The L

^{2}–norm is defined as $\sqrt{\sum _{i=1}^{n}{\left({\left|{\mathit{x}}_{\mathit{\alpha},\mathit{d}\mathit{i}\mathit{a}\mathit{s}}^{\mathit{e}\mathit{x}\mathit{p}}-{\mathit{x}}_{\mathit{\alpha},\mathit{d}\mathit{i}\mathit{a}\mathit{s}}^{\mathit{e}\mathit{s}\mathit{t}\mathrm{*}}\right|}_{i}\right)}^{2}}$ for all nodes $=1,2\dots n$; and represented as ${\left|\right|{\mathit{x}}_{\mathit{\alpha},\mathit{d}\mathit{i}\mathit{a}\mathit{s}}^{\mathit{e}\mathit{x}\mathit{p}}-{\mathit{x}}_{\mathit{\alpha},\mathit{d}\mathit{i}\mathit{a}\mathit{s}}^{\mathit{e}\mathit{s}\mathit{t}\mathrm{*}}\left|\right|}_{2}$. New scaling factor ${\gamma}^{\mathrm{*}}$ was determined as an outcome of the second-stage optimisation (Steps 6 and 7; Figure 3):

#### 2.4. Method Validation

#### 2.4.1. Synthetically Created Reference Data

#### 2.4.2. Cylindrical Geometry Reference Data

#### 2.4.3. Aortic Geometry Reference Data

#### 2.5. Simulations and Analyses

#### 2.5.1. Assessment of Algorithm Accuracy

^{−6}.

#### 2.5.2. Pre-Stresses

^{2}–norms of the principal stresses of the reference diastolic configuration geometry (${\sigma}_{cylinder,dias}^{exp}$) and the estimated diastolic configuration (${\sigma}_{cylinder,dias}^{est}$) were calculated using:

#### 2.5.3. Effect of Different Initial Conditions

## 3. Results

## 4. Discussion

#### 4.1. Methodological Differences with Respect to BI

#### 4.2. Basic Numerical Validation

#### 4.3. Residual Stresses

#### 4.4. Fitting Procedure

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Schematic representation of computational domain, $\Omega $, of the cylindrical geometry. The finite element system of equations are solved by a structural solver ($S)$ by applying prescribed displacements at the opening boundaries (${\Gamma}_{d})$ and pressure ($p)$ at the inner surface (${\Gamma}_{n})$. The general notation for nodes in the undeformed reference configuration and the deformed configuration are represented as $\mathit{X}$ and $\mathit{x}$, respectively, with corresponding stresses as $0$ and $\stackrel{\u033f}{\sigma}$.

**Figure 2.**(

**A**) Inner-wall aortic boundaries obtained from 3D DIXON MRI scan and centre-line connecting the lumen at six plane locations. The six plane locations were at root, ascending aorta, before brachiocephalic artery (pre-arch), after left sub-clavian artery (post-arch), descending aorta and at diaphragm levels. (

**B**) Solid mesh ($\Omega ({\mathit{X}}^{\mathit{M}\mathit{R}\mathit{I}},0)$) of the aortic geometry, including the branches. Boundary conditions were applied on the nodes located on ${\Gamma}_{d1}$ to ${\Gamma}_{d4}$ (fixed), ${\Gamma}_{root}$ (fixed/displacement) and ${\Gamma}_{n}$ (pressure).

**Figure 3.**Workflow of the algorithm depicting the iterative process of estimating the mechanical parameter $c$ and the scaling factor $\gamma $. The subscript $\mathit{\alpha}$ in ${\mathit{x}}_{\mathit{\alpha},\mathit{d}\mathit{i}\mathit{a}\mathit{s}}^{\mathit{e}\mathit{x}\mathit{p}}$ and ${\mathit{U}}_{\mathit{\alpha}}^{\mathit{d}\mathit{i}\mathit{s}\mathit{p}}$, depict the type of geometry under consideration. In this paper, $\mathit{\alpha}$ = cylinder and aorta.

**Figure 4.**Steps involved to generate reference data for cylindrical geometry for a given unloaded configuration. (

**A**) Schematic representation of the unloaded configuration $({\mathsf{\Omega}}_{0}$ = $\Omega ({\mathit{X}}_{\mathit{c}\mathit{y}\mathit{l}\mathit{i}\mathit{n}\mathit{d}\mathit{e}\mathit{r}}^{\mathit{u}\mathit{n}\mathit{l}\mathit{o}\mathit{a}\mathit{d}},0))$, diastolic configuration $({\mathsf{\Omega}}_{dias}=\Omega ({\mathit{x}}_{\mathit{c}\mathit{y}\mathit{l}\mathit{i}\mathit{n}\mathit{d}\mathit{e}\mathit{r},\mathit{d}\mathit{i}\mathit{a}\mathit{s}}^{exp},{\stackrel{=}{\sigma}}_{dias}^{exp}))$ and systolic configuration ${\mathsf{\Omega}}_{sys}=\Omega ({\mathit{x}}_{\mathit{c}\mathit{y}\mathit{l}\mathit{i}\mathit{n}\mathit{d}\mathit{e}\mathit{r},\mathit{s}\mathit{y}\mathit{s}}^{exp},{\stackrel{=}{\sigma}}_{sys}^{exp})$. (

**B**) Finite element simulations depicting the nodes in unloaded (${\mathit{X}}_{\mathit{c}\mathit{y}\mathit{l}\mathit{i}\mathit{n}\mathit{d}\mathit{e}\mathit{r}}^{\mathit{u}\mathit{n}\mathit{l}\mathit{o}\mathit{a}\mathit{d}}$), diastolic (${\mathit{x}}_{\mathit{c}\mathit{y}\mathit{l}\mathit{i}\mathit{n}\mathit{d}\mathit{e}\mathit{r},\mathit{d}\mathit{i}\mathit{a}\mathit{s}}^{\mathit{e}\mathit{x}\mathit{p}}$) and systolic configurations (${\mathit{x}}_{\mathit{c}\mathit{y}\mathit{l}\mathit{i}\mathit{n}\mathit{d}\mathit{e}\mathit{r},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{e}\mathit{x}\mathit{p}}$) associated with ${\mathsf{\Omega}}_{0}$, ${\mathsf{\Omega}}_{dias}$ and ${\mathsf{\Omega}}_{sys}$.

**Figure 5.**(

**A**) Four-step procedure to generate reference data for aortic geometry: (i) initial MRI geometry ${\mathsf{\Omega}}_{MRI}$ is inflated (till systolic pressure) to obtain an intermediate geometry ${\mathsf{\Omega}}_{int}$; (ii) unloaded configuration ${\mathsf{\Omega}}_{0}$ is obtained by subtracting the scaled nodal displacements from MRI geometry ${\mathsf{\Omega}}_{MRI}$; (iii) subsequently pressurising ${\mathsf{\Omega}}_{0}$ with diastolic and (iv) systolic pressures, resulted in ${\mathsf{\Omega}}_{dias}$ and ${\mathsf{\Omega}}_{sys}$. (

**B**) Finite element simulations depicting the nodes in unloaded (${\mathit{X}}_{\mathit{a}\mathit{o}\mathit{r}\mathit{t}\mathit{a}}^{\mathit{u}\mathit{n}\mathit{l}\mathit{o}\mathit{a}\mathit{d}}$), diastolic (${\mathit{x}}_{\mathit{a}\mathit{o}\mathit{r}\mathit{t}\mathit{a},\mathit{d}\mathit{i}\mathit{a}\mathit{s}}^{\mathit{e}\mathit{x}\mathit{p}}$) and systolic configurations (${\mathit{x}}_{\mathit{a}\mathit{o}\mathit{r}\mathit{t}\mathit{a},\mathit{s}\mathit{y}\mathit{s}}^{\mathit{e}\mathit{x}\mathit{p}}$) associated with ${\mathsf{\Omega}}_{0}$, ${\mathsf{\Omega}}_{dias}$ and ${\mathsf{\Omega}}_{sys}$.

**Table 1.**Initial guesses and estimated outcomes of $c$ and $\gamma $ parameters for reference values of $c$ for cylindrical and aortic geometries.

Geometry | $\mathbf{Reference}\mathit{c}$ (MPa) | Initial Guesses | Estimated Values | $\mathsf{\Delta}\mathbf{c}$ (MPa) | $\mathbf{Accuracy}\mathbf{\left(}\mathit{\zeta}\mathbf{\right)}$ | ||
---|---|---|---|---|---|---|---|

$\mathit{c}$ (MPa) | $\mathit{\gamma}$ | $\mathit{c}$ (MPa) | $\mathit{\gamma}$ | ||||

0.5 | 1 | 0.930 | 1.260 | 0.030 | |||

0.9 | 0.75 | 0.75 | 0.908 | 1.289 | 0.008 | 98.43% | |

1 | 0.5 | 0.911 | 1.285 | 0.011 | |||

1.5 | 0.25 | 0.914 | 1.281 | 0.014 | |||

0.5 | 1 | 1.032 | 1.293 | 0.032 | |||

Cylinder | 1 | 0.75 | 0.75 | 1.008 | 1.322 | 0.008 | 98.50% |

1 | 0.5 | 1.010 | 1.320 | 0.010 | |||

1.5 | 0.25 | 1.010 | 1.317 | 0.010 | |||

0.5 | 1 | 1.134 | 1.320 | 0.034 | |||

1.1 | 0.75 | 0.75 | 1.107 | 1.350 | 0.007 | 98.50% | |

1 | 0.5 | 1.109 | 1.347 | 0.009 | |||

1.5 | 0.25 | 1.110 | 1.345 | 0.010 | |||

0.5 | 1 | 0.895 | 0.303 | 0.005 | |||

0.9 | 0.75 | 0.75 | 0.897 | 0.302 | 0.003 | 99.15% | |

1 | 0.5 | 0.879 | 0.313 | 0.021 | |||

1.5 | 0.25 | 0.905 | 0.297 | 0.005 | |||

0.5 | 1 | 0.996 | 0.302 | 0.004 | |||

Aorta | 1 | 0.75 | 0.75 | 0.997 | 0.302 | 0.003 | 99.22% |

1 | 0.5 | 0.979 | 0.312 | 0.021 | |||

1.5 | 0.25 | 1.003 | 0.298 | 0.003 | |||

0.5 | 1 | 1.096 | 0.302 | 0.004 | |||

1.1 | 0.75 | 0.75 | 1.098 | 0.301 | 0.002 | 99.20% | |

1 | 0.5 | 1.079 | 0.311 | 0.021 | |||

1.5 | 0.25 | 1.105 | 0.297 | 0.005 |

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**MDPI and ACS Style**

Parikh, S.; Moerman, K.M.; Ramaekers, M.J.F.G.; Schalla, S.; Bidar, E.; Delhaas, T.; Reesink, K.; Huberts, W.
Biomechanical Characterisation of Thoracic Ascending Aorta with Preserved Pre-Stresses. *Bioengineering* **2023**, *10*, 846.
https://doi.org/10.3390/bioengineering10070846

**AMA Style**

Parikh S, Moerman KM, Ramaekers MJFG, Schalla S, Bidar E, Delhaas T, Reesink K, Huberts W.
Biomechanical Characterisation of Thoracic Ascending Aorta with Preserved Pre-Stresses. *Bioengineering*. 2023; 10(7):846.
https://doi.org/10.3390/bioengineering10070846

**Chicago/Turabian Style**

Parikh, Shaiv, Kevin M. Moerman, Mitch J. F. G. Ramaekers, Simon Schalla, Elham Bidar, Tammo Delhaas, Koen Reesink, and Wouter Huberts.
2023. "Biomechanical Characterisation of Thoracic Ascending Aorta with Preserved Pre-Stresses" *Bioengineering* 10, no. 7: 846.
https://doi.org/10.3390/bioengineering10070846