Fig 1.
Viscosity of Newtonian and Carreau-Yasuda Constitutive Blood Models.
The Carreau-Yasuda model of blood shows the viscosity changing as a function of the shear-rate. The Newtonian model has constant viscosity at all shear-rates.
Fig 2.
Numerical Method Verification via a 3-Dimensional Idealized Bifurcating Artery.
(A) The 3-dimensional idealized bifurcating artery proposed by Chung [45] was used for verifying our method. The parent artery where the inflow is prescribed is 2 mm in diameter, 12.23 mm in length up to the bifurcation, and 13.95 mm in length to the bifurcation apex. The daughter arteries are both 1.2 mm in diameter and 12.72 mm in length from the bifurcation. The radius of curvature (Rc) is 0.1 mm, and the angle between the daughter artery walls is 60°. (B) Pressure was extracted along the center of the parent artery from the inflow to the apex of the bifurcation.
Fig 3.
3-Dimensional Idealized Femoral Artery Tree and Inflow Conditions.
(A) Dimensions of the idealized femoral artery tree. The bifurcations are defined by the circled numbers, with the first bifurcation defined as the 9 mm artery bifurcating into 8.55 mm arteries and the second bifurcation defined as an 8.55 mm artery bifurcating into 8.1 mm arteries. In both bifurcations, centerline to centerline of the daughter arteries = 60° and the Rc = 2.25 mm. Dimensions for only one side of the idealized femoral artery tree are shown due to symmetry. (B) The inflow velocity waveform applied at the geometric center of the parent artery. Marked peaks on the inflow velocity waveform are time points where comparisons of the two models are drawn. (C) Example inlet boundary condition across the parent artery at TA. X-axis is shown as squared distance from the vessel center (r) over the squared vessel radius (R). The origin (0,0,0) of the idealized femoral artery tree is defined as the geometric center of the parent artery, and all spatial locations given for this geometry are based on this origin in Cartesian coordinates.
Fig 4.
Daughter Artery Velocity Profiles.
Daughter artery velocity profiles immediately following the idealized femoral artery tree bifurcations at TB. (A) Locations along the idealized femoral artery tree where the contours were extracted. Velocity profiles in daughter arteries following (B) the first bifurcation and (C) the second bifurcation. Artery walls adjacent to bifurcations are portrayed as the top wall, and artery walls opposite bifurcations are portrayed as the bottom wall. Velocity slices were extracted from planes intersecting the idealized femoral artery tree. The (x,y,z) coordinates in mm are given for each velocity slice indicates the origin of the plane, which were oriented perpendicular to the normal vectors () given for each daughter artery.
Fig 5.
Velocity Profiles within the Idealized Femoral Artery Tree Bifurcations at TB.
(A) Velocity profiles plotted over the cross sections shown at TB. Velocity slices across first bifurcation were extracted at (B) the bifurcation start, (C) bifurcation midpoint, and (D) the bifurcation apex. Velocity slices were extracted at respective locations (E-G) across the second bifurcation. X-axis is shown as squared distance from the vessel center (r) over the squared vessel radius (R). For slices within the bifurcations (C, D, F, G) the vessel is not cylindrical, so R is chosen as half the wall to wall length. The solid black line is the SDF velocity profile and the dashed blue line is the Newtonian velocity profile. Velocity profiles were extracted from slices originated at the (x,y,z) coordinates in mm given, and slices were oriented perpendicular to the normal vectors () given for each bifurcation.
Fig 6.
Velocity Streamlines through the Idealized Femoral Artery Tree Bifurcations.
Velocity streamlines through the idealized femoral artery tree bifurcations for both models. (A) The first bifurcation at TA, (B) the first bifurcation at TB, (C) the second bifurcation at TA, and (D) the second bifurcation at TB. Due to symmetry, only one half of the idealized femoral artery tree is shown. The left half is the Newtonian model and the right half is the SDF model, separated across the black line.
Fig 7.
The Pressure Drop Across the Idealized Femoral Artery Tree.
The pressure drop (Pinflow—Poutflow) across the idealized femoral artery tree given by both models across two inflow waveforms. Due to symmetry, the pressure at all four outlets is identical, and thus any outlet artery can be chosen.
Table 1.
The pressure drop across the idealized femoral artery tree.
Fig 8.
WSS Contours across the Idealized Femoral Artery Tree.
WSS contours at instantaneous time points (A) TA, (B) TB, and (C) TC the idealized femoral artery tree. Due to symmetry, only one half of the idealized femoral artery tree is shown for each model, where the left half is the Newtonian model and the right half is the SDF model, separated across the black line.
Fig 9.
WSS Versus Time Across the Idealized Femoral Artery Tree.
WSS versus time at different spatial locations on the idealized femoral artery tree given by the (x,y,z) coordinates in mm. WSS is shown at the inflow wall (P-1), the wall opposite the first bifurcation (P-2), the wall opposite the second bifurcation (P-3), and the outflow wall (P-4). The solid black line is the SDF WSS and the blue dashed line is the Newtonian WSS.
Fig 10.
WSS Across the Wall Opposite the First Bifurcation Versus Time and Spatial Location.
(A) WSS was extracted along the wall opposite the first bifurcation shown by the black line. The (x,y,z) coordinates in mm are given for the three points shown. (B) The WSS magnitude difference between the two models (|SDFWSS−NewtonianWSS|) versus time and spatial location. Spatial location is given as arc length distance, where arc length of zero corresponds to the lowest spatial location along the artery wall.
Fig 11.
WSS at Varying Degrees of Atherosclerosis.
Atherosclerotic lesions of varying degrees were induced in the idealized femoral artery tree at the inflow artery or at the first bifurcation. The (x,y,z) coordinates in mm show were the atherosclerotic lesion endpoints were prescribed. WSS from the SDF and Newtonian models were extracted along the wall opposite the plaque and plotted at TA, TB, and TC. The x-axis indicates location along the wall, where zero corresponds to the lowest y-coordinate of the plaque and one as the highest y-coordinate.