# Evaluation of Biaxial Mechanical Properties of Aortic Media Based on the Lamellar Microstructure

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Surgical Procedures

**Figure 1.**(

**a**) Removal of adventitial tissue; (

**b**) resulting intima-media composite and loose adventitial layer.

#### 2.2. Tissue Preparation

#### 2.3. Biaxial Testing

**Figure 2.**(

**a**) A schematic of the sample under biaxial stretch; (

**b**) Stained aortic media. Dark regions in this tissue section denote elastic fibers (Layer I), and light colored regions denote interlamellar zones, including collagen bundles, fine elastic fibers and smooth muscle cells (SMCs) (Layer II); (

**c**) Volume fraction computation steps, including radial strip extraction, conversion to black and white and, finally, counting the number of pixels of the dark (Layer I) and light (Layer II) areas, as well as finding their proportion regarding the total number of pixels.

#### 2.4. Histological Staining

#### 2.5. Constitutive Equations and Parameter Estimation

#### 2.5.1. Theoretical Framework

_{1}) and circumferential (λ

_{2}) stretch ratios are calculated from planar biaxial tests, and the radial component (λ

_{3}) is given by the incompressibility constraint ($J=\text{det}(F)={\mathrm{\lambda}}_{1}{\mathrm{\lambda}}_{2}{\mathrm{\lambda}}_{3}=1$):

_{i}is used instead of E

_{ii}to denote diagonal elements of the Green–Lagrange strain tensor for simplicity) and the first invariant of right Green–Cauchy strain tensor (${I}_{1}={\mathrm{\lambda}}_{1}^{2}+{\mathrm{\lambda}}_{2}^{2}+\frac{1}{{\mathrm{\lambda}}_{1}^{2}{\mathrm{\lambda}}_{2}^{2}}$) were calculated for the time increments during the tests. Furthermore, the measured force per unit of the initial orthogonal cross-section of the tissue was calculated as the first Piola stress (P); then, it is converted to the second Piola stress:

_{exp}will denote the stress obtained from the experiments.

#### 2.5.2. Strain Energy Function

_{I}and f

_{II}represent volume fractions of Layers I and II of the overall non-liquid phase, respectively, and W denotes the SEF for Layer I (W

_{I}), Layer II (W

_{II}) and the whole media (W

_{media}). Additionally, E

_{i}stands for the Green–Lagrange strain and ${I}_{1}$ for the first invariant of the right Green–Cauchy strain tensor. Furthermore, c

_{1}, c

_{2}, a

_{1}, a

_{2}, a

_{3}are unknown material parameters. Considering the concentric and parallel configurations of two layers and the fact that layers do not detach during stretch within the physiological range, it is assumed that the deformations of Layers I and II are equal to the deformation of the whole tissue, which is recorded during biaxial tests.

_{media}) can be obtained in terms of layer stresses using Equations (3) and (8). Then, the following set of equations can be written for the computational axial (${S}^{a}$) and circumferential (${S}^{c}$) stresses of layers and the whole wall:

#### 2.5.3. Parameter Estimation

_{i}, I

_{1}). A MATLAB code was developed to search for the optimized set of unknown material parameters that can simultaneously best interpolate experimental data in circumferential and axial directions. Nonlinear regression was utilized to evaluate the error function as the squared difference of computational (S

_{comp}) and experimental (S

_{exp}) stresses in a full range of experimental deformations, Equation (10). The material parameters were updated in each iteration, till the minimum difference criterion was met.

## 3. Results

^{2}) and are reported in the same Table. The computational stress predicted by our model is plotted together with respective experimental data for one of the cases (M25) in Figure 4 to visualize the ability of the model to describe the anisotropic behavior of the aortic media. The same trend was reported for other cases. In addition, Figure 4 illustrates the contribution of Layers I and II on the stress-strain response of the media. In lower strains, Layer I bears higher stresses. By further straining, Layer II exceeds Layer I in stress. This pattern is observed in both the axial and circumferential directions. The intersection point of layer stresses denotes the equal contributions of them to the mechanical behavior of the whole media.

**Figure 3.**Stress-strain response obtained by biaxial tests: (

**a**) axial and (

**b**) circumferential responses. At low strain ranges, similar responses of the three cases in the axial and circumferential directions indicate that components leading to anisotropy do not contribute significantly in this region.

**Figure 4.**Experimental stresses plotted together with computational stresses for M25 (male donor, aged 25), based on the evaluated material parameters. The contribution of Layers I and II on the mechanical behavior of the media is depicted, as well, for the range of experimental strains: (

**a**) axial direction; (

**b**) circumferential direction.

Parameters | Cases | ||
---|---|---|---|

M25 | M28 | M42 | |

f_{I} (-) | 0.2803 ± 0.0113 | 0.2711 ± 0.0100 | 0.2172 ± 0.0067 |

f_{II} (-) | 0.7197 ± 0.0113 | 0.7289 ± 0.0100 | 0.7828 ± 0.0067 |

c_{1} (kPa) | 33.894 | 43.795 | 43.428 |

c_{2} (kPa) | 35.877 | 15.692 | 46.627 |

a_{1} (-) | 1.4655 | 1.9984 | 1.1188 |

a_{2} (-) | 1.85712 | 2.2043 | 1.2865 |

a_{3} (-) | 0.0473 | 0.0080 | 0.2265 |

R^{2} (-) | 0.9946 | 0.9761 | 0.9959 |

## 4. Discussion

## 5. Limitations

## 6. Conclusions

- Visualizing the contributions of Layers I and II for the range of physiologic and supraphysiological deformations.
- Providing an appropriate fit to biaxial test data of the aortic samples in both the axial and circumferential directions for different functional phases, i.e., initial crimp and gradual activation of collagen fibers. The model predictions were in good agreement with the experimental data in both the circumferential and axial directions.

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Taghizadeh, H.; Tafazzoli-Shadpour, M.; Shadmehr, M.B.; Fatouraee, N.
Evaluation of Biaxial Mechanical Properties of Aortic Media Based on the Lamellar Microstructure. *Materials* **2015**, *8*, 302-316.
https://doi.org/10.3390/ma8010302

**AMA Style**

Taghizadeh H, Tafazzoli-Shadpour M, Shadmehr MB, Fatouraee N.
Evaluation of Biaxial Mechanical Properties of Aortic Media Based on the Lamellar Microstructure. *Materials*. 2015; 8(1):302-316.
https://doi.org/10.3390/ma8010302

**Chicago/Turabian Style**

Taghizadeh, Hadi, Mohammad Tafazzoli-Shadpour, Mohammad B. Shadmehr, and Nasser Fatouraee.
2015. "Evaluation of Biaxial Mechanical Properties of Aortic Media Based on the Lamellar Microstructure" *Materials* 8, no. 1: 302-316.
https://doi.org/10.3390/ma8010302