Integrating blood cell mechanics, platelet adhesive dynamics and coagulation cascade for modelling thrombus formation in normal and diabetic blood

Abstract

Normal haemostasis is an important physiological mechanism that prevents excessive bleeding during trauma, whereas the pathological thrombosis especially in diabetics leads to increased incidence of heart attacks and strokes as well as peripheral vascular events. In this work, we propose a new multiscale framework that integrates seamlessly four key components of blood clotting, namely transport of coagulation factors, coagulation kinetics, blood cell mechanics and platelet adhesive dynamics, to model the development of thrombi under physiological and pathological conditions. We implement this framework to simulate platelet adhesion due to the exposure of tissue factor in a three-dimensional microchannel. Our results show that our model can simulate thrombin-mediated platelet activation in the flowing blood, resulting in platelet adhesion to the injury site of the channel wall. Furthermore, we simulate platelet adhesion in diabetic blood, and our results show that both the pathological alterations in the biomechanics of blood cells and changes in the amount of coagulation factors contribute to the excessive platelet adhesion and aggregation in diabetic blood. Taken together, this new framework can be used to probe synergistic mechanisms of thrombus formation under physiological and pathological conditions, and open new directions in modelling complex biological problems that involve several multiscale processes.

1. Introduction

Normal haemostasis and pathological thrombosis in a flowing bloodstream can occur on damaged vessels as well as on the surfaces of artificial internal organs [1]. A thrombus is predominantly composed of platelets, which are non-deformable ellipsoid shaped blood cells in their passive state circulating in blood with a concentration of 150 000–$400\hspace{0.17em}000\hspace{0.17em}{\mathrm{per}\hspace{0.17em}\hspace{0.17em}\mathrm{mm}}^3$. The formation and growth of a thrombus at a site of injury on a blood vessel wall is a complex biological problem, involving a number of multiscale simultaneous processes including: (1) margination of the platelets from the core to the periphery of the blood vessel by red blood cells (RBCs) [2], contributing to platelets’ quick activation and accumulation on the injured area of vessel wall; (2) transport of the coagulation factors to the injured sites of the vessel; (3) a series of chemical reactions in the coagulation cascade occurring in plasma around the injured sites and on the platelet’s surface; (4) activation of platelets and adherence to the thrombogenic sites [3]. Comprehensive studies involving all these processes are challenging as they occur at different spatio-temporal scales [4]. For example, platelet–wall and platelet–platelet interactions through receptor–ligand bindings occur at a sub-cellular nanoscale, whereas the blood flow dynamics in the vessel around the developing thrombus is described as a macroscopic process from several micrometres to millimetres.

Numerous computational models [5–15] have been developed to address the underlying blood clotting processes and predict thrombus growth despite making assumptions and simplifications in the models, see reviews in [16,17]. It is worth noting that Xu et al. [18] outlined a multiscale model that included macroscale dynamics of the blood flow using the continuum Navier–Stokes (NS) equations, and microscale interactions between platelets and the wall using a stochastic discrete cellular Potts model. The enzymatic reactions of the coagulation pathway at the injured wall in plasma as well as on the platelet’s surface were incorporated through coupling advection–diffusion–reaction (ADR) equations to blood flow. These numerical simulations, however, were limited to two-dimensions due to the significant cost of solving many ADR equations. Fogelson & Guy [19] used a continuum Eulerian–Lagrangian approach, where the NS equations were solved along with the ADR equations for the transport of coagulation factors, whereas an immersed boundary method was used to couple platelet (treated as rigid spherical particles) interactions with blood plasma. However, neither Tosenberger et al. [13] or Fogelson & Guy [19] simulated RBCs explicitly. The effect of RBCs on the platelet motion was not considered through enhancing platelet diffusive motion toward the wall either, in their works. Although computational models without explicit representations of RBCs and platelets can produce results in good agreement with experimental data in simple channel flows, they may not perform well in complex flow geometries such as vessels containing aneurysms or stenoses.

The high level of complexity and heterogeneity in the existing models as well as the computational cost have limited their applications in simulating a three-dimensional physiological clotting process. In this work, we propose a multiscale numerical framework that seamlessly integrates different sub-processes involved in platelet aggregation and adhesion. This framework, which is developed based on dissipative particle dynamics (DPD) method, a mesoscopic particle-based hydrodynamics approach [20], can concurrently simulate the transport and interaction between RBCs and platelets, transport of coagulation factors involved in the coagulation, chemical reactions and adhesion of platelets to the thrombogenic sites, the four key sub-processes of blood clotting. Prior studies have demonstrated that DPD is an effective method to study cell biomechanics [21,22], cell motions and interactions in blood flow [23–25] as well as chemical species transport [26]. The advantage of this framework is the seamless integration of hydrodynamics, cell mechanics and chemical reactions, thereby significantly reducing the computational cost induced by the unnecessary communication overhead between different solvers. To demonstrate the new capabilities, we employ the proposed framework to simulate the platelet activation and adhesion to thrombogenic sites in a three-dimensional microchannel in normal and diabetic blood, and dissect the effects of the biomechanics and biochemistry on platelet adhesion and aggregation under physiological and pathological conditions.

The rest of the paper is organized as follows. In §2, we introduce the methods to describe the four key components in this framework, namely blood flow transport, coagulation kinetics, blood cell mechanics and platelet adhesion dynamics as well as how these four components are integrated in this framework. In §3, we implement this framework to simulate platelet adhesion to the injury sites in a micro-channel in normal and diabetic blood, respectively. In §4, we summarize the simulation results and discuss their clinical implications.

2. Multiscale numerical methodology

The foundation of our multiscale framework is the DPD method established by Espanol & Warren [20], upon which other modules are added. The multiscale framework includes four main modules: the haemodynamics, coagulation kinetics and transport of chemical species, blood cell mechanics and platelet adhesive dynamics. Each of these modules is introduced in the following sections below. Nomenclature is listed in Table 1.

Table 1.

Nomenclature.

n	number density of fluid DPD particles
rc	cut-off radius (μm)
η	dynamic viscosity of fluid (N s m−2)
kBT	thermal energy unit (J)
Ci	concentration per unit mass of species i (M kg−1)
Qi	total concentration flux of species i (M)
$Q_{\mathrm{ij}}^D$	Fickian concentration flux between species i, j (M)
$Q_{\mathrm{ij}}^R$	random concentration flux between species i, j (M)
Di	diffusion coefficient of species i
$Q_i^S$	source term of concentration i resulting from to the enzymatic reactions (M)
kP	the lowest clot permeability (m2)
Nv	number of particles in the cell model
Vs	elastic energy of worm-like-chain bonds in the blood cell model (J)
Vb	bending energy of the blood cell model (J)
Va	surface energy of the blood cell model (J)
Vv	volume energy of the blood cell model (J)
μ0	shear modulus of the normal RBC (μN m−1)
k0	bending rigidity of the RBC (J)
$A_0^{\mathrm{tot}}$	surface area of the RBC (μm2)
$V_0^{\mathrm{tot}}$	volume of the cell (μm3)
AR	aspect ratio of the cell
Pf	probability of the bond formation
Pr	probability of the bond breakage
kf	rate of the bond formation (1 s−1)
kr	rate of the bond breakage (1 s−1)
$k_{\hspace{0.17em}f}^0$	intrinsic rate of the bond formation (1 s−1)
$k_r^0$	intrinsic rate of the bond breakage (1 s−1)
kb	stiffness of the GPIbα–vWF bond (N m−1)
lb	equilibrium bond length (μm)
xb	receptor–ligand distance (μm)
[i]	concentration for species i (M)

2.1. Haemodynamics

The blood plasma in the simulation is modelled using DPD method following our previous paper [23]. Detailed information of the DPD method and model parameters can be found in electronic supplementary material.

2.2. Transport of chemical species and coagulation kinetics

Transport of chemical species involved in the coagulation cascade is modelled by transport dissipative particle dynamics (tDPD), an extension of the classical DPD framework [26] with extra variables for describing concentration fields. To consider the coagulation cascade, we follow the mathematical model of Anand et al. [27], which contains a set of 21 coupled ADR equations for describing the evolution of 23 biological reactants involved in both intrinsic and extrinsic pathways of blood coagulation. Detailed information of transport of chemical species and the model of Anand et al. [27] can be found in the supporting information (electronic supplementary material).

2.3. Cell mechanics

In this framework, we employ the DPD approach introduced in the work of Fedosov et al. [28] to represent RBCs and use its extension for platelets [29]. As shown in figure 1a, in this study, we model a normal RBC with N_v = 500, shear modulus μ₀ = 4.73 μN m⁻¹ and bending rigidity k₀ = 2.4 × 10⁻¹⁹ J. The cell surface area is selected to be $A_0^{\mathrm{tot}}=132.9\hspace{0.17em}\mu {\mathrm{m}}^2$, and cell volume $V_0^{\mathrm{tot}}=92.5\hspace{0.17em}\mu {\mathrm{m}}^3$, which give surface to volume ratio S/V = 1.44. All parameters used in our normal or diabetic RBCs model are calibrated based on existing experimental data and validated with simulations, from single RBC mechanics to blood flow dynamics [21,30–34].

Figure 1.

Schematics of the simulation set-up. (a) Models for normal and diabetic RBCs and platelets. (b) A three-dimensional rectangular microchannel with the size of 110 μm × 20 μm × 20 μm is filled with RBCs, platelets and solvent particles which carry 22 species. Left: front view, right: side view. Solvent particles are omitted in visualization for clarity. Two injury sites (highlighted in blue) with a length of 40 μm are located at the upper and bottom channel wall. The mass flux of four species listed in electronic supplementary material, table S7 are injected into the channel to initiate and drive the coagulation cascade.

For platelets, which are nearly rigid in their passive form, we choose shear modulus and bending rigidity sufficiently large (100 times larger than the normal RBCs) to ensure its mechanical behaviour as a more rigid cell. The number of vertices in the platelet’s membrane network is N_v = 48 and the aspect ratio of the cell is AR = 0.38. Based on our previous analysis on the patient-specific data [22], a normal platelet has cell volume $V_0^{\mathrm{tot}}=6\hspace{0.17em}\mu {\mathrm{m}}^3$. Detailed information of the RBC and platelet model and the model parameters can be found in electronic supplementary material.

2.4. Platelet adhesive dynamics

The platelet adhesive dynamics model describes the adhesive dynamics of receptors on the platelet membrane binding to their ligands. In this work, we employ the platelets adhesive dynamics model proposed by Mody & King [35]. This model uses the Monte Carlo method to determine each bond formation/breakage event based on specific receptor-ligand binding kinetics. We estimate the probability of bond formation P_f and probability of bond breakage P_r using the following equations:

$$P_{\hspace{0.17em}f}=1-\exp (-k_{\hspace{0.17em}f}\mathrm{\Delta }t)$$ 2.1a

and

$$P_r=1-\exp (-k_r\mathrm{\Delta }t),$$ 2.1b

where k_f and k_r are the rates of formation and breakage, respectively. Following the catch bond formula by Mody & King [35], the force-dependent k_f and k_r are evaluated by

$$\begin{array}{cc}& k_{\hspace{0.17em}f}=k_{\hspace{0.17em}f}^0\exp (\mid F_b\mid \frac{({\sigma }_{\hspace{0.17em}f}-0.5\mid x_b-l_b\mid )}{k_{B}T}),\\ & k_r=k_r^0\exp \left(\frac{{\sigma }_r\mid F_b\mid }{k_{B}T}\right),\end{array}\}$$ 2.2

where $k_{\hspace{0.17em}f}^0$ and $k_r^0$ are the intrinsic bond formation and breakage rates, respectively, and σ_f and σ_r are the corresponding reactive compliance. Detailed information of the platelet adhesive dynamics model and the model parameters can be found in electronic supplementary material.

2.5. Problem set-up

For simplicity, we set up a three-dimensional straight microchannel with the size of $110\times 20\times 20\hspace{0.17em}\mu {\mathrm{m}}^3$ filled with RBCs, platelets and solvent particles which carry 22 chemical species, as shown in figure 1b. The channel is periodic in the x- and z-directions. The initial concentration of the species are listed in electronic supplementary material, table S5. The haematocrit of the blood and platelet density are selected to be $25\text{\%}$ and $500\hspace{0.17em}000\hspace{0.17em}{\mathrm{mm}}^{-3}$, respectively. The platelet density is taken somewhat larger than the physiologic value such that the adhesion of platelets to vessel injury site can be more pronounced. In addition, all platelets are initially placed on the upper and lower sides of the channel to accelerate margination and the adhesion process. The wall boundaries in the simulation domain are treated as pseudo-planes, following the work of Lei et al. [36]. Specifically, particles within one cut-off distance from the wall are reflected in a bounce-forward scheme. Further, in order to impose the no-slip condition, a predefined force that compensates for the missing particles on the opposite side of the wall is applied to the particles both in wall normal and tangential directions. Following our work in [26], we assume that coagulation occurs at the site of injury. As shown in figure 1b, the process is initiated and sustained by imposing a boundary flux for the four species listed in electronic supplementary material, table S7. All the simulations were performed at Stampede2’s Knights Landing (KNL) compute nodes at Texas Advanced Computing Center. Each node has 68 cores. Each simulation runs 1 000 000 timesteps (corresponding to 45 s in the physical time) on five nodes (340 CPU cores) running for 24 h.

People with type 2 diabetes mellitus (T2DM), the most common type of diabetes, are prone to experience thrombotic events and develop cardiovascular complications [37]. It has been determined that multiple factors induced by T2DM contribute to the prothrombotic status, such as elevated coagulation, impaired fibrinolysis, endothelial dysfunction and platelet hyperreactivity [38]. Our previous studies have demonstrated that the altered morphology and biomechanics of RBCs and platelets in diabetic blood lead to enhanced platelet margination [25] and abnormal haemorheology [22], both of which potentially aggravate the platelet aggregation and thrombus formation. In particular, diabetic RBCs, which exhibit a near-oblate shape with a reduced S/V ratio (see electronic supplementary material, table S3), are larger in size and less deformable than normal RBCs [39]. The reduced deformability of diabetic RBCs leads to increase in blood viscosity up to approximately 20% higher than normal samples [40]. On the other hand, clinical data have showed that the mean cell volume of platelets measured from the blood samples of diabetic patients is larger than that of the non-diabetic control, which is likely caused by osmotic swelling as a result of hyperglycaemia [25].

In this work, we simulate both the normal and diabetic blood flow. Diabetic RBCs have a shear modulus of μ_s = 2μ₀ = 9.46 μN m⁻¹ and a reduced S/V = 1.04. A normal platelet has cell volume $V_0^{\mathrm{tot}}=6\hspace{0.17em}\mu {\mathrm{m}}^3$ whereas a diabetic platelet has $V_0^{\mathrm{tot}}=12\hspace{0.17em}\mu {\mathrm{m}}^3$. The parameters of diabetic RBCs and platelets are selected based on patient-specific data and they were validated in our previous work [22], see electronic supplementary material, table S3.

3. Results

3.1. Platelet adhesion in normal blood

We start the simulation without turning on the coagulation cascade and allow the blood flow first to reach equilibrium. To drive the blood flow in the microchannel, we apply a constant body force to each DPD particle, which yields a plug-like profile for blood. In figure 2a, we plot the plug-like velocity profile and the mean blood velocity is computed as $\overline{u}=2.05\hspace{0.17em}{\mathrm{mm}\hspace{0.17em}\mathrm{s}}^{-1}$. The Reynolds number, $Re=\overline{u}H/\nu$, defined based on the channel width H = 20 μm, is approximately 0.015. The characteristic shear rate is defined as the wall shear rate of a parabolic Poiseuille flow with the same mean velocity and is $\dot{{\gamma }_{\omega }}=6\overline{u}/H=615\hspace{0.17em}{\mathrm{s}}^{-1}$. Furthermore, the Peclet number, a measure of ratio of advection to diffusion in transport processes, is defined by Pe = Re Sc, which is ∼231 for thrombin (assuming D_IIa = 6.47 × 10⁻⁷ cm² s⁻¹) in our simulations. We also plot the average distribution of the RBCs and platelets across the channel height in figure 2b, which shows the presence of a cell-free layer adjacent to the walls with a thickness of ≈3 μm.

Figure 2.

(a) Flow velocity profiles across the channel height (y-direction) for normal and diabetic blood. (b) Platelet count density distribution profiles across the channel height (y-direction) (blue lines) and haematocrit profiles (red lines). Solid lines represent normal blood and dashed lines represent diabetic blood. NRBC denotes normal RBCs and DRBC denotes diabetic RBCs. NPLT denotes normal platelets and DPLT denotes diabetic platelets.

Next, we initiate the coagulation cascade by injecting the four species into the blood flow from the injured wall region, as shown in figure 1b. A sequence of snapshots of the whole blood simulation in the channel is plotted in figure 3, which shows that the concentration boundary layer has been fully developed (shown by contours of thrombin concentration [IIa] in the $x-y$ plane). We note that we set a threshold value of 1 nM for thrombin-mediated platelet activation [41] for which the platelets in the boundary layer become instantaneously activated and ready to form bonds with the particles representing vWF ligands on the site of injury. Figure 3a–c shows that platelets, after activation, are gradually adhered to the site of injury (highlighted with green particles), attempting to cover the injury sites. Meanwhile, the adhered platelets can release ADP that also contributes to the activation of passive platelets flowing the blood. The thickness of thrombin boundary layer is approximately 2 μm, which is reasonable under the current flow conditions and the value of the Peclet number.

Figure 3.

(a–c) Three sequential snapshots of normal RBCs and platelets flowing in the microchannel and adhered to the injury sites at simulation times of t = 10 s, 20 s, 40 s, respectively. Coagulation reactions generate thrombin that activates passive platelets when its concentration is larger than 1 nM. An increased number of platelets are adhered to the injury sites (highlighted in green) after being activated by thrombin.

Figure 4a presents a closer look at the time-evolution of thrombin concentration at three different axial positions, i.e. x = 40, 50, 60 μm. The results show a rapid burst of thrombin concentration occurring at x = 40 μm, followed by a gradual decrease before the concentration stabilizes. Similar variations of thrombin concentration are observed at x = 50 μm and x = 60 μm, but with reduced magnitudes, illustrating the effect of blood flow on the transport of chemical species generated from the coagulation cascade. This effect also can be observed from figure 5 where we plot three sequential snapshots of the concentrations of three coagulation factors, including factor X, antithrombin III and fibrin. These figures show that at the early stage of the coagulation (t = 5 s), the factor X and antithrombin III are consumed and the fibrin is generated around the injury sites. As the coagulation reactions continue, the coagulation factors flow in from upstream to supply the reactions, whereas the generated fibrin flow downstream.

Figure 4.

Thrombin concentration evolution over time in the $y-\mathrm{z}$ cross-section at three axial locations of x = 40, 50, 60 μm in case of (a) normal and (b) diabetic blood. The injury sites range from x = 25 to 65 μm. The surface concentration of TF–VIIa complex is taken as 0.25 nM for both cases and the reaction rate is accelerated 100 times. The error bars are computed based on five stochastic realizations of the simulations.

Figure 5.

Three sequential snapshots of the concentration boundary layers of (a) factor X (b) antithrombin III and (c) fibrin in the $x-\mathrm{y}$ plane at the middle cross-section of the microchannel. Results are computed at simulation times of t = 5 s, 20 s, 40 s, respectively (from left to the right).

To quantify the adhesion dynamics of the platelets to the injured vessel wall, we plot the total number of bonds between activated platelets and the ligands at the injury sites in figure 6a with respect to time for five independent simulations under the same flow conditions. In addition, we also recorded the number of adhered platelets with time, which is shown in figure 6c. These results show that after an initial time lag of approximately 5 s, the number of bonds and adhered platelets both start to rise sharply, and the growth of these numbers slows down as more platelets adhered to the injured area. We note that in our simulations platelets only form bonds with the injury sites on the wall, which leads to formation of a monolayer, but the trend of platelet aggregation is consistent with an in vitro study [42] and an in vivo measurements in mice arterioles [43], where the authors observed a time lag in the initiation of platelet aggregation followed by the increase in fluorescence intensity (i.e. number of adhered platelets) that eventually reaches a plateau. Next, we present the sensitivity of the results to several important parameters in the model, i.e. the boundary concentration level of TF–VIIa complex as well as the platelet’s adhesive parameters. First, we vary the TF–VIIa concentration, which represents the extent of injury on the vessel wall, through increasing and reducing the control value by a factor of 5, respectively, and investigate how these variations affect the bond formation between the platelets and the injured vessel wall. Figure 7a shows that as the TF–VIIa concentration is reduced to 0.05 nM, the number of formed bonds is reduced by approximately 40%. On the other hand, when the TF–VIIa concentration is increased to 1.25 nM, the number of bonds is comparable to the case of [TF–VIIa] = 0.25 nM. These results suggest that under our simulation conditions, decreased [TF–VIIa] leads to a less pronounced platelet adhesion whereas a higher level of [TF–VIIa] by a factor of 5 does not boost the platelet adhesion, which is likely attributed to the lack of other necessary species to increase the generation of thrombin.

Figure 6.

The evolution of the number of bonds formed between platelets and injury sites in case of (a) normal and (b) diabetic blood as well as the number of adhered platelets for (c) normal and (d) diabetic blood. The surface concentration of TF–VIIa complex is taken as 0.25 nM for both cases and the reaction rate is accelerated 100 times. Each of the plots consists of five stochastic realizations and the black curves highlight the average of the five cases.

Figure 7.

Sensitivity analysis of TF–VIIa concentration, the bond breakage constant k_r and platelet receptor binding capacity on the number of bonds formed between platelets and injury sites. Platelet-wall bond formation comparisons are performed at (a) three levels of TF–VIIa concentration (b) three values of k_r and (c) number of the bonds that can form on a platelet particle.den.rep. is short for density representation.

Then, we vary the intrinsic bond breakage rate $k_r^0$, which can be associated with the genetic bleeding disorder von Willebrand disease (VWD), such as 2B-type VWD or the platelet-type VWD, where the alterations in the binding kinetics allow longer bond lifetimes and enhanced bond formation rates, leading to undesired platelet and vWF binding [44]. Here, we increase and reduce the physiological value of $k_r^0=5\times {10}^{-7}\hspace{0.17em}{\mathrm{s}}^{-1}$ by 10 times, respectively. Figure 7b shows that a higher rate of breakage reduces the number of bond formations and thus diminishes the platelet adhesion whereas a lower rate of breakage promotes the bond formation. Next, we vary the binding site density on the platelets (number of bonds can be formed on each platelet particle) and examine its effect on the bond formation. We note that although there is a significant number of GPIbα receptors on the platelet’s surface, not all of them are able to form bonds. In particular, under pathological conditions, such as in the Bernard–Soulier syndrome, a type of glycoprotein receptor deficiencies, platelets are characterized by loss-of-function mutations in GPIbα [45]. To consider the reduced binding capability of GPIbα in diseased states, we vary the binding site density of the platelet model from the control value of 5 to 1 and 2. As shown in figure 7c, as the binding site density is reduced, the number of bonds formed between the platelets and injured vessel wall is decreased due to the compromised binding capacity of the GPIbα receptors.

3.2. Platelet adhesion in diabetic blood

In this section, we simulate the platelet adhesion in diabetic blood and investigate how the altered biomechanics of diabetic RBCs and platelets as well as the changes in the amount of species involved in the coagulation affect the platelet adhesion. Following our previous work [22,25], the structure of the diabetic RBC model is similar to the normal RBCs except that an increased shear modulus of μ_s = 9.46 μN m⁻¹ and a reduced S/V ratio of 1.04 are employed to the diabetic RBC model, as shown in figure 1a. On the other hand, the diabetic platelet model has the same stiffness as the normal platelet model but with an increased volume of 12 μm³, which is twice as large as the normal one. These morphological and biomechanical changes are made based on our previous study of patient-specific data in [22,25]. The simulation set-up is the same as the case for normal RBCs shown in figure 1b. We apply a constant body force to drive the blood flow and we allow the flow to reach equilibrium before we start the coagulation cascade, following the same approach for normal RBCs.

The velocity profile of the diabetic blood flow in figure 2a shows that under the same pressure drop, the maximum velocity of the diabetic blood is approximately $15\text{\%}$ smaller than the normal blood. This difference is caused by the increased blood viscosity as a result of increased stiffness and size of diabetic RBCs, in agreement with our previous work [22]. The average distribution of the diabetic RBCs and platelets across the channel height are plotted in figure 2b, which shows enhanced platelet margination due to increase mean platelet volume, consistent with our previous study in [25]. A sequence of snapshots of the diabetic blood in the channel after the concentration boundary layer of thrombin has been fully developed (figure 8) show that an increased number of platelets are adhered to the site of injury (highlighted with green particles). The number of bonds formed between activated diabetic platelets and injury sites as well as the number of the adhered diabetic platelets are shown in figure 6b,d, respectively. Both of these two metrics show a rapid increase after an initial time lag of approximately 5 s, and then gradually slows down after platelets cover most of the injured area. We note that both the number of formed bonds and the adhered platelets for the diabetic RBCs are higher than those of normal RBCs (figure 6a,c), due to the enhanced platelet margination. Since the thickness of the thrombin boundary layer is only approximately $2\hspace{0.17em}\mu \mathrm{m}$, an increased margination of platelets can expose more platelets to the high concentration of thrombin and increase the number of activated platelets, thereby promoting platelet adhesion.

Figure 8.

(a–c) Three sequential snapshots of diabetic RBCs and platelets flowing in the microchannel and adhered to the injury sites at simulation times of t = 10 s, 20 s, 40 s, respectively. Coagulation reactions generate thrombin that activates passive platelets when its concentration is larger than 1 nM. An increased number of platelets are adhered to the injury sites (highlighted in green) after being activated by thrombin.

In addition to the alterations in the biomechanics of blood cells, changes in coagulation factors in the diabetic blood could promote thrombus formation in T2DM. For example, the concentration of plasma fibrinogen, a key source of fibrin, is reported to be approximately $50\text{\%}$ higher than the normal blood samples [46]. During coagulation, fibrinogen is converted into fibrin through a thrombin-mediated proteolytic process. Fibrin molecules then form long fibrin threads and networks that trap platelets and RBCs, building up a spongy mass that subsequently hardens and contracts to form a stable clot. Since fibrin is an essential component in a thrombus, it has been widely used to approximate the extent of a thrombus. Here, we increase the initial concentration of fibrinogen ([I]) in our simulation by 50% to represent the impact of elevated coagulation factors in diabetic blood and investigate how this change affects fibrin generation. Following the work of Anand et al. [27], we assume that a thrombus occurs when fibrin concentration equals or exceeds a concentration of 350 nM at the injury regions. The concentration boundary layers of fibrin ([Ia]) in the $x-y$ plane at the middle cross-section of the channel are plotted in figure 9a,b for normal and diabetic blood, respectively. These results show that the excessive fibrinogen in the plasma of diabetic blood could cause increased extent and density of the thrombus formed at the injury sites of the channel wall. Quantitative results of the fibrin generation in figure 9c show that fibrin generated in the $y-z$ plane at the three x-locations at the injury sites in diabetic blood is approximately $45\text{\%}$ higher than the case of normal blood due to the increased amount of plasma fibrinogen.

Figure 9.

The concentration boundary layers of fibrin ([Ia]) in the $x-y$ plane in the middle cross-section of the microchannel (z = 10 μm) in case of normal (a) and (b) diabetic blood. (c) Evolution of fibrin concentration over time in the $y-z$ cross-section at locations of x = 40, 50, 60 μm for normal blood (solid lines) and diabetic blood (dashed lines). The injury sites range from x = 25 μm to x = 65 μm. Initial concentration [I]₀ = 7 μM is given in normal blood whereas [I]₀ = 10.5 μM is given in diabetic blood. In the figure legend, (N) stands for normal blood, (D) for diabetic blood.

4. Discussion and conclusion

Computational modelling of thrombus initiation and development at the injury sites of blood vessel wall is challenging due to the multiscale and multiphysics nature of this process, which involves chemical species and blood cell transport, coagulation cascade reactions and platelet adhesion. Diverse studies have been conducted to address this complex problem at different scales, such as cellular, mesoscopic and continuum levels or from different perspectives. In order to reduce the model complexity and increase the computational efficiency, in this work, we develop a multiscale numerical framework based on the transport DPD method that can seamlessly integrate the multiple sub-processes of thrombus formation. To demonstrate the capability of the proposed framework, we implemented this framework to simulate platelet adhesion and aggregation triggered by exposure of tissue factor in a three-dimensional microchannel. We show that the proposed framework can simulate thrombin-mediated platelet activation and the subsequent adhesion to the injury site of the channel wall, a key process in haemostasis. In particular, our results on transport of chemical species illustrate that the concentration boundary layers of the thrombin that is sufficient to activate the platelets in the flowing blood are very thin (approx. 2 μm), implying that platelet margination is essential for platelet activation, adhesion and subsequent clot formation in a blood vessel.

To show the versatility of the model, we simulate platelet adhesion under several pathological conditions involving different levels of vessel injury and abnormalities in the platelet adhesion, which occur in various genetic haemostasis disorders, such as VWD and the Bernard–Soulier syndrome. Our model predictions from these simulation are qualitatively consistent with prior clinical findings [44,45]. More importantly, our model also can provide quantitative prediction of the relation between parameters obtained from experimental measurements at cellular level and the extent of thrombus formation under in vivo flow conditions. In addition, we simulate the platelet adhesion in diabetic blood and we find that the alterations in RBC and platelet biomechanics enhance the margination of the platelets and thus promote platelet adhesion to the injury sites on the blood vessel, whereas the changes in the amount of the species involved in the coagulation boost generation of fibrin, a key component for the thrombus growth and development after the initial platelet adherence to injury sites. These results suggest that both the elevated coagulation and altered properties of blood cells contribute to the prothrombotic status in T2DM.

In the current work, we examine the performance of the proposed model in a straight channel for simplicity as the focus of this work is to introduce this novel multiscale modelling framework. We note that different vessel geometries such as stenosed-arteries [47,48], curved vessels [49,50], bends [51,52], bifurcations [53–56], could change the blood cells and coagulation factors transport and thus impact the thrombus formation. We also note that several biological assumptions have been made in this framework. In order to observe the impact of the coagulation cascade on the platelet activation and adhesion, we assume that only activated platelets can adhere to the injury sites. However, a number of studies have shown that the platelets could adhere to the injury site or growing thrombus before activation [57–61]. As a result, our simulation results could underestimate the number of platelets adhered to the injured vessel wall. In addition, platelets become more deformable upon activation through reordering of actin network in their membrane and undergo drastic morphological changes [62,63]. In this work, we focus on the initial steps of thrombus formation triggered by coagulation cascade and thus did not consider the shape change of activated platelets to reduce the complexity of the model. A more sophisticated platelet model, such as [9,64], is required to address this complex process.

Taken together, we develop a computational framework that is able to effectively address different multiscale biological processes involved in thrombosis and predict thrombus formation under physiologic and pathological conditions. An important feature of this proposed framework is that each of its components can be refined by future experimental studies to improve the accuracy and capability of the model predictions. Therefore, this framework can be potentially extended to model more complicated vascular systems and geometries and make predictions based on organ-specific flow conditions and patient-specific blood coagulation factors. As the proposed framework is capable of dissecting complex conditions when multiple factors play a role in triggering thrombosis in human circulation, it also can be potentially used to identify the key factors in thrombus formation and test the efficacy of anti-thrombosis drugs, thereby accelerating the development of thrombosis medication.

Ethics

This article does not present research with ethical considerations.

Data accessibility

All data and codes used in this manuscript are publicly available on GitHub [65].

Authors' contributions

A.Z, Y.D., H.L., Z.L., J.D.H., G.E.K. designed research; A.Z, Y.D, H.L., E.J., Z.L. performed research; A.Z, Y.D, H.L. E.J. Z.L., S.J., C.L., J. D.H., C.S.M., G.E.K. analysed data; A.Z, Y.D, H.L., E.J., Z.L., S.J., J. D.H., C.S.M. and G.E.K. wrote the paper.

Competing interests

We declare we have no competing interests.

Funding

This work was supported by National Institute of Healthgrant nos. U01 HL1163232, U01 HL142518 and R01HL154150. High Performance Computing resources were provided by the Center for Computation and Visualization at Brown University and the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundationgrant nos. ACI-1053575, TG-DMS140007, TG-MCB190045 and COVID-19 HPC Consortium TG-BIO200088.