Three-stage dynamics of a thrombus shell in silico is the consequence of thrombus core confinement, granule pool depletion and the irreversible platelet activation by a low-dose thrombin

To get new insights into the specific dynamics of both thrombus core and shell observed in vivo, we built a novel 3D computational model that describes formation of a thrombus in a vessel using the continuum approach. The model takes into account irreversible platelet activation by thrombin, reversible platelet activation by ADP, platelet dense granule secretion in response to thrombin, heterogeneous structure of a thrombus consisting of the core and shell zone with different porosity values, as well as transport of thrombin and secreted ADP molecules (Fig 1, see also Methods section for details). The crucial model parameters were taken from the in vitro experiments on mouse platelets.

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Fig 1. Illustration of the continuum model and its basic concepts.

A) A continuum model considers a 3D vessel with an ellipsoidal injury site (with a zone of thrombin generation characterized by the time-dependent thrombin flux). Brinkman approach is used to calculate flow inside the thrombus and in the vessel. Thrombus shell and core zones are represented as porous domains with different porosities. Their location depends on the local concentrations of platelet activators- thrombin and ADP and degree of degranulation, as thrombin also triggers platelet dense granule secretion, which results in ADP release. Transport of thrombin and ADP is described by the convection-diffusion equation. B) In a continuum model the local platelets’ state depends on the local concentration of thrombin and ADP. ADP induces reversible platelet activation; low concentration of thrombin induces irreversible platelet activation; high concentration of thrombin additionally induces release of platelet dense granules with ADP, and transformation of a shell platelet into a core platelet if the particular degranulation threshold has been reached.

https://doi.org/10.1371/journal.pcbi.1014062.g001

Using this new model, we simulated thrombus formation after the laser-induced injury of cremasteric mouse arterioles. Two external model parameters- equilibrium thrombin flux and characteristic time of thrombin generation (see Eq. 10) were inferred based on the comparison between model simulations and experimental video from [20](see details in Supplemental Text A in S1 Text). Localization of the thrombin generation zone in the upstream half of the injury zone was also chosen based on the comparison between model simulations with different possible locations of thrombin generation zone and the same experimental video from [20] (see Fig 2A, and details in Supplemental Text A in S1 Text). The calibrated model recapitulated both spatial and temporal dynamics of the thrombus core and shell observed in vivo (Fig 2 and S7, S1 and S2 Movies). Fast initial growth of the thrombus shell (Fig 2B) was the result of a burst-like secretion of dense granules with ADP from thrombin-activated platelets (Fig 3C and 3E).

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Fig 2. Comparison between the continuum model simulations and in vivo experiments on the laser-induced thrombosis in the wild type mice.

A) Model geometry. 3D computational domain consisted of a cylinder representing the vessel and an injury site zone. Vessel diameter was 36 microns, while vessel length was 3060 microns. Injury zone was represented as an oblate ellipsoid with semiaxes of 13.5, 13.5 and 6 microns. Center of the ellipsoid was located on the vessel wall. Blue color marks the zone of thrombin generation. B) Temporal dynamics of thrombus core area and total thrombus area in vivo (data from Movie A [20], orange and red dots) and in the model simulation (green and blue dots). C) Temporal dynamics of the thrombus core area in vivo and in the model simulation (high resolution). D) Images of thrombus showing the core and a shell in the model simulation (vessel lumen is brown, shell is blue, core is dark blue). Flow direction was from top to the bottom. On each image color bar shows the values of porosity corresponding to the colors on this image. Note that thrombus core efficiently occupied the injury site zone.

https://doi.org/10.1371/journal.pcbi.1014062.g002

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Fig 3. Continuum model profiles of thrombin and ADP, dense granule density and platelet activation state.

On A)-D) color bar shows the values of concentration (units [nM]) corresponding to colors on this image. A), B)-Profiles of thrombin at 60 s and 130 s. C), D) - Profiles of ADP at 60 s and 130 s. Note the significant drop of ADP concentration in thrombus at 130 seconds. E), F)- Profiles of dense granule density inside the thrombus at 60 s and 130 s. Color bar shows the values of dense granule density (arbitrary units) corresponding to colors on the image. G), H)- Profiles of platelet activation state at 60 s and 130 s. Platelets reversibly activated by ADP are shown in green, platelets irreversibly activated by thrombin are shown in brown, vessel lumen is blue. Note the depletion of dense granules in a thrombus core at 130 s.

https://doi.org/10.1371/journal.pcbi.1014062.g003

The limitation of thrombus core propagation in the model (Fig 2B and 2D) was the consequence of two effects: 1) the presence of thrombin concentration gradient due to diffusion and convection-mediated transport of thrombin from the injury site (Fig 3A and 3B) and 2) the threshold-like response of platelets to thrombin that was recapitulated from the in vitro data [21] (S12 Fig). Importantly, these effects were responsible for several phenomena in silico: 1) the confinement of thrombus core growth in space 2) the pronounced localization of the region with depleted granule pool at the later stages of thrombogenesis (Fig 3F) – another prediction by the continuum model and, finally, 3) the decrease in ADP secretion rate, which was the consequence of the first two phenomena. As far as the core zone is by definition (see Methods section) the region where platelets actively secrete (or secreted) their granule contents, granule pool depletion and the confinement of a thrombus core resulted in a decrease of the total ADP secretion rate and hence – the drop in the overall ADP concentration (as ADP is a subject to both diffusion and convection) (Fig 3C and 3D). The decrease of ADP concentration due to both dense granule pool depletion and the confinement of thrombus core propagation resulted in a decrease of thrombus size, in line with the results obtained earlier with a particle-based model [19].

Formation of a large residual thrombus in the model (Fig 2D) occurred in a two-step process: 1) large initial thrombus shell formed due to ADP-induced platelet activation and 2) thrombin partly propagated inside this large thrombus shell and irreversibly activated platelets there (Figs 3G and S11). This part of a thrombus shell remained stable (Fig 3H) after the drop of ADP concentration in a thrombus (Fig 3D) but was passive in terms of granule secretion, because thrombin concentration was too low (Fig 3B and 3F).

Spatial localization of the thrombus core, which fully filled the injury site zone both in the model and experiment (Fig 2D) was the result of localization of thrombin generation zone in the upstream part of the injury site zone (Fig 2A and Supplemental Text A in S1 Text), and thrombin transport in a thrombus (Fig 3A and 3B).

Thrombus core size and residual thrombus size are decreased in mouse models of Hermansky-Pudlak syndrome due to the lack of hydrodynamic protection of thrombin by the large initial thrombus

To better validate our model, we performed 3D simulations of thrombus formation following laser-induced injury in mice with Hermansky-Pudlak syndrome (also referred to as HPS mice).

The structure of the model and its parameters were the same as in the original model described above. The only change was to the initial value of releasable ADP in platelet N_ADP, which was set to zero, because HPS mice lack normal secretion of dense granules containing ADP [20]. As a result, in the model simulations platelets did not release ADP. Model simulations were compared to the data from experimental video of laser-induced thrombosis in congenic B6.C3-Pde6brd1 Hps4le/J (light ear) mice (Movie D from [20]).

The model calculations showed quantitative agreement with the experimental dynamics of thrombus formation, with the exception of the short initial phase of growth (Fig 4A and S3 and S4 Movies). The position of the zones of the thrombus core and the thrombus shell relative to the injury site were also similar in the experiment and the model (Fig 4B).

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Fig 4. Comparison between the continuum model simulations and in vivo experiments on laser-induced thrombosis in mice with Hermansky-Pudlak syndrome.

A) Temporal dynamics of thrombus core area and total thrombus area in vivo (light ear mouse, data from Movie D from [20], orange and red dots) and in the model simulation (green and blue dots). B) Upper image: image of a thrombus in the model simulation for HPS mouse at 130 s (vessel lumen is brown, shell is blue, core is dark blue). Flow direction was from top to the bottom. Color bar shows the values of porosity corresponding to colors on this image. Lower image: profile of thrombin in the model simulation for HPS mouse at 130 s. Color bar shows values of thrombin concentration (units [nM]) corresponding to colors on this image. Note the significant decrease of overall thrombus size and thrombus core size in HPS mouse compared to the wild type mouse. C) Overall data on temporal dynamics of thrombus core area and thrombus area in vivo (wild type mouse and mouse with Hermansky-Pudlak syndrome denoted as WT and HPS; data from Movie A (orange and red dots) and Movie D (light magenta and robin egg blue dots) from [20]; and in the model simulations (green and dark blue dots for WT mice; dark purple and dark olive dots for HPS mice).

https://doi.org/10.1371/journal.pcbi.1014062.g004

As expected, calculations showed that in the absence of ADP release, the formation of a large thrombus shell did not occur. The lack of hydrodynamic protection from the thrombus shell led to a change in the thrombin profile in the thrombus and a decrease in its maximum concentration in the thrombus (compare Figs 4B and 3B). As a result, both the thrombus core area and the residual thrombus area were significantly reduced compared to the calculation results for the wild-type mice (Fig 4C).

Taken together, our simulations predicted that both thrombus core size and residual thrombus size are decreased in case of an abrogated dense granule secretion due to the lack of a hydrodynamic protection of thrombin by the large initial thrombus.

Thrombin flux from the injury site defines the scenario of thrombus growth in a FeCl₃-induced thrombosis in a carotid artery

To investigate the regimes of thrombus formation in the large vessels, we performed simulations of the FeCl₃-induced thrombosis in mouse carotid artery using a 2D version of the continuum model (Fig 5; see details in Methods section, ‘2D model of thrombogenesis’). In experiments, the severity of the injury and the dynamics of thrombus formation strongly depend on the concentration of the applied FeCl₃ [22,23]. To reproduce the effects of different degrees of the injury, we performed simulations with different values of the thrombin flux.

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Fig 5. Mechanism of thrombus growth and occlusion in the in silico simulations of FeCl₃-unduced thrombosis.

A) Model geometry. 2D computational domain consisted of a rectangle representing the vessel and a segment representing the injury site zone. Vessel height was 0.5 mm, vessel length was 42.5 mm. Injury site length was 1 mm. Injury site zone coincided with the zone of thrombin generation. Green color marks the injury site zone. B)-F) show the data from the same simulation which ended in vessel occlusion at 248.6 second. Maximal thrombin flux was 2 pmol/(m² ⋅ s). At all of the images flow direction was from top to the bottom. B) Image of the thrombus at the time of occlusion. Vessel lumen is brown, thrombus core is dark blue, while thrombus shell is blue. Color bar shows the values of porosity corresponding to colors on the image. C) Platelet activation state at the time of occlusion. Platelets reversibly activated by ADP are shown in green, platelets irreversibly activated by thrombin are shown in brown, vessel lumen is blue. D) The profile of thrombin at the time of occlusion. Color bar shows the values of concentration (unit [nM]) corresponding to colors on the image E) Profile of ADP at the time of occlusion. Color bar designations correspond to panel “D”. F) Profile of dense granule density inside the thrombus at the time of occlusion. Color bar shows the values of dense granule density (arbitrary units) corresponding to the colors on the image. G) Thrombus growth at intermediate value of maximal thrombin flux. Image of a non-occlusive thrombus at 600 seconds. Vessel lumen is brown, thrombus core is dark blue, while thrombus shell is blue. Maximal thrombin flux was 1.5 pmol/(m² ⋅ s). Color bar designations correspond to panel “B”. H) The effect of turning off the ADP release from platelet dense granules on thrombus growth. Image of a thrombus at 600 seconds. Vessel lumen is brown, thrombus core is dark blue, while thrombus shell is blue. Maximal thrombin flux was 3 pmol/(m² ⋅ s). Color bar designations correspond to panel “B”. I) Temporal dynamics of a blood flow in the vessel in silico. Red line: simulation with maximal thrombin flux of 3 pmol/(m² ⋅ s). Orange line: simulation with maximal thrombin flux of 2 pmol/(m² ⋅ s). Blue line: simulation with maximal thrombin flux of 1.5 pmol/(m² ⋅ s). Green line: simulation with maximal thrombin flux of 3 pmol/(m² ⋅ s), when the ADP release from platelet dense granules was turned off. J) Temporal dynamics of the thrombus area in the model simulations. Color designations correspond to panel “I”. K) Temporal dynamics of the thrombus core area in the model simulations. Color designations correspond to panel “I”.

https://doi.org/10.1371/journal.pcbi.1014062.g005

At high values of thrombin flux (2–3 pmol/(m² ⋅ s)), the model predicted vessel occlusion (Fig 5B and 5I and S5 and S6 Movies). Expectedly, time to occlusion decreased with increasing thrombin flux (Table 1). The model predicted that thrombus penetration into the vessel and subsequent occlusion were the result of platelet activation in response to ADP (Fig 5C). The high concentration of ADP in the thrombus was due to the secretion of dense granules by thrombin-activated platelets (Fig 5E, 5D, and 5F).

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Table 1. Scenarios of FeCl3-induced thrombosis in 2D model simulations. * Simulations were stopped at the moment of occlusion or, if no occlusion was observed during 600 seconds, simulation lasted 600 seconds.

https://doi.org/10.1371/journal.pcbi.1014062.t001

Interestingly, model predicted expansion of thrombus core after the occlusion (S8 Fig and Supplemental Text B in S1 Text), however, these simulation results should be interpreted with caution.

At intermediate values of thrombin flux (1.5 pmol/(m² ⋅ s)), a subocclusive thrombus regime was observed (S7 Movie and Fig 5G). The thrombus grew large, almost reaching the opposite wall of the vessel. However, after reaching the peak, the thrombus size began to decrease (Fig 5J). Eventually, vessel occlusion was not achieved, and the flow in the vessel was gradually restored (Fig 5I).

At low values of thrombin flux (1 pmol/(m² ⋅ s)), thrombus growth was absent (S12 Fig). This was explained by the fact that the maximum concentration of thrombin did not reach the threshold of platelet activation in the model (see Table 1).

Turning off the secretion of ADP by platelets led to the formation of a small thrombus near the injury site zone (Fig 5H and 5J).

All these predictions of the model are in qualitative agreement with the experimental data [22–24].

Large thrombus shell might be important for reaching the occlusion in response to the penetrating injury

Our simulations show that the transient build-up of a large thrombus shell is the result of the interplay between thrombin activity and spatiotemporal dynamics of dense granule secretion. However, the physiological role of such transient and massive thrombus shell is not clear.

We hypothesized that a burst-like release of the secondary messengers (like ADP) and consequent build-up of a massive though transient structure allows the system to scan the environment and ultimately select between a non-occlusive response (preferred in case of a non-penetrating injury that can be associated with thrombosis) and occlusive response (optimal in case of a penetrating injury and associated with a hemostasis). If the surface of a growing platelet aggregate does not “touch” itself due to the geometry of the injury, it will eventually decrease in size due to the decrease of ADP concentration (Fig 6A, top). However, in case of a penetrating injury the hemostatic aggregate that grows in several directions can close on itself if it’s big enough, resulting in a vessel or injury closure (Fig 6A, bottom).

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Fig 6. Simulation of thrombus growth after the circular vessel injury in 3D.

A) Illustration of the idea suggesting the importance of the large thrombus shell for hemostasis in response to the penetrating injury. Top row: three stages of thrombus formation after the non-penetrating injury. Thrombus becomes smaller due to the decrease in ADP concentration. Thrombus shell is shown in orange, while the core is shown in green. Bottom row: three stages of the hemostatic plug formation upon complete vessel dissection: formation of a large shell allows the thrombus to reach itself from the opposite sides and thus completely occlude the injured vessel. B)-E) 2D axisymmetric computational domain consisted of a cylinder representing the vessel and the injury site zone. Vessel diameter was 36 microns, vessel length was 3060 microns. Injury zone was represented as a torus with an elliptic section with semiaxes of 13.5 and 6 microns. B), C)- Images of thrombus in the model simulation at 40 seconds and at the moment of occlusion (42.123 s). Vessel lumen is brown, shell is blue, while the core is dark blue. Flow direction was from top to the bottom. On each image color bar shows the values of porosity corresponding to colors on this image. D) 3D view of the thrombus at the moment of occlusion. Vessel lumen is brown, shell is blue, core is dark blue. E) Simulation where ADP release from platelet dense granules was turned off. Image of the thrombus in the model simulation at 200 seconds. Vessel lumen is brown, shell is blue, while the core is dark blue. Flow direction was from top to the bottom. Color bar shows values of porosity corresponding to colors on this image.

https://doi.org/10.1371/journal.pcbi.1014062.g006

To illustrate this idea we performed simulation of thrombus formation in response to the circular injury of the vessel, which mimics the hemostasis scenario, i.e., when a thrombus can “connect” to itself (see details in Methods section, ‘2D axisymmetric model of thrombogenesis’). In this case thrombus became occlusive already at the 42nd second of the simulation as it rapidly closed the vessel growing towards itself (Fig 6B, 6C, and 6D). Importantly, in this simulation we used the same value of thrombin flux as in the simulations describing non-penetrating laser-induced thrombosis model (Fig 2), where thrombus was not occlusive and displayed a three-stage dynamics. In line with our idea, turning off granule secretion and hence ADP-dependent thrombus shell formation resulted in small circular thrombus that could not occlude the vessel (Fig 6E). These results reinforce the hypothesis suggesting that formation of a large thrombus shell is crucial in the settings of hemostasis.

Model limitations

The current version of the model has several important limitations: it does not describe platelet transport to the thrombus (that may lead to the overestimated values of thrombus growth rate when ADP is released in a thrombus core), does no account for thrombus shape change due to the mechanical effects (both thrombus plasticity and disruptions). It also postulates certain dynamics of the thrombus flux from the injury site (see Eq. 10) and does not account for thrombus contraction that leads to increase in platelet packing density (a mechanism through which platelets in the outer layers can be dragged to the core region with high thrombin concentration and start to secrete their granule content). It also does not account for thrombin binding to platelets and fibrin within the thrombus and the hindered diffusion of molecules in a thrombus core [4] that might quantitatively affect thrombin distribution within the thrombus. Model also does not account for vessel elasticity.

Since hindered thrombin diffusion and burst-like dynamics of thrombin flux from the injury site were discussed in the literature as major possible regulators of thrombus dynamics [14,15,27], we performed several additional simulations which considered these effects (see Supplemental Text A in S1 Text). These simulations demonstrated quantitative, but not qualitative effect on the dynamics of thrombus formation comparing to the basic model.

Our model did not reproduce the initial 30-second period of thrombus growth in laser-induced injury. Importantly, similar period of growth is observed in the presence of thrombin inhibitor hirudin [2], which suggests that this process is controlled via thrombin-independent pathways. These pathways may include release of platelet agonists from injured endothelial cells or collagen-induced platelet activation, which were not considered in the model.

In general, we believe that all these limitations are not critical for achieving the main goal of this study: investigate the new potential mechanism that regulates thrombus dynamics.

Predictions and conclusions

In our model the confinement of the core was the result of plateau-like kinetics of thrombin generation, localization of thrombin generation zone, thrombin transport within the thrombus, and a threshold-like response of platelets to thrombin concentration. In contrast, Stalker and colleagues suggested that limited propagation of thrombin inside the thrombus was the result of the action of the thrombin inhibitors [2]. We performed additional simulation and compared the dynamics of thrombus growth in the presence and in the absence of thrombin inactivation by plasma inhibitors. The results were almost identical (S4 Fig), because thrombin retention time within thrombus is much less than the characteristic time of thrombin inhibition. We conclude that thrombin inactivation by plasma inhibitors does not affect significantly the dynamics of arterial thrombus growth in this particular model. However, inactivation of factors Va and VIIIa by activated protein C was shown to be an important factor limiting thrombus formation in cremaster laser-induced injury model [30] that likely affects both the magnitude and kinetics of thrombin flux from the injury size and should be analyzed separately using a more detailed model.

Meng and colleagues suggested that decrease of the thrombus core size in mice with Hermansky-Pudlak syndrome reflects the decreased sensitivity of a granule secretion to thrombin that they observed ex vivo [20]. In contrast, in our model we did not suggest any change of platelet sensitivity to thrombin in mice with Hermansky-Pudlak syndrome compared to the wild type mice. Significant decrease of thrombus core size was due to the decrease of thrombin propagation inside thrombus in the absence of hydrodynamic protection by large thrombus shell. This idea also allowed to explain why thrombus core is significantly decreased in HPS mice, but not in mice treated with P2Y12 antagonist cangrelor (compare data from [20] and [31]). In mice treated with cangrelor, the maximal thrombus area was significantly larger than in HPS mice. As a result, sufficient level of hydrodynamic protection of thrombin was likely present, which possibly allowed thrombin to form a larger thrombus core.

In FeCl₃-induced thrombosis of mouse carotid artery the scenario of thrombus growth and time to occlusion depend on the degree of injury induced by different concentration of the applied FeCl3 [22,23]. In mice with Hermansky-Pudlak syndrome, which lack normal platelet secretion of ADP, the thrombus does not grow at all [24]. In wild type mice, when platelet activation in response to ADP is inhibited by clopidogrel, the thrombus grows to a very small size [22]. Our computations of thrombus growth in FeCl₃ – induced thrombosis have qualitatively reproduced all of these effects.

Our model predicts rather low peak thrombin concentration in a thrombus of about 2–3 nM. Cornelissen and colleagues demonstrated that mouse platelets are very sensitive to thrombin, and have low thrombin EC50 value of 0.1 nM, compared to about 1 nM for human platelets [32]. High concentrations of thrombin are usually observed in the absence of flow in the in vitro tests (like TGA) – however, under arterial shear rate conditions of 1000–2000 s^-1 it is generally recognized that local thrombin concentrations can be in nanomolar range both in vitro and in vivo. This provides a possible explanation for the physiological relevance of such a high sensitivity of platelets to thrombin.

Our simulations reinforce the idea that large thrombus shell is important in the settings of hemostasis: during primary response to the penetrating injury the thrombus may grow towards itself due to the geometry of the injury. Formation of a large shell may allow thrombus to connect to itself like in our simulations (Fig 6), thus sealing the breach. In line with this hypothesis, dense granule pool deficiency results in bleeding complications in both humans [33] and the mouse models of HPS [24]. Another important argument is the failure to form hemostatic plug under the effect of P2Y12 antagonist upon the severe injury in a mouse jugular vein puncture model [34].

There are several model predictions that can be tested experimentally. First, our computations predict that thrombus core platelets are degranulated in their dense granule content at 2 minutes after laser-induced injury, while shell platelets preserve their dense granules. Second, our computations predict that both core and shell platelets are irreversibly activated by thrombin (i.e., their PAR receptors are cleaved) at 2 minutes after laser-induced injury. Finally, our computations estimate maximal thrombin concentration in thrombus at 1–3 nM both in laser-induced and FeCl₃-induced injury.

Taken together, our results give several new insights into the potential mechanisms that drive the dynamics of arterial thrombus in both micro- and macrovasculature, offer a modification to the basic core-and-shell concept through emphasizing the possible role of the irreversible activation of platelets by the low-dose thrombin in a thrombus shell and also suggest that large ADP-dependent thrombus shell is crucial in the settings of hemostasis.

3D continuum model of thrombogenesis

In this work we describe a novel 3D computational model for analyses of the spatiotemporal evolution of a thrombus in a mouse cremaster arteriole.

The model takes into account the following phenomena: heterogeneous structure of the thrombus consisting of a core and a shell; reversible activation of platelets in response to ADP; irreversible activation of platelets in response to thrombin; secretion of dense granules with ADP from thrombin-activated platelets. This new model allows us to take into account the relationship between thrombus formation and the distribution of soluble platelet activators within the thrombus.

The continuum model is based on a set of partial differential equations. The thrombus was considered as a porous medium consisting of 2 compartments with different porosity values corresponding to the core and shell of the thrombus. Flow in the vessel and inside the thrombus was described using Brinkman approach (see below). Thrombin and ADP were considered as the primary platelet agonists and their concentration dynamics were computed via convection-diffusion transport models. Agonist concentrations were used to determine the evolution of local platelet activation status and thrombus porosity as well as the dynamics of ADP release from platelets in a two-compartment model approach. The key activation process in the model is platelet dense granule secretion: the degree of platelet degranulation determines platelet activation status, which, in turn, impacts local porosity values (see below). Solving the system of model equations allows to infer the dynamics of thrombus formation as a function of both external and internal model parameters. All model equations were solved using the COMSOL Multiphysics software package [35]. The model predictions were compared with experimental data in the form of previously published videos of thrombosis in mouse arteriole models.

General description

Computational fluid dynamics.

Model geometry and hydrodynamics: The model geometry reproduced the vessel with the injury site zone from a representative video of laser-induced thrombosis in microvasculature [20]. The computational domain consisted of a cylindrical vessel with a diameter of 36 microns and a length of 3060 microns and an injury site zone. Such vessel geometry has the diameter/length ratio of 0.0117, in line with the geometry described in [36] to account for the correct hemodynamic conditions with constant pressure drop boundary conditions. The injury site zone was described as the difference between an ellipsoid with semi-axes of 13.5, 6 and 13.5 microns and a cylindrical vessel (Fig 2A). The center of the ellipsoid was located on the wall of the vessel. The pressure drop was 1100 Pa to ensure a wall shear rate of 1000 s^-1 in a vessel without thrombus.

Thrombus permeability: The thrombus was considered as a porous medium consisting of 2 compartments with different porosity values corresponding to the core and shell of the thrombus [2]. The hydraulic permeability of the core and shell zones were calculated based on their porosity values according to the Kozeny-Carman equation [37]. We used the following equation:

(1)

where is the permeability, is the porosity, is the average platelet diameter (2 µm). The values of all model parameters are given in Table 2.

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Table 2. Model parameters. Designations: MPV - mean platelet volume, WSR - wall shear rate.

https://doi.org/10.1371/journal.pcbi.1014062.t002

Brinkman equations: The Brinkman approach was used to calculate the flows in a vessel with a thrombus [47].

We used the following set of equations:

(2)(3)(4)

where ρ is the fluid density (1000 kg/m³), μ is the dynamic fluid viscosity, u is the flow velocity, p is the pressure, is the porosity, is the hydraulic permeability, I is the identity (unity) tensor.

The vessel walls and the surface of the injury site zone were considered as no-slip boundaries.

To solve Brinkman equations it is necessary to know the values of viscosity (μ), porosity () and permeability () at each moment in time and at each point in the vessel. In our model, the permeability value is completely determined by the porosity value (Eq. 1). Importantly, in our model local porosity and viscosity values depended on the platelet activation state, as discussed below.

Platelet responses in the model: Our model implies three types of platelet response to activation stimuli (Fig 1). The first response involves integrin activation which mediates platelet aggregation. In the model platelet aggregation process is described as formation of the region with high value of porosity that we term shell. This process is triggered by both thrombin and ADP. The second response is dense granule secretion that in our model is triggered by thrombin. The model describes this process as a localized flux of ADP molecules that depends on local thrombin concentration. The third response is platelet contraction that in our model is represented as localized transition to low porosity value. This region with this low porosity value we term the core. Importantly, the second and the third processes are considered to be correlated in the model: we imply that platelets that secreted some critical percentage of their dense granules, also participate in contraction process, hence lowering local porosity value. In order to compare simulation results with experimental data wherein core region is defined as thrombus area where platelet secreted their alpha-granules, we assume that the contracted platelets also secreted their alpha granules.

Relationship between agonists concentration, platelet activation and local porosity: In our model, thrombus formation is driven by thrombin and ADP. These activators play an essential role in thrombus formation in both laser-induced thrombosis and FeCl₃-induced thrombosis [2,22].

To describe the formation of thrombus shell, we suggested that stable initial platelet attachment to the thrombus requires a significant level of agonist-induced αIIbβ3 integrin activation, and is only possible if the local concentration of thrombin or ADP exceeds threshold values. We considered that the thrombus shell zone instantaneously forms where agonist concentrations exceed these threshold values. Since the characteristic time of platelet integrin activation is about 1 second [11], which is significantly less than the characteristic time of thrombus growth in the laser-induced thrombosis model (about 1 minute [2]), we consider this assumption acceptable. In addition, for platelets activated only by ADP, we considered that if the local concentration of ADP falls below the threshold value, these platelets will be deactivated and will instantly detach from the thrombus, since ADP reversibly activates platelets [25]. In contrast, for thrombin-activated platelets, we assumed that platelets would remain activated and attached to the thrombus even if the thrombin concentration fell below a threshold value, since thrombin irreversibly activates platelets [26]. To describe this, we assumed that platelet integrin activation in response to thrombin depends on the maximum local concentration of thrombin over a time interval [0,t], where t is the current moment in time.

In agreement with the core-shell paradigm [2], we considered that the formation of the thrombus core zone is the result of platelet activation by high doses of thrombin (see below). Strong activation of platelets by thrombin is accompanied by granule release. Based on this, we postulated that the formation of the thrombus core zone occurs where the (normalized) platelet granule content (w) becomes less than the threshold value.

Taking all this into account, we used the following system of equations:

(5)

Where shPor is the thrombus shell porosity (0.65); corePor is the thrombus core porosity (0.45); [ADP] is the local concentration of ADP; [maxII_a] is the maximal local concentration of thrombin in the interval [0,t], where t is the current moment in time; is the threshold of the platelet activation by thrombin (0.1 nM); is the threshold of the platelet activation by ADP (90 nM); w is the normalized platelet granule content; is the threshold value (0.368).

Formation of the thrombus core zone: As platelet granule secretion is an irreversible process and occurs in response to potent platelet activators – such as thrombin and collagen, it is usually considered as a marker of strong platelet activation. As in vivo data demonstrates that low porosity core region of thrombus possesses platelets that secreted their alpha-granules [2,4], we postulated that transition to low porosity value occurs only in the region where degree of granule release (w) is below some critical value. Platelet response to thrombin was described by sigmoidal function, based on in vitro data on thrombin-induced αIIbβ3 receptor activation and the secretion of dense granules [21,32] by mouse platelets. For simplicity, we assumed that in the wild type mice both platelet dense granule release and alpha granule release share the same kinetics.

To describe the kinetics of shell-to-core transformation we used the following equation:

(6)

Where w is the normalized platelet granule content, w (t = 0)=1; [II_a] is the local thrombin concentration; k_t([II_a]) is rate of the process, which is the sigmoidal function describing platelet response to thrombin. We assumed that the transformation takes time and occurs when the normalized platelet granule content (w) falls below a threshold value = 1/e = 0.368.

Maximal rate of transformation K_max was set to the rate of dense granule release in response to high doses of thrombin [41]. Parameters of the sigmoid function describing dynamics of transformation were recovered from in vitro data on the dependence of total platelet ATP release on thrombin concentration [21] (see details in Supplemental Text C in S1 Text). Taking all this into account, we used the following equation:

(7)

Where c₁ = 0.4 nM, c₂ = 0.8 nM, q=(([II_a]-(c₁ + c₂)/2)/(c₂-c₁))+0.5, K_max is the maximal rate of the process (0.15 s^-1).

Due to the sigmoid nature of the thrombin response, the equations Eq. 6, Eq. 7 guarantee that the transformation of the shell zone into the core zone will not occur where the thrombin concentration is too low. This is consistent with the existence of large thrombus shell and experimental data on thrombin activity in the thrombus [2,4].

Equations Eq. 6, Eq. 7 were used to describe the formation of the thrombus core in both wild-type and mice with Hermansky-Pudlak syndrome, since the thrombus core is formed in both cases [2,20].

Relationship between agonist concentration and local viscosity: In our model, we considered two fluids with different viscosity values - blood plasma inside the thrombus, and whole blood in the vessel. To introduce this into the model we used the following equations:

(8)

where μ is the local viscosity, is the blood plasma viscosity (1.25 mPa ⋅ s), is the whole blood viscosity (3.4 mPa ⋅ s).

For implementation purposes, instead of equation Eq. 8 we used an equivalent equation in which the porosity was expressed based on the concentrations of the agonists (using Eq. 5):

(9)

Generation and transport of thrombin and ADP

Generation of thrombin.

In the cremaster model thrombin has the potential to be generated both extravascularly, where TF is present [3], and intravascularly on the injured endothelium [48]. Thrombin generation on the surface of activated platelets does not seem to play the major role in cremaster model [48], and was not considered in our model, in line with another recent work [30].

To describe thrombin generation we modified the boundary condition and introduced an inward time-dependent thrombin flux from a zone on the surface of the injury site (blue zone on Fig 2A). This approach allowed us to take into account extravascular sources of thrombin. We used the following equation:

(10)

Where t₀ is the characteristic time of thrombin generation (30 s), is equilibrium thrombin flux (12 pmol/(m² ⋅ s)) (we will refer to it simply as ‘thrombin flux’). These parameters were calibrated based on the comparison of simulation results with experimental videos of thrombus formation (S1 Fig).

We tested several options for the spatial location of the thrombin generation zone (S2 Fig). Thrombin generation in the upstream half of the injury site zone showed better agreement with experimental data compared to other alternatives (see Supplemental Text A in S1 Text).

We also performed additional simulations with a different dynamics of thrombin generation, see details in Supplemental Text A in S1 Text.

Inactivation of thrombin.

Thrombin is inactivated by several plasma proteins. The main pathway consists of complex formation with antithrombin III, but several other inhibitors, such as α1 -antitrypsin and α2 -macroglobulin, are active as well [49].

Computation of the constant of thrombin inactivation was performed based on data from the detailed computational model of coagulation by Dashkevich and colleagues [40]. We used the following equation:

(11)

where [IIa]- local thrombin concentration, 0.025 s^-1. Details are described in Supplemental Text C in S1 Text.

Thrombin transport equation.

To describe the transport of thrombin, we used the following convection-diffusion equation, which takes into account the inactivation of thrombin by blood plasma inhibitors (Eq. 11):

(12)

Where D_IIa is the thrombin diffusion coefficient, [IIa] is the local thrombin concentration. In the basic version of the model, thrombin diffusion coefficient did not depend on porosity. Computations with porosity-dependent thrombin diffusion coefficient are discussed in Supplemental Text A in S1 Text.

ADP release model.

The secretion of cargo molecules from platelet granules in response to activation is well described as a first-order reaction:

(13)

where N is the number of cargo molecules remaining in the platelet at time t, k_ex is the rate of secretion which depends on the cargo molecule, potency and dosage of the platelet agonist [41].

In out continuum approach, we used the following form of this equation:

(14)

Where N_ADP is the local concentration of releasable ADP in platelet (amount of ADP in platelet dense granules divided by mean platelet volume).

In our model, we considered only thrombin-induced platelet secretion, in line with our previous study [19]. The dependence of the ADP secretion rate k_ex on the thrombin concentration was described by a sigmoid function. We used the same sigmoidal function describing platelet response to thrombin that we used to describe the process of thrombus core formation (see details in Supplemental Text C in S1 Text). Maximal rate of secretion K_max was set to the rate of dense granule release in response to high doses of thrombin [41]. Taking all this into account, we used the following equation:

(15)

Where c₁ = 0.4 nM, c₂ = 0.8 nM, q=(([II_a]-(c₁ + c₂)/2)/(c₂-c₁))+0.5, K_max is the maximal rate of secretion (0.15 s^-1).

In our continuum approach, volumetric flux of ADP was described using the following equation:

(16)

Where is the local porosity value, and the factor corresponded to the fraction of the local volume occupied by platelets.

No-flux boundary condition was applied for ADP on the vessel walls and the surface of the injury site zone.

In our model, kinetics of thrombin-induced granule release (Eq. 15) is identical to the kinetics of thrombin-induced transformation of thrombus shell zone into thrombus core zone (Eq. 7). This feature of the model corresponds to the presence of two parallel processes - changes in porosity and granule secretion, which occur in the thrombus core on the same time scale [4].

ADP transport equation.

To describe the transport of ADP, we used the following convection-diffusion equation, which takes into account the ADP release from thrombin-activated platelets (Eq. 16):

(17)

Where D_ADP is the ADP diffusion coefficient, [ADP] is the local ADP concentration.

Model implementation

Simulation details and data processing.

Each simulation lasted 130 seconds of the biological time. The model predictions were compared to 3 experimental videos from 2 papers [2,20]. For this purpose, we calculated the dynamics of the thrombus area and the area of the thrombus core - an approach widely used in the original articles to analyze the dynamics of thrombus formation [2,4,20]. The dynamics of the thrombus area and the thrombus core area in the experiment (that is, the area occupied by CD 41 positive platelets and P-selectin positive platelets) was calculated based on the analysis of frames of experimental videos using a Python script. The dynamics of the thrombus area and the thrombus core area in the model simulations were inferred based on the 3D data by 2D integration over a plane passing through the center of the injury site zone and the horizontal axis of the vessel. To calculate the dynamics of the thrombus area, the following expression A_t was integrated:

(18)

To calculate the dynamics of the thrombus core area, the following expression A_c was integrated:

(19)

Choice of the parameter values

In total, our model had 39 parameters. There were 2 parameters related to thrombin generation: characteristic time of thrombin generation t₀ (30 s) and equilibrium thrombin flux J_max. In 3D version of the model both values were fitted by comparing results of model simulations with thrombus growth dynamics in a representative experimental video of laser-induced thrombosis [20].

For the thrombus structure, model had 2 parameters: thrombus shell porosity and thrombus core porosity. For the thrombus shell porosity, we used shPor = 0.65. For the thrombus core porosity, we used corePor = 0.45. These values were taken from in vivo data on mouse thrombi (Fig 4 from [2]).

Platelets were described by 2 parameters: mean platelet volume, and average platelet diameter. Value of mean mouse platelet volume MPV = 4 fl was taken from in vitro data [43]. This value was also used to estimate average platelet diameter.

The model had 2 key parameters that described platelet activation by agonists. For the value of the platelet activation threshold by thrombin, we used . This value is equal to experimentally measured EC50 value for thrombin-induced integrin αIIbβ3 activation in mouse platelets [32]. For the value of the platelet activation threshold by ADP, we used . This value is significantly lower than the experimentally measured EC50 value 590 nM for ADP-induced αIIbβ3 activation in mouse platelets, but it is close to the ADP concentration 100 nM, which causes ~10% of the maximal platelet response [32].

Two diffusion coefficients were used to describe the transport of agonists. The diffusion coefficient of thrombin, D_IIa = 67 µm²/s, was estimated based on experimental value for bovine thrombin [38] corrected for blood plasma viscosity. The diffusion coefficient of ADP D_ADP = 237 µm²/s was taken from published data [39].

Secretion of ADP from thrombin-activated platelets was described by 5 parameters. For the maximal ADP secretion rate, we used K_max = 0.15 s^-1. This value was reported for the secretion of serotonin, a marker of ADP-containing dense granules, in response to activation by high doses of thrombin in human platelets [41]. In our model, the dependence of secretion kinetics on thrombin concentration was described by a sigmoid function recovered from in vitro data [21]. The sigmoid was described by 2 parameters: lower threshold of thrombin concentration, c₁ = 0.4 nM, which corresponds to zero secretion rate, and upper threshold of thrombin concentration c₂ = 0.8 nM, which corresponds to maximal secretion rate. These values were fitted to describe the experimentally observed relationship between total ATP secretion from mouse platelet dense granules and thrombin concentration [21](see details in Supplemental Text C in S1 Text). For the total amount of releasable ADP in mouse platelet, we used ADP₀ = 0.46 ∙ 10^-8 nmol. This value corresponds to the total amount of ADP in a mouse platelet [42], which means that it may be overestimated. Based on this value we calculated concentration of releasable ADP in mouse platelet N₀ as the ratio between the total amount of releasable ADP and the platelet volume.

Transformation of shell zone into core zone was described by 4 parameters, 3 of which (the maximal rate of transformation and parameters of the sigmoid describing response to thrombin) were also used for ADP release model and they were described above. The threshold of transformation of the shell platelet into core platelet was set to 1/e = 0.368. According to equations Eq. 6, Eq. 7 this means that minimal time of transformation equals 1/K_max = 6.67 s. Since in experiments core starts to form about 25 seconds after the injury [4] and platelet activation time is about 1 second [11], we suggest that such minimal time of transformation is in the realistic range.

Parameters related to model geometry are discussed in the corresponding sections.

2D model of thrombogenesis

To analyze FeCl₃ –induced thrombosis in mouse carotid artery we used 2D version of the continuum model described above. The structure of the model and its parameters were the same as in the 3D case. The only changes were introduced to the model geometry, hydrodynamics, thrombin generation and simulation procedures as described below.

2D axisymmetric model of thrombogenesis

To describe the axisymmetric 3D case for simulations of thrombus growth in response to the circular injury, we used 2D axisymmetric version of the 3D model described above. Model structure and model parameters were the same as in the 3D model. The only changes were related to model geometry, hydrodynamics and thrombin generation as discussed below.