## 2 Materials and Methods  
### The Holzapfel-Gasser-Ogden (HGO) Constitutive Model  
The aortic wall is a biological soft tissue with biodynamic properties such as anisotropy, hyperelasticity, nonlinearity, and incompressibility [20]. The HGO constitutive model, based on hyperelasticity theory, formulates anisotropic material mechanics equations, making it the most effective method for studying aortic wall biomechanics [21,22].  
Let the aortic wall strain energy function be composed of isotropic \(\Psi_{iso}\) and anisotropic \(\Psi_{aniso}\) components, expressed as:  
\[\Psi=\Psi_{iso}+\Psi_{aniso} \tag{1}\]  
According to the HGO model theory,  
\[\Psi_{iso}=C_{10}\big{(}\overline{I}_{1}-3\big{)}+\frac{1}{D}\bigg{(}\frac{I_ {el}^{2}-1}{2}-\ln J_{el}\bigg{)} \tag{2}\]  
\[\Psi_{aniso} =\frac{k_{1}}{2k_{2}}\Big{(}\exp\Big{\{}k_{2}\big{[}k_{2}\big{[}k \overline{I}_{1}+(1-3\kappa)\overline{I}_{4}-1\big{]}^{2}-1\Big{\}}\Big{)} \tag{3}\] \[+\frac{k_{1}}{2k_{4}}\Big{(}\exp\Big{\{}k_{4}\big{[}k\overline{I} _{1}+(1-3\kappa)\overline{I}_{6}-1\big{]}^{2}-1\Big{\}}\Big{)}\]  
\(\Psi_{iso}\) is associated with the mechanical response of elastin and smooth muscle cells in the passive state, and \(\Psi_{aniso}\) is related to the reaction of collagen fibers to the load on the tissue specimen. The TL wall, intimal flap, and FL wall in AD are assumed to be incompressible, meaning tissue shape changes do not affect volume. The Jacobian matrix of the deformation gradient is \(J_{el}=1\), indicating that the material is incompressible. A simplification of the formula yields the following:  
\[\Psi=C_{10}\big{(}\overline{I}_{1}-3\big{)}+\Psi_{aniso} \tag{4}\]  
\(C_{10}\) is the material constant, \(C_{10}=\mu/2\), \(\mu\) is the shear modulus. \(\overline{I}_{1}\) is the first invariant of the Cauchy-Green tensor; \(\overline{I}_{1}=C:I\), \(C=F^{T}F\), where \(F\) is the deformation gradients and \(I\) is the unit tensor. \(\overline{I}_{4},\overline{I}_{6}\geq 1\) characterizes the mechanical response of the fiber in the preferred direction. \(k_{1},k_{3}>0\) are stress-type parameters, and \(k_{2},k_{4}>0\) are uncaused parameters. The orientation of the fibers is represented by the invariants \(\overline{I}_{4}\) and \(\overline{I}_{6}\). The anisotropic orientation in the tissue is assumed to be a helical orientation of \(\pm\theta\) degrees with respect to the longitudinal direction, as shown in Figure 1a. Therefore, the \(\overline{I}_{4},\overline{I}_{6}\) invariants are equal.  
\[\overline{I}_{4},\overline{I}_{6}=\lambda_{\theta}^{2}\cos^{2}\theta+ \lambda_{z}^{2}\cos^{2}\theta \tag{5}\]  
where \(\lambda_{\theta}\) is the circumferential tensile ratio, and \(\lambda_{Z}\) is the longitudinal tensile ratio.  
Figure 1: (**a**) Vascular mechanics model structure diagram. (**b**) AD before expansion. (**c**) AD after expansion.  
The fiber parameter is \(\kappa\in[0,\,1/3]\) in the HGO model, with values of \(\kappa\) close to 0 indicating a concentration of fibers along the preferential orientation \(\theta\), while values close to 1/3 indicate a dispersion of fibers. Using \(\rho(\theta)\) to represent the risk density function, the distribution of the number of fibers is normalized over the range \([\theta,\theta+d\theta]\), and the value of \(\kappa\) is denoted as:  
\[\kappa=\frac{1}{4}\int_{0}^{\pi}\rho(\theta)\sin\theta^{3}d\theta \tag{6}\]  
Each family of fibers is assumed to have the same mechanical response along the same \(\theta\)-angle direction, so that \(k_{1}=k_{3}\) and \(k_{2}=k_{4}\). Because \(\Psi_{aniso}\) is only valid when the tissue is stretched, it is zero when \(\overline{I}_{4}\) or \(\overline{I}_{6}<1\).  
In summary, the basis for constructing parametric equations for the human aortic wall model of HGO lies in obtaining the \(C_{10}\), \(k_{1}\), \(k_{2}\), \(\theta\), and \(\kappa\) in the above equations.