Force balance ratio is a robust predictor of arterial thrombus stability

Abstract

Thrombus formation on a damaged vessel wall can lead to the formation of a stable occlusive/subocclusive clot or unstable embolizing thrombus. Both outcomes can cause significant health damage. The mechanisms that regulate maximum thrombus size, its stability, and embolization in both micro- and macrocirculation are poorly understood. To investigate the impact of flow and intrathrombus forces on the stability of homogeneous and heterogeneous platelet thrombi in a wide range of thrombus geometries, critical interplatelet forces, vessel diameters, and hydrodynamic conditions, we took advantage of the recently developed in silico models. To perform analysis of thrombus stability/embolization in arterioles, we used our previously developed particle-based 2D model with a single-platelet resolution. Its results and predictions were further extended to a 3D case and the large spatial scales of arteries using novel particle-based and continuum 3D models. We found a robust quantitative parameter, termed force balance ratio, which quantifies the balance between destabilizing hydrodynamic and stabilizing interplatelet forces. This parameter predicts whether a homogeneous thrombus (or the shell of a heterogeneous thrombus) with a particular value of critical interplatelet forces will embolize under given hydrodynamic conditions. Our simulations also predict that, for a given magnitude of critical interplatelet forces, the longer thrombi are more stable than the shorter ones. Furthermore, the aggregates formed on top of the severe stenosis are more stable than thrombi formed at moderate stenosis. Taken together, our results give new insights into the interplay between critical interplatelet forces, local hydrodynamics, and overall thrombus stability against the flow.

Significance

Arterial thrombus can cause occlusion of arteries, which will significantly damage the supplied tissues. Mechanisms regulating the maximal possible size of arterial thrombus are still unclear. Hydrodynamic force acting on a thrombus can be one of the factors, which destabilize thrombus and determine its maximal possible size. We analyzed the interplay between the critical interplatelet forces, local hydrodynamics, and overall thrombus stability against the flow and found a robust quantitative parameter, termed a force balance ratio, which predicts whether the thrombus will embolize or stay stable under given hydrodynamic conditions. Hence, our work offers a useful link between the critical value of interplatelet forces and overall mechanical stability of the thrombus against the flow.

Introduction

Stable (sub)occlusive thrombus may cause prolonged reduction of a blood flow in a vessel, potentially leading to the significant damage of the supplied tissues. On the other hand, if a significant portion of a thrombus will embolize, such a large emboli may cause occlusion of peripheral arteries and also lead to hypoxia in corresponding downstream tissues. Mechanical stability of a thrombus depends on its structure. Recent in vivo studies of the hemostatic response to the laser-induced and penetrating injuries revealed the heterogeneous structure of arterial thrombus, consisting of loosely packed outer shell with slightly activated platelets and a densely packed stable inner core consisting of highly activated platelets and fibrin (1,2). In the normal response to the injury, the outer shell first increases in size and then becomes significantly smaller due to the loss of platelets and embolization. The dynamics of the external thrombus shell, driven by the balance between the hydrodynamic and interplatelet forces, determines the overall dynamics of thrombus size.

Two types of interplatelet interactions are considered to be of the most importance for thrombus growth and stability: interaction via integrins αIIbβ3-fibrinogen and interaction via GPIb-von Willebrand factor (vWF). Integrin αIIbβ3 is the dominant integrin on platelets and is essential for normal platelet function, including stable adhesion and aggregation (3). There are more than 80,000 αIIbβ3 receptors on the surface of platelets and forces required to break a single αIIbβ3-fibrinogen bond are in the range of 60–150 pN (4). Interaction via GPIb-vWF is crucial for both primary platelet adhesion to the injured vessel wall and the primary attachment to the growing thrombus under high shear rates (5) and can drive biomechanical platelet aggregation at the sites of flow disturbances (6,7,8).

Individual platelets can generate contractile force of 29 nN and form adhesions stronger than 70 nN (9). Rupture forces between individual platelets were shown to be in the range 1–3 nN and depended on the extent of platelet activation (10). Blocking interplatelet interaction via αIIbβ3-fibrinogen with abciximab significantly reduced rupture forces (10). Integrin αIIbβ3-transmitted platelet force distribution parameters were also sensitive to treatment with the anti-platelet drug tirofiban (11): at low concentration, the drug altered distribution of integrin-mediated forces (with values higher than 12 pN), but did not significantly influence total platelet adhesion force. At high drug concentrations, integrin αIIbβ3-mediated total adhesion force was significantly reduced. Another experimental study (4) showed that clinically used anti-platelet drugs (eptifibatide, abciximab, tirofiban) in proper doses result in a 10-fold and more decrease of the probability of a high-yield (>60 pN) integrin αIIbβ3-fibrinogen bond formation. Obviously, these drugs significantly affect the mechanical stability of a thrombus against the flow by reducing the magnitude of the integrin-mediated interplatelet forces. However, quantitative estimation of the effect of these drugs on thrombus stability in vivo, which could be used for effective patient-specific therapeutic strategy, remains a challenge. Moreover, the link between the mechanical parameters of interplatelet interactions and overall stability of the platelet aggregate against the flow is currently missing.

Recent in silico study (12) suggested that, for a given composition and thrombus size, embolization extent and rate increase with increase in wall shear rate. Results also indicated that increased density of interplatelet bonds makes thrombus more stable against the flow and that the kinetics of the bonds strongly influence their ability to sustain drag forces.

Another in silico study (13) demonstrated that blood clots with higher permeability are more prone to embolization with enhanced disintegration under increasing shear rate. In contrast, less permeable clots are more resistant to rupture due to shear-rate-dependent clot stiffening originating from enhanced platelet adhesion and aggregation. Our recent in silico study suggested the important role of stochasticity of the primary interplatelet interactions for the observed fluidity of the outer layers of microvascular thrombus and allowed us to estimate the critical interplatelet forces (i.e., the maximal value of integrin-mediated forces acting between a pair of platelets) in the “shell” to be in the 600–1200 pN range (14).

Here, we aimed to analyze the interplay between microscopic parameters of the interplatelet interactions and the overall macroscopic stability of the platelet-rich thrombus. To investigate the mechanical stability of a platelet-built thrombus, we adapted the recently developed 2D particle-based models (14,15) and managed to find a key parameter, the force balance ratio, which allows one to predict whether homogeneous thrombus (or shell of a heterogeneous thrombus) will be mechanically stable or embolize under given hydrodynamic conditions. To get deeper insight into the problem of the mechanical stability of thrombus shell in 3D, we developed several other 3D models with different complexity. These models allowed us to analyze the interplay between thrombus mechanical stability, hydrodynamic forces acting on thrombus, thrombus geometry, thrombus porosity, and reduction of the flow in a vessel both in micro- and macrocirculation.

Materials and methods

2D particle-based model of a thrombus

To analyze the dynamics of a platelet thrombus under the impact of flow in microcirculation, we adapted a recently developed 2D particle-based model of thrombus formation (14,15) (see supplemental text A in the Supporting Material for more details). Model consists of three modules: initiation module, platelet dynamics, and hydrodynamics modules. Initiation module generates starting platelet positions and radii, which are used as an input for hydrodynamic and platelet dynamics modules. Initiation module uses Python and Numpy software (16). Hydrodynamics module generates mesh in the vessel and inside the thrombus, solves Navier-Stokes equation to calculate fluid flows and pressure fields in the vessel and in a thrombus, and calculates the hydrodynamic forces acting on platelets. Blood plasma in the vessel and inside the thrombus is modeled as Newtonian incompressible fluid. Vessel walls and platelet surfaces were no-slip boundaries. To allow fluid motion inside the aggregate, hydrodynamic radii of platelets were two times smaller than geometrical ones (which were used to calculate the interplatelet forces). Hydrodynamic module uses OpenFOAM and Gmsh software (17,18). Platelet dynamics module calculates forces acting on platelets and computes dynamics of platelets by solving Newton equations via the Verlet method (19). Module takes into account interplatelet interaction via GPIb-vWF, which is described using the model of stochastic springs, and interplatelet interaction via αIIbβ3-fibrinogen, which is described using deterministic Morse potential. This third module also considers the forces of elastic repulsion between the intersecting platelets, platelet-wall repulsion forces, and takes into account hydrodynamic forces acting on platelets (which are calculated by the hydrodynamics module). Platelet dynamics module is implemented using C++ language. The details and simulation procedures are described in the supplemental text A.

3D particle-based model of thrombus dynamics

To analyze the dynamics of thrombus under the impact of flow in microcirculation, we also developed a novel 3D particle-based model, which represents a natural extension of a 2D model described earlier (14,15) to a 3D case (see Fig. 3 a; supplemental text B and C; Videos S6 and S7). Model consists of the platelet dynamics module, which generates new platelets on the inlet of the computational domain, calculates forces acting on platelets, and computes dynamics of platelets solving Newton equations via the Verlet method (19). Module takes into account interplatelet interaction via αIIbβ3-fibrinogen, which is described using the deterministic Morse potential. Hydrodynamic forces acting on platelets in a thrombus were calculated using a simplified approach: these forces were considered nonzero only for the platelets on the surface of thrombus (which were detected using the Breadth-first search algorithm), whereas constant parabolic velocity approximation was used to calculate local velocities and apply Stokes’ law to these surface platelets. This single module also considers forces of the elastic repulsion between intersecting platelets and platelet-wall repulsion forces. Model limitations include simplified computation of the hydrodynamic forces acting on platelets, simplified description of the process of platelet activation as a time-dependent process, neglecting the impact of the growing aggregate on the flow field, and neglecting stochastic interplatelet interactions, which described GPIb-vWF interactions in a 2D model. This model was implemented using C++ language.

Figure 3.

The mechanical stability of a thrombus in a 3D particle-based model. (a) 3D computational domain, showing the dynamic platelets in the aggregate (purple and blue), platelets in the flow (white), and anchored platelets, imitating injury site (green). The “wall” of white platelets to the left represents the buffer zone with platelets that currently do not participate in the simulation. Color shows the extent of platelet activation (its increase is shown through the blue to purple transition). Upper image: stable thrombus (at 2 s). Middle image: a beginning of the embolization process. Lower image: embolizing piece of a thrombus (white) flow away from the injury site. (b) The dependence between the critical thrombus height, CIF, WSR, and the geometry of injury site. Data are represented as mean ± SEM, n ≥ 30 embolization events for each set of model parameters. Red dots: injury site 10 × 10 μm, WSR 400 s⁻¹. Orange dots: 10 × 10 μm, 800 s⁻¹. Blue dots: 14 × 14 μm, 800 s⁻¹, (c) The dependence between the surface coverage of the injury site, СIF, WSR, and the geometry of the injury site for the same set of simulation data as in (b). (d) The dependence between the FBR and CIF, WSR, and geometry of the injury site for the same set of data as in (b). To see this figure in color, go online.

The details and simulation procedures are described in supplemental texts B and C.

3D model with explicit hydrodynamics and single-platelet resolution

To calculate the hydrodynamic forces acting on a homogeneous thrombus in a microcirculation, we developed a novel 3D particle-based model, which represents another extension of the 2D particle-based model of a thrombus (Fig. 4 a and b; see also supplemental text D and E). Model consists of three modules: initiation module, platelet dynamics module, and hydrodynamic module. The first module generates the initial platelet positions and radii, which are used as an input for platelet dynamics module. Platelet dynamics module performs the process of thrombus relaxation: it calculates interplatelet forces and forces between platelets and the wall, and it computes dynamics of platelets by solving Newton equations via the Verlet method (19). Module takes into account interplatelet interaction via αIIbβ3-fibrinogen, which is described using deterministic Morse potential, and elastic repulsion between intersecting platelets. Module does not take into account the hydrodynamic forces. Platelet dynamics module is implemented using Python and Numpy software (16). The third hydrodynamic module is used only once at the end of the simulation performed by platelet dynamics module. It generates mesh in a vessel and inside the thrombus, solves the Navier-Stokes equation to calculate fluid flows, and calculates hydrodynamic forces acting on the platelets as well as overall hydrodynamic force. Module considers two fluids: whole blood in a vessel and blood plasma inside the thrombus. Both fluids are modeled as Newtonian incompressible fluids. Vessel walls and platelet surfaces were no-slip boundaries. Hydrodynamic module uses Comsol Multiphysics 5.4 software (20). Model limitations include considering platelet as spheres, neglecting stochastic interplatelet interactions, and neglecting the effect of hydrodynamic forces on platelet dynamics.

Figure 4.

Analyses of the total hydrodynamic forces acting on a porous thrombus in microcirculation using a 3D model with explicit hydrodynamics and single-platelet resolution. (a) Central part of the 3D computational domain, consisting of a vessel subdomain and a thrombus with spherical platelets. Vessel subdomain is represented as a difference between cylindrical vessel and a thrombus subdomain. Vessel diameter was 34 μm, vessel length 2900 μm. Pressure drop was 1170 Pa, corresponding to a WSR of 1000 s⁻¹ in a thrombus-free vessel. (b) Velocity profile in the central section of the vessel for a thrombus with radius of 20 μm and porosity 0.53. (c) The dependence between average intrathrombus plasma velocity in a thrombus, its radius, and thrombus input porosity. Red, orange, and blue dots correspond to porosity values of 0.7, 0.6, and 0.53, respectively. Thrombus input porosity was an input parameter that determined real average porosity of a thrombus. (d) The dependence between contact force acting on 3D thrombus, thrombus radius, and thrombus input porosity. Red, orange, and blue dots correspond to porosity values of 0.7, 0.6, and 0.53, respectively. Green dots correspond to impermeable thrombus model. Thrombus radius was defined as maximal distance between platelet center and thrombus base center. (e) The dependence between flow in vessel, thrombus radius, and thrombus input porosity. Color designations correspond to (d). (f) The dependence between minimum CIF corresponding to a stable 3D thrombus, thrombus radius, and thrombus input porosity. Color designations correspond to (d). Minimum CIF was calculated using the formula: minimum CIF = contact force/0.135, where 0.135 is the value of the critical FBR, obtained with a 3D particle-based model of thrombus dynamics. Data on porous thrombi are represented as mean ± SEM, with n = 3 simulations for each set of model parameters. To see this figure in color, go online.

The details and simulation procedures are described in supplemental texts D and E.

Impermeable thrombus 3D model

To estimate hydrodynamic forces acting on 3D thrombus in macrocirculation, we developed an impermeable thrombus model (Fig. 5 a). Model uses Comsol Multiphysics 5.4 software to generate mesh, solve Navier-Stokes equations, and compute hydrodynamic forces acting on a thrombus (20). The 3D computational domain was represented as a difference between the vessel domain and impermeable thrombus domain. Vessel domain is either a cylinder or a cylinder with cosine-shaped stenosis. Cylindrical vessel radius was 1.7 mm, whereas the length was 290 mm. Such vessel geometry has diameter/length ratio of 0.0117, in line with geometry described in (21). Impermeable thrombus is semi-spherical, semi-ellipsoidal, or toric. Constant pressure drop boundary condition with pressure drop 1170 Pa is applied, corresponding to the wall shear rate in the thrombus-free vessel of 1000 s⁻¹. Whole blood is modeled as Newtonian noncompressible fluid with dynamic viscosity 0.0034 Pa^∗s (22) and density 1060 kg/m³ (23). Vessel walls, thrombus surface, and stenosis surface were no-slip boundaries. For details see supplemental text F.

Figure 5.

Analyses of the hydrodynamic forces acting on a thrombus in micro- and macrocirculation using 3D impermeable thrombus model. (a) Velocity profile in the central section of the vessel for thrombus with radius 24 μm. 3D computational domain for computational fluid dynamics is represented as a difference between cylindrical vessel domain and spherical thrombus. Vessel diameter was 34, 340, or 3400 μm; vessel length was 2900, 29,000 or 290,000 μm, respectively. Pressure drop was 1170 Pa, corresponding to a WSR of 1000 s⁻¹ in thrombus-free vessel for all vessel diameters (as vessel length and diameter were scaled equally). (b) The dependence between contact force acting on 3D impermeable thrombus, thrombus radius, and vessel diameter. Red, orange, and blue dots correspond to vessel diameters of 34, 340, or 3400 μm, respectively. (c) The dependence between minimum CIF corresponding to stable impermeable 3D impermeable thrombus, thrombus radius, and vessel diameter. Minimum CIF was calculated using the formula: minimum CIF = contact force/0.135, where 0.135 is the critical FBR value, obtained with 3D particle-based model of thrombus dynamics. (d) The dependence between normalized flow in the vessel, thrombus radius, and vessel diameter. For each vessel diameter, the normalized flow was calculated by dividing the current volumetric flow by the value obtained when no thrombus was present. (e) The dependence between contact force acting on 3D impermeable thrombus, normalized flow in vessel, and vessel diameter. To see this figure in color, go online.

Stenotic geometry

We applied impermeable thrombus model to analyze stability of thrombi growing on initial stenosis (atherosclerotic plaques) in coronary artery. We used axisymmetric cosine-shape initial stenosis geometry based on geometry described in (24):

$$\frac{r}{R}=1-\frac{d}{2\ast 100\%}[1-\cos \left(\pi \left(\frac{x}{x0}+1\right)\right)]\text{,}$$ (1)

where r is the radius at a given axial coordinate, R is the vessel radius, d is the stenosis severity in percent stenosis by diameter, x is the axial distance from the stenosis apex, and x0 is half of the stenosis length. We used $x0=3R$ which we referred to as short stenosis, or $x0=6R$, which we termed long stenosis.

The source code for the new models can be accessed here:

https://github.com/ErmDaniel/3D-model-with-explicit-hydrodynamics-and-single-platelet-resolution; https://github.com/ErmDaniel/3D-particle-based-model-of-thrombus-dynamics.

All the simulations were performed using computational facilities of CTP PCP RAS (Moscow, Russia), and Lomonosov-2 supercomputer resources at Lomonosov Moscow State University (Moscow, Russia) (25).

Results

Force balance ratio is a predictor of the homogeneous thrombus stability under flow

First, to investigate the critical factors that affect platelet thrombus stability, we turned to a recently developed 2D model of the microvascular thrombus formation (14,15,26). As a first step, the stability of a homogenous platelet aggregate with an equal state of platelet activation and (hence critical interplatelet forces) was considered under constant pressure drop conditions. The critical interplatelet force is the maximal integrin-mediated force acting between a pair of platelets and is close to the maximal value of the external separating force that the two interacting platelets can withstand. The small difference between the critical interplatelet force and the maximal external separating force in this model is due to the effects of the stochastic springs-dependent forces that describe the GPIb-vWF-mediated interactions (14).

Aggregates were generated to fill the given ellipsoid geometry, followed by relaxation and turning on the flow, as described in the “materials and methods” section. Depending on the parameters of the simulation, these in silico platelet aggregates either remained stable during the simulation time or embolized (Fig. 1 a; Videos S1 and S2). Analysis of the dependency of the model outcome on the initial conditions and model parameters allowed us to determine the predictors of thrombus embolization in our model. For given pressure drop and thrombus length parameters, aggregate height was a robust predictor of the embolization (Fig. 1 c): thrombus embolized only if its height exceeded some critical value.

Figure 1.

Mechanical stability of the homogeneous thrombus in 2D model. (a) 2D computational domain, showing dynamic platelets (red) and anchored platelets, imitating injury site (green). Vessel diameter was 34 μm, vessel length was 2900 μm in all the simulations. Scale bar 10 µm. Upper and lower images show unstable thrombus at 1 and 6 s after the start of the simulation, respectively. (b) Upper image: map of the magnitude of the integrin-mediated forces in a stable thrombus. For each couple of interacting platelets, force is depicted as a segment of the line passing through the centers of corresponding two platelets. Magnitude of the force is shown by the color: when ratio force/f_max increases, color changes from blue to green. Thrombus length was 100 μm, whereas pressure drop was set to 40 Pa, f_max = 250 pN. Lower image: map of the magnitude of the integrin-mediated force for ensemble of different thrombi. Ensemble of n = 7 of the stable thrombi with a close value of thrombus height was considered. All thrombi had a length of 100 μm. Pressure drop was 40 Pa, f_max = 250 pN. (c) Interplay between 2D thrombus height and thrombus stability for homogeneous thrombus with a fixed level of platelet activation (fixed critical interplatelet force (CIF) of 290 pN). Each dot represents the result of a single simulation in which thrombus either embolized or was stable. Thrombus base length was 100 μm, whereas pressure drop was 40 Pa (corresponding to wall shear rate (WSR) of 1000 s⁻¹ in case when no thrombus is present). Thrombus height (c), bulk hydrodynamics force (d), and the inlet velocity (e) were calculated at 40 ms after the end of thrombus relaxation process. (d) The dependence between bulk hydrodynamic force acting on a thrombus and thrombus height for homogeneous aggregates with the same parameters as in (c). Each dot represents a result of a single simulation in which thrombus either embolized (red) or was stable (blue). (e) The dependence between the inlet velocity and thrombus height for homogeneous aggregate with a fixed level of platelet activation (same parameters as in (c)). (f) The interplay between normalized hydro force, thrombus geometry, hydrodynamic conditions, and thrombus stability for homogeneous thrombus with a fixed level of platelet activation (CIF of 290 pN). Each dot represents the result of a single simulation in which thrombus either embolized or was stable. Thrombus base length was 50, 100, and 100 μm for red, orange, and blue dots, whereas the pressure drops were 40, 40, and 80 Pa, respectively. For each set of dots, the initial thrombus height was varied in the 20%–90% range of the vessel diameter: the chosen heights evenly covered this range for all three sets of simulations. (g) Interplay between force balance ratio (FBR), CIFs, and thrombus stability for a homogeneous thrombus. Each dot represents the result of a single simulation in which thrombus either embolized or was stable. Red dots: thrombus base length 50 or 100 μm, pressure drop 40 or 80 Pa, CIF was 290 pN. Orange dots: thrombus base length 50 or 100 μm, pressure drop 80 or 160 Pa, CIF was 650 pN. Blue dots: thrombus base length 50 or 100 μm, pressure drop 270 or 480 Pa, CIF was 1800 pN. Initial thrombus height was varied in the same manner as described in (f). To see this figure in color, go online.

Video S1. Stable homogeneous 2D thrombus in 2D particle-based model of a thrombus

Injury site represented by anchored platelets is shown in green. Platelets are red disks. The critical interplatelet force is 0.3 nN. Direction of flow: from left to right. Time is shown in seconds.

Video S2. Dynamics of embolization of 2D homogeneous thrombus in 2D particle-based model of a thrombus

Injury site represented by anchored platelets is shown in green. Platelets are shown as red disks. The critical interplatelet force is 0.3 nN. Direction of flow is from left to right. Time is shown in seconds.

When thrombus height increases, hydrodynamic force acting on a thrombus also increases, which in turn can result in thrombus embolization. To get a deeper insight into the mechanics of embolization in the model, we calculated the projection of the total hydrodynamic force acting on a thrombus on the vessel axis (we term this entity a bulk hydrodynamic force). The bulk hydrodynamic force increased steeply when thrombus height increased (Fig. 1 d), despite the fact that flow in the vessel decreased significantly (Fig. 1 e) when thrombus approached the opposite vessel wall. Importantly, when thrombus height was 80% of the vessel diameter, the bulk hydrodynamic force was about nine times higher than bulk hydrodynamic force acting in thrombus with the height corresponding to 40% of the vessel diameter, suggesting a pronounced nonlinear increase of the bulk hydrodynamic force as thrombus became subocclusive.

As both the hydrodynamic force and critical thrombus height depend on the pressure drop and thrombus length, we next aimed to find a universal parameter, which could predict whether thrombus will embolize under given conditions. As a first step, we performed a series of simulations with thrombi of different geometry (length and height) and under different pressure drops conditions (determining the wall shear rate (WSR) in the absence of a thrombus) but with the same extent of platelet activation and, hence, a fixed value of the critical interplatelet forces (Fig. 1 f). We found that the ratio of bulk hydrodynamic force to the thrombus length (further termed normalized hydro force) represents a robust predictor of the embolization process: when a normalized hydro force of a given thrombus is higher than some critical value, thrombus embolized (Fig. 1 f).

Suggesting the linear relationship between the critical normalized hydro force (the maximal value of normalized hydro force corresponding to the stable thrombus in a given series of simulations) and critical interplatelet forces, we next introduced a force balance ratio R in the following manner:

$$R=d·F_b/\left(f_{cr}·L\right)\text{,}$$ (2)

where d is the average platelet diameter, F_b is a projection of the total hydro force on the direction of the flow in a vessel, f_cr is a critical (maximal integrin-mediated) interplatelet force, and L is the thrombus length. This ratio shows by how much the force required to counteract the external hydrodynamic force (reduced to a single-platelet level) is stronger than the critical interplatelet force (see supplemental text G “physical basis of the force balance ratio” and Fig. S7 in the Supporting Material for the detailed justification of this value).

To check whether the force balance ratio was a good predictor of thrombus stability/embolization, we changed the critical interplatelet force in the range 290–1800 pN (Fig. 1 g) and analyzed aggregate stability. In this analysis, we considered critical interplatelet force as an independent parameter, because its value depends on the platelet activation state, which, in turn, depends on multiple processes triggered by the vessel wall injury in vivo and is generally expected to vary in various types/degrees of the vascular damage. For each value of the critical interplatelet force, thrombi embolized when a force balance ratio R exceeded some critical value. This critical value only moderately increased from ∼0.2 to 0.3 when the critical interplatelet force increased more than sixfold from 290 to 1800 pN (Fig. 1 g). We conclude that the force balance ratio is a robust predictor of the homogeneous 2D platelet aggregate’s stability in a wide range of relevant parameters, namely the critical interplatelet forces, thrombus height, length, and pressure drops. The formalism of the force balance ratio suggests the overall homogeneity of the interplatelet force’s distribution in a thrombus on a large scale, which is supported by the analysis performed for the 2D thrombi (Figs. 1 b and S15).

Force balance ratio in a thrombus shell is a robust predictor of shell embolization in a heterogeneous thrombus

Our next aim was to analyze the stability of a preformed heterogeneous thrombus consisting of a highly activated core (platelets with activation level corresponding to the high values of the critical interplatelet forces) and a moderately activated homogeneous shell (platelets with activation states corresponding to the moderate or low values of the critical interplatelet forces). Typical embolization events resulted in the detachment of one or two pieces of the thrombus shell (Fig. 2 a; Videos S4 and S5, see also stable heterogeneous thrombus on Video S3). Thrombus core and the monolayer of the platelets belonging to the shell, which firmly attached to the core platelets, were always stable (Fig. 2 a). In some simulations, a few holes occurred in that monolayer of platelets (Videos S4 and S5). In line with the case of homogeneous thrombus, we calculated force balance ratio in a thrombus shell $R_{shell}$ in the following way:

$$R_{shell}=d·F_{b_{shell}}/\left(f_{c{r}_{shell}}·L\right)\text{,}$$ (3)

where d is an average platelet diameter, $F_{b_{shell}}$ is a projection of the total hydro force acting on a thrombus shell on the direction of flow in vessel, $f_{c{r}_{shell}}$ is a critical (maximal integrin-mediated) interplatelet force in a shell, and L is the length of the segment of ellipse that separates the shell and the core.

Figure 2.

Analyses of the mechanical stability of a heterogeneous thrombus using a 2D model. (a) 2D computational domain, showing the dynamic platelets of a thrombus shell (red), the dynamic platelets of a thrombus core (blue), and the anchored platelets, imitating injury site (green). Vessel height was 34 μm, whereas vessel length was 2900 μm in all the simulations. Scale bar 10 µm. Upper image: stable thrombus at 1 s after the start of the simulation. Middle image: the beginning of embolization. Lower image: embolized thrombus at sixth second after the start of the simulation. (b) Interplay between the CIF in a thrombus shell and thrombus stability for heterogeneous thrombus with highly activated platelets in a thrombus core and moderately activated platelets in a thrombus shell. Each dot represents result of a single simulation in which thrombus either embolized or was stable. Red dots: thrombus base length 50 or 100 μm, pressure drop 40 or 80 Pa, CIF in a thrombus shell 290 pN. Orange dots: thrombus base length 50 or 100 μm, pressure drop 40 or 80 Pa, CIF in the shell 510 pN. Blue dots: thrombus base length 50 or 100 μm, pressure drop 80 or 160 Pa, CIF in the shell 800 pN. Initial thrombus height was varied in the range of 20%–90% of the vessel diameter. Core height was constant and set to 20% of the vessel diameter with CIF of 1800 pN. (c) Interplay between CIF in a thrombus core and thrombus stability for heterogeneous thrombus with highly or moderately activated platelets in a thrombus core and moderately activated platelets in a thrombus shell. Each dot represents the result of a single simulation in which thrombus either embolized or was stable. Red dots: thrombus base length 50 or 100 μm, pressure drop 40 or 80 Pa, CIF in the core 290 pN. Orange dots: thrombus base length 50 or 100 μm, pressure drop 40 or 80 Pa, CIF in the core 800 pN. Blue dots: thrombus base length 50 or 100 μm, pressure drop 80 or 160 Pa, CIF in the core 1800 pN. Initial thrombus height was varied in a range 20%–90% of the vessel diameter. Core height was set to 20% of vessel diameter. CIF in the shell was set to 290 pN. To see this figure in color, go online.

Video S3. Stable heterogeneous 2D thrombus with core and shell in 2D particle-based model of a thrombus

Injury site represented by anchored platelets is shown in green. Shell platelets are shown as red disks. Core platelets are shown as blue disks. The critical interplatelet force in shell is 0.3 nN. The critical interplatelet force in core is 1.8 nN. Direction of flow is from left to right. Time is shown in seconds.

Video S4. Dynamics of embolization of 2D heterogeneous thrombus with core and shell in 2D particle-based model of a thrombus (case 1)

Injury site represented by anchored platelets is shown in green. Shell platelets are shown as red disks. Core platelets are shown as blue disks. The critical interplatelet force in shell is 0.3 nN. The critical interplatelet force in core is 1.8 nN. Direction of flow is from left to right. Time is shown in seconds.

Video S5. Dynamics of embolization of 2D heterogeneous thrombus with core and shell in 2D particle-based model of a thrombus (case 2)

Injury site represented by anchored platelets is shown in green. Shell platelets are shown as red disks. Core platelets are shown as blue disks. The critical interplatelet force in shell is 0.3 nN. The critical interplatelet force in core is 1.8 nN. Direction of flow is from left to right. Time is shown in seconds.

To understand how platelet activation level in a thrombus shell affects its stability, we performed a series of computational experiments with various extents of platelet activation, which corresponded to the critical interplatelet forces in the shell in the range of 290–800 pN. In these simulations, we fixed the activation level of the platelets belonging to the core (the critical interplatelet force in the core was 1800 pN) (Fig. 2 b). Similar to the homogeneous aggregates, thrombi embolized when force balance ratio in a thrombus shell exceeded the critical value, which was in a range of 0.2–0.3 for all tested activation levels of platelets in a thrombus shell. To analyze how core activation level affects thrombus stability, we also performed a series of simulations with various activation levels of platelets in the core (corresponding to the critical interplatelet forces in a thrombus core in the range of 290–1800 pN) and fixed the activation level of platelets in a shell (corresponding to the critical interplatelet force of 290 pN) (Fig. 2 c). Expectedly, changing activation level of platelets in the core had little effect on the critical value of the force balance ratio in a thrombus shell. Therefore, although in real thrombus there are multiple factors that stabilize the core (including fibrin), here the stability of the core has been reached solely by the increased level of the critical interplatelet forces in this region compared to that of a shell. Importantly, Eq. 3 considers only the parameters of the platelets in the shell region of the thrombus, which were shown to retain the discoid shape (1), and hence allowed us to consider platelet geometry parameters to be constant in further analysis.

Hence, our results demonstrate that force balance ratio in a thrombus shell is a robust predictor of the embolization of the heterogeneous platelet thrombi in a 2D case.

Force balance ratio is a predictor of the 3D thrombus stability under flow

To test whether the force balance ratio might be naturally extended to a 3D case, we analyzed the mechanical stability of the 3D thrombus against the flow using a novel 3D particle-based model, which is a straightforward but simplified extension of the 2D model described above. We termed this model the 3D particle-based model of thrombus dynamics, as it enables us to analyze the process of thrombus growth and embolization in a 3D case (Videos S6 and S7; Fig. 3 a).

Video S6. Dynamics of growth and embolization of thrombus in 3D particle-based model of thrombus dynamics

Computational domain is rectangular parallelepiped. Injury site is 14 × 14 μm and represented by the anchored platelets shown in green. Platelets in flow are shown in white. Platelets in a thrombus are represented by spheres with different color intensity, which changes from blue for low-activated platelets to purple for highly activated platelets. Five-hundred frames correspond to 1 s of simulation time. Wall shear rate is 800 1/s and critical interplatelet force is 326 pN.

Video S7. Dynamics of growth and embolization of 3D thrombus in 3D particle-based model of thrombus dynamics

Computational domain is rectangular parallelepiped. Injury site is 10 × 10 μm and represented by the anchored platelets shown in green. Platelets in flow are shown in white. Platelets in a thrombus are represented by spheres with different color intensity, which changes from blue for low-activated platelets to purple for highly activated platelets. Five-hundred frames correspond to 1 s of simulation time. Wall shear rate is 800 1/s, and critical interplatelet force is 326 pN.

As the analyses of the preformed thrombi of regular shapes performed using the 2D model might represent oversimplification, we decided to analyze the irregular-shaped thrombi that formed and embolized during the simulation in 3D.

Simulated thrombi with chosen model parameters grew rather small (Fig. 3 b) and had small “injury” surface coverage (Fig. 3 c), likely due to the absence of the stochastic interactions, which were earlier shown to be critical for covering the whole injury site in a 2D case (14). Typical embolization events resulted in the detachment of a single piece of the thrombus (Figs. 3 a and S8 c; Videos S6 and S7).

In accordance with a 2D case, we calculated the force balance ratio R, using the formula

$$R=s·F_b/\left(f_{cr}·A\right)\text{,}$$ (4)

where s is an average platelet cross-section area, $F_b$ is projection of the bulk hydro force acting on a thrombus on the direction of the flow in a vessel, f_cr is a critical (maximal integrin-mediated) interplatelet force, and $A$ is area of contact between the thrombus and the monolayer of platelets, which represent the injury site.

For each set of the crucial model parameters (critical interplatelet force, wall shear rate, and the area of the injury site), we performed a series of simulations and calculated the critical force balance ratio, corresponding to the state of a thrombus just before the embolization. In line with our 2D results, there was no significant dependence of the critical force balance ratio on the critical interplatelet force, wall shear rate, and the size of the injury site (Fig. 3 d).

Therefore, our results suggest that the force balance ratio is a robust predictor of thrombus embolization in both 2D and 3D cases.

Direct calculation of the total hydrodynamic forces acting on a homogeneous thrombus and critical interplatelet forces as a function of thrombus geometry and porosity in 3D

Based on the 2D and 3D particle-based model simulations, we conclude that the mechanical stability of the 3D thrombus depends on the force balance between the hydrodynamic forces acting on a thrombus and the critical interplatelet forces. More specifically, we suggest that, for a fixed level of maximal interplatelet forces, the mechanical stability of a thrombus is determined by the 3D generalization of the 2D normalized hydro force described above. We termed this 3D generalization a contact force, which represents the value of the projection of the total hydro force reduced to a single-platelet level. By definition, the contact force is calculated using the formula

$$F_c=s·F_b/A\text{,}$$ (5)

where $F_c$ is a contact force acting on the thrombus, s is average platelet cross-section area, $F_b$ is projection of the total hydro force acting on a thrombus on the direction of the flow in a vessel, and $A$ is area of the contact between the thrombus and a vessel wall. From the continuum mechanics point of view, the contact force is proportional to the average mechanical shear stress on the thrombus-vessel contact surface. For simplicity, from now on we shall consider only homogeneous thrombi, suggesting that the size of the core is significantly smaller than the size of the external shell.

Importantly, for a given value of the contact force F_c, we can estimate the magnitude of the minimal critical interplatelet forces for a stable thrombus $f_{cr}$ using the value of the critical force balance ratio R_crit, derived from the 3D particle-based model (Fig. 3 d):

$$f_{cr}=F_c/R_{crit}\text{,}$$ (6)

To calculate the contact force acting on a 3D platelet thrombus, we developed a 3D particle-based model, which is another natural extension of the 2D model described above (Fig. 4). We called this model the 3D model with explicit hydrodynamics and single-platelet resolution. This model considers hydrodynamic equations in a vessel and inside stationary thrombus with platelets, which makes it suitable for accurate computation of the hydrodynamic forces acting on a microvascular thrombus (see “materials and methods” section for more details). The basic version of the model treats blood as a Newtonian fluid, but additional computations showed that nearly the same results were obtained when the nonlinear Carreau model for blood viscosity was used (see supplemental text G, “additional computations of contact forces acting on thrombi using Carreau model of blood as a non-Newtonian fluid”; Fig. S12 c).

We applied this model to calculate the hydrodynamic force acting on semi-spherical thrombi of different porosity in the range of 0.53–0.7 in an arteriole-sized vessel. The lower value of porosity, 0.53, is the minimal possible porosity that could be achieved in our model. The higher value of porosity, 0.7, was chosen based on the experimental estimates of porosity for the thrombus shell (1). Our model predicted that the contact force in a thrombus steeply increased when thrombus radius increased (Fig. 4 d), in line with a 2D case (Fig. 1 d). Contact force acting on a subocclusive thrombus with the porosity of 0.7 was over 250 pN. In line with our 2D results, we observed that flow in the vessel steeply decreased when thrombus radius increased (Fig. 4 e) due to the constant pressure drop boundary conditions; in contrast, average plasma velocity inside the thrombus steeply increased when the thrombus radius increased (Figs. 4 c and S3 e).

Using estimation of the critical force balance ratio obtained using 3D particle-based model of thrombus dynamics, we also calculated the minimum critical interplatelet forces in a thrombus, which correspond to a stable thrombus using Eq. 6 with R_crit = 0.135 (Fig. 4 f). For a subocclusive thrombus with porosity of 0.7, the minimum critical interplatelet force was over 1800 pN. Interestingly, this value is higher than the critical interplatelet force in a nonocclusive thrombus shell estimated earlier using a simple 2D model (14).

In general, the 3D model with explicit hydrodynamics and single-platelet resolution gives results that are in line with 2D particle-based model and allows us to perform detailed computation of hydrodynamic forces acting on 3D thrombus as well as flow velocities in a vessel and inside thrombus. Moreover, the obtained values of the contact force allowed us to estimate minimum critical interplatelet forces in the wide range of microvascular thrombus geometries and porosities.

Nonstenotic arteries are more protected against the thrombosis-induced flow drop compared to arterioles

To analyze the mechanical stability of thrombus in arteries, we developed a simplified impermeable 3D thrombus model that formally corresponds to the case when thrombus porosity is set to zero (Fig. 5 a). The basic model treats blood as a Newtonian fluid; however, our analysis showed that almost the same results were obtained when a nonlinear Carreau model of blood viscosity was used (see supplemental text G, “additional computations of contact forces acting on thrombi using Carreau model of blood as a non-Newtonian fluid”; Fig. S12 a and b).

To validate this basic model, we compared its predictions to a more complex 3D model with explicit hydrodynamics and single-platelet resolution in microcirculation (Fig. 4 d). We found that the ratio between contact force acting on porous thrombus and contact force acting on impermeable thrombus with the same radius was in the range of 0.2–0.6 when thrombus porosity was 0.7 and in the range of 0.35–0.6 when thrombus porosity was 0.53 (Fig. 4 d). We conclude that the impermeable thrombus model overestimates hydrodynamic forces acting on a porous thrombi but still can be used to estimate them. Importantly, the impermeable thrombus model has a short computational time and can be applied to the wide range of vessel sizes (Fig. 5).

We used this stationary model to analyze the potential thrombus stability (here and below we use the term potential stability to emphasize that our analysis did not include the explicit modeling of thrombus dynamics) and compare contact forces acting on the semi-spherical thrombi in microcirculation and in macrocirculation. As thrombus size increased, the contact force acting on thrombus behaved differently in microcirculation and macrocirculation (Fig. 5 b). For medium-sized thrombi with the height in a range of 20%–80% of a vessel diameter, the contact force acting on the thrombi in arterioles was significantly less than the contact force acting on scaled thrombi in arteries. Our results indicate that this difference was due to the difference in the Reynolds number between micro- and macrocirculation, specifically the role of the inertial term in the Navier-Stokes equations (Fig. S6).

Based on a significant difference in a contact force, we suggest that thrombi in arterioles are generally more stable than scaled thrombi in arteries. To get deeper insight, we analyzed the dependence between contact force acting on thrombus and flow drop in vessel due to partial occlusion by the thrombus (Fig. 5 d and e). We observed that, for the same level of the contact force acting on a thrombus, thrombi in arteries caused significantly less flow drop in a vessel compared to thrombi in arterioles. Stable thrombi exposed to the contact force of 400 pN (and, hence, having the critical interplatelet forces exceeding ∼3000 pN) caused a reduction of the blood flow by 35% in arterioles and only by 15% in arteries.

This indicates that, if both thrombi in artery and arteriole have the same strength limit, artery is more protected against the thrombus-induced flow drop compared to arterioles, which might be important from the physiological point of view.

Using the estimation of the critical force balance ratio obtained with 3D particle-based model of thrombus dynamics, we also calculated minimum critical interplatelet forces in impermeable thrombus, which correspond to the potentially stable thrombus (Fig. 5 c). For stable subocclusive impermeable thrombus, the minimum critical interplatelet force was over 6400 pN both in micro- and macrocirculation.

Longer subocclusive thrombi are mechanically more stable

To understand how thrombus length affects its potential mechanical stability, we performed a series of simulations using impermeable thrombus model in macrocirculation and microcirculation (Fig. 6). Thrombi were ellipsoidal, with circular cross-section (Fig. 6 a). Changing thrombus length had a significant impact on contact force acting on a thrombus but smaller effect on the flow in a vessel (Figs. 6 b–d and S14). We also observed that, when thrombus length varied in the range of one to four vessel diameters, contact force in subocclusive thrombus was inversely proportional to the thrombus length (Fig. 6 e). This happened because bulk hydrodynamic force acting on a subocclusive thrombus was nearly constant when thrombus length was varied in the range of one to four vessel diameters (Fig. S6 f), whereas the area of contact between thrombus and vessel wall increased proportionally to the thrombus length. Similar dependence between the bulk hydrodynamic force and thrombus length was also observed for an average-sized thrombus (Fig. 6 f). We conclude that, for a given level of the critical interplatelet forces, long subocclusive thrombi are potentially more stable than the short ones in both micro- and macrocirculation.

Figure 6.

Analyses of the effect of thrombus length on thrombus mechanical stability in micro- and macrocirculation. (a) Velocity profile in the central section of the vessel for thrombus with length 13.2 mm and height 2.4 mm. 3D computational domain is represented as a difference between cylindrical vessel domain and ellipsoidal thrombus domain with circular cross-section. Vessel diameter was 3.4 mm, whereas vessel length was 290 mm. Pressure drop was 1170 Pa, corresponding to a WSR of 1000 s⁻¹ in a thrombus-free vessel. (b) The dependence between contact force acting on 3D impermeable thrombus, thrombus height, and thrombus length in macrocirculation. Red, orange, and blue dots correspond to thrombus length 3.4, 6.8, and 13.6 mm, respectively. Thrombus length was defined as the length of z axis of the thrombus ellipsoidal domain. Thrombus height was defined as the length of x semi-axis (or, equivalently, y semi-axis) of the ellipsoid. (c) The dependence between contact force acting on impermeable thrombus, thrombus height, and thrombus length in microcirculation. Red, orange, and blue dots correspond to thrombus length 34, 68, and 136 μm, respectively. (d) The dependence between flow in the vessel, thrombus height, and thrombus length in macrocirculation. (e) The dependence between contact force acting on 3D impermeable subocclusive thrombus and thrombus length in macrocirculation. Thrombus height was 3.3 mm. (f) The dependence between bulk hydrodynamic force acting on the thrombus, its height, and length in macrocirculation. To see this figure in color, go online.

Subocclusive thrombi growing in conditions of long, severe stenosis are much more stable than thrombi formed at moderate stenosis

To understand how stenosis geometry might affect the mechanical stability of the subocclusive thrombus in coronary arteries, we performed a series of simulations using impermeable thrombus model in macrocirculation, with a vessel diameter of 3.4 mm. We used axisymmetric geometry with cosine stenosis, similar to simulations of thrombus growth in stenosed coronary artery performed in (24) (Fig. 7 a and b; see also “materials and methods” section).

Figure 7.

Analyses of the hydrodynamic forces acting on a thrombus in the stenosed artery. (a) Axisymmetric computational domain is represented as a difference between cylindrical vessel domain and the union of the cosine-shaped initial stenosis (plaque, blue) and thrombus (orange). Vessel diameter was 3.4 mm, whereas vessel length was 290 mm. Pressure drop was 1170 Pa, corresponding to a WSR 1000 s⁻¹ when no thrombus and stenosis is present. Initial stenosis severity was 40% or 80%, initial stenosis length was 10.2 or 20.4 mm (the latter termed long stenosis). Thrombus is represented as a difference between the torus and initial stenosis. Torus disk radius was used as a model parameter and termed thrombus radius. At the top of the thrombus, the overall stenosis (with thrombus) was always 95% to describe the subocclusive thrombus. Left image: initial stenosis 40%, initial stenosis length 10.2 mm, thrombus radius 3 mm. Right image: initial stenosis 80%, initial stenosis length 10.2 mm, thrombus radius 3 mm. (b) Velocity profiles in the central section of the artery for the vessel geometries shown in (a). (c) The dependence between contact force acting on a thrombus, thrombus radius, and stenosis severity. Red dots: thrombus on moderate initial stenosis, initial stenosis length 10.2 mm. Orange dots: thrombus on moderate initial stenosis, initial stenosis length 20.4 mm. Blue dots: thrombus on severe initial stenosis, initial stenosis length 10.2 mm. Green dots: thrombus on severe initial stenosis, initial stenosis length 20.4 mm. To see this figure in color, go online.

Expectedly, we observed that the contact force changed significantly when the effective thrombus radius increased both in moderate and severe initial stenosis (Fig. 7 c). Contact force acting on a thrombus in conditions of long initial stenosis was lower than the contact force acting on a thrombus in conditions of short initial stenosis. Contact force acting on thrombus in conditions of long severe stenosis was minimal among all four tested stenosis geometries (see also additional data for bulk hydrodynamic force acting on thrombus and area of contact between thrombus and stenosis in Fig. S16 a and b). Hence, for a given level of the interplatelet forces, subocclusive thrombi growing in conditions of long severe stenosis are potentially much more stable than subocclusive thrombi growing in conditions of moderate stenosis, which may be important from the pathophysiological point of view.

Discussion

Using several in silico models, in this work we demonstrate that the mechanical stability of both homogeneous and heterogeneous platelet aggregates depends on the specific ratio of hydrodynamic and interplatelet forces. We found a robust predictor—termed force balance ratio—of thrombus stability, which depends on both macroscopic parameters of the thrombus and microscopic parameters of interplatelet interactions. The critical level of the force balance ratio, estimated using the 3D models with single-platelet resolution, allowed as to predict the critical sizes of thrombi in both micro- and macrocirculation for a wide range of the critical interplatelet forces (which generally depend on the level of platelet integrin activation and can significantly vary depending on platelet activation state). This formalism also allowed us to estimate the critical interplatelet forces required to generate occlusive thrombi in nonstenotic arteries and analyze the potential stability of thrombi formed in stenotic arteries with various degrees of stenosis.

Our work has several important limitations. We did not consider the effect of fibrin formation on thrombus stability, as we focused on the stability of the external platelet-rich and low-packed areas of a growing thrombus, which correspond to the shell in a core-and-shell model of arterial thrombus heterogeneity. We also considered both core and shell as homogeneous domains with constant level of critical interplatelet forces that do not change in space and time. In the absence of the experimental data on the distribution of the interplatelet forces inside in vivo/in vitro thrombus and their spatiotemporal dynamics, we suggest that our simplification is acceptable.

We considered platelets as 3D spherical particles or 2D discs, whereas real nonactivated platelets are ellipsoidal. Using the Kozeny-Carman model of a porous medium (27), we estimated that the hydrodynamic force acting on a thrombus with ellipsoidal platelets is likely in the range 1.0–1.3 of the hydrodynamic force acting on a thrombus with spherical platelets (see supplemental text G, “analyses of the effect of platelet shape on the hydrodynamic forces acting on a thrombus”).

In a force balance ratio formula ((2), (3), (4)) we considered only the effect of bulk hydrodynamic force (projection of the hydrodynamic force acting on thrombus on the direction of flow) on the mechanical stability of the thrombus. Still, the stability can also be compromised by the hydrodynamic lift force (normal to the flow direction) and the hydrodynamic torque. Our analyses showed that, in microcirculation, the normal component of the hydrodynamic force acting on a thrombus has a minor effect on its mechanical stability (see Fig. S11 a). On the contrary, hydrodynamic torque might have a significant effect on thrombus stability comparable with an effect of bulk hydrodynamic force (see Fig. S11 c–g). However, the significant correlation between these entities (Fig. S11 d) likely explains why the force balance ratio turned out to be a robust predictor of thrombus stability.

The basic version of the 3D thrombus dynamics model does not take into account interplatelet interaction via GP1b-vWF. To estimate the effect of this limitation on the model predictions, we developed a new 3D model with stochastic springs, describing platelet-platelet interactions through GP1b-vWF(see supplemental text B, “GP1b-VWF module”; supplemental text G, “the effect of the interaction via GP1b-vWF on the mechanical stability of the 3D thrombus”; and Fig. S18). Simulations showed that, for this model, the values of the critical force balance ratio were in the range of 0.12–0.32. These values were higher than the critical force balance ratios in the basic 3D model (critical force balance ratio 0.12–0.14) but were close to the range of values obtained using the 2D particle-based model of a thrombus, which included stochastic springs describing GP1b-vWF interactions (where critical force balance ratio was in range 0.2–0.3).

Relatively low value of the critical force balance ratio, which was less than unity in all our models, is likely due to the vector nature of interplatelet forces (see the map of forces distribution inside the thrombus in Fig. 1 b). Interplatelet forces, which compensate the hydrodynamic force acting on a thrombus, have different directions; hence, their average projection is lower than the critical interplatelet force, which enters our equations for the force balance ratio.

As platelets in a thrombus shell have been shown to be weakly activated and do not undergo significant shape change (1), we used platelet cross-section area as a constant parameter in (3), (4), (5). However, strongly activated platelets can spread upon activation, which can cause significant change in platelet area (28). Thus, for strongly activated platelets (for example, platelets in a thrombus core), Eqs. 4 and 5 should be used with different values of the platelet cross-section area parameter.

Another limitation of our study is neglecting the pulsatility of the flow. Pulsatility in microcirculation is generally considered to be of less significance; still, the existing data show that, in human eye arterioles, the wall shear rate peak systolic value exceeds end diastolic value two to three times (29), whereas pulsatility in arteries is even more significant. In our models, we used boundary conditions corresponding to a constant wall shear rate of 1000 s⁻¹ in thrombus-free arteries; such values correspond to the typical peak wall shear rate in arteries (30). This indicates that our estimations of the hydrodynamic forces acting on the thrombi in arteries correspond to peak hydrodynamic forces.

Our results here are generally in line with a recently published work (12): expectedly, increasing the flow in a vessel destabilizes arterial thrombus, whereas increasing the density of interplatelet bonds (which corresponds to increasing interplatelet forces) stabilizes the clot against the flow. However, the quantitative comparison of the predictions between the two papers is impossible due to different dimensionality (2D vs. 3D), as well as different stenosis and thrombus geometries.

Our results also predict that, in the nonstenotic vessels, longer thrombi are mechanically more stable, thus suggesting that longer injury size is more likely to generate occlusive thrombi, in line with conclusions from another theoretical work (21), albeit suggesting another underlying mechanism.

Our computations show that contact forces acting on thrombi growing on long, severe stenosis are in the range of 250–500 pN. Importantly, these values are likely two to three times overestimated due to thrombus impermeability approximation (see “results” section and Fig. 4 d). We can conclude that subocclusive thrombi growing on severe stenosis in coronary artery will be mechanically stable if the critical interplatelet force is in the range of 1 nN or higher. Interestingly, this value is within a range recently estimated for a nonstable shell in the murine arteriole in the presence of hirudin (14), suggesting that an aggregate consisting of platelets with rather low level of integrin activation might be sufficient to occlude the severely stenosed artery.

Our simulations also suggest that thrombus stability can be estimated in flow-induced thrombus embolization experiments in vitro similar to those described in a previous study (31). Given thrombus geometry before embolization and hydrodynamic conditions, one can calculate the critical contact force acting on a thrombus using an impermeable thrombus model and use this value as a patient-specific characteristic of thrombus stability. Such patient-specific tests can also characterize the effect of anti-platelet drugs on thrombus stability and can be used to choose the optimal drug and dosage for each patient.

In conclusion, our results offer a simple mechanical formalism to analyze both homogeneous and heterogeneous platelet aggregates under the flow conditions applicable to a wide range of physiologically relevant conditions. We believe that the proposed “bridge” between microscopic parameters (such as critical interplatelet forces and effective platelet size/area) and overall thrombus stability offers a fruitful approach that can be further extended to a wider range of applications.

Author contributions

E.S.B. and D.A.E. developed the models, carried out the calculations, analyzed the results, and wrote the manuscript. V.A.K. developed and validated the models, carried out the calculations, and wrote the manuscript. M.A.P. outlined the scientific problem. D.Y.N. outlined the scientific problem, planned the research, analyzed the results, and wrote the manuscript.

Editor: Karin Leiderman.

Supporting citations

References (32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50) appear in the Supporting Material.