Fig 1.
Schematic of Morse potential and the resulting adhesive force.
Morse potential is used in this study to mimic inter-platelet attractive/repulsive forces. Passive and triggered platelets only generate repulsive forces to prevent overlap, whereas activated platelets attract each other as well.
Fig 2.
Plot of the Morse potential’s well depth De.
Following Eq (10), De is calibrated as a function of λ2, the second invariant of the flow strain rate tensor, where defines the lower bound of the platelet’s adhesive force.
Fig 3.
Snapshots of low-shear blood clotting in a 50μm circular tube.
(a) schematic of the simulation setup; (b), (c) and (d) velocity at the inlet is parabolic with the mean velocity equal to 100, 400, 800 μm/s, respectively. Green particles represent the seeded platelets at the site of injury, whereas blue and red particles are passive and activated platelets, respectively. Blue particles are plotted smaller for clarity. Activated particles can form thrombus and adhere to the injured wall. The circular plots on the left column present the side views of each tube showing the clot that is formed by activated platelets, whereas the circular plots on the right column are cross-sections taken at the center of clots showing the contours of λ2 on those planes.
Fig 4.
Low-shear simulation results of blood clotting in a 50μm circular tube.
(a) A typical example of the number of platelets aggregated in the thrombus vs. time, plotted in semi-log axes. Exponential growth is achieved after a few seconds. The exponential growth rate is computed by fitting the data (red line). (b) Exponential growth rates (normalized by the maximum value) computed from the simulations and plotted as a function of blood flow velocity (−□−). Here, the size of injury is 30μm and platelet concentration is taken as 300,000mm−3; experimental data extracted from Begent and Born [19] (○). (c) Exponential growth rates derived from simulations for three different conditions: platelet concentration taken as 500,000mm−3 (−△−); increased size of injury to 60μm (−▽−); and the inclusion of shear-induced platelet’s drift according to Eq (6) (−○−). Results from (b) replotted here for comparison (−□−).
Fig 5.
Schematic of a microchannel with constriction representing a stenosis and used for modeling platelet aggregation at high shear rate.
(a) view normal to the flow direction; (b) side view along the flow direction; green particles are seeded uniformly on the left wall to represent vWF-coated regions similar to the experimental device in Westein et al. [14].
Fig 6.
Simulation results for platelet aggregation at high shear rates with occlusion levels of 20–60% corresponding to the undisturbed maximum wall shear rates 2,000 − 6,000 s−1.
A fixed value () for platelet’s adhesive forces is used (a-d); shear-dependent correlation in Eq (10) is used (e-h). (a), (b) and (c) Snapshots of platelet aggregation inside 60, 40 and 20% stenoses, respectively. (e), (f) and (g) Snapshots of platelet aggregation inside 60, 40 and 20% stenoses, respectively. No aggregation is found for 20% stenosis; (d) and (h) density of adhered platelets inside the stenosis vs. simulation time. Color coding for particles is the same as in Fig 3. Here, the activation delay time is τact = 0s.
Fig 7.
Simulation results for platelet aggregation at high shear rates inside an 80% stenosis, where the undisturbed maximum wall shear rate is 8,000 s−1.
(a) View normal to the flow direction, and (b) view from above. Color coding for particles is the same as in Fig 3. (c) Normalized density of adhered platelets throughout the stenosis along the flow direction vs. normalized axial location. The density is normalized by the number of adhered platelets at the inlet, and axial distance is normalized by the length of the stenosis. Simulation results (−□−) are based on the activation delay time τact = 6 ± 3ms, and the error bars are computed based on 5 simulations with the same τact; experimental data (−○−) are extracted from Westein et al. [14] and plotted for comparison.
Fig 8.
Simulation results for platelet aggregation at high shear rates in a fixed geometry with 60% stenosis, whereas the increase in flow rate develops high shear rates in the stenosis.
Printed number in each figure is the undisturbed maximum wall shear rates encountered in each stenosis (before aggregation occurs). Full occlusion is observed in (b) and (c). Color coding for particles is the same as in Fig 3, and green particles are seeded on the circular arc only. Here, the activation delay time is τact = 0 s. (d) and (h) Wall shear rate contours plotted on the opposite wall of the arc for the simulations with undisturbed wall shear rate values of 15,000 and 28,000 s−1, respectively.
Fig 9.
Blood coagulation and thrombus formation in a venous flow.
(a) Schematic of the simulation setup with the seeded particles (green) placed circumferentially to represent the subendothelial matrix (150 − 200 μm). The hexahedral elements show the structured grid used to solve the N-S and ADR equations. (b) Time course of platelet aggregation density on the injured area at 64 s−1 shear rate.
Fig 10.
Blood coagulation and thrombus formation in a venous flow ().
(a) Snapshots of platelet aggregation at different time instants superposed on the contours of thrombin ([IIa]). Color coding for particles is the same as in Fig 3. (b), (c) and (d) Concentration profiles of thrombin ([IIa]), fibrin ([Ia]) and [ADP], respectively, at three axial positions on the site of injury: x = 157 μm (−−), x = 177 μm (− ⋅ −) and x = 193 μm (—).