^{1}

^{*}

^{2}

^{2}

^{2}

^{3}

The elastic behavior of arteries is nonlinear when subjected to large deformations. In order to measure their anisotropic behavior, planar biaxial tests are often used. Typically, hooks are attached along the borders of a square sample of arterial tissue. Cruciform samples clamped with grips can also be used. The current debate on the effect of different biaxial test boundary conditions revolves around the uniformity of the stress distribution in the center of the specimen. Uniaxial tests are also commonly used due to simplicity of data analysis, but their capability to fully describe the in vivo behavior of a tissue remains to be proven. In this study, we demonstrate the use of inverse modeling to fit the material properties by taking into account the non-uniform stress distribution, and discuss the differences between the three types of tests. Square and cruciform samples were dissected from pig aortas and tested equi-biaxially. Rectangular samples were used in uniaxial testing as well. On the square samples, forces were applied on each side of edge sample attached with hooks, and strains were measured in the center using optical tracking of ink dots. On the cruciform and rectangular samples, displacements were applied on grip clamps and forces were measured on the clamps. Each type of experiment was simulated with the finite element method. The parameters of the Mooney-Rivlin constitutive model were adjusted with an optimization algorithm so that the simulation predictions fitted the experimental results. Higher stretch ratios (>1.5) were reached in the cruciform and rectangular samples than in the square samples before failure. Therefore, the nonlinear behavior of the tissue in large deformations was better captured by the cruciform biaxial test and the uniaxial test, than by the square biaxial test. Advantages of cruciform samples over square samples include: 1) higher deformation range; 2) simpler data acquisition and 3) easier attachment of sample. However, the nonuniform stress distribution in cruciform samples requires the use of inverse modeling adjustment of constitutive model parameters.

Due to the difficulty of testing soft tissues such as arteries in vivo, mechanical testing of arteries in vitro is often preferred. Uniaxial and biaxial testing has been used to characterize anisotropic materials such as arteries, although methodological aspects are still debated [

Inverse modeling has been used to obtain material properties from experiments where an analytical solution is difficult to obtain [

Seven thoracic aortas were harvested within the day of death of pigs from a local slaughterhouse. Upon arrival, any visible connective tissue was dissected away from the external wall of the artery. Then the artery was cut open along its length, and cut out in rectangular, square and cruciform shapes (

The biaxial test bench (Bose Corporation, Minnetonka, MN), shown in _{11}, ε_{22}) and the shear strain (ε_{12}).

Eight rectangular samples were tested uniaxially: four in the circumferential direction and four in the axial direction. Preliminary destructive tests revealed that failure rarely occurs below a nominal stretch ratio of 2.00.

Therefore a 1.80 nominal stretch ratio was applied (8.00 mm displacement by each actuator). The forces were measured with one load cell. The original length of the sample was taken as the distance between the pair of grips (

Four cruciform samples were tested biaxially. An equal displacement on each of the four grips was applied. The forces at the grips were measured with the two load cells. In cruciform samples, the maximum displacement allowed by each actuator (12 mm) was applied, creating a non uniform strain distribution in the sample. This corresponds to an average nominal stretch ratio of about 1.60, calculated using the distance between facing grips.

Four square samples were also tested biaxially. Threads were used to attach four fishing hooks to each side of the sample. The threads were connected to the arms of the actuators with a pulley system that allowed maintaining an equal tension in each thread. A specially designed jig was used to ensure that the hooks were always placed at the same position on the sample. The forces at the arms of the actuators were measured with the two load cells. Four dots, each one located 8.5 mm apart of the axis center, were printed in a square pattern on the center of the top surface of the sample using a water resistant oil-based quick drying ink. The deformation at the center of the sample was measured using the optical strain extensometer.

In preliminary tests, failure at the hook puncture hole due to the high stress concentration occurred between 18 and 22 mm displacement. Therefore, the displacement was limited to 16 mm, corresponding to an average nominal stretch ratio of 1.64, calculated from the distance between the hooks on the parallel edges of the sample.

For all tests, a triangular wave form displacement at a one Hertz frequency was applied for 10 pre-condition- ing cycles [

In order to adjust the parameters of this model, the inverse modeling technique was used. In this approach, finite element simulations of the experiment are performed by applying either displacement or force conditions, on a mesh of the same size and shape as the sample. In the present work, IMI’s large deformation finite element software was used, with a convergence criterion of 0.01 mm on nodal displacements, of 1% on nodal forces. This software has been validated for predicting molten polymer forming processes such as blow molding and thermoforming [

A set of initial values for the material properties is input. The simulation predicts the reaction forces at the boundaries or the displacement of the nodes corresponding to the ink dots in the center of the sample. The material properties are then adjusted using an optimization technique until a set of force-displacement experimental data matches the values calculated by the model. In this work, the sequential quadratic programming (SQP), a second order optimization method of Design Optimization Tools (DOT) software [

For the case of biaxial cruciform extension (

where c is the vector of unknown material properties; L_{x} and L_{y} are the applied displacements; F_{x} and F_{y} are the experimentally measured reaction forces at the grips; and f_{x}(L_{x}, c) and f_{y}(L_{y}, c) are the reaction forces predicted by the finite element model. N is the total number of data points (L, F) gathered in the experiment. Mooney- Rivlin parameters (a_{10}, a_{01}, a_{11}, a_{20}, a_{30}) are the components of vector c. In order to obtain f_{x}(L_{x}, c) and f_{y}(L_{y}, c), first, an initial guess of material parameters is made; second, the model predicts Total (T), Cauchy (σ), and second Piola-Kirchhoff (S) stresses by using the first and second invariants of the Green-Cauchy tensor defined for

Mooney-Rivlin Strain Energy Density Function (SEDF). Finally, the reaction forces are estimated using sample’s area.

Therefore, different material properties were obtained for the circumferential and for the axial directions, from the uniaxial extension experimental data.

In biaxial tests with square samples, forces were applied to each one of the nodes corresponding to the location of the hook sites, and strains were predicted at nodes corresponding to the location of the ink dots. Hence, the following objective function was minimized:

where

The finite element meshes used for simulation of uniaxial, biaxial cruciform and biaxial square tests are shown in

Type of test | Sample shape | Sample attachment | Applied conditions | Predicted values |
---|---|---|---|---|

Uniaxial | Rectangular | Grips | Displacement | Reaction forces at grips |

Biaxial | Cruciform | Grips | Displacement | Reaction forces at grips |

Biaxial | Square | Hooks and sutures | Forces | Strain at center of sample |

The mean thickness of all tested specimens tested was 2.4 ± 0.4 mm.

^{4} kPa to 10^{4} kPa value range. Repeated optimization with different initial parameter values consistently converged towards the same solution. The computed circumferential and axial forces obtained using the Mooney-Rivlin fitted parameters from

Mooney-Rivlin | Parameters (kPa) | |||
---|---|---|---|---|

Uniaxial circumferential | Uniaxial axial | Biaxial cruciform | Biaxial square | |

a_{10} | 18.15 | 8.13 | 1.32 | 127.52 |

a_{01} | 16.80 | 8.94 | 0.63 | −95.24 |

a_{11} | 2.48 | 0.0054 | 2.90 | 0.08 |

a_{20} | 1.96 | 0.0045 | 3.85 | 49.73 |

a_{30} | 12.27 | 9.32 | 30.94 | 1.60 |

and biaxial square are shown in _{30}, using three different initial guesses.

In

In

Small sample variability between the axial force-stretch curves of the samples tested uniaxially and biaxially was found in this work. Circumferential data had a slightly higher variability, particularly in the biaxial square data. Prendergast et al. [

The inverse modeling technique was demonstrated to be a useful tool to tune constitutive models to experimental data. However, Mooney-Rivlin parameters obtained from each type of test vary greatly in this study. They are also very different from those reported by Prendergast et al. [

model fitted to the cruciform biaxial data predicted well the uniaxial behavior up to stretch ratios of 1.5. This suggests that additional information on the material behaviour at higher stretch ratios than can be reached in biaxial tests might be obtained from uniaxial tests. Therefore, the inverse modeling method could be further improved by simultaneously fitting the results of complementary uniaxial and biaxial tests.

The agreement between biaxial and uniaxial simulations using the same material parameters was better when the parameters were obtained from the biaxial experiments with cruciform samples, than from the biaxial experiments with square samples. This might be due to the fact that the biaxial tests with the square samples do not take into account the high stretch ratios outside of the square area defined by the four ink dots. Because of stress concentration, failure occurred at the hook puncture sites before a stretch ratio of 1.5 could be reached at the center of the specimen. In contrast, the inverse modeling technique used with cruciform biaxial tests allows taking into account the whole behavior of the sample, including high stresses on curved borders, therefore allowing to reach maximum stretch ratios up to 1.64 in the sample.

Waldman and Lee [

Advantages of cruciform samples over square samples include: 1) characterization over a higher deformation range, hence better prediction of behavior at large deformations; 2) simpler data acquisition through the use of actuator displacement rather than extensovideometry; 3) easier attachment of sample with grip clamps rather than sutures. However, the nonuniform stress distribution in cruciform samples requires the use of inverse modeling adjustment of constitutive model parameters.

We have reported uniaxial and biaxial experimental data of porcine aortas in vitro. We found that the highly nonlinear behavior of the arterial wall under large deformation was more easily and precisely measured biaxially using cruciform samples with grip clamps, than using square samples with hooks and sutures. Inverse modeling can be used to fit the material parameters of constitutive models, such as Mooney-Rivlin, to biaxial data measured on cruciform samples. More work is required in order to define the optimal set of experiments, and thus, to use the inverse modeling method to fit anisotropic, viscoelastic or active (muscle tone) models of soft tissues.

This research was possible thanks to the PhD scholarship given by the Mexican Council of Science and Technology (Consejo Nacional de Ciencia y Tecnología), CONACYT; and the support of the Industrial Materials Institute (IMI), CNRC-NRC and the Department of Biological & Chemical Engineering of the University of British Columbia. The authors would also like to thank Marc-Andre Rainville for his guidance and assistance in sample preparation and mechanical testing.

Jorge O. ViruesDelgadillo,SebastienDelorme,FrancisThibault,RobertDiRaddo,Savvas G.Hatzikiriakos, (2015) Large Deformation Characterization of Porcine Thoracic Aortas: Inverse Modeling Fitting of Uniaxial and Biaxial Tests. Journal of Biomedical Science and Engineering,08,717-732. doi: 10.4236/jbise.2015.810069

In the last years, several SEDF (Strain Energy Density Function), which are a measure of the energy stored in the material as a result of deformation, were developed and used to predict the mechanical properties of soft tissues. Most of these models make the assumption that the material is isotropic, instead of considering the anisotropy (due to the organized arrangement of the collagen components) of the arterial wall. Nevertheless, they can be used to simulate the deformation of the arterial wall in special cases, such as that corresponding to axial extension of arterial samples. SEDF selected is used to estimate Second Piola-Kirchhoff stress tensor (S) as follows:

where W is the Strain Energy Density Function;

In finite deformations the second Piola-Kirchhoff stress tensor is used to relate forces in the reference configuration to areas in the same reference configuration. In contrast, the Cauchy stress tensor (σ) is used to express the stress relative to the current configuration. Under infinitesimal deformations both stresses are identical. Second Piola-Kirchhoff stress tensor is related to Cauchy and total stress tensors as follows:

where σ_{ij} are the components of the Cauchy stress tensor (σ); T_{ij} are the components of the total stress (T); the Jacobian (J) is defined as the determinant of the finite strain deformation tensor; and p is the hydrostatic pressure. In incompressible materials the Jacobian is equal to one (J = detF = 1).

Subject to the regularity assumption that W is continuously differentiable infinitely many times with respect to the 1^{st} (I_{1}), 2^{nd} (I_{2}), and 3^{rd} (I_{3}) invariants of the Green-Cauchy tensor (right), we may write W as an infinite series in powers of

The Mooney-Rivlin model selected in this investigation is a function of the 1^{st} (I_{1}) and 2^{nd} (I_{2}) invariants of the Green-Cauchy tensor (right). Its strain energy function is written as:

Moreover, I_{1} and I_{2} can be written as a function of the principal stretch ratios:

The Mooney-Rivlin model selected has five parameters (a_{10}, a_{01}, a_{11}, a_{20}, a_{30}), all of them in MPa. The corresponding stress component of the stress tensor can be obtained when the strain density function is differentiated with respect to the corresponding strain component. This procedure is done by the finite element method by solving the equations described above.