## 2 Materials and Methods
### The Analysis and Validation  
Nonlinear static calculations are performed for the problems defined above. The Newton-Raphson method is used to solve the nonlinear equilibrium equations. Therefore, the result is obtained as a series of subsequent increments, with iterations to obtain equilibrium within each increment. Material and geometrical nonlinearities are included in the calculations. All analyses are conducted in two stages. First, each cylinder is stretched along its axis. Then, it is loaded with internal pressure. Such a sequence of computational steps enables the efficient recreation of the conditions that the artery is subjected to. The values of the longitudinal axial stretches assigned to the computational model are taken from the literature [19; 23]. It is worth mentioning that various stretches of mice carotid arteries have been measured experimentally for different mouse models and are available in the literature. In [23], one may find that the in vivo stretches \(\lambda_{z}\)in-vivo range from 1.71 to 1.84 for the wild-type, mdx, and sgcd\(-\)/\(-\) mice and are rather high. In [19], the \(\lambda_{z}\)in-vivo stretches, resulting from image artery analysis, are noticeably lower and lie between 1.04 and 1.53 (for Eln+/+, Eln+/\(-\), Fbln5+/+, and Fbln5\(-\)/\(-\) models); thus, some of them are very small. The longitudinal stretch will affect the mechanical response of the artery. If a higher initial longitudinal stretch is applied, then a larger initial reduction in the external diameter of the carotid artery will be observed. The initial longitudinal stretch also influences buckling pressures. If the carotid artery is subjected to a greater initial stretch, then buckling pressure loads will also become greater, due to an additional pretension effect. Therefore, characteristic initial stretch values are assigned for each individual mouse model under consideration, in accordance with the values recommended in the reference paper [19; 23] to appropriately reflect the mechanical behavior of the artery and to compare the current results with the reference values.  
The results of the current FEM nonlinear static analysis are compared with the experimental values collected in the reference articles, and the validation of the constitutive laws is conducted. To make this process as simple and reliable as possible, raw data coming from the measurement devices are analyzed. Especially, information is studied and compared about the change in the external diameter artery of the artery under increasing pressure. A set of constants is treated as perfectly valid when the root mean square error (RMSE) between the FEM predictions and the experimental response is less than 20 \(\upmu\)m. This threshold value is proposed because it is 5% of a typical outer diameter of the mouse carotid artery, which we find to be approximately 400 \(\upmu\)m according to the works [19; 23]. Moreover, the RMSE is normalized by the mean observation value and thus the nRMSE parameter is introduced to ease the process of the FEM estimations' accuracy analysis. Therefore, not only the average difference between values predicted by the FEM model and the observed experimental ones is measured (RMSE, which unit in this case is [\(\upmu\)m]), but its non-dimensional variant (nRMSE) is added. The nRMSE enables the fast assessment of the correctness of results.  
In the case of the buckling analysis, the buckling pressures are derived from equilibrium paths. The pressure value at the moment when the lateral displacement of the deformed artery axis equals the radius of the unloaded artery is also checked, since it has been reported and used in [19] to determine experimental buckling pressures and is a good reference for the calculations performed here.