# Hemodynamic Interference of Serial Stenoses and Its Impact on FFR and iFR Measurements
## 2 Materials and Methods  
The generalized coronary artery with one eccentric stenosis (A) followed by a concentric one (B) was considered, as shown in Figure 1. This was based on the morphological observations of five hundred coronary arteries, in which 365 (73%) were eccentric and 185 (27%) were concentric [18]. The angiography in Figure 1 illustrated the clinical case with one eccentric (Stenosis A) stenosis followed by a concentric one (Stenosis B). The artery diameter (D) and total length were chosen as 2.5 mm and 70 mm, respectively [19]. The diameter stenosis was 72% for both stenoses. The eccentricity index (EI) of the stenosis A was 0.875, measured by the ratio of the subtraction of the maximal (\(l_{1}\)) and minimal (\(l_{2}\)) thicknesses of the vessel wall to the maximal thickness (A) [20]. The spacing ratio, defined as the ratio of the distance (L) between two serial stenoses to the artery diameter (D), was adopted as 1, 3, 5, 7, and 10 [21]. Transverse planes \(\alpha\) and \(\beta\) represented locations distal to each stenosis, where the pressure was measured as in clinical settings. The model was meshed with 2.69 million tetrahedral elements and 0.48 million nodes. The minimum and maximum volumes are 1.50 \(\times\) 10\({}^{-14}\) m\({}^{3}\) and 2.78 \(\times\) 10\({}^{-13}\) m\({}^{3}\), as shown in Figure 1. A pulsatile inlet flow approximation by a sinusoidal profile for a cardiac cycle (T), which was 120 beats per minute [22], was adopted, as shown in Figure 2. The outlet pressure was adopted as the averaged blood pressure of 100 mmHg, i.e., 13.3 kPa for simplicity [23]. Five cardiac cycles (5T) were performed to ensure the stability of the flow. A no-slip boundary condition was implemented at the rigid arterial wall. Blood was considered as an incompressible Newtonian fluid with density, \(\rho\), as 1035 kg/m\({}^{3}\) and dynamic viscosity, \(\mu\), as 0.0035 Pa-s [24; 25; 26]. Figure 1: Top: Schematic diagram of serial stenoses with one eccentric stenosis (**A**) followed by one concentric stenosis (**B**) (unit, mm); Bottom: Definition of eccentricity Index (EI); (**C**) Angiography of eccentric (Stenosis A) and concentric stenosis (Stenosis B); (**D**) enlarged view of mesh at inlet; (**E**) 3-Dimensional view of mesh for coronary artery. The incompressible blood flow was governed by the Navier-Stokes equations, i.e., conservation of momentum, along with the conservation of mass equation as shown below [27]. \[\rho\frac{\partial\overrightarrow{V}}{\partial t}+\rho\left(\overrightarrow{V} \cdot\
abla\right)\overrightarrow{V}=-\
abla P+\mu\
abla^{2}\overrightarrow{V} \tag{1}\]  
\[\
abla\cdot\overrightarrow{V}=0 \tag{2}\]  
where pressure is denoted as \(P\), the velocity vector as \(\overrightarrow{V}\), fluid density as \(\rho\), and fluid viscosity as \(\mu\). SIMPLE (Semi-Implicit Method for Pressure Linked Equations) algorithm was adopted for solving the Navier-Stokes equations in the commercial package ANSYS FLUENT 17.2 (ANSYS Inc., Canonsburg, PA, USA). The first order implicit scheme was adopted for the temporal discretization for stability. The second order upwind scheme was used for the spatial discretization for high-order accuracy. The Reynolds number (Re) was estimated as 202.05. A mesh independence study was carried out to ensure that the solution was independent of the mesh density. The results with low Re and the adopted schemes could provide a reasonably accurate solution [28]. For the vortex identification, the vortex dominant region was defined with a positive second invariant (\(Q\)) of the velocity gradient tensor (\(T\)), i.e., \(Q>0\)[17]. The velocity gradient tensor (\(T\)) can be decomposed into a symmetric tensor (\(S\)) and a skew-symmetric part (\(\Omega\)), i.e., \(T=S+\Omega\) where \(S\) = \(\frac{1}{2}[\left(\
abla\overrightarrow{V}\right)+\left(\
abla\overrightarrow {V}\right)^{T}]\) is the rate of strain tensor and \(\Omega=\frac{1}{2}[\left(\
abla\overrightarrow{V}\right)-\left(\
abla \overrightarrow{V}\right)^{T}]\) is the vorticity tensor.