# Determination of the Material Parameters in the Holzapfel-Gasser-Ogden Constitutive Model for Simulation of Age-Dependent Material Nonlinear Behavior for Aortic Wall Tissue under Uniaxial Tension

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## Abstract

**:**

## 1. Introduction

^{2+}channel-mediated contractions differed between the smaller, muscular arteries (femoral and mesenteric arteries) and the larger, elastic conduit vessels (aorta and carotid artery) of mice. As a result, they assumed that the different physiological behavior of elastic and muscular arteries of young adults were linked to the well-known observation that arterial stiffness develops differently with aging.

## 2. Constitutive Model

#### 2.1. Strain-Based Formula

**F**is the deformation gradient, which is expressed in terms of the displacement vector,

**u**(

**F**= ∇

**u**+

**I**), and

**I**is the identity matrix. Without loss of generality, the strain energy function can be written as follows:

#### 2.2. Invariant-Based Formula

**C**and vectors

**A**

_{α}:

#### 2.3. Anisotropic Hyper-Elastic Materials

_{10}, D, k

_{1}, k

_{2}, and κ are temperature-dependent material parameters; N is the number of families of fibers (N ≤ 3); ${\overline{I}}_{1}$ is the first invariant of $\overline{C}$; ${J}^{el}$ is the elastic volume ratio; and ${\overline{I}}_{4\left(\alpha \alpha \right)}$ are the pseudo-invariants of $\overline{C}$ and ${A}_{\alpha}$. This model presumes that the orientation of the collagen fibers in each family is distributed rotationally symmetrically with respect to the mean preferred orientation. The parameter, κ (0 ≤ κ ≤ 1/3), expresses the level of dispersion in the fiber directions. If $\rho \left(\mathsf{\Theta}\right)$ is the orientation density function that identifies the distribution (it implies the normalized number of fibers with orientations in the range $\left[\mathsf{\Theta},\text{}\mathsf{\Theta}+d\mathsf{\Theta}\right]$ associated with the mean orientation) [24], then the parameter, κ, is defined as follows:

## 3. Experiments and Simulations

#### 3.1. Experiment Details for the Material Test

#### 3.2. Simulation Details for the Material Test

#### 3.3. Results for the Material Test

## 4. Results

#### 4.1. Experiment Details and Trend Lines

#### 4.2. Simulation Results Based on Age

_{1}and k

_{2}, according to age were estimated based on the failure stress and stretch because the parameters were associated with the stress and strain of soft tissue. The material parameter, C

_{10}, according to age was estimated by comparing the stress–stretch curves obtained from the numerical simulation, the porcine experiment, and the reference for the experiment of the aorta [12,15,16,19,29,30]. These simulation results obtained from estimating the material parameters according to age were compared with the trend lines of the maximum and minimum values, and it was clear that the maximum error rates in the trend lines of the maximum and minimum values were 0.0013 and 0.0082, respectively. In addition, according to the line graph, the average failure stress and stretch of the healthy human abdominal aorta specimens decreased from 1.44 MPa–0.85 MPa and from 1.67–1.34, respectively, as the age increased from 46 years–89 years. The trend line ranges of the failure stress and stretch at the age of 46 years were 0.79–2.09 MPa and 1.47–1.87, respectively. At the age of 89 years, the trend line ranges of the failure stress and stretch were 0.20–1.50 and 1.14–1.54, which were 28.2–74.7% and 17.6–22.4% lower than the results at the age of 46 years, respectively.

#### 4.3. Parametric Study for the Material Constants of the HGO Model

_{10}, k

_{1}, and k

_{2}, which are associated with the Young’s modulus, stress, and strain of the material, were examined from the results based on the age and loading direction, as shown in Figure 10. Generally, the maximum and minimum trend lines of the material parameters increased consistently for the ages of 46–89 years. According to the line graph, the values of the material parameter, C

_{10}, increased from 0.190–0.233 and from 0.001–0.005 on the maximum and minimum trend lines, respectively. Moreover, the values of the material parameter, k

_{1}, increased nonlinearly from 0.081–0.282 and from 1.040–8.050 on the maximum and minimum trend lines, respectively. The values of the material parameter, k

_{2}, also increased nonlinearly from 2.530–6.486 and from 8.4–72.0 on the maximum and minimum trend lines, respectively. In particular, for the results for above 78 years old, the value of the material parameter, k

_{2}, on the minimum trend line increased rapidly from 35–72. This is attributable to the samples attaining the failure stress at lower stretch values as the age increased.

## 5. Discussion

_{10}, k

_{1}, and k

_{2}were identified using the uniaxial test for each layer or unified layers of soft biological tissue including aorta wall tissue.

## 6. Concluding Remarks

- According to the uniaxial tensile test results with respect to the loading direction, the circumferentially-oriented strip samples exhibited a maximum tensile strength value of 2.49 MPa, which was 18.3% higher than that for the longitudinally-oriented strip samples, based on the average value of the experimental results. Therefore, the uniaxial tensile stress and stretch in the arterial tissue were identified to be dependent on the fiber orientation.
- In addition, the failure stress and stretch were investigated as a function of age, and the material constants for age were calculated based on the maximum and minimum trend lines. The HGO model was applied to the numerical model for anisotropic hyper-elastic materials during the numerical simulations.
- In the parametric study, the formulae associated with the value of the material constant for the ages of 46–89 years were proposed, and the proportion variance in the dependent variable that was predictable from the independent variable was examined.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 2.**Schematic of the uniaxial tensile test specimens and experimental apparatus. (

**a**) Dimensions of the test specimens; dimensions in millimeters. (

**b**) Closer view of the specimen fixation. (

**c**) Universal testing machine.

**Figure 3.**(

**a**) Boundary and loading conditions and (

**b**) orientations of the fiber and mesh in the tensile test specimen.

**Figure 4.**Circumferential orientation of the collagen fiber in the layer [26].

**Figure 6.**Stress–stretch curves of the porcine arterial tissue according to the loading direction determined from the tensile tests and simulations on (

**a**) circumferentially- and (

**b**) longitudinally-oriented strips.

**Figure 9.**Simulation results under tensile loading along (

**a**) circumferential and (

**b**) longitudinal directions.

**Figure 10.**Variation in the material constants as a function of age in the trend line range ((

**a**) maximum and (

**b**) minimum).

**Figure 11.**Comparison between material parameters ((

**a**) C

_{10}, (

**b**) k

_{1}, and (

**c**) k

_{2}) and formulae based on age.

**Table 1.**Tissue specimens of the porcine abdominal aorta in the tensile test (AC: Aorta and Circumferentially (Load direction), AL: Aorta and Longitudinally (Load direction)).

Specimen | Sample Dimension (in mm) | Tensile Load Direction Relative to Oriented Strip | |||
---|---|---|---|---|---|

Width (Top) | Width (Middle) | Length | Thickness | ||

AC1 | 14.96 | 9.98 | 24.97 | 1.98 | Circumferentially |

AC2 | 15.10 | 10.04 | 25.05 | 2.01 | Circumferentially |

AC3 | 14.98 | 10.01 | 25.03 | 2.01 | Circumferentially |

AC4 | 15.05 | 10.01 | 25.01 | 1.99 | Circumferentially |

AC5 | 15.01 | 9.99 | 25.00 | 2.03 | Circumferentially |

AL1 | 15.02 | 10.02 | 24.98 | 2.01 | Longitudinally |

AL2 | 15.04 | 10.02 | 24.99 | 2.02 | Longitudinally |

AL3 | 14.97 | 10.03 | 25.01 | 1.97 | Longitudinally |

AL4 | 14.99 | 9.98 | 24.98 | 2.01 | Longitudinally |

AL5 | 15.04 | 9.97 | 25.03 | 2.01 | Longitudinally |

**Table 2.**Coefficient and R-squared in the correlation between the values of the material parameter and formulae.

Material Constant | α_{1} | α_{2} | R^{2} | |||

C_{10,max} | 0.001 | 0.144 | 1.0 | |||

C_{10,min} | 0.0001 | 0.0035 | 1.0 | |||

Material Constant | β_{1}, γ_{1} | β_{2}, γ_{2} | β_{3}, γ_{3} | β_{4}, γ_{4} | β_{5}, γ_{5}, γ_{6} | R^{2} |

k_{1,max} | 8.26 × 10^{−9} | −5.11 × 10^{−7} | −2.30 × 10^{−5} | 0.0038 | −0.0315 | 1.0 |

k_{1,min} | 8.58 × 10^{−7} | −0.0001 | 0.0099 | −0.2863 | 3.6079 | 0.9997 |

k_{2,max} | −2.59 × 10^{−7} | 8.92 × 10^{−5} | −0.0092 | 0.4205 | −4.9386 | 0.9999 |

k_{2,min} | 1.97 × 10^{−6} | −0.0006 | 0.0742 | −4.4980 | 135.35, −1611.33 | 0.9988 |

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**MDPI and ACS Style**

Huh, U.; Lee, C.-W.; You, J.-H.; Song, C.-H.; Lee, C.-S.; Ryu, D.-M. Determination of the Material Parameters in the Holzapfel-Gasser-Ogden Constitutive Model for Simulation of Age-Dependent Material Nonlinear Behavior for Aortic Wall Tissue under Uniaxial Tension. *Appl. Sci.* **2019**, *9*, 2851.
https://doi.org/10.3390/app9142851

**AMA Style**

Huh U, Lee C-W, You J-H, Song C-H, Lee C-S, Ryu D-M. Determination of the Material Parameters in the Holzapfel-Gasser-Ogden Constitutive Model for Simulation of Age-Dependent Material Nonlinear Behavior for Aortic Wall Tissue under Uniaxial Tension. *Applied Sciences*. 2019; 9(14):2851.
https://doi.org/10.3390/app9142851

**Chicago/Turabian Style**

Huh, Up, Chung-Won Lee, Ji-Hun You, Chan-Hee Song, Chi-Seung Lee, and Dong-Man Ryu. 2019. "Determination of the Material Parameters in the Holzapfel-Gasser-Ogden Constitutive Model for Simulation of Age-Dependent Material Nonlinear Behavior for Aortic Wall Tissue under Uniaxial Tension" *Applied Sciences* 9, no. 14: 2851.
https://doi.org/10.3390/app9142851