## 4 Discussion
### Buckling
#### 4.3.1 Eln+/+ Mouse Model  
The FEM calculations reveal that all the laws analyzed for the Eln+/+ mice are capable of predicting buckling.  
The loss of stability of the arteries, which have been initially stretched before inflation, typically has the form of a single half-wave mode, as shown in Figure 7a. In such a case, additional prestress forces are introduced to the analyzed system before the inflation process is started. In effect, the relatively soft carotid artery becomes stiffer and buckles globally due to increasing pressure. This behavior is in accordance with the research from [19] and our preliminary experimental observations conducted on stretched mouse carotid arteries, as shown in Figure 1c. However, when there is no initial stretch (\(\lambda_{z}=1.0\)) and the Fung law is used, a slightly different response is observed and the artery deforms in such a way that three half-waves are observed along the carotid artery, as presented in Figure 7b. When the analysis is started and the internal pressure is increased, the artery is also compressed in the longitudinal direction, as it is tied tightly to the cannulas and no stretch has been applied. In consequence, due to the low stiffness of the material, local buckling along the artery in the form of wrinkles is observed at low internal pressure values. As internal pressure is increased and also due to the fact that the artery is slightly curved, the artery finally experiences a form of deformation which resembles a three half-wave mode but which is mainly caused by the initial aforementioned local buckling. This has been observed in the FEM analysis. The deformations of the artery shortly before their rapid growth is started are shown in Figure 7c. The deformations in Figure 7c have been multiplied by a scale factor of 4 to better depict the local buckling waves and the process of the artery final shape formation. Although such a deformation has not been reported in [19], we claim that it may occur and, in addition, we observed a similar one during the preliminary testing of the CB57 BL/6 mice carotid arteries that were not initially stretched. This is depicted in Figure 7d.  
Comparison of the values of \(p_{cap}\) and \(p_{uA,FEM}\) for the Eln+/+ mouse model indicates that the FEM predictions underestimate the buckling pressures rather noticeably for \(\lambda_{z}\) being 1.0 or 1.2, when the Fung law is used, since the relative errors calculated between these values are 29.9% and 42.3%, respectively. This FEM response is similar to the one determined in analytical calculations presented in [19]. The FEM analysis with the Fung law is accurate when the initial stretch is 1.4, because the relative error is only 6.7%. This prediction is also better than the one calculated analytically and presented in [19], which overestimated the buckling pressure by -17%. In the case of the application of the proposed set of constants of the Holzapfel law, the actual computational results seem to be more accurate than the ones obtained with the help of the Fung law. The buckling pressure is overestimated by 14.9%, when the artery is not initially stretched (\(\lambda_{z}=0\)). The relative error in the buckling analysis of the carotid artery when the longitudinal stretch is \(\lambda_{z}=1.2\) is even lower and equals only 6.3%. The FEM result is almost the same as the experimental measurement for the case of initial stretch of 1.4. This suggests that the Holzapfel constitutive relation may be a better choice than the Fung law, when the buckling of the carotid arteries is to be analyzed. Furthermore, the analysis proves that the new set of Holzapfel constants for the Eln+/+ mice is effective for the description of complex responses of the carotid artery subjected to a change in internal pressure.