Comprehensive reviews of computational modeling in the field of hemostasis and thrombosis can be found in [8–13]. Among such models, those addressing arterial thrombosis come in many varieties including analytical approximations of platelet adhesion and continuum models [14, 15], particle-based discrete models [16, 17], and fully three-dimensional simulations of platelet attachment with deformable cells [18, 19] that attempt to fully resolve the structure of flowing blood.

Each of these approaches has its specific advantages and shortcomings. Three-dimensional models have the most potential to fully grasp all intricacies of thrombus formation, but they cannot yet be used for this task: even with the help of supercomputers, such models require weeks to simulate several real-world seconds of a single experiment. Therefore, three-dimensional models, and even many two-dimensional ones, are predominantly used not for the simulation of thrombosis but rather for the analysis of single cell attachment [19]. Two-dimensional models without cell deformation can describe the entire process of thrombus growth [16]; however, they do not allow qualitative analysis of the adhesion, only numerical experiments.

Finally, the continuum-based models are an approximation that overlooks the fates of single platelets and that adresses platelet concentrations and mechanical properties of the thrombus medium. The main shortcomings of this approach include poor small-scale accuracy and the need to include phenomenological mechanical properties of a thrombus. However, these models allow analytical investigations of the thrombus formation laws and the development of theories. For this purpose, however, one would need to develop such a model in a way that would be suitable for analysis, i.e., in the form of simple reaction-diffusion equations with smooth functions or another analyzable form; complex rule-based continuum models suitable for computational simulations only [14] would not be sufficient. In the present study, we aimed at the development of a model to overcome this problem.

To obtain a continuous model describing thrombus development, let us first consider the one-dimensional attachment of suspended particles to each other, as depicted in Fig 1. This figure is a representation of a region at the boundary of a thrombus, which is sufficiently small such that we can assume the local rate of platelet adhesion (in general depending on the local hematocrit and shear rate of the fluid) to be constant throughout this region. In other words, we locally approximate platelet attachment using a “chemical adhesion” model [20]. This approximation in the 1D case does not take into account platelet re-arrangement and rolling along the thrombus (which will be added later for the 2D case). For simplicity, in the initial derivation, we will only consider irreversible attachment.

Let us denote the mean concentration of adhered particles in the aggregate with the variable C. Because we consider the one-dimensional case here, we assume that the thrombus can only grow in the direction x (see Fig 1); however, the concentration C of the particles at each value of x can vary from 0 to C_max, where C_max corresponds to the situation in which the channel is fully packed by the particles in the vertical direction.

Although real platelets are distributed non-uniformly in the vessel, we are only interested in their concentration in the region close to the thrombus. Therefore, we assume that the volume concentration of the un-attached particles in the suspension is constant, C_na. Let us now consider the evolution of the thrombus within a small time interval Δt. The concentration of the adhered platelets C at each spatial point along the x axis over this time interval increases according to the following rules:

It is proportional to the mean concentration of the non-attached particles. In time, the mean concentration C_na changes, because some fraction of the volume (C/C_max) is occupied by the already aggregated particles. Therefore, the mean concentration of the non-aggregated platelets is C_na(1 − C/C_max); i.e., it is zero where an aggregate is formed and C_na everywhere else.

Let us analyze the structure of the binding sites in more detail. For a better understanding, let us replace the concentration of platelets by the number of particles N and the concentration of free sites by the number of these sites. We can easily exchange these quantities, because they have the same nature. We also represent attaching particles as rectangles and redraw Fig 1 in an orderly fashion, displaying layers as depicted in Fig 2.

The particles arrive from the solution and form layer upon layer in a manner not dissimilar to the Tetris game. There is no requirement for a layer to be completed when the following layer begins to be built. Fig 3 clearly illustrates the fact that there are two types of binding sites for particle attachment. The first type is provided by the ‘side planes’ of each attached rectangle. Attachment to these sites does not increase the local aggregate height but rather increases its density by ‘filling out’ cavities on the thrombus surface. The second type is provided by the ‘free ends’ of each column, and this type of attachment provides a possibility of initiating a new layer. This is the mechanism that actually propagates a thrombus in space. These two types of particle attachment are schematically depicted in Fig 3. In reality, there is no such clear distinction because real particles or platelets are not square bricks; nevertheless, these two types of attachments are present.

Let us now consider a small interval Δx along the x axis (from x to x + Δx) and determine the increase in the number of attached platelets in this interval within the time interval Δt (from t to t + Δt).

Let us consider the second type of particle attachment (Fig 3 left). The number of ‘free ends’ can be defined as the change of N within the interval Δx, i.e., ∣N(x) − N(x + Δx)∣. Next, we consider the first type of attachment. The number of ‘side planes’ available for attachment can be defined as the average (arithmetic average) number of particles corresponding to the columns in the interval multiplied by the number of layers within the interval [x, x + Δx], which equals Δx/λ, where λ is the characteristic size of the particle, i.e., N_avΔx/λ. Uncertainty of the geometric model might change this several-fold with no effect on the overall validity; this can be corrected by adjustment of λ.

All the above reasoning remains the same for the concentrations of platelets and ‘free attachment sites’. Therefore, taking into account that we considered a small interval Δx, we obtain: C | t + Δ t - C | t ∼ C λ + | d C d x | . (1)

By combining assumptions 1–3 and model Eq (1), we arrive at the equation of thrombus propagation: ∂ C ∂ t = k e f f ( 1 - C C m a x ) ( C λ + | ∂ C ∂ x | ) , (2) where C_max is the maximum value of the concentration and k_eff is the coefficient of platelet adhesion efficiency. After dividing both parts of the equation by C_max we can obtain the dimensionless form of the equation and denote the now unitless concentration varying within the range [0, 1] again by C.

Note that Eq (2) is not completely novel in the field of adsorption models. Similar models have already been obtained based on the Brunauer-Emmett-Teller (BET) adsorption theory [20]. These types of models include the first spatial derivative of the concentration of the adhered particles (because adsorption is determined by the ‘fractional surface’ of each layer) and do not have a diffusion term (because adsorbed material does not diffuse). However, this theory assumes the presence of a single binding site provided by each particle for the formation of the uppermost layer. In our case, this would lead to a simplified (but more obvious) model, where the increase of particle concentration only depends on the concentration gradient: ∂ C ∂ t = k e f f ( 1 - C C m a x ) ( - ∂ C ∂ x ) . (3)

Note that the term ∂ C ∂ x is always negative in the 1D case if the concentration propagates in the positive direction of x; therefore, the equation can also be rewritten in a form similar to Eq (2): ∂ C ∂ t = k e f f ( 1 - C C m a x ) | ∂ C ∂ x | . (4)

The latter form is more convenient because we want to extend the model to the 2D case.

The simplest model similar to that in the present study is the one by Rosen [21] that does not consider the saturation of binding sites: ∂ C ∂ t = k e f f ( - ∂ C ∂ x ) . (5)

Regardless, most studies based on the BET theory for gases do not use partial differential equations but rather focus on ordinary equations assuming equilibrium of the outermost layer. Removal of these two limitations constitutes the main difference between our approach and the previous approaches.

Eq (2) describes the spatial propagation of a one-dimensional platelet thrombus. To characterize and evaluate the dynamics of the thrombus growth, we introduce the following quantity that shows how fast the concentration is changing in the direction normal to the wall of the channel; we call this quantity velocity. V = d ∫ 0 L C ( x ) d x d t , (6) where L is the length of the computational domain.

The velocity and the profile of the wave depend on the model parameters. The equation was solved numerically on a uniform mesh using an explicit finite difference method. The solution resembles a traveling wave (Fig 4), although the equation does not have a diffusion term and is not a typical chemical traveling wave in the true sense of the word. The simulation was performed for the following parameter values: λ = 10⁻⁶ m, k_eff = 5 ⋅ 10⁻⁷ m/sec.

As shown in Fig 5, the steepness of the concentration profile depends on the coefficient λ: it increases with increasing λ. Regarding the velocity of the wave propagation, Fig 6 shows that there is a linear dependence of the velocity on the coefficient k_eff.

Because the behavior of a platelet clot in the blood flow strongly depends on the flow properties, we propose a two-dimensional model that combines the blood flow and the platelet clot dynamics description. ∂ C ∂ t = k a d h γ ( 1 - C C m a x ) ( C λ + | ∇ C | ) - k r o l ∇ · ( v ¯ C ) , (7) where v ¯ is the blood velocity, γ = γ ( v ¯ ) is the wall shear rate of the fluid, k_adh is the coefficient of platelet adhesion efficiency, and k_rol is the coefficient of platelet movability within a thrombus. The second term on the right-hand side of the equation models the convection. By introducing this term, we allow the concentration of the platelets to be adjusted by the flow and the intensity of such adjustment is defined by the constant k_rol. We assume that the movement of platelets by the flow is one of the key processes in the thrombus formation. We further assume that the concentration propagation depends on the wall shear rate γ, and as we do not assume it to be constant in the two-dimensional case, k_eff is replaced by k_adh γ.

We consider the blood flow in a two-dimensional rectangular domain (see Fig 7a) with height M = 5 ⋅ 10⁻⁵ m and length L = 15 ⋅ 10⁻⁵ m. The following boundary conditions were chosen for Eq (7): C | A = 0 , ∂ C ∂ x | B = 0 , ∂ C ∂ y | C , D = 0 . (8)

As an initial condition for Eq (7) we set the platelet concentration to be equal to 1 on a part of the boundary D of length l₀ (Fig 7a), which imitates an injury on the vessel wall.

We assume that blood plasma is a Newtonian incompressible homogeneous fluid with constant viscosity, which can be modeled using the Navier-Stokes equations in the following form: ∂ v ¯ ∂ t + ( v ¯ · ∇ ) v ¯ = ν Δ v ¯ - 1 ρ ∇ p , (9) ∇ · v ¯ = 0 , (10) where v ¯ = ( u , v ) is the flow velocity, p is the pressure, ν is the viscosity of the fluid, and ρ is the density of blood.

Eqs (9) and (10) are supplemented by no-slip boundary conditions on the boundaries C and D (see Fig 7a), and there are two options for the boundary conditions on sides A and B: pressure gradient or velocity profiles.

To couple Eqs (7), (9) and (10), we set the boundary D to be varying in time and define an additional parameter of the model C_B, which determines a new boundary for the flow. If the value of the concentration C at some point in the domain (x′, y′) reaches this value, i.e., C(x′, y′) = C_B, this point becomes a part of the new boundary D, i.e., v ¯ ( x ′ , y ′ ) = 0. This particularly implies that there is no fluid flow inside the thrombus, i.e., that the thrombus is a rigid obstacle for the fluid; the fluid can penetrate the thrombus only in those areas where the concentration is less than C_B.

The two-dimensional problem was also solved numerically on an orthogonal uniform mesh. The explicit finite difference method was used for the numerical solution of Eq (7). The projection method introduced by Chorin [22] was used to solve the Navier-Stokes Eqs (9) and (10). For all simulations, we set ρ = 1060kg/m³ and ν = 5 ⋅ 10⁻⁶ m²/sec, which corresponds to the parameters of real blood. We further set the value Δp = 30Pa in the case of pressure gradient boundary conditions or the wall shear rate equal to 1000 1/sec for the velocity profile boundary conditions. These values create approximately equivalent velocity fields at the initial moment of time and correspond to the conditions of real blood.

The model developed above is designed specifically for arterial thrombus formation caused by platelet adhesion to the damaged vessel wall. This model is continuous; thus, it is not applicable at characteristic spatial scales comparable to the size of a platelet. It does not consider fibrin formation because the overall dynamics of such a thrombus is defined by the “shell” of weakly activated, discoid platelets with little fibrin at the core ([23], see also the analysis in [24]). Therefore, the model cannot be applied to venous thrombosis. The model does not explicitly include platelet activation, although it is assumed to occur upon platelet attachment. There is no explicit detachment of platelets, although it is assumed to affect the overall adhesion constant.

We assumed the platelet concentration to be uniformly distributed along the vessel width, which is only a rough approximation because it is known that platelet concentration increases near vessel walls; however, the concentration near the wall is the only concentration of importance for thrombus formation. In future works, the fluid flowing inside the thrombus according to the Darcy’s law could be considered. In the current work, we neglected the fluid flow inside the thrombus (because the velocities are expected to be small) and approximated the flow at the boundary of the thrombus using the Navier-Stokes equations.