Viscoelastic Behavior of Porcine Arterial Tissue: Experimental and Numerical Study

Abstract

Background

The viscoelastic properties of aortic tissue dictate vessel behavior in certain disease states, injury modalities, and during some endovascular procedures.

Objective

We characterized the viscoelastic response of porcine abdominal aortic tissue via test and simulation to demonstrate the utility of a viscoelastic anisotropic (VA) constitutive model.

Methods

In this study, the measured stress relaxation response for five samples and uniaxial tensile testing for one sample measured with the digital image correlation (DIC) technique were used to identify material parameters for the VA model using an inverse method through finite element analysis (FEA).

Results

Based on the stress relaxation test, the values of the stress-like parameter $\mu$, relative stiffness of the fibers ${k}_{1}$, dimensionless parameter ${k}_{2}$, angle of fibers $\gamma$, dispersion parameter $\kappa$, relaxation times for the ground matrix ${\mathsf{T}}_{g1}$ and collagen fibers ${\mathsf{T}}_{f1}$ and the dimensionless parameters for the ground matrix ${\beta }_{g1}$ and collagen fibers ${\beta }_{f1}$ for 0 degree specimen orientation were 12.1 ± 8.96 kPa, 77.3 ± 46.4 kPa, 0.032 ± 0.043, 30.25 ± 6.81°, 0.19 ± 0.06, 0.028 ± 0.016 s, 92.76 ± 26.51 s, 3.46 ± 3.78, 0.24 ± 0.08 and for 90 degree specimen orientation were 13.7 ± 7.7 kPa, 72.6 ± 35.4 kPa, 2.18 ± 4.12, 55.35 ± 7.12°, 0.22 ± 0.06, 23.51 ± 38.90 s, 81.52 ± 29.16 s, 5.14 ± 8.72, 0.21 ± 0.05, respectively. The validation revealed an overall good agreement from cycles 2 and 3 based on uniaxial tensile tests and surface strains data from DIC measurements with the material parameters from inverse analysis using FEA for the response in cycle 1.

Conclusions

The identified material model and numerical simulations provide a comprehensive description of the viscoelastic behavior of the aortic wall tissue and a quantitative understanding of the spatial and directional variability underlying aortic tissue mechanical behavior.

Appendix: Verification of Viscoelastic Anisotropic (VA) Model

The VA model is verified by comparing numerical and analytical results under the assumption that the material is under plane strain deformation. A unit-cubic model of fiber-reinforced material was implemented with two families of fibers distributed 47° with respect to the circumferential direction (Fig. 2(a)). The surfaces at the x-axis and z-axis are fixed along the x and z directions, respectively. The assumption of material incompressibility, implying that ${\uplambda }_{\mathrm{x}}{\uplambda }_{\mathrm{y}}{\uplambda }_{\mathrm{z}}=1$. When the cubic model was stretched along the x-axis with a stretch ratio ${\lambda }_{x}$, the deformation gradient was calculated as follows:

$${\varvec{F}}=\left[\begin{array}{ccc}{\lambda }_{x}& 0& 0\\ 0& {1/\lambda }_{x}& 0\\ 0& 0& 1\end{array}\right]$$

(28)

Substituting equation (9) into equation (21), the Cauchy stress was calculated,

$${{\varvec{\sigma}}}_{m+1}={\left({{{\varvec{\sigma}}}_{ }}_{vol}^{\infty }{{+{\varvec{\sigma}}}_{ }}_{g}^{\infty }+{{+{\varvec{\sigma}}}_{ }}_{f}^{\infty }+{{\varvec{\sigma}}}_{visf1 }+{{\varvec{\sigma}}}_{visg1 }\right)}_{m+1}$$

(29)

For the volumetric component of the Cauchy stress tensor,

$${{{\varvec{\sigma}}}_{ }}_{vol}^{\infty }=\frac{1}{D}\left(J-\frac{1}{J}\right)J{\varvec{I}}=-p{\varvec{I}}$$

(30)

Since the surfaces perpendicular to the y-axis are traction free,

$$0={\sigma }_{m+1 (\mathrm{yy})}=-p+{\left({{\sigma }_{ }}_{g}^{\infty }+{{+\sigma }_{ }}_{f}^{\infty }+{\sigma }_{visf1 }+{\sigma }_{visg1 }\right)}_{m+1\left(\mathrm{yy}\right)}$$

(31)

where $p={\left({{\sigma }_{ }}_{g}^{\infty }+{{+\sigma }_{ }}_{f}^{\infty }+{\sigma }_{visf1 }+{\sigma }_{visg1 }\right)}_{m+1 (\mathrm{yy})}$.

With the material parameters of specimen #6–0 from Table 3, the comparison of Cauchy stresses in x direction between analytical and numerical results is shown in Fig. 8. The results show that the numerical predictions of stress vs. stretch ratio curves match the analytic outputs.

Fig. 8

Illustrative plots of Cauchy stress-stretch relationship in x-axis direction for analytical (empty circle) and numerical results (red line)