Continuous Modeling of Arterial Platelet Thrombus Formation Using a Spatial Adsorption Equation

In this study, we considered a continuous model of platelet thrombus growth in an arteriole. A special model describing the adhesion of platelets in terms of their concentration was derived. The applications of the derived model are not restricted to only describing arterial platelet thrombus formation; the model can also be applied to other similar adhesion processes. The model reproduces an auto-wave solution in the one-dimensional case; in the two-dimensional case, in which the surrounding flow is taken into account, the typical torch- like thrombus is reproduced. The thrombus shape and the growth velocity are determined by the model parameters. We demonstrate that the model captures the main properties of the thrombus growth behavior and provides us a better understanding of which mechanisms are important in the mechanical nature of the arterial thrombus growth

Behavior of a thrombus in time regarding the dependence on the length of an injury for <i>λ</i> = 3 ⋅ 10⁻⁶<i>m</i>, <i>k</i>_<i>adg</i> = 8 ⋅ 10⁻¹¹<i>m</i>, <i>k</i>_<i>rol</i> = 7 ⋅ 10⁻⁵ and <i>C</i>_<i>B</i> = 0.9.

<p>The length of the injury does not considerably influence the behavior.</p

Numerical solution of Eq (2).

<p>Dependence of concentration profiles on the parameter <i>λ</i> for <i>k</i>_<i>eff</i> = 5 ⋅ 10⁻⁷<i>m</i>/<i>sec</i>. The steepness of the concentration profiles increases with increasing <i>k</i>_<i>eff</i>.</p

Two types of particle attachment.

<p>One is determined by the ‘free space’ in the aggregate, and the other depends on the number of ‘free ends’.</p

2D model.

<p>Dependence of the wave velocity in time on the boundary conditions; the pressure gradient boundary condition gives a realistic behavior, as the thrombus expansion slows down in time.</p

Numerical solution of Eq (2).

<p>Propagation of a traveling-wave solution of <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0141068#pone.0141068.e002" target="_blank">Eq (2)</a> for <i>λ</i> = 1 ⋅ 10⁻⁶<i>m</i> and <i>k</i>_<i>eff</i> = 5 ⋅ 10⁻⁷<i>m</i>/<i>sec</i>. Concentration increases in time in the positive direction of <i>x</i>.</p

Multi-layer particle attachment to a substrate from a solution.

<p>The substrate (collagen or stable region of the thrombus) is on the left. Adhesion of particles from the suspension leads to an increase of the particle aggregate on the surface of the substrate such that it grows to the right.</p

Dependence of the fluid velocities on the concentration for the following parameter values: <i>λ</i> = 3 ⋅ 10⁻⁶<i>m</i>, <i>k</i>_<i>adh</i> = 8 ⋅ 10⁻¹¹<i>m</i>, <i>k</i>_<i>rol</i> = 7 ⋅ 10⁻⁵, <i>C</i>_<i>B</i> = 0.9 and <i>l</i>₀ = 15 ⋅ 10⁻⁶<i>m</i> and the pressure gradient boundary condition.

<p>Velocities increase with the decreasing concentration values.</p

Multi-layer particle attachment re-drawn to illustrate individual layers.

<p>In all aspects, this Figure illustrates the same mechanism as <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0141068#pone.0141068.g001" target="_blank">Fig 1</a>.</p