## 2 Methods
### Governing Equations  
In this study, we aimed to compute the flow of blood in the aorta and to investigate the deformation of the cardiac muscle due to blood pressure and the behavior of the vessel walls. The numerical study consisted of two parts.  
**Part 1:**: Fluid dynamics analysis which required to solve the Navier-Stokes equations. It included the calculation of the velocity field and the distribution of blood pressure (variable over time and in space).
**Part 2:**: Mechanical analysis of the deformation of the cardiac tissue and artery.
**Navier-Stokes equations:**: The blood flow fluid dynamics was described using the time-dependent Navier-Stokes equations and the equations for the conservation of mass. The fluid is assumed to be incompressible and homogeneous, with a Newtonian behavior. The blood was assumed to be Newtonian in this study, as its viscosity is almost constant in arteries with relatively large diameters (5 mm); therefore, non-Newtonian effects were neglected [28].  
The Navier-Stokes equations may be written as follows:  
\[\
abla.\mathbf{u}=0, \tag{1}\]  
\[\rho\bigg{(}\frac{\partial\mathbf{u}}{\partial t}+\mathbf{u}.\
abla\mathbf{u}\bigg{)}= \
abla\mathbf{\sigma} \tag{2}\]  
where \(\rho\) is the blood density, \(\mathbf{u}\) is the blood velocity, and \(\mathbf{\sigma}\) is the stress tensor [29]. The relationship describing the stress tensor and the strain rate tensor is defined as:  
\[\mathbf{\sigma}=-p\mathbf{I}+\mu\Big{(}\
abla\mathbf{u}+(\
abla\mathbf{u})^{T}\Big{)} \tag{3}\]  
where \(p\) is the blood pressure, \(\mu\) is the dynamic viscosity of the blood, \(\mathbf{I}\) is the identity tensor, and \({}^{T}\) denotes the transpose tensor.  
The fluid properties such as the density and the dynamic viscosity were assumed to be constant. Their values were, respectively \(\rho_{blood}=1060\) kg/m\({}^{3}\) and \(\mu_{blood}=0.005\) Pa.s.  
Figure 1: Schematic showing a view of the aorta, its ramifications (branching vessels), and the cardiac muscle.  
**Structural mechanics:** The motion of the vessel wall can be mathematically described by the following equation [30]:  
\[\rho_{w}\frac{\partial^{2}\mathbf{u}_{solid}}{\partial t^{2}}-\
abla\mathbf{\sigma}_{s}= \rho_{w}\mathbf{f}_{s} \tag{4}\]  
where \(\rho_{w}\) is the wall density,\(\mathbf{f}_{s}\) represents the external body forces acting on the solid, and \(\mathbf{\sigma}_{s}\) is the Cauchy stress tensor. The wall can be the artery or the cardiac muscle. The value of the wall density is \(\rho_{artery}=960\) kg/m\({}^{3}\) for the artery and \(\rho_{muscle}=1200\) kg/m\({}^{3}\) for the cardiac muscle. It should be noted that \(\mathbf{u}\) is a velocity (unit m/s); however, \(\mathbf{u}_{solid}\) is a displacement (unit meter).  
The artery is assumed to be subjected to large strains and undergo large deformations. Its behavior is non-linear. More precisely, it behaves as a hyperelastic neo-Hookean material. The constitutive law of this material involves the material shear modulus \(\mu^{s}\) and the bulk modulus \(\kappa^{s}\). The values of these parameters are provided in Table 1. In the present study, the material is assumed to be isothermal, and no effect of the temperature is considered.  
For a three-dimensional, compressible, neo-Hookean material such as the artery, the strain energy density is given by [24]:  
\[W=\frac{1}{2}\mu^{s}\bigg{(}\frac{1}{J^{2/3}}tr(\mathbf{C})-3\bigg{)}+\frac{1}{2} \kappa^{s}\bigg{[}\frac{1}{2}\big{(}J^{2}-1\big{)}-\ln(J)\bigg{]} \tag{5}\]  
where \(J=\text{detF}\), F is the deformation gradient tensor, and \(\mathbf{C}=\mathbf{F}^{T}\mathbf{F}\) is the Cauchy-Green deformation gradient tensor. From the above constitutive law, the expressions of various tensors can be deduced by means of more or less arduous calculations.