Grow with the flow: a spatial-temporal model of platelet deposition and blood coagulation under flow

Abstract

The body's response to vascular injury involves two intertwined processes: platelet aggregation and coagulation. Platelet aggregation is a predominantly physical process, whereby platelets clump together, and coagulation is a cascade of biochemical enzyme reactions. Thrombin, the major product of coagulation, directly couples the biochemical system to platelet aggregation by activating platelets and by cleaving fibrinogen into fibrin monomers that polymerize to form a mesh that stabilizes platelet aggregates. Together, the fibrin mesh and the platelet aggregates comprise a thrombus that can grow to occlusive diameters. Transport of coagulation proteins and platelets to and from an injury is controlled largely by the dynamics of the blood flow. To explore how blood flow affects the growth of thrombi and how the growing masses, in turn, feed back and affect the flow, we have developed the first spatial-temporal mathematical model of platelet aggregation and blood coagulation under flow that includes detailed descriptions of coagulation biochemistry, chemical activation and deposition of blood platelets, as well as the two-way interaction between the fluid dynamics and the growing platelet mass. We present this model and use it to explain what underlies the threshold behaviour of the coagulation system's production of thrombin and to show how wall shear rate and near-wall enhanced platelet concentrations affect the development of growing thrombi. By accounting for the porous nature of the thrombus, we also demonstrate how advective and diffusive transport to and within the thrombus affects its growth at different stages and spatial locations.

Fig. 1.

Schematic of coagulation reactions. Dashed magenta arrows show cellular or chemical activation processes. Blue arrows indicate chemical transport in the fluid or on a surface. Green segments with two arrowheads depict binding and unbinding from a surface. Rectangular boxes indicate surface-bound species. Solid black lines with open arrows show enzyme action in a forward direction, while dashed black lines with open arrows show feedback action of enzymes. Red disks indicate chemical inhibitors.

Fig. 2.

Left: Platelet density-dependent mobility limitation function. Right: Binding affinity function g(η).

Fig. 3.

Platelet inlet concentration profiles. Peak-to-centre ratios are set to approximately 1, 5, 7.5 and 10 for platelet profiles P₀, P₁, P₂ and P₃, respectively.

Fig. 4.

Time sequence of growing thrombus at times 50, 100, 150, . . . , 600 s (from left to right, top to bottom) for initial TF density of 15 fmol cm⁻², shear rate of 1500 s⁻¹ and platelet profile of P₃. The arrows show the fluid velocity and have a uniform scaling throughout the sequence. Bound platelet concentrations vary from 0 (dark blue) to P_max (dark red).

Fig. 5.

The top-left plot shows the spatial maximum and mean thrombin concentrations within the thrombus 10 min after TF exposure as a function of the density of exposed TF. The bottom-left plot shows the relative simultaneous availability of e₉ and $e_8^{\text{m}}$ as a function of time for TF densities (bottom to top) 1, 3, 6, 9, 12, 15 and 30 fmol cm⁻². The plots in columns 2 and 3 are the spatial concentrations of fluid-phase thrombin (e₂) in nanomolar and bound platelets (P^b,a + P^se,a) that vary from 0 (dark blue) to P_max (dark red). From top to bottom, the density of TF exposed is 1, 3, 15 and 30 fmol cm⁻². For all cases, the wall shear rate is 500 s⁻¹ and the platelet profile is P₁.

Fig. 6.

Left: Maximum prothrombinase and tenase concentrations at any point within the entire domain 10 min after TF exposure to plasma as a function of the density of TF exposed. Wall shear rate of the flow was 500 s⁻¹. Right: Spatial concentration (nanomolar) of tenase, prothrombinase and available binding sites for factor X/Xa. The TF density is 15 fmol cm⁻² and the wall shear rate is 500 s⁻¹.

Fig. 7.

Close-up of approximately 188 μm long by 38 μm high region around thrombus. Bound platelet concentrations after 10 min vary from 0 (dark blue) to P_max (dark red). Columns 1, 2 and 3 represent wall shear rates of 500, 1000 and 1500 s⁻¹, respectively. The experiments in rows 1 and 2 use platelet profiles P₀ and P₁, respectively. The experiments across the bottom row use the platelet profiles P₁, P₂ and P₃.

Fig. 8.

Clot ‘area’ for each set of experiments: uniform platelet profiles (top row, a–c), fixed nonuniform platelet profiles (middle row, d–f) and varying nonuniform profiles (bottom row, g–i). In all cases, the dotted line, dash-dot line and solid line represent wall shear rates of 500, 1000 and 1500 s⁻¹, respectively.

Fig. 9.

Activation of platelets by ADP (left) and thrombin (right) in the previous 10-s interval. The rows from top to bottom are for times 10, 60, 150, 240, 480 and 600 s. The shear rateis 500 s⁻¹, the platelet profile is P₁ and the TF density is 15 fmol cm⁻². Platelet activation is shown in densities that range from 0 (dark blue) to 4 (dark red) times 10⁴ platelets per mm³ for ADP and 0 (dark blue) to 4 (dark red) times 10⁵ platelets per mm³ for thrombin.

Fig. 10.

Flow speed (cm s⁻¹) inside the thrombus approximately 1 μm from the vessel wall. The curves left, middle and right correspond to locations about 8 μm from the upstream end, at the middle and from the downstream end of the injury, respectively. The wall shear rate is 500 s⁻¹, the platelet profile is P₁ and the TF density is 15 fmol cm⁻².

Fig. 11.

Close-up of near thrombus region at times 200, 400 and 600 s (top to bottom). The left column shows platelet velocities and the middle column shows fluid velocities along with bound platelet concentrations. All velocity vectors are scaled relative to the 2 μm s⁻¹ vector shown in the upper-right corner of the plots. The velocity vectors only appear in regions where the bound platelet concentration exceeds 1×10⁷ platelets mm−3 and the nondimensional concentration of the virtual substance, η, exceeds 0.27. The right column shows the spatial distribution of 1/Pe. Bound platelet concentrations vary from 0 (dark blue) to P_max (dark red), and 1/Pe varies from near 0 (dark blue) to 2 (dark red).

Fig. 12.

Close-up of 120 μm long by 15 μm high region around thrombus at times 140, 180 and 250 s in left, middle and right columns, respectively. In each column are shown (top to bottom) spatial concentrations (nanomolar) of platelet-bound chemical species thrombin $e_2^{\text{m}}$, $z_8^{\text{m}}$, $e_9^{\text{m}}$ and TNE (tenase) and $z_5^{\text{m}}$, $z_{10}^{\text{m}}$, $e_{10}^{\text{m}}$ PRO (prothrombinase) as well as the bound platelet concentration (fraction of P_max) and intrathrombus velocity (velocity scale vector 2 μm s⁻¹). The wall shear rate is 500 s⁻¹, the platelet profile is P₁ and the TF density is 15 fmol cm⁻².