# Numerical Models Can Assist Choice of an Aortic Phantom for In Vitro Testing

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Description of the Numerical Models

^{3}$\mu $ = 3.5 × 10

^{−3}Pa·s, respectively.

- Dirichlet conditions in portions of the solid domain, i.e., null displacements of the nodes;
- Neumann conditions at the inlet of the fluid domain, i.e., flow condition at the aortic inlet;
- Dirichlet conditions at the outlet of the fluid domain, i.e., pressure condition at the phantom outlet.

_{max}is chosen so that the stroke volume ejected in each cycle is SV = 60 mL. At the outlet of the descending aorta, a constant pressure of 90 mmHg is set, whereas the fluid is initially at rest. We also prevent any displacement along the flux direction of the inlet and outlet surface. The overall duration of the simulation is 5.0 s, to achieve regular periodic flow into the vessel.

#### 2.2. Materials Characterization

^{2}samples are performed, and the averaged values of measured stress and strain are used to compute the Ogden coefficients through the algorithm provided by Abaqus.

_{0}= 35.340 × 10

^{3}mm

^{3}), the fluid pressure is increased from ${p}_{0}$ up to 120 mmHg, i.e., within the physiological aortic pressure range. The pipe compliance, $C$, is then calculated as C = $\Delta V/\Delta p$, $\Delta V$ being the difference between the final and the initial inner volume. Finally, the wall distensibility $AD=C/{V}_{0}$ is computed.

## 3. Results and Discussion

_{max}= 40–45%, whereas in the PSS aorta ε

_{max}= 30%. Moreover, it is interesting to note that the parametric silicone aorta (PMS) shows an average strain closer to the PSB aorta (porcine tissue) than the PSS case (silicone). This may be likely due to the most complex geometry of the arch of the PS case, which hinders the vessel deformation.

^{2}. Some differences in the spatial distribution can be observed, even if to a more minor extent than before. Presumably, they are due to the anatomy of the arch: both PSB and PSS show high helicity along the intrados of the arch up to the beginning of the descending aorta, while in the PMS model, it is this latter region the one that results to be mainly interested.

#### Study limitations and Future Developments

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Sketch of the four aortic arch models: patient-specific anatomy with rigid wall, biological tissue and silicone, and parametric aorta in silicone.

**Figure 4.**Constitutive model characterization of the Ogden (dashed green line, panel (

**a**)) and Holzapfel (red and blue solid line for the longitudinal and circumferential direction, respectively, panel (

**b**)) models. (

**a**) The white circles show the average experimental data of biaxial stress tests of 5 samples of PROCHIMA Cristal Rubber silicone. (

**b**) The white circles and white diamonds show the averaged experimental data of 4 tensile stress tests of porcine tissues along the circumferential and longitudinal directions, respectively. Red and blue bars show the maximum and minimum stress measured in the experiments to the longitudinal and circumferential direction, respectively.

**Figure 5.**Comparison of numerical aortic distensibility between the two constitutive models; black diamonds and black circle are computed $AD$ respectively, for the Ogden (silicone) and Holzapfel (porcine tissue). Grey area shows the range of in vivo human $AD$ measured in young healthy patients [20].

**Figure 6.**Maximum principal strain computed for the PSB model (column on the

**left**), the PSS model (column at

**center**), and PMS model (column on the

**right**) at three time instants: the systolic peak (first row), the backflow peak (second row), and at the end diastole (third row).

**Figure 7.**Principal stress corresponding to the strain field reported in Figure 6.

**Figure 8.**Cross-sectional velocity distribution of the systolic peak in the inlet (first column), in the middle (second column), and in the outlet (third column) of the aortic arch. The blue, red, and green color indicates the PSB case, the PSS case, and the PMS case, respectively.

**Figure 10.**Global performance of the three aortas during the cardiac cycle. (

**a**) Total volume variation; (

**b**) mean pressure at the aortic valve inlet; and (

**c**) mean flow kinetic energy.

**Table 1.**Estimates of the parameters for the silicone (Ogden model, Equation (1)) and the biological tissue (Holzapfel model, Equations (2) and (3)).

Ogden | |||||

μ_{1} | α_{1} | D_{1} (MPa^{−1})
| |||

1.73 × 10^{−1} | 4.39 | 1.193 | |||

Holzapfel | |||||

C (MPa) | k_{1} (MPa)
| k_{2} | k | D (MPa^{−1})
| θ° |

2.89 × 10^{−2} | 1.20 × 10^{−1} | 0.4 | 0.25 | 0.7 | 27 |

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**MDPI and ACS Style**

Comunale, G.; Di Micco, L.; Boso, D.P.; Susin, F.M.; Peruzzo, P.
Numerical Models Can Assist Choice of an Aortic Phantom for In Vitro Testing. *Bioengineering* **2021**, *8*, 101.
https://doi.org/10.3390/bioengineering8080101

**AMA Style**

Comunale G, Di Micco L, Boso DP, Susin FM, Peruzzo P.
Numerical Models Can Assist Choice of an Aortic Phantom for In Vitro Testing. *Bioengineering*. 2021; 8(8):101.
https://doi.org/10.3390/bioengineering8080101

**Chicago/Turabian Style**

Comunale, Giulia, Luigi Di Micco, Daniela Paola Boso, Francesca Maria Susin, and Paolo Peruzzo.
2021. "Numerical Models Can Assist Choice of an Aortic Phantom for In Vitro Testing" *Bioengineering* 8, no. 8: 101.
https://doi.org/10.3390/bioengineering8080101