Mesoscale Simulations of Blood Coagulation in Flow and Quiescent Domains Using SDPD

ABSTRACT

Blood coagulation is governed by tightly regulated reaction networks whose activation and early evolution are influenced by transport processes. While reduced kinetic models of the intrinsic and extrinsic pathways have been validated in well‐mixed experimental settings, their behavior in spatially resolved domains remains insufficiently characterized. In this work, two established reduced coagulation networks are embedded within a thermodynamically consistent mesoscale particle‐based framework that resolves fluid momentum transport together with multispecies advection–diffusion–reaction dynamics. Unlike conventional continuum coagulation solvers, the proposed formulation captures coupled transport and biochemical interactions through interacting particles within a unified hydrodynamic description. The framework is used to investigate the initiation phase of coagulation under controlled microvascular‐like flow conditions in simplified channel geometries. The simulations reproduce characteristic thrombin generation curves (TGCs) across physiologically relevant parameter ranges while additionally revealing spatial transport effects that are not captured by outlet‐averaged measures alone. In particular, the results show that transport–reaction coupling induces pronounced spatial heterogeneities in thrombin concentration, with transitions between localized activation, wall‐aligned accumulation, and advective washout depending on Reynolds and Péclet regimes. Injury geometry is further shown to modulate coagulation amplification, with capped configurations producing enhanced thrombin accumulation due to localized surface‐mediated activation. Complementary quiescent‐domain simulations additionally reproduce qualitative trends observed in thrombodynamics assays, including fibrinogen‐dependent variations in fibrin formation. Overall, the study introduces a unified mesoscale computational framework for analyzing how flow, transport, and biochemical kinetics jointly regulate early coagulation dynamics in spatially resolved domains.

Mesoscale simulations show how flow conditions and domain type (flow vs. quiescent) influence thrombin and fibrin dynamics. Changes in Reynolds and Péclet numbers, together with fibrinogen levels, affect both spatial patterns and temporal evolution, highlighting the importance of transport–reaction coupling beyond well‐mixed conditions.

1. Introduction

Blood clotting, or coagulation, is a complex process driven by a cascade of chemical reactions and cellular interactions within the bloodstream. It is broadly categorized into three stages: initiation, amplification, and propagation [1]. A critical outcome of this process is the formation of a fibrin network, which stabilizes the clot and prevents excessive bleeding. This involves the enzymatic conversion of soluble fibrinogen into insoluble fibrin by thrombin, a key protease generated through a sequence of proteolytic reactions involving clotting factors such as prothrombin and Factor X. These reactions occur primarily on the surface of activated platelets, which provide a localized environment for the clotting factors to interact.

While the initiation phase of coagulation is relevant across vessels of all sizes, its dynamics take on particular significance in the microvasculature [2, 3]. Here, smaller vessel dimensions, slower flow rates, and higher surface‐to‐volume ratios amplify the localized effects of tissue factor (TF) exposure and platelet activation. These features create conditions where clotting must be tightly regulated to balance hemostasis and thrombosis. During the initial stages of clotting (Figure 1), TF, exposed due to vessel injury or inflammation, triggers the extrinsic pathway, leading to Factor VIIa activation and subsequent Factor X activation. Similarly, the intrinsic pathway, initiated by Factor XII activation on collagen or negatively charged surfaces, also converges at Factor X activation [1, 4]. Factor Xa cleaves prothrombin (II) to thrombin (IIa), which not only catalyzes fibrin (Ia) formation but also amplifies the cascade through feedback loops.

FIGURE 1.

Flowchart of the coagulation network during the initiation phase. The extrinsic pathway (blue) is triggered by tissue trauma, while the intrinsic pathway (pink) begins with Factor XII activation. Both converge at Factor X activation to Factor Xa, leading to thrombin (IIa) generation. Thrombin cleaves fibrinogen (I) into fibrin (Ia) and initiates three positive feedback loops (Factors V, VIII, XI; green dashed lines). Major inhibitors, including TFPI and Antithrombin, are shown in red.

Mathematical and computational models of coagulation have expanded greatly, with a large number of publications indexed in PubMed since the 1970s [5]. These models aim to capture the interplay of coagulation factors, platelet activity, fluid flow, and vessel wall interactions [6, 7, 8, 9]. Yet, reproducing the full multiscale physiology remains challenging [10, 11], as it requires coupling reaction kinetics with diffusion, advection, and cellular processes. Early ODE‐based kinetic models described thrombin generation but neglected spatial transport [12, 13]. Reaction–diffusion (RD) models later addressed this by adding diffusion, though at the cost of greater computational load, reduced reproducibility, and difficulty in validation [14, 15]. Numerical stability can also be compromised by stiff reaction terms [16]. Reduced models based on physiologically motivated simplifications have been proposed, retaining essential dynamics while limiting parameter complexity [15, 17]. Sensitivity analyses have also been used to identify the influence of reaction rates and initial concentrations on thrombin generation metrics [18]. To better capture in vivo conditions, RD models were further extended to include flow, resulting in advection–diffusion–reaction (ADR) frameworks [11, 19]. These models enable the simulation of clot formation under physiologically relevant flow regimes by accounting for the influence of transport [14, 20] and shear forces [14, 21, 22].

The large biochemical scope of the coagulation cascade has motivated substantial efforts toward subsystem or reduced‐order formulations that isolate the key drivers of thrombin generation. Early kinetic models, including those by Hockin et al. [12], Anand et al. [8], and Chatterjee et al. [9], provided important insights while keeping parameter counts tractable. Building on these foundations, Fogelson and colleagues incorporated flow and surface effects [6, 7, 14], highlighting the importance of spatial and hemodynamic regulation in clotting. More recent reduction strategies include multistep model‐compression approaches [23], thrombin‐focused schemes by Chen and Diamond [24], patient‐specific calibrations proposed by Ratto et al. [25], and automated or genetic algorithm‐based reduction methods [26, 27]. These efforts collectively highlight the continuing need for models that capture essential biology while remaining computationally tractable and interpretable.

In parallel, particle‐based methods have played an important role in advancing spatially resolved models of thrombosis, complementing continuum and hybrid approaches. Reactive DPD frameworks have been used to explore fibrin polymerization dynamics [28], while hybrid particle–continuum models have captured flow–thrombus interactions at mesoscale resolution [29, 30]. Related approaches include SPH formulations in which particles transition from fluid to solid phases based on coagulation species and mechanical triggers [31]. Smoothed dissipative particle dynamics (SDPD) provides a particle discretization of fluctuating hydrodynamics consistent with the Navier–Stokes equations, in contrast to conventional DPD formulations based on a soft equation of state, making it particularly suitable for studying the coupling between fluid transport and biochemical reaction networks in coagulation systems. Furthermore, increasingly realistic multiphysics frameworks have integrated biochemical kinetics with image‐based platelet aggregation, RD descriptions of fibrin polymerization, and fluid‐mechanical characterizations of shear‐dependent clot deformation. These multiscale approaches [32, 33, 34, 35] provide a comprehensive view of thrombosis mechanisms, but often rely on extensive reaction networks or tightly coupled modules whose calibration remains challenging. Even focused sensitivity analyses, such as the 88‐equation system examined by Stobb et al. [36], illustrate the inherent trade‐off between biological realism and computational tractability. Despite this progress, important questions remain regarding how flow, geometry, and spatially evolving biochemical fields jointly regulate the initiation phase of clot formation; existing ADR models typically rely on continuum formulations that do not capture mesoscale hydrodynamics, while particle‐ or agent‐based approaches often incorporate only limited reaction networks [37, 38].

Compared with conventional continuum ADR formulations based on finite‐volume, finite‐element, or finite‐difference discretizations [39, 40], the SDPD framework adopted here represents the fluid and transported biochemical species through interacting particles. This allows transport, diffusion, and hydrodynamic fluctuations to be captured through local interparticle interactions [41, 42]. Continuum coagulation models have successfully described reaction–transport coupling in blood clotting and have been extended through adaptive‐mesh techniques to resolve steep gradients and moving activation fronts [40]. In contrast, the SDPD formulation does not rely on a fixed computational mesh and naturally accommodates evolving spatial heterogeneities, complex geometries, and coupled transport–reaction effects within a thermodynamically consistent hydrodynamic framework. This makes it particularly suitable for microscale coagulation problems where multiple physical processes interact across scales. Importantly, this mesoscale description also provides a natural bridge toward future extensions involving clot microstructure and mechanics. In particular, recent fibrin gel models that resolve fibrin network formation and rheology explicitly [43, 44] can be directly coupled to the present framework, enabling a unified description of biochemical activation and fibrin structural evolution. Further extensions may also incorporate thixotropic multiphase formulations and multiscale blood rheology models including red blood cell dynamics [45].

The primary aim of this work is to assess how established reduced kinetic models of the intrinsic [25] and extrinsic [24] pathways behave when embedded in a spatially resolved mesoscale flow framework capable of resolving coupled transport and reaction dynamics. To this end, these validated reaction networks are integrated within a thermodynamically consistent SDPD particle‐based formulation [46], which captures fluid momentum transport, multispecies advection–diffusion, and their nonlinear biochemical interactions. Accordingly, the present SDPD formulation is intended as a mesoscale Lagrangian fluctuating extension of continuum Navier–Stokes and ADR approaches rather than a direct replacement. Its motivation lies in representing hydrodynamics, species transport, and future discrete clot constituents within a unified particle‐based framework, which is particularly relevant for problems involving spatial heterogeneity, evolving interfaces, and transport–reaction coupling at the microscale. The associated increase in computational cost and timestep restrictions relative to continuum formulations is discussed in Section 2.

The resulting framework is used to analyze thrombin and fibrin formation in both channel‐flow configurations (including capped and uncapped geometries) and quiescent domains. Thrombin generation curves (TGCs) [47, 48] are used to characterize thrombin dynamics, while selected biochemical perturbations, such as variations in fibrinogen concentration, are used to probe model response. The focus is on evaluating how established coagulation models interact with transport under controlled conditions, with emphasis on transport–reaction coupling and its impact on spatial activation dynamics.

2. Computational Model

2.1. Compositional SDPD

We model blood flow and species transport using a compositional SDPD framework for mesoscale ADR processes [46]. In this approach, the fluid is represented by interacting particles carrying mass, momentum, and biochemical species, and the governing dynamics are formulated directly at the particle level within a thermodynamically consistent fluctuating‐hydrodynamics framework [41, 49].

The SDPD formulation is constructed such that, in the hydrodynamic limit, it is consistent with the weakly compressible Navier–Stokes equations, $\mathit{d}\rho /\mathit{dt}=-\rho \nabla \cdot \mathbf{v}$ and $\rho \mathit{d}\mathbf{v}/\mathit{dt}=-\nabla p+\eta {\nabla }^2\mathbf{v}+(\zeta +\eta /\mathcal{D})\nabla (\nabla \cdot \mathbf{v})$, ¹ together with advection–diffusion–reaction balances for transported species. In the numerical implementation, these continuum equations are directly discretised by means of pairwise interacting particles. The fluid is represented by particles with position ${\mathbf{r}}_i$, velocity ${\mathbf{v}}_i$, and volume $V_i$, such that the particle number density is defined as $d_i=1/V_i={\sum }_{j}W\left(r_{\mathit{ij}},h\right)$, where $W\left(r_{\mathit{ij}},h\right)$ is a normalized smoothing kernel with compact support $h$, and $r_{\mathit{ij}}=|{\mathbf{r}}_i-{\mathbf{r}}_j|$. Pairwise interactions are constructed from the kernel gradient $F_{\mathit{ij}}=-\nabla W\left(r_{\mathit{ij}},h\right)/r_{\mathit{ij}}$. In this work, we adopt the Lucy quartic kernel [50]. The system evolution is therefore described in terms of particle velocities (${\mathbf{v}}_i$) and concentrations of the species $\alpha$ ($C_i^{\alpha }$), governed by conservative, dissipative, and stochastic interactions. At the particle level, the momentum equation includes reversible pressure forces, viscous dissipation, and thermal fluctuations consistent with the fluctuation–dissipation theorem (FDT) [51]:

$$\begin{array}{ll}m\frac{\mathrm{d}{\mathbf{v}}_i}{\mathrm{d}t}& =-\sum _j\left[\frac{p_i}{d_i^2}+\frac{p_j}{d_j^2}\right]F_{\mathit{ij}}{\mathbf{r}}_{\mathit{ij}}-\sum _j\left[a{\mathbf{v}}_{\mathit{ij}}+b\left({\mathbf{v}}_{\mathit{ij}}\cdot {\mathbf{e}}_{\mathit{ij}}\right){\mathbf{e}}_{\mathit{ij}}\right]\frac{F_{\mathit{ij}}}{d_i{d}_j}\\ & +\sum _j\left(A_{\mathit{ij}}\mathrm{d}{\bar{\mathbf{W}}}_{\mathit{ij}}+B_{\mathit{ij}}\frac{1}{\mathcal{D}}\mathrm{tr}\left[\mathrm{d}{\mathbf{W}}_{\mathit{ij}}\right]\right)\cdot \frac{{\mathbf{e}}_{\mathit{ij}}}{\mathrm{d}t},\end{array}$$ (1)

where $p_i$ and $p_j$ are the pressure of particles $i$ and $j$, respectively, ${\mathbf{v}}_{\mathit{ij}}={\mathbf{v}}_i-{\mathbf{v}}_j$ is the relative velocity between particles, ${\mathbf{e}}_{\mathit{ij}}={\mathbf{r}}_{\mathit{ij}}/|{\mathbf{r}}_{\mathit{ij}}|$ is the unit vector, $a$ and $b$ are friction coefficients related to the shear $\eta$ and bulk $\zeta$ viscosities of the fluid through $a=(\mathcal{D}+2)\eta /\mathcal{D}-\zeta$ and $b=(\mathcal{D}+2)(\zeta +\eta /\mathcal{D})$, with $\mathcal{D}$ the dimension of the system. The last term in (1), compactly written as $m_i\mathrm{d}{\tilde{\mathbf{v}}}_i/\mathrm{d}t$ [41] introduces thermal fluctuations into the momentum balance through stochastic pairwise interactions. More details of coefficients, amplitudes $A_{\mathit{ij}}$ and $B_{\mathit{ij}}$ are provided in Hydrodynamic Coefficients and Kernel Definitions. Regarding species transport, the evolution of a concentration field, $C^{\alpha }(\mathbf{r},t)$, of the species $\alpha$ (reactants or macromolecules) transported within the Lagrangian fluid particle, in a velocity field $\mathbf{v}(\mathbf{r},t)$, and accounting for fluctuating compositional fluxes [52], can be expressed in terms of the chemical potential ${\mu }^{\alpha }$ as described by Ellero et al. [49] as $\mathrm{d}{C}^{\alpha }/\mathrm{d}t=\nabla \cdot \left(\left({\mathit{TC}}^{\alpha }/{\xi }^{\alpha }\right)\nabla {\mu }^{\alpha }/T\right)+S^{\alpha }+\mathrm{d}{\tilde{C}}^{\alpha }/\mathrm{d}t$, where ${\xi }^{\alpha }$ is a transport friction coefficient, T is the temperature, $S^{\alpha }$ denotes the local reaction source term associated with biochemical production and consumption of species, and $\mathrm{d}{\tilde{C}}^{\alpha }/\mathrm{d}t$ represents stochastic compositional fluctuations. In the present framework, these fluctuations are neglected under the assumption that each SDPD particle $i$ represents a sufficiently large number of molecules, such that the fluctuation contribution remains small compared to the mean concentration evolution, that is, $\mathrm{d}{\tilde{C}}_i^{\alpha }/\mathrm{d}t\ll \mathrm{d}{C}_i^{\alpha }/\mathrm{d}t$. This approximation is consistent with the mesoscopic concentration regimes considered here; further details and discussion can be found in [46]. In SDPD, this continuum balance is discretized into conservative pairwise exchanges between neighboring particles, yielding the following evolution equation for species $\alpha$ in particle $i$:

$$\frac{\mathrm{d}{C}_i^{\alpha }}{\mathrm{d}t}=\sum _j{d}_{\mathrm{eq}}{D}_{\mathit{ij}}^{\alpha }{C}_{\mathit{ij}}^{\alpha }\frac{F_{\mathit{ij}}}{d_i{d}_j}+S_i^{\alpha }+\frac{\mathrm{d}{{\tilde{C}}_i}^{\alpha }}{\mathrm{d}t},$$ (2)

where $D_{\mathit{ij}}^{\alpha }=4D_i^{\alpha }{D}_j^{\alpha }/\left(D_i^{\alpha }+D_j^{\alpha }\right)$, and $C_{\mathit{ij}}^{\alpha }=C_i^{\alpha }-C_j^{\alpha }$. Here, $d_{\mathrm{eq}}$ is the equilibrium particle number density, $D_i^{\alpha }$ and $D_j^{\alpha }$ corresponds to the diffusion of the $\alpha$‐th species on particles $i$ and $j$. Notice that if the diffusion of the species is the same among particles, $D_{\mathit{ij}}^{\alpha }=2D^{\alpha }$. $F_{\mathit{ij}}$ is the kernel‐gradient term introduced in the momentum equation, and $S_i^{\alpha }$ is the particle‐wise counterpart of the reaction source term. Throughout this document, we use superscripts $\alpha$ and $\beta$ to denote species or components of the system, while subscripts $i$ and $j$ refer to SDPD fluid particle indices. Under the assumptions of negligible compositional fluctuations (${\tilde{C}}_i^{\alpha }\approx 0$), absent cross‐species diffusive coupling, and local reaction kinetics, this discrete formulation preserves species mass for the diffusive transport contribution, remains thermodynamically consistent, and reproduces Fickian diffusion.

We adopt a system of dimensionless (reduced) units by defining characteristic scales for length ($l$), mass ($m$), and energy ($\epsilon$). These scales yield a natural unit of time, $\tau =l\sqrt{m/\epsilon }$, and all quantities are expressed relative to them. For example, a domain length $L_x=50$ corresponds to $50l$, and an interparticle spacing $\mathit{dx}=0.2$ provides a resolution of $0.2l$. Transport and reaction parameters are likewise nondimensionalized: diffusion coefficients in $l^2/\tau$, reaction rates in $1/\tau$, and velocities in $l/\tau$. This consistent framework facilitates comparison across all test cases.

To characterize transport regimes, dimensionless numbers are defined using the channel height $H$ as the characteristic length scale. The Reynolds number, $\mathrm{Re}=\rho \mathit{uH}/\eta =\mathit{uH}/\nu$, measures the ratio of inertial to viscous forces, where $\rho$ is the fluid density, $u$ a characteristic velocity, $\eta$ the dynamic viscosity, and $\nu$ the kinematic viscosity. The Péclet number, $\mathrm{Pe}=\mathit{uH}/D$, quantifies the relative importance of advection and diffusion, while the Schmidt number, $\mathrm{Sc}=\nu /D$, compares momentum and mass transport. Finally, the Damköhler number, $\mathrm{Da}=H^2/\left(D{\tau }_{\mathrm{r}}\right)$, relates diffusion and reaction timescales, with ${\tau }_{\mathrm{r}}$ denoting a characteristic reaction time. Additional methodological details, including stability criteria, are summarized in Appendix A.

In this work, we focus on the microvascular regime, characterized by vessel diameters of 5–100 $\mathrm{\mu m}$, low Reynolds numbers (Re $\lesssim 1$), and moderate‐to‐high Peclet numbers (Pe $\sim {10}^1-{10}^4$). Such conditions typify arterioles and venules and are often reproduced in flow‐based in vitro assays of coagulation and thrombus formation [53, 54, 55]. This regime is physiologically and experimentally relevant and well suited for mesoscale modeling, where transport and reaction dynamics are strongly coupled. The corresponding parameter estimates for our simulations and detailed description range of Pe and Re simulated are provided in Estimates for Microvascular Regimes. The choice of characteristic scales and non‐dimensional parameters ensures that the simulated flow remains within the physiological range of arteriolar conditions. The ADR–SDPD scheme is implemented in the open‐source Large‐scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) [56] (stable release: October 29, 2020), which provides a robust parallel framework for particle‐based simulations and compatibility with existing molecular and rigid‐body models. The computational cost of the SDPD formulation is primarily governed by short‐range interparticle interactions, leading to an approximately linear scaling with the number of particles. The simulations presented here involve approximately $1.5\times {10}^4$ fluid particles, providing sufficient spatial resolution to resolve the benchmark transport–reaction mechanisms studied [46]. Compared with standard SDPD formulations, the present model incurs additional computational cost due to the simultaneous evolution of multiple reacting species, reflecting the increased physical complexity of the coagulation problem. For larger or fully three‐dimensional domains, this motivates the use of multiscale strategies, in which SDPD resolves local clot microstructure and biochemical activation while continuum models capture transport at larger scales [57].

2.2. Mathematical Models for the Coagulation Pathways

We next introduce the sets of ordinary differential equations (ODEs) used to describe the biochemical kinetics of the coagulation pathways. These equations correspond to the source term $S^{\alpha }$ in the compositional balance Equation (2), and provide the kinetic component of the SDPD framework. To simplify the notation, the compositions of each species are denoted by the abbreviation of the species (e.g., $C^{\mathrm{IIa}}=\mathrm{IIa}$).

2.2.1. Intrinsic Coagulation Pathway

Ratto et al. [25] reduced the intrinsic coagulation pathway to a minimal system of three species: activated Factor X (Xa), prothrombin (II), and thrombin (IIa). Their dynamics are described by nine rate constants that capture initiation ($k_1$), propagation (positive terms), and termination (negative terms). The resulting system of ODEs reads

$$\frac{d\mathrm{Xa}}{\mathit{dt}}=\left(k_1+k_2\mathrm{IIa}+k_3{(\mathrm{IIa})}^2\right)\left({\mathrm{X}}^0-\mathrm{Xa}\right)-k_4\mathrm{Xa},$$ (3a)

$$\frac{d\mathrm{II}}{\mathit{dt}}=-\left(k_5\mathrm{Xa}+k_6\mathrm{IIa}+k_7{(\mathrm{IIa})}^2+k_8{(\mathrm{IIa})}^3\right)\mathrm{II},$$ (3b)

$$\frac{d\mathrm{IIa}}{\mathit{dt}}=\left(k_5\mathrm{Xa}+k_6\mathrm{IIa}+k_7{(\mathrm{IIa})}^2+k_8{(\mathrm{IIa})}^3\right)\mathrm{II}-k_9\mathrm{IIa},$$ (3c)

$$\frac{d\mathrm{fbg}}{\mathit{dt}}=-{\alpha }_1\mathrm{IIa}\mathrm{fbg},$$ (3d)

$$\frac{d\mathrm{Fbn}}{\mathit{dt}}={\alpha }_1\mathrm{IIa}\mathrm{fbg}-{\alpha }_2\mathrm{Fbn},$$ (3e)

$$\frac{d{\mathrm{Fbn}}_P}{\mathit{dt}}={\alpha }_2\mathrm{Fbn}.$$ (3f)

Equations ((3a), (3b), (3c)) constitute the minimal system employed in preliminary tests. To capture fibrin formation, the model is extended with Equations ((3d), (3e), (3f)), where fibrinogen ($\mathrm{fbg}$) is cleaved by thrombin (IIa) to fibrin ($\mathrm{Fbn}$), which then polymerizes into stable fibrin (${\mathrm{Fbn}}_P$), as a polymer which does not undergo transport. In Reduced Extrinsic Coagulation Model, we provide a breakdown of the rate constants and parameters for reaction ($k_1$–$k_9$, ${\alpha }_1$, ${\alpha }_2$), and the diffusivities $D^{\alpha }$ of the species.

2.2.2. Extrinsic Coagulation Pathway

We adopt the extrinsic pathway model of Chen and Diamond [24], which captures thrombin and fibrin formation triggered by TF. The system tracks eight chemical species: TF, Xa, IXa, XIa, fibrin (Fbn), free thrombin (IIa), and IIa bound to Fbn at either weak () or strong () sites. Their dynamics are described by the ODE system in Equation (4), with fibrin availability determining binding capacity ().

$$\frac{d\mathrm{TF}}{\mathit{dt}}=-k_{i,\mathrm{TF}}\cdot \mathrm{TF},$$ (4a)

$$\frac{d\mathrm{Xa}}{\mathit{dt}}=a_1\cdot \mathrm{TF}+a_3\cdot \mathrm{IXa}-k_i\cdot \mathrm{Xa},$$ (4b)

$$\frac{d\mathrm{IXa}}{\mathit{dt}}=a_2\cdot \mathrm{TF}+a_7\cdot \mathrm{XIa}-k_i\cdot \mathrm{IXa},$$ (4c)

$$\frac{d\mathrm{XIa}}{\mathit{dt}}={\eta }_6\cdot a_6\cdot \mathrm{IIa}-k_{\text{elute}}\cdot \mathrm{XIa}-k_i\cdot \mathrm{XIa},$$ (4d)

$$\frac{d\mathrm{Fbn}}{\mathit{dt}}={\eta }_5\cdot a_5\cdot \mathrm{IIa},$$ (4e)

$$\begin{array}{ll}\frac{d\mathrm{IIa}}{\mathit{dt}}& ={\eta }_4\cdot a_4\cdot \mathrm{Xa}-\left(d^{\mathrm{E}}\mathrm{S}/\mathit{dt}+d^{\gamma }\mathrm{S}/\mathit{dt}\right)\\ & -k_{\text{elute}}\cdot \mathrm{IIa}+k_i\cdot \mathrm{IIa},\end{array}$$ (4f)

(4g)

(4h)

Reaction rates (${\alpha }_i$), binding/unbinding constants, inhibition ($k_i$), and tuning parameters (${\eta }_i$) are listed in Reduced Extrinsic Coagulation Model.

2.3. Computational Domain and Scaling Considerations

Simulations were first performed in homogeneous domains without flow or geometric features, providing a minimal setting to test the reaction kinetics in (3) and (4) against reference thrombin generation assays. We then extended the framework to heterogeneous domains including localized injury sites, wall‐bound particles, and capped regions, representing simplified spatial configurations in which coagulation is locally initiated. In this way, homogeneous cases serve as a baseline for model validation, while heterogeneous domains enable analysis of the interplay between flow, transport, and localized biochemical activation. Geometric ratios and flow conditions are selected following established dimensional scaling guidelines for coagulation flow models. In particular, the choice of channel aspect ratio ($H/L_x<0.2$) and relative injury size ($L_{\mathrm{inj}}/H=1$) is consistent with recommendations from the ISTH Scientific and Standardization Committee [58], which define parameter ranges for transport–reaction systems under shear flow. These choices promote well‐developed laminar flow and ensure that coagulation dynamics are governed by surface‐mediated activation within physiologically relevant transport regimes.

Within the heterogeneous group, two main configurations were distinguished. The first configuration consists of rectangular channel domains (Figure 2a) in which transport is governed by convection and diffusion, characterized by the Re and Pe ranges introduced in Section 2.1. The computational domain length and height are $L_x=55l$ and $H=10l$, respectively, corresponding to a systematic exploration of characteristic length scales in the range 5–100 $\mathrm{\mu m}$, (Estimates for Microvascular Regimes). Fluid particles (blue) represent the plasma phase. Coagulation is initiated by a set of “injury” particles (red) localized at the lower boundary. Quiescent particle clusters are represented as a semicircular cap (pink) mimicking preexisting surface‐bound structures. The dashed outline in the schematic highlights the corresponding homogeneous reference domain. The second configuration considers quiescent, closed domains without imposed flow. A cuvette‐like geometry (Figure 2b) is used to analyse diffusion‐driven clot precursor propagation under transport conditions representative of macroscopic assays. The domain and parameters preserve the relevant dimensionless groups, ensuring diffusion‐dominated conditions. Simulations are performed under quiescent conditions to examine the spatial evolution of the fibrin front.

FIGURE 2.

Simulation domains for channel‐flow and cuvette‐like studies. (a) Two rectangular channel domains ($L_x=$ 55, $H=$ 10), with a dashed outline marking the homogeneous (hom.) domain. The heterogeneous domain includes “injury” particles (red) to trigger coagulation; a semicircular cap (pink) represents quiescent particle clusters. 13,260 is the total count of fluid particles (blue). (b) Cuvette for in vitro tests, with 6426 fluid particles; TF (red) initiates coagulation, static cuvette walls (gray).

To provide well‐controlled settings for systematically exploring transport, reaction kinetics, and flow intensity, we consider simplified two‐dimensional geometries. These domains enable the investigation of coagulation dynamics under heterogeneous flow and in microfluidic‐like configurations [59], without aiming at anatomical reconstruction.

2.3.1. Boundary and Source Term Conditions

Different boundary conditions are applied depending on the configuration (channel‐flow or cuvette‐like), as detailed below. For particle motion, no‐slip conditions were imposed at solid walls, while periodic boundaries were applied along the streamwise direction [60] in the channel‐flow configurations. This choice leads to statistically stationary flow conditions and removes inlet–outlet and entrance‐length effects, which allows a controlled study of transport–reaction coupling. Clot initiation is represented through localized reactive surfaces and static geometric features, rather than dynamically evolving occlusive growth. This keeps the focus on biochemical activation, advection, and diffusion, without the additional complexity associated with imposed inlet velocity profiles or developing flow conditions.

In the channel‐flow configurations, prothrombin (II) was continuously supplied at the inlet at reference physiological concentrations (see Table A3), whereas homogeneous Neumann conditions, $\mathbf{n}\cdot \nabla C^{\alpha }=0$, were applied at the outlet for all species. Coagulation was locally triggered through a prescribed Factor Xa flux applied to particles above the injury region, defined as $\text{flux}=\gamma \left(C^{{\mathrm{X}}^0}-C^{\mathrm{Xa},\mathrm{avg}}\right)$ where $\gamma$ and $C^{{\mathrm{X}}^0}$ are given in Table A3, and $C^{\mathrm{Xa},\mathrm{avg}}$ denotes the spatial average Xa concentration over the fluid particles in the injury region. For semicircular cap geometries, the total Xa supply increased naturally with the size of the cap region, as more particles were subjected to the prescribed flux. To achieve target Re and Pe numbers representative of microscale flows, a body force was applied to drive the flow, with the fluid viscosity $\eta$ and characteristic velocity $u$ adjusted to systematically vary Re and Pe (see Estimates for Microvascular Regimes). In quiescent, cuvette‐like configurations, coagulation was initiated by TF immobilized at a localized wall region under zero‐flow conditions. The red region in Figure 2b indicates the activator surface, where a prescribed nonzero TF concentration locally triggered the reaction cascade. No‐flux boundary conditions were imposed at all remaining domain boundaries.

3. Results

In the following, we present the results in increasing order of complexity. We begin with a consistency check of the reduced coagulation models in homogeneous domains, where the SDPD formulation is compared against the corresponding ODE‐based reference solutions. In this context, system (3) is used for the intrinsic pathway and system (4) for the extrinsic pathway, serving as independent benchmark reaction networks rather than a coupled model.

Building on this verification step, we extend the simulations to channel‐flow configurations, where the intrinsic pathway is used to examine the influence of transport parameters, injury size, and variations in fibrinogen concentration on thrombin generation and fibrin formation. Finally, we perform in vitro‐inspired simulations motivated by experimental assays such as thrombodynamics [59], highlighting the role of diffusion in reaction kinetics under controlled conditions.

Time is nondimensionalized as $t^{*}=t/t_{\mathrm{ref}}$, with $t_{\mathrm{ref}}=1\mathrm{s}$, and concentrations as $C^{*}=C^{\alpha }/C_{\mathrm{ref}}$, with $C_{\mathrm{ref}}=1\mu \mathrm{M}$ representing a typical inlet fibrinogen concentration. The SDPD solver operates in its own mesoscopic units, while $t_{\mathrm{ref}}$ provides a consistent reference to express reaction rates in dimensionless form. This normalization is applied throughout all figures unless otherwise noted.

3.1. Consistency Check Against ODE‐Based Reference Models

Simulations to test our SDPD formulation (1) and (2) were first conducted in a homogeneous square domain (dashed region in Figure 2) for initial consistency check and performance assessment. We used the ODE systems described in Section 2.2 for the intrinsic (Ratto et al. [25]) and extrinsic (Chen and Diamond [24]) pathways. Figure 3a shows the concentration profiles of key species in the intrinsic pathway, including Factor Xa, prothrombin (II), and thrombin (IIa), averaged over all fluid particles. Averaged concentration profiles for the species involved in the extrinsic pathway, together with the temporal evolution of IIa, both in its individual forms and total concentration, are shown in Figure 3b,c, respectively.

FIGURE 3.

(a) Intrinsic pathway: Evolution of mean concentration profiles for Factor Xa, prothrombin (II), and thrombin (IIa) [25]. (b, c) Extrinsic pathway: (independent reduced coagulation network [24]). Panel (b) shows all species (log scale), and (c) focuses on thrombin forms: Strongly bound (), weakly bound (), free, and total. Solid lines show MATLAB ode15s results, and circles denote SDPD simulations; the inset highlights Xa and IIa peaks.

The SDPD simulations successfully reproduce the trends predicted by the ODE systems (3) and (4), supporting the robustness of our framework for modeling both coagulation pathways. Preliminary tests using the intrinsic pathway model [25] under homogeneous conditions (see Figure A1) showed negligible differences between simulations with uniform and species‐specific diffusivities. This indicates that, under such conditions, coagulation kinetics dominate over diffusive transport (Da $\gg$ 1), and the system is expected to remain primarily reaction‐controlled even when spatial transport is introduced in the channel configuration.

3.2. Flow‐Driven Simulation Results

We analyzed blood coagulation in platelet‐free plasma under flow conditions using system (3) describing the reduced intrinsic pathway. The results are reported as peak thrombin values and local outlet‐averaged thrombin concentrations. In the spatially resolved channel configuration, thrombin concentrations vary along the flow direction due to advection and local reaction dynamics. To obtain a representative global signal comparable to TGCs, concentrations are averaged over fluid particles crossing the outlet section of the channel. These quantities can be directly compared to TGCs as standard benchmarks for the temporal evolution of thrombin. From this point forward, prothrombin and thrombin will be referred to as II and IIa, respectively.

Capped configurations are used as idealized representations of injury or partially occlusive geometries in microvascular flow, allowing systematic control of the procoagulant surface area and local flow constriction (see Section 3.2.2). In these simulations, the imposed obstacles perturb the local hydrodynamic field and thereby affect advection and transport of reacting species. However, the evolving coagulation state is not coupled back to the flow and does not alter either the geometry or the hydrodynamic field. This configuration isolates transport–reaction coupling during clot initiation, without accounting for feedback between coagulation dynamics, flow modification, and structural evolution.

3.2.1. Transport Regime Effects on Coagulation Dynamics

To assess the role of transport in coagulation dynamics, we examined channel domains at low Re (0.02) while varying Pe (10, 70), consistent with the microvascular regimes introduced in Section 1. The transport–reaction system comprised Factor Xa, II, and IIa, coupled through the kinetic scheme in ((3a), (3b), (3c)) and subject to advection–diffusion.

Figure 4a shows the peak concentration profiles of Factor Xa (blue) and IIa (yellow) for the two Pe values. At lower Pe, Factor Xa stabilizes more rapidly and the global IIa peak ($t_{\text{peak}}^{*}$) is reduced. These differences become less evident in the outlet‐averaged profiles (Figure 4b), underscoring that TGCs alone may not fully reflect spatially localized dynamics. The scatterplots (Figure 4c,d) show particle‐resolved IIa concentrations normalized to the largest peak, that of Pe $=$ 70, while the histograms (Figure 4e,f) report the number of fluid particles binned by IIa concentration. All data are taken at the times of maximal absolute IIa (cf. Figure 4a) and peak outlet IIa (cf. Figure 4b). At Pe = 10, IIa forms compact, localized structures (Figure 4c) with narrow concentration distributions (Figure 4e). At Pe = 70, IIa becomes more elongated and flow‐aligned (Figure 4d), accompanied by a broader spread of local concentrations (Figure 4f). The broader histogram at Pe = 70 (Figure 4f) coincides with the global maximum at $t^{*}\approx 260$ (Figure 4a), indicating stronger spatial variability associated with faster transport.

FIGURE 4.

Influence of Pe (10,70) under low Re = 0.02. (a) Maximum concentrations of species Xa (blue) and IIa (yellow) over time, with solid lines for Pe $=$ 10 and filled squares for Pe $=$ 70. The red vertical lines highlight the time of maximal absolute IIa peak occurrence, $t_{\text{peak}}^{*}$. (b) Mean Xa and IIa concentration profiles at the channel outlet; magenta lines indicate $t_{\max }^{*}$. (c–d) Normalized IIa concentration scatter plots at $t_{\text{peak}}^{*}$ for Pe $=$ 10 and Pe $=$ 70, respectively, with matched axis ranges. The SDPD length scale is nondimensionalized by the channel height $H$. (e–f) IIa concentration histograms: Lilac for values at $t_{\text{peak}}^{*}$ in (c–d), red for values at $t_{\max }^{*}$.

Overall, these results indicate that higher Pe not only enhances advective transport but also alters the spatial distribution of coagulation species, increasing heterogeneity in local IIa concentrations. Such heterogeneity cannot be inferred from TGCs, which provide only global outlet‐averaged kinetics. Complementary spatially resolved descriptors are therefore required to fully characterize coagulation in the microvasculature.

We next examined the influence of Re on coagulation dynamics under microvascular conditions. Simulations were performed at Pe $=100$, with Re $=0.02$, $0.08$, $0.1$ and $0.5$, corresponding to diffusion–advection balances relevant to arterioles and venules (see Estimates for Microvascular Regimes).

The outlet‐averaged IIa profiles (Figure 5a) show a sharp rise followed by a decay and plateau, except for the highest Re ($0.5$, $\mid$), where IIa reaches a steady value without a pronounced peak. These temporal trends are consistent with the spatial maps at $t^{*}=5000$ (Figure 5b), where the faded regions at the channel outlet indicate the slabs used for spatial averaging, corresponding to the plateau concentrations shown in Figure 5a.

FIGURE 5.

(a) Evolution of thrombin (IIa) concentration at the channel outlet for different Reynolds numbers (Re $=0.02$, $0.08$, $0.1$, and $0.5$) at Pe $=100$. Markers indicate Re $=0.02$ (square), Re $=0.08$ (circle), Re $=0.1$ (diamond), and Re $=0.5$ (vertical bar). (b) Spatial distribution of IIa in the channel‐flow domain at $t^{*}=5000$ for each Re value. The outlet $x$‐slab used to compute local averages is shown faded. A consistent rainbow color scale is used across all cases for direct comparison. Concentrations are reported in normalized form, $C^{*}/C_{\max }^{*,\mathrm{Re}=0.08}$, where $C_{\max }^{\mathrm{Re}=0.08}$ denotes the maximum IIa concentration in the case Re $=0.08$.

The maps reveal that IIa remains concentrated near the bottom‐wall injury at intermediate $\mathrm{Re}$ ($0.08$ and $0.1$, circles and diamonds), consistent with a stronger convective washout with larger Re, and reflected at the larger magnitude for mean profiles in the plateaus of Figure 5a, while at Re $=0.5$, stronger advection enhances downstream transport but limits local accumulation intensity near the injury. Across all cases, however, the concentration front extends downstream and arches upward, reaching the upper wall despite the absence of any source there. This apparent symmetry results from the left‐to‐right inflow of reactive species and their rapid vertical homogenization, driven by the combined effects of diffusion and flow. The observed arch‐like pattern indicates a regime with high Da $\gg$ 1. Computed timescales (Table A2) give ${\mathrm{Da}}^{\mathrm{IIa}}\approx 2.3$, confirming a reaction‐dominated regime in our parameter window, where fast reaction kinetics further promote homogenization across the channel height. These findings are consistent with previous microfluidic studies under Re $<1$, which also reported laminar, reaction‐dominated conditions [61, 62, 63]. Likewise, confinement effects have been shown to modulate activation thresholds in coagulation networks [64], supporting our interpretation that in microvascular‐like channels, subtle variations in Re, Pe, or aspect ratio can shift the balance between localized activation and homogenized downstream propagation. While geometric confinement is not varied explicitly in the present simulations, prior microfluidic work has demonstrated that confinement can modulate activation thresholds by altering the balance between advection, diffusion, and reaction [64]. The present results isolate transport–reaction effects under controlled conditions, without addressing confinement effects explicitly.

3.2.2. Impact of Injury Size

We investigated how the size of the procoagulant region affects thrombin generation and flow patterns. Three cap sizes were simulated in a channel with radii $r=1.5$, $4.5$, and $8.5$, corresponding to $r/H=0.15$, $0.45$, and $0.85$ (denoted S, M, and L). Figure 6a shows particle velocity fields at $t^{*}=5000$, illustrating that cap size strongly influences local hydrodynamics. Small caps allow faster flow, whereas the largest cap (L) substantially slows the fluid, consistent with previous studies treating growing clots as partially occluding porous regions [65]. Figure 6b presents outlet profiles of Xa and IIa at Re $=0.02$, confirming that larger caps sustain higher Xa fluxes and stronger, more persistent thrombin responses. IIa peaks early in all cases but persists most effectively for the large cap. Finally, Figure 6c compares the small cap (S) across flow regimes: higher Re (0.37) maintains IIa levels after the initial peak, while at low Re (0.02) IIa declines toward zero. Xa dynamics mirror these trends, exhibiting sharper but less sustained responses at lower Re.

FIGURE 6.

Impact of injury size on coagulation dynamics across three cap sizes. (a) Particle velocity fields at the end of the simulation for small (S), medium (M), and large (L) cap sizes. (b) Mean concentration profiles of Factor Xa (blue) and thrombin (yellow) at the channel outlet for each cap size under low Re (0.02). (c) Comparison of mean concentration profiles of Xa and thrombin for the small cap (S) with varying Re: Low (0.02) versus high (0.37). The results highlight the effects of injury size and flow conditions on coagulation dynamics.

These results indicate that injury size governs coagulation severity by combining geometric and biochemical effects: a larger procoagulant surface both slows local flow and increases Xa flux, amplifying thrombin generation. All capped configurations produce IIa levels roughly 50% higher than the uncapped case, highlighting the importance of localized, concentrated activation in driving thrombin formation. The absolute IIa magnitude is less critical than the demonstrated trend: geometric confinement and surface‐mediated fluxes strongly modulate the coagulation response.

3.2.3. Effect of Fibrinogen Concentration on Fibrin Formation

We simulated the full intrinsic coagulation cascade (Equation 3) to examine how the initial fibrinogen concentration modulates thrombin‐mediated fibrin formation. No advection was imposed, thereby isolating diffusion–reaction dynamics.

Increasing fibrinogen input leads to sustained fibrin (Fbn) accumulation beyond reference threshold values ($\gtrsim 20$), whereas lower initial levels produce only transient and low‐amplitude responses (Figure 7a). Outlet concentrations remain below elevated levels only for ${\mathrm{fbg}}_0=10$, indicating limited downstream propagation under reduced substrate availability (Figure 7b). Variations in ${\mathrm{fbg}}_0$ primarily affect fibrinogen and Fbn concentrations, while thrombin (IIa) and other intermediates remain largely unchanged (Figure 7c), reflecting the localized impact of substrate availability on the terminal conversion step. Spatial distributions (Figure 7d–j) further demonstrate progressively stronger Fbn accumulation throughout the channel with increasing ${\mathrm{fbg}}_0$. At the representative time point $t^{*}=250$, Figure 7e shows the corresponding Fbn profiles. The ${\mathrm{fbg}}_0=30$ case exhibits a sharply localized, high‐amplitude peak; ${\mathrm{fbg}}_0=20$ produces a moderate, spatially confined elevation; and ${\mathrm{fbg}}_0=10$ remains uniformly low across the domain.

FIGURE 7.

Effect of the initial fibrinogen concentration (${\mathrm{fbg}}_0$) on fibrin (Fbn) dynamics under diffusion–reaction conditions. (a) Maximum fibrin (Fbn) concentration across all fluid particles (logarithmic scale). Dashed horizontal lines indicate reference concentration levels used for comparison across cases. The inset highlights early‐time dynamics ($t^{*}\le 1000$), including the representative time point $t^{*}=250$ corresponding to the spatial profiles shown in (e). (b) Mean Fbn concentration at the channel outlet for ${\mathrm{fbg}}_0=10$, 20, and 30. (c) Mean outlet concentrations for all simulated species, showing that variations in input primarily affect fibrinogen and Fbn, while other species remain largely unchanged. (d–j) Spatial distributions of Fbn at selected time points for the three initial fibrinogen levels.

Both IIa and Fbn evolve in a reaction‐dominated regime ($\mathrm{Da}>1$), consistent with the rapid vertical homogenization of the concentration fields. IIa exhibits moderate reaction dominance ($\mathrm{Da}\approx 2.3$), whereas Fbn formation is characterized by a much larger Damköhler number ($\mathrm{Da}\approx 355$), leading to rapid local stabilization once thrombin becomes available. Overall, altering ${\mathrm{fbg}}_0$ does not modify the transport–reaction balance but shifts the available substrate pool, resulting in proportional changes in Fbn production. This behavior is consistent with experimental observations that clot density and growth scale with fibrinogen concentration [66, 67].

3.3. Cuvette‐Like Simulation Results

We simulate a cuvette‐like in silico domain, computationally inspired by the thrombodynamics (TD) assay [59], to investigate the influence of fibrinogen on clot growth. In this section, the reduced extrinsic pathway model (ODE system (4)) is used exclusively as the governing reaction system. The localized TF source is implemented as an immobile region in the upper layer of the cuvette domain (Figure 2b). Simulations were performed with initial fibrinogen concentrations (${\mathrm{fbg}}_0$) of 1, 10, 20, and 30. The values 10, 20, and 30 correspond to the regimes explored in the preceding subsection, while ${\mathrm{fbg}}_0=1$ represents an extreme low‐fibrinogen case included solely to probe model sensitivity to substrate availability. The simulation setup is intended for controlled, idealized investigation and does not aim to reproduce the experimental device.

Clot size is quantified as the $y$‐position where Fbn reaches $C^{*}=20$. At low ${\mathrm{fbg}}_0$, clot formation is delayed ($t^{*}\sim 1000$), whereas higher ${\mathrm{fbg}}_0$ produces rapid, sustained Fbn accumulation (Figure 8a,b). These trends indicate that fibrinogen availability regulates both polymerization and effective diffusion: increasing ${\mathrm{fbg}}_0$ promotes localized Fbn deposition while reducing mobility, consistent with diffusion‐limited dynamics [68]. Simulations extended to $t^{*}=1200$ showed continued Fbn accumulation, eventually causing numerical overflow; beyond $t^{*}\approx 800$, results should be interpreted with caution, as the reduced kinetic network lacks feedback mechanisms to limit long‐term Fbn growth.

FIGURE 8.

(a) Clot size (mm) as a function of dimensionless time $t^{*}$, defined by the furthest $y$‐coordinate of fluid particles where the normalized Fbn concentration ($C^{\mathrm{Fbn}}$) reaches 20, for four initial fibrinogen levels ($C^{{\mathrm{fbg}}_0}=\mathrm{1,10,20},$ and $30$). (b) Final spatial distribution of $C^{\mathrm{Fbn}}$ at $t^{*}=1200$ for the same four initial fibrinogen levels.

The observed behavior underscores the dominant role of fibrinogen in diffusion‐limited clot growth and spatial propagation, illustrating the ability of the SDPD framework to resolve transport–reaction interactions in spatially structured, idealized domains.

4. Conclusions

Using a particle‐based fluctuating‐hydrodynamics framework [46], we investigated how transport processes interact with biochemical kinetics during the initiation of thrombin generation under controlled microvascular‐like flow conditions. Validated reduced coagulation networks were embedded within the SDPD method to systematically examine the influence of flow intensity, dispersion, and injury geometry on the spatial organization and temporal evolution of clotting reactions.

The results indicate that transport–reaction coupling plays a dominant role in shaping coagulation dynamics. Variations in the Péclet number influence not only transport rates but also the development of pronounced spatial heterogeneities in thrombin concentration, which are not captured by outlet‐averaged measures such as TGCs. Changes in Reynolds number (Re $=$ 0.02–0.5) further modify dispersion regimes, from localized activation near the injury at low Re, to wall‐aligned accumulation at intermediate Re, and reduced thrombin levels under stronger advective washout. Injury geometry introduces an additional regulatory mechanism: capped configurations with concentrated surface‐mediated Xa fluxes significantly enhance thrombin production compared with diffuse injuries, highlighting the impact of geometric confinement on reaction amplification. Complementary cuvette‐like simulations reproduced qualitative features observed in in vitro thrombodynamics assays [59], including the sensitivity of clot propagation to fibrinogen concentration [67, 68], illustrating the flexibility of the computational framework.

These findings demonstrate that the behavior of reduced coagulation networks can differ markedly once embedded in spatially resolved flow fields, emphasizing the importance of resolving transitions between diffusion‐dominated and advection‐enhanced regimes. While the present study focuses on plasma‐based flow and idealized geometries to isolate transport–reaction mechanisms, the framework provides a solid numerical foundation for further developments. At present, the framework resolves the influence of flow on coagulation dynamics but does not yet account for feedback of the developing clot on the flow field, making it most appropriate for studying coagulation initiation and early transport–reaction coupling. Future extensions incorporating thrombus permeability, fibrin mechanics, and cellular interactions will enable tighter coupling between biochemical reaction networks and emergent clot structure [44, 69]. Within the present SDPD framework, these extensions could be incorporated through additional particle populations and modified transport interactions. For example, thrombus permeability could be represented through locally varying drag or resistance effects that alter momentum and species transport within clot regions, while platelets or red blood cells could be introduced as discrete mesoscale entities coupled to the surrounding fluid particles [45]. In addition, recent mesoscale fibrin‐clot models [43, 44] already provide explicit descriptions of fibrin gel mechanics and microstructure, offering a natural route for coupling the present biochemical transport framework with emergent clot rheology. At larger scales, these developments could further be integrated into heterogeneous multiscale or thixotropic multiphase formulations [70], extending the applicability of the model across multiple spatial scales. With targeted validation in microfluidic settings, this mesoscale approach offers a robust building block for multiscale numerical models of coagulation.

Author Contributions

M.E.F. and N.M. designed the research project. M.E.F. performed the simulations and analyzed the resulting data. N.M. developed the software and implemented the open‐source plugins. M.E. supervised the project and provided overall guidance throughout the study. M.E.F. and N.M. jointly drafted the original version of the manuscript, and all authors contributed to revisions. All authors discussed the results, provided critical feedback, and approved the final manuscript.

Funding

We acknowledge funding by the Basque Government through the BERC 2022–2025 program and by the Ministry of Science, Innovation and Universities: BCAM Severo Ochoa accreditation CEX2021‐001142S/MICIN/AEI/10.13039/501100011033.

Ethics Statement

The authors have nothing to report.

Consent

All authors have read and approved the final manuscript and consent to its publication.

Conflicts of Interest

The authors declare no conflicts of interest.

Appendix A. Smoothed Dissipative Particle Dynamics Formulation

This appendix summarizes the particle‐level formulation of the compositional Smoothed Dissipative Particle Dynamics (SDPD) model used in Section 2. This appendix summarizes the particle‐level formulation of the compositional Smoothed Dissipative Particle Dynamics (SDPD) model used in Section 2. SDPD is a mesoscale particle‐based method that combines the Lagrangian discretization of Smoothed Particle Hydrodynamics (SPH) with thermodynamically consistent thermal fluctuations as in Dissipative Particle Dynamics (DPD), providing a discrete representation of the fluctuating Navier–Stokes equations [41, 71]. Its main advantages are the direct specification of transport properties (e.g., speed of sound, viscosity, and diffusivity) and consistency with the fluctuation–dissipation theorem and the GENERIC framework for nonequilibrium thermodynamics [72]. In the macroscopic limit, SDPD recovers the continuum hydrodynamic behavior of SPH and deterministic Navier–Stokes flow.

Hydrodynamic Coefficients and Kernel Definitions

In SDPD, the fluid is represented by interacting particles carrying mass, momentum, and transported species. The particle mass is defined as $m_i={\rho }_i\mathrm{\Delta }{x}^{\mathcal{D}}$, where ${\rho }_i$ is the local density, $\mathrm{\Delta }x$ the equilibrium interparticle spacing, and $\mathcal{D}$ the spatial dimension. The pressure $p_i$ of the $i$‐th particle is estimated using an equation of state of the form $p_i=c^2{\rho }_0/7\left[{\left({\rho }_i/{\rho }_0\right)}^7-1\right]$, where $c$ is the artificial speed of sound on the fluid, and ${\rho }_0$ is the reference density. The term $c^2{\rho }_0/7$ correspond to the reference pressure of the system, where ${\left.c^2=\partial p/\partial \rho \right|}_{\rho ={\rho }_0}$. Pairwise interactions are constructed using a compact‐support smoothing kernel $W\left(r_{\mathit{ij}},h\right)$ with cutoff radius $h$. In this work, we adopt Lucy's Quartic kernel [50], which ensures smoothness and exact normalization within its support:

$$W\left(r_{\mathit{ij}},h\right)=\left\{\begin{array}{ll}\frac{w_0}{h^{\mathcal{D}}}\left(1+\frac{3r}{h}\right){\left(1-\frac{r}{h}\right)}^3,& r/h<1,\\ 0,& r/h\ge 1,\end{array}\right.$$ (A1)

where $w_0=5/\pi$ and $w_0=105/(16\pi )$ in two and three dimensions, respectively. Spatial derivatives are obtained through the kernel gradient, entering all interaction forces via $F_{\mathit{ij}}=-\nabla W\left(r_{\mathit{ij}},h\right)/r_{\mathit{ij}}$. This yields a symmetric SPH‐consistent discretization of pairwise interactions [41].

Thermal fluctuations are introduced through a stochastic stress contribution expressed in terms of pairwise Wiener increments $\mathrm{d}{\mathbf{W}}_{\mathit{ij}}$. These are decomposed into isotropic and traceless symmetric parts,

$$\mathrm{d}{\bar{\mathbf{W}}}_{\mathit{ij}}=\frac{1}{2}\left(\mathrm{d}{\mathbf{W}}_{\mathit{ij}}+\mathrm{d}{\mathbf{W}}_{\mathit{ij}}^T\right)-\frac{1}{\mathcal{D}}\mathrm{tr}\left(\mathrm{d}{\mathbf{W}}_{\mathit{ij}}\right)\mathbf{I},$$ (A2)

ensuring angular momentum conservation at the pair level [41].

The corresponding noise amplitudes $A_{\mathit{ij}}$ and $B_{\mathit{ij}}$ control shear and bulk fluctuations and are related to the friction coefficients ($a=(\mathcal{D}+2)\eta /\mathcal{D}-\zeta$ and $b=(\mathcal{D}+2)(\zeta +\eta /\mathcal{D})$) through the FDT. The amplitudes are given by.

$$A_{\mathit{ij}}={\left[4k_{B}Ta\frac{F_{\mathit{ij}}}{d_i{d}_j}\right]}^{1/2},$$ (A3)

$$B_{\mathit{ij}}={\left[4k_{B}T\left(b-a\frac{\mathcal{D}-2}{\mathcal{D}}\right)\frac{F_{\mathit{ij}}}{d_i{d}_j}\right]}^{1/2}.$$ (A4)

where $d_i={\sum }_{j}W\left(r_{\mathit{ij}},h\right)$ is the particle number density and $k_B$ is the Boltzmann constant and $\mathcal{D}$ the dimension of the system. These relations ensure thermodynamic consistency of the SDPD formulation and guarantee convergence to equilibrium statistical mechanics [41]. A complete summary of SDPD units, material parameters, and numerical constants is provided in Table A1.

TABLE A1.

Units, simulation parameters, characteristic timescales, and dimensionless numbers used in the SDPD models.

Basic units
Length	$l$
Energy	$k_{B}T$
Mass	$m_f$

Derived units
Time	$\tau =l\sqrt{m_f/\left(k_{B}T\right)}$
Velocity	$u=l/\tau =\sqrt{k_{B}T/m_f}$
Diffusion constant	$D=l^2/t_0=l\sqrt{k_{B}T/m_f}$
Dynamic viscosity	$\eta =m_f/\left(t_{0}l\right)=\sqrt{m_f{k}_{B}T}/l^2$
Kinematic viscosity	$\nu =l^2/t_0=l\sqrt{k_{B}T/m_f}$

Parameter	Value (non‐dimensional)
Mass density, $\rho$	1.0
Dynamic viscosity, $\eta$	3.0
Kinematic viscosity, $\nu$	3.0
Thermal energy, $k_{B}T$	${10}^{-4}$
Speed of sound, $c_s$	10
Time step, $\mathrm{\Delta }\tau$	${10}^{-3}\tau$
Reference length, $l$	1 (unit length)
Reference time, $\tau$	1 (unit time)
Simulation box length	$L_x$ (varied)
Discretization spacing, $\mathit{dx}$	0.2
Cutoff radius, $h$	4 $\mathit{dx}$

Characteristic timescales and dimensionless numbers
Diffusion timescale	${\tau }_{\text{diff}}=H^2/D$
Advection timescale	${\tau }_{\mathrm{adv}}=H/U$
Reaction timescale	${\tau }_{\text{react}}=1/k_{\mathrm{eff}}{C}_{\mathrm{ref}}$
Péclet number	$\mathrm{Pe}=\mathit{UH}/D=\frac{{\tau }_{\text{diff}}}{{\tau }_{\mathrm{adv}}}$
Reynolds number	$\mathrm{Re}=\rho \mathit{UH}/\eta$
Damköhler number	$\mathrm{Da}={\tau }_{\mathrm{adv}}/{\tau }_{\text{react}}=k_{\mathrm{eff}}{C}_{\mathrm{ref}}H/U$
Schmidt number	$\mathrm{Sc}=\nu /D$
Mach number	$\mathrm{Ma}=U/c_s$

Numerical Implementation

The SDPD equations are integrated in time using a stochastic velocity–Verlet scheme [73], which preserves weak second‐order accuracy for the deterministic dynamics while consistently incorporating stochastic forcing. The fluid is discretized using particles with equilibrium spacing $\mathit{dx}=0.2$ and kernel cutoff $h=4\mathit{dx}$. The timestep $\mathrm{\Delta }t={10}^{-3}\tau$ is selected to ensure numerical stability of the coupled hydrodynamic and diffusive dynamics, satisfying the characteristic viscous and acoustic stability constraints of the SDPD discretization [74].

At each timestep, the coupled transport–reaction problem is advanced sequentially. First, particle positions and velocities are updated from the momentum balance. Second, diffusive transport is evaluated using the updated particle configuration. Third, local reaction source terms are integrated according to the reduced coagulation networks given in Tables A3 and A4. This operator‐splitting scheme provides a consistent coupling between hydrodynamics, diffusion, and reaction kinetics.

Macroscopic viscosity and diffusivity emerge from the microscopic SDPD interaction parameters and kernel choice. Flow is driven by a uniform body force per particle:

$$f_b=\frac{8\eta \mathit{v}}{\left(1/{\mathit{dh}}^2\right)H^2},$$ (A5)

where $\eta$ is the fluid viscosity, $v$ the mean flow velocity, $H$ the channel height, and $\mathit{dh}$ the particle spacing. This produces a steady Poiseuille‐like profile consistent with the selected Re and Pe numbers.

To avoid compressibility artifacts, the maximum velocity is constrained by the fixed SDPD speed of sound $c_s$, ensuring a low Mach number ($\mathit{Ma}=v/c_s\ll 1$). Consequently, Pe and Re values were restricted to ranges compatible with $\mathit{Ma}\lesssim 0.1$, preventing unphysically high velocities even if larger Pe might be physiologically relevant.

Appendix B. Characteristic Timescales and Dimensionless Numbers

To assess the transport and reaction regimes in the simulations, characteristic diffusion (${\tau }_{\text{diff}}$), advection (${\tau }_{\mathrm{adv}}$), and reaction (${\tau }_{\text{react}}$) timescales were computed using the dimensionless channel height ($H=$ 10) and diffusivities consistent with the reference scales ($H_{\mathrm{ref}}=100\mathrm{\mu m}$) and ($t_{\mathrm{ref}}=1\mathrm{s}$). From these timescales, the corresponding Péclet ($\mathrm{Pe}$) and Damköhler ($\mathrm{Da}$) numbers were derived to quantify the relative importance of transport and biochemical kinetics. The effective rate constants ($k_{\mathrm{eff}}=k/k_{\mathrm{ref}}$) were obtained using ($k_{\mathrm{ref}}={10}^{-3}{\mathrm{nM}}^{-1}{\mathrm{s}}^{-1}$). Parameters for thrombin (IIa) and fibrin (Fbn) formation were calculated as summarized in Table A2.

TABLE A2.

Characteristic timescales and nondimensional numbers used to describe diffusive transport and reaction regimes in the simulated channel.

Species	$D^{\alpha }$	$k^{\alpha }$	${\tau }_{\text{diff}}=H^2/D^{\alpha }$	${\tau }_{\mathrm{adv}}=L/U$	${\tau }_{\text{react}}=1/k^{\alpha }$	$\mathbf{Da}={\tau }_{\text{diff}}/{\tau }_{\text{react}}$
Thrombin (IIa)	0.00647	$3.76\times {10}^{-3}$	$1.55\times {10}^4$	$8.50\times {10}^2$	$2.66\times {10}^2$	$2.33$
Fibrin (Fbn)	0.00150	$1.33\times {10}^{-1}$	$6.67\times {10}^4$	$3.67\times {10}^3$	$7.52$	$354.6$

Note: $k^{\mathrm{IIa}}=k_6$ and $k^{\mathrm{Fbn}}={\alpha }_2$, from the model parameters in Table A3. All quantities are expressed in dimensionless simulation units.

The Damköhler numbers confirm that both IIa and Fbn operate in a reaction‐dominated regime (${\mathrm{Da}}^{\mathrm{IIa}}\gtrsim 1$, ${\mathrm{Da}}^{\mathrm{Fbn}}\gg 1$), consistent with the rapid vertical homogenization observed in the spatial maps in Figure 7.

Estimates for Microvascular Regimes

We report here physiological estimates of the Reynolds and Péclet numbers in the microcirculation, which motivate the choice of nondimensional parameters used in this study. Values are derived from canonical plasma and vessel properties, consistent with experimental measurements [75].

Re Number

Using plasma density $\rho =1.03\times {10}^3$ kg/m and viscosity $\eta =1.3\times {10}^{-3}$ Pa$\cdot$s ($\nu =1.26\times {10}^{-6}$ m/s), together with velocities $u={10}^{-3}$–$2\times {10}^{-2}$ m/s and vessel diameters $H=5$–$100\mathrm{\mu }$m [55, 75], we obtain

$$\mathrm{Re}=\frac{\rho \mathit{uH}}{\eta }\approx 0.008-1.6,$$

spanning diffusion‐dominated to weakly inertial regimes. These values reflect the microvascular regime (arterioles and venules), where viscous effects dominate and flow remains laminar.

Pe Number

The effective transport of solutes in blood is shaped by both molecular diffusion and shear‐enhanced dispersion. Reported effective diffusivities for macromolecules and clotting factors lie in the range $D_{\mathrm{eff}}=(1.3$–$21.8)\times {10}^{-10}$ m/s, depending on hematocrit and shear rate [75, 76, 77]. Inserting the same velocity and vessel size scales yields

$$\mathrm{Pe}=\frac{\mathit{uH}}{D_{\mathrm{eff}}}\approx 5-1.5\times {10}^4.$$

These values indicate predominantly convection‐driven transport, with the lower bound corresponding to molecular diffusion and the upper bound to red blood cell–mediated dispersion.

Implications

These estimates place the microvascular regime firmly within low–Reynolds‐number flows ($\mathrm{Re}\lesssim 1$), where inertia is negligible, and moderate‐to‐high Péclet numbers ($\mathrm{Pe}\sim {10}^1-{10}^4$), where advection competes with or dominates over molecular diffusion. Such conditions are characteristic of arterioles and venules with diameters of 5–100 $\mathrm{\mu m}$ [55]. In vivo, shear‐enhanced transport by red blood cells further increases effective diffusivity [62, 78], and comparable ranges are reproduced in microfluidic and flow‐based in vitro assays [53, 79].

Appendix C. Blood Coagulation Benchmarks

Overview of Reduced Coagulation Models

Before running our simulations, we employed established reduced coagulation models [24, 25], representing either the intrinsic or extrinsic pathways shown in Section 1. These models reproduce characteristic differences in clot growth across healthy and pathologic conditions, providing a basis for exploring treatment effects.

Reduced Intrinsic Coagulation Model

Parameter adjustments in the Ratto model [25] allow calibration to patient‐specific thrombin generation curves (TGCs), demonstrating its adaptability. Figure A1 summarizes the model behavior under different conditions. In Figure A1a, the main species—Xa, II, and IIa—are shown for a reaction‐dominated healthy system. The near overlap between simulations using species‐specific diffusivities ($D^{\alpha }$; square markers) and a uniform diffusivity confirms that such variations have negligible impact on thrombin kinetics, consistent with the diffusion‐dominated homogenization seen in channel simulations. Figure A1b illustrates how increasing initial prothrombin (II) accelerates and amplifies thrombin formation, reproducing experimental trends observed in hyperprothrombinemic plasma. Finally, Figure A1c compares healthy and thrombotic parameter sets, showing a sharper and earlier thrombin peak in the latter. These results demonstrate how modest parameter changes, particularly in Xa‐mediated reactions, can reproduce clinically relevant TGC profiles, highlighting the reduced Ratto model's ability to capture diverse coagulation phenotypes. Table A3 lists the parameters.

TABLE A3.

Values of reaction kinetics, diffusion coefficients, and initial concentrations adopted in the intrinsic coagulation model.

Symbol	Healthy	Thrombotic	UM	References
Biochemical reactions kinetic constants
$k_1$	$4.082\times {10}^{-05}$	$4.082\times {10}^{-05}$	s	[80]
$k_2$	$1.111\times {10}^{-05}$	$1.111\times {10}^{-05}$	nM s	[80]
$k_3$	$2.4482\times {10}^{-07}$	$2.4482\times {10}^{-07}$	nM s	[80]
$k_4$	$0.000379$	$0.000479$	nM s	[80]
$k_5$	$1.859\times {10}^{-05}$	$3.859\times {10}^{-05}$	nM s	[80]
$k_6$	${3.76210}^{-06}$	$4.762\times {10}^{-06}$	nM s	[80]
$k_7$	$1.285\times {10}^{-10}$	$1.285\times {10}^{-10}$	nM s	[80]
$k_8$	$4.066\times {10}^{-10}$	$4.066\times {10}^{-10}$	nM s	[80]
$k_9$	$0.02$	$0.009$	nM s	[80]
${\alpha }_1$	$1.33\times {10}^{-04}$	$1.117\times {10}^{-03}$	nM s	[80]
${\alpha }_2$	$1.52\times {10}^{-04}$	$1.67\times {10}^{-03}$	nM s	[80]
Diffusion coefficients
$D^{\mathrm{Xa}}$	$7.37\times {10}^{-07}$		${\mathrm{cm}}^2{\mathrm{s}}^{-1}$	[8]
$D^{\mathrm{II}}$	$5.21\times {10}^{-07}$		${\mathrm{cm}}^2{\mathrm{s}}^{-1}$	[8]
$D^{\mathrm{IIa}}$	$6.47\times {10}^{-07}$		${\mathrm{cm}}^2{\mathrm{s}}^{-1}$	[8]
$D^{\mathrm{fbg}}$	$2.47\times {10}^{-07}$		${\mathrm{cm}}^2{\mathrm{s}}^{-1}$	[8]
$D^{\mathrm{Fbn}}$	$1.50\times {10}^{-07}$		${\mathrm{cm}}^2{\mathrm{s}}^{-1}$	[8]
$D^{{\mathrm{Fbn}}_p}$	$1.00\times {10}^{-10}$		${\mathrm{cm}}^2{\mathrm{s}}^{-1}$	[8]
Initial concentrations
${\mathrm{C}}^{{\mathrm{X}}^0}$	135	175	nM	[80]
${\mathrm{C}}^{\mathrm{II}}$	1400	1900	nM	[80]
${\mathrm{C}}^{\mathrm{fbg}}$	$\le$10,000	>10,000	nM	[80]

Note: The healthy and thrombotic columns list values for normal and pathological conditions, respectively, reported by Ratto et al. [25].

FIGURE A1.

Thrombin Generation Curves (TGCs) from Ratto's model under different conditions. Concentrations are expressed in nM. (a) Mean concentrations of Xa (blue), II (red), and IIa (yellow) in a reaction‐dominant healthy system; square markers indicate species‐specific diffusivities ($D^{\alpha }$), consistent with uniform diffusivity results. (b) Effect of varying initial prothrombin (II) while keeping healthy reaction rates. (c) Comparison of healthy versus thrombotic TGCs, showing differences in thrombin peak magnitude and timing.

Initialization at Steady‐State

To avoid artificial transients and nonphysical reaction delays, all simulations are initialized from steady‐state concentrations. For Ratto's model, steady‐state values of Factor Xa ($\mathrm{Xa}$), Prothrombin ($\mathrm{II}$), and Thrombin ($\mathrm{IIa}$) are obtained by solving the reaction system (3) in Section 2.2.1 with all time derivatives set to zero.

Starting from initial guesses (${\mathrm{Xa}}^0=135\mathrm{nM}$, $\mathrm{fbg}=\mathrm{10,000}\mathrm{nM}$) and using the parameter set in Table A3 for a healthy subject, MATLAB's fsolve yields the following equilibrium concentrations:

$$\begin{array}{ll}& \mathrm{Xa}\approx 14.99\mathrm{nM},\mathrm{II}\approx 1.11\mathrm{nM},\mathrm{IIa}\approx 0.00\mathrm{nM},\\ & \mathrm{fbg}\approx \mathrm{9,999.96}\mathrm{nM},\mathrm{Fbn}\approx 0.20\mathrm{nM},{\mathrm{Fbn}}_{\mathrm{P}}\approx 7.66\mathrm{nM}.\end{array}$$

These concentrations define the reference steady state for the kinetic parameters considered.

Reduced Extrinsic Coagulation Model

We use the reduced extrinsic model from Chen and Diamond (2019) [24]. Reaction rates $a_i$ follow Michaelis–Menten kinetics:

$$a_i=\frac{k_{\mathrm{cat}}{[\mathrm{S}]}_i}{K_m+{[\mathrm{S}]}_i},$$ (A6)

where $k_{\mathrm{cat}}$ is the catalytic constant, ${[\mathrm{S}]}_i$ is the substrate concentration, and $K_m$ is the Michaelis constant. Species‐specific parameters are defined in Table A4. Authors therein reported that three effectiveness factors (${\eta }_4=0.18$, ${\eta }_5=0.05$, and ${\eta }_6=0.36$) needed calibration to match experimental data. The system remains well‐defined up to $t=800\mathrm{s}$; beyond this point, the numerical solution diverges and non‐finite (NaN) values appear.

TABLE A4.

Parameters for the reduced model of the extrinsic coagulation pathway used in the Michaelis–Menten formulation of Equation (A6).

Enzyme	[S] ($\mu \mathrm{m}$)	$k_{\mathrm{cat}}$ (s)	$K_m$ ($\mu \mathrm{m}$)	$\alpha$ (s)	$\eta$
TF/VII	${\mathrm{X}}_0=0.17$	1.15	0.24	0.48 [19]	1.00
TF/VIIa	${\mathrm{IX}}_0=0.09$	1.8	0.42	0.32 [9]	1.00
IXa/VIIIa	${\mathrm{X}}_0=0.17$	8.2	0.082	5.53 [9]	1.00
Xa/Va	${\mathrm{II}}_0=1.4$	30	0.3	24.7 [9]	0.18
IIa	$\alpha$‐fbg = 18	80	6.5	58.8 [9, 81]	0.05
IIa/p	${\mathrm{XI}}_0=0.031$	0.00013	0.05	$5\times {10}^{-5}$ [9]	0.36
XIa/p	${\mathrm{IX}}_0=0.09$	0.21	0.2	0.065 [9]	1.00

Diffusion coefficients $C^{\alpha }$ (${\mathrm{cm}}^2{\mathrm{s}}^{-1}$)
$D^{\mathrm{TF}}$	$2.00\times {10}^{-07}$
$D^{\mathrm{Xa}}$	$7.37\times {10}^{-07}$
$D^{\mathrm{IXa}}$	$7.00\times {10}^{-07}$
$D^{\mathrm{XIa}}$	$7.00\times {10}^{-07}$
$D^{\mathrm{Fbn}}$	$1.00\times {10}^{-07}$
$D^{\mathrm{IIa}}$	$6.47\times {10}^{-07}$
$D^{E\mathrm{S}}$	$1.00\times {10}^{-07}$
$D^{\gamma \mathrm{S}}$	$1.00\times {10}^{-07}$

Data Availability Statement

This study did not generate new experimental or observational datasets. The coagulation reaction networks, kinetic parameters, and ODE formulations used in the simulations are drawn entirely from previously published models. All inputs are publicly available in the cited articles. The custom simulation and analysis codes used in this study are available from the corresponding author upon reasonable request.