# Physics of Within-Tissue Wave Propagation Generated by Pulse Propagation in the Carotid Artery
## 3 Materials and Methods
### Common Carotid Artery (CCA) Model
#### 3.1.2 Wave Propagation and Mesh Sensitivity Analysis  
Numerical settings such as time increment size, numerical damping and mesh size were carefully chosen to ensure computationally efficient and reliable wave propagation simulations. The equations of motion for wave propagation analyses were solved in Abaqus using an explicit direct time integration scheme, which is computationally the most efficient for our considered problem. This method is conditionally stable, in which the size of the stable time increment is automatically calculated as the minimum time that a longitudinal wave takes to propagate across any mesh element in the model. This means that the magnitude of the time increment is dependent on the chosen mesh size and material parameters. When approximating soft tissue as a linear elastic material, the longitudinal wave speed expression (Equation (2)) is highly sensitive to the choice of Poisson's ratio \(v\): if \(v\) approaches 0.5 (incompressibility-as is often assumed for soft tissue), the numerical time increment size gets smaller leading to longer simulation times. Therefore, Poisson's ratio should be cautiously selected based on a trade-off between realistically modeled compressional speeds for soft tissue (typically 1540 m/s) and reasonable computational time. The finally chosen material model together with its parameters is further discussed in Section 3.1.3. In order to improve the modeling of high-speed dynamic events, the explicit integration scheme in Abaqus automatically introduces bulk viscosity as a numerical damping term. The bulk viscosity model consists of two parameters, linear and quadratic bulk viscosity, which generates a bulk viscosity pressure that is linear and quadratic with the volumetric strain rate. Linear bulk viscosity damped \"ringing\" at the highest mesh element frequency, while quadratic bulk viscosity introduced a resisting pressure that prevented mesh elements from collapsing. For more information about the implementation and physical meaning of the bulk viscosity, we refer to the Abaqus manual [21], but it is important to note that the bulk viscosity is intended as a numerical effect only and it is thus not part of the material's constitutive response. Both parameters retained their default value in the simulations (0.06 and 1.2 for linear and quadratic bulk viscosity respectively). Figure 1: (**a**) Geometry of the common carotid artery (CCA) embedded within a cuboidal domain. The free (skin) surface is in dark grey, while infinite mesh elements placed on the outer boundaries are in light grey. In red are highlighted the node sets at the skin and lumen surface, where displacement data were extracted for the wave analysis. (**b**) Geometry of the patient-specific carotid artery CA embedded within a cuboidal domain. Sub regions common carotid artery, CCA; bifurcation, BIF; internal carotid artery, ICA and external carotid artery, ECA are indicated. Again, the top external surface (dark grey) is a free surface, while all the other boundaries in light grey are infinite elements. The lines in red show the paths along which the data for the wave analysis were exported. For both panel (**a,b**), the coordinate system in red indicates the actual position of the model, but for visualization purposes a view triad (located in the right corner of each panel) will be used further in the manuscript to display the view orientation. A mesh sensitivity analysis was performed to determine the optimal mesh size, taking into account the literature-suggested criterion of 10 to 15 elements per wavelength \(\lambda\)[22]. As longitudinal wavelengths (order of meters) are much larger than shear or Rayleigh wavelengths (both in the order of centimeters) in biological materials, the mesh criterion is more restrictive for shear/Rayleigh waves than for longitudinal waves. Therefore, we considered only shear and Rayleigh wave modeling in the mesh sensitivity analysis.