# Heterogeneity of Ex Vivo and In Vivo Properties along the Length of the Abdominal Aortic Aneurysm
## 2 Materials and Methods
### Ex Vivo Analysis  
#### 2.2.1. _Biaxial Testing and Data Analysis_  
Biaxial testing was carried out on a four-motor biaxial testing system (ElectroForce Systems, TA Instruments, Springfield, MO, USA, Figure 1b), designed to allow independent control of each motor. Full-thickness tissue samples were cut into squares (target dimensions--10 \(\times\) 10 mm) and mounted to the four linear motors via sutures and hooks (Figure 1b). Four hooks per side were used to minimize edge effects and ensure homogeneous strain distribution at the center of specimens [24]. In the case of samples with dimensions too small to accommodate four hooks, two hooks per side were used as recommended in [PERSON] et al., 2016 [24]. Samples were carefully oriented with direction (11) corresponding to the direction along the circumference of the aorta (circumferential direction), and direction (22) along the length of the aorta (longitudinal direction). The thickness and side-to-side distances (where sides were assumed along the points of hooks attachment) were measured using a caliper. Five dots were drawn on the sample central region using a surgical skin marker to provide distinguishable marks that can be tracked by the overhead camera. A high-resolution digital video extensometer (DVE) camera (640 \(\times\) 480 pixel resolution, 55 m focal length, 200 frames per second) mounted above the test specimen was used to track the dots, producing deformation measures at the central region of the sample. When testing a tissue's response, the in vivo environment must be mimicked. As such, the sample was fully immersed in a (PBS) solution at 37 \({}^{\circ}\)C and pH 7.4. Prior to testing, a pre-load of 0.05 N was applied to avoid sagging effects due to sutures slack. Loading of the sample was recorded using two load cells (22 N). At first, due to AAA tissue fragility and a high risk of premature tearing, displacements resulting in a simultaneous 20% increase of the distances between parallel sides of the sample were assigned. Then, to achieve a wider range of mechanical responses, the sample was extended with different ratios of these initial displacements. Next, the procedure was repeated for larger loadings, namely 40% and 60%. For each test, samples were conditioned by running the test for 5 cycles at a rate of 0.6 mm/s. Over 30% of the specimens failed prematurely due to disruption of tissue by hook tear-through or delamination (Figure 1b). For the surviving specimens, the data was collected from the last cycles of the tests associated with the 40% displacement protocol. The principal stretches \(\lambda_{11}\) and \(\lambda_{22}\) were calculated from the deformations of the center of the specimen enclosed by the tracked markers [25]. The Cauchy stresses \(\sigma_{11}\) and \(\sigma_{22}\) were determined from the forces required to displace the specimen's sides along two directions multiplied by the corresponding principal stretch and divided by the undeformed area (i.e., the corresponding side length times thickness). The three-dimensional stress-stretch curves derived from the biaxial tests for a representative specimen are shown in Figure 1c. We specify that the assigned displacements do not necessarily produce equivalent deformation data points (\(\lambda_{11},\lambda_{22}\)) for different specimens because we do not control \(\lambda_{11}\):\(\lambda_{22}\) ratios [25]. In order to achieve mutual comparison between specimens, one option is to apply constitutive modelling and produce comparable stress-stretch data points, for which different effective mechanical properties such as stiffness and transition strains could be determined. To bypass the constitutive modelling stage, we reached the same result by fitting a surface to the data in the three-dimensional space and interpolating the stresses \(\sigma_{11}^{\text{eq}}\) and \(\sigma_{22}^{\text{eq}}\) at the equi-stretch deformation state (i.e., \(\lambda_{11}=\lambda_{22}=\lambda^{\text{eq}}\)). This analysis was conducted in MATLAB (version R2018a; MathWorks, Natick, MA, USA).