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Multiscale Modeling of Blood Flow-Mediated Platelet Thrombosis

  • Alireza Yazdani
  • Peng ZhangEmail author
  • Jawaad Sheriff
  • Marvin J. Slepian
  • Yuefan Deng
  • Danny Bluestein
Living reference work entry
  • 227 Downloads

Abstract

The blood coagulation cascade that leads to thrombus formation may be initiated by flow-induced platelet activation, which prompts clot formation in prosthetic cardiovascular devices and in arterial disease processes. Upon activation, platelets undergo complex morphological changes of filopodia formation that play a major role in aggregation and attachment to surfaces. Numerical simulations based on continuum approaches fail to capture such molecular-scale mechano-transduction processes. Utilizing molecular dynamics (MD) to model these complex processes across the scales is computationally prohibitive. We describe multiscale numerical methodologies that integrate four key components of blood clotting, namely, blood rheology, cell mechanics, coagulation kinetics and transport of species, and platelet adhesive dynamics across a wide range of spatiotemporal scales. Whereas mechanics of binding/unbinding for single-molecule receptor-ligand complexes can be simulated by molecular dynamics (MD), the mechanical structure of platelets in blood flow and their interaction with flow-induced stresses that may lead to their activation can be efficiently described by a model at coarser scales, using numerical approaches such as coarse-grained molecular dynamics (CGMD). Additionally, CGMD provides an excellent platform to inform other coarser-scale models in a bottom-up approach in the multiscale hierarchy. The microenvironment of most biological systems such as coagulation normally involves a large number of cells, e.g., blood cells suspended in plasma, limiting the utility of CGMD at the larger transport scales of blood flow. However, dissipative particle dynamics (DPD), along with its sub-models such as energy conserving and transport DPD, provides a very flexible platform for scaling up these mesoscopic systems. At the macroscopic top scales of the vasculature and cardiovascular devices, simulating blood and tissues using continuum-based methods becomes viable and efficient. However, the challenge of interfacing these larger transport scales with the orders of magnitude smaller spatiotemporal scales that characterize blood coagulation, and given the issue of the slow-dynamic timescales of biological processes, makes long-term simulations of such systems computationally prohibitive. In this chapter, we describe various numerical remedies based on these methodologies that facilitate overcoming this multiscale simulation challenge.

Keywords

Multiscale Modeling Coarse-grained Molecular Dynamics (CGMD) Dissipative Particle Dynamics (DPD) Mechano-transduction Process Flow-induced Stresses 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1 Introduction

Hemostasis (thrombus formation) is the normal physiological response that prevents significant blood loss after vascular injury. The resulting clots can form under different flow conditions in the veins as well as the arteries. The excessive and undesirable formation of clots (i.e., thrombosis) in our circulatory system may lead to significant morbidity and mortality. Some of these pathologies are deep vein thrombosis and pulmonary embolism and atherothrombosis (thrombosis triggered by plaque rupture) in coronary arteries, to name a few. The process of clot formation and growth at a site on a blood vessel wall involves a number of simultaneous processes including multiple chemical reactions in the coagulation cascade, species transport, and platelet adhesion, all of which are strongly influenced by the hydrodynamic forces.

Platelets are fundamental to both hemostasis and thrombosis in many vascular diseases. Normal platelets do not interact with the healthy arterial wall. In cases of endothelial injury or exposure of extracellular matrix to blood flow, however, platelets can quickly activate and cover the injured area to stop bleeding. The initial adhesion of platelets on the thrombogenic area can be attributed to a variety of platelet membrane receptor-ligand interactions, such as glycoprotein Ib (GPIb)-V-IX with immobilized von Willebrand factor (vWF), GPIIb–IIIa (αIIbβ3) with vWF, GPVI with collagen, α2β1 with collagen, αIIbβ3 with fibrinogen, and so on, depending on the nature of the lesion and the local shear rate of blood flow. At low shear rates ( \(\dot {\gamma } < 1000\,\mathrm {s}^{-1}\)), platelets adhere to the thrombogenic area through different pathways, relying on the exposed extracellular matrix (ECM) proteins. On the other hand, as shear rate increases, interactions between immobilized vWF and GPIb become exclusive in initializing platelet aggregation, while other interactions are broken down due to high bond failure rates. The reason that vWF-GPIb interactions persist at such high shear rates (25, 000 s−1 shown in in vitro experiments) is that the vWF proteins, which are normally in a coiled state, tend to extend severalfold in high-shear environments. The conformational change of vWF exposes the repeating functional A1 domains in multimeric vWF, leading to enhanced adhesive interactions between GPIb and vWF (Schneider et al. 2007) (Fig. 1).
Fig. 1

Bio-chemo-mechanics of platelet aggregation and coagulation. (a) Platelet adhesion receptors and their ligands. Each platelet surface bears ≈25, 000 GPIb receptors that bind to surface-bound von Willebrand factor; ≈50, 000 integrin αIIbβ3 receptors that bind to fibrinogen and vWF; ≈4, 000 GPVI receptors; and 1,000–4,000 integrin α2β1 receptors that bind to several types of collagen. The integrins must be activated to form strong long-lived bonds. Collagen is a major constituent of the subendothelial (SE) matrix; fibrinogen is an abundant plasma protein; and vWF is adsorbed to SE collagen, circulates in plasma, and is secreted by endothelial cells. (b) Schematic illustration of coagulation reactions. Most coagulation proteins exist in an inactive (FVIII) and active form (FVIIIa). FIIa (also known as thrombin), FVIIa, FIXa, FXa, and FXIa are enzymes; their inactive precursors are called zymogens. FVa and FVIIIa are cofactors for the enzymes FXa and FIXa, respectively, and must be activated from their precursors FV and FVIII. Tissue factor (TF) is a cofactor for FVIIa. The coagulation enzyme-cofactor complexes form on SE and platelet surfaces and have enzymatic efficiencies 105−106-fold those of the enzymes alone. The activation of a coagulation protein is by proteolysis (i.e., cutting) of the precursor by another enzyme. Thrombomodulin (TM) on ECs is a cofactor for thrombin in producing the inhibitor activated protein C (APC). Other major inhibitors are antithrombin (AT) and tissue factor pathway inhibitor (TFPI). Surface-bound enzyme complexes TF:VIIa, VIIIa:IXa, Va:Xa, and TM:IIa and other surface-bound species are shown in boxes. Also shown are cell or chemical activation (purple lines), movement in fluid or along a surface (dark blue lines), enzyme action in a forward direction (solid gray lines), the feedback action of enzymes (dashed gray lines), binding to or unbinding from surface (light blue double-headed arrows), and chemical inhibitors (red circles). (Reprinted with permission from Fogelson and Neeves 2015)

While platelet activation may be induced by biochemical agonists, shear stresses arising from pathological flow patterns enhance the propensity of platelets to activate and initiate coagulation cascade, causing thrombosis. Flow stresses acting on the platelets can be represented down to the order of microns by continuum models. However, processes such as the morphological changes in platelets upon activation happen at the molecular scales. Continuum approaches fail to capture such processes. Mesoscopic particle-based methods were developed to model blood cells and thrombosis in flowing blood, using dissipative particle dynamics (DPD) for modeling the blood, blood flows in microvessel, interactions of cells, and thrombosis growth. However, complex processes such as platelet activation require drastically different modeling approaches in order to achieve highly resolved microstructural mechanics and the mechano-transduction process of hemodynamic stresses that may induce platelet activation. Such approaches are needed for more accurately describing the platelets intracellular structure and the mechano-transduction processes of platelet activation by hemodynamic forces.

Incorporating all these processes that occur at different spatiotemporal scales, however, remains a rather challenging task. For example, platelet-wall and platelet-platelet interactions through receptor-ligand bindings occur at a subcellular nanoscale, whereas the blood flow dynamics in the vessel around the developing thrombus is described as a macroscopic process from several micrometers to millimeters. Here, we describe a hierarchical multiscale numerical approach that may seamlessly unify and integrate different subprocesses within the clotting process.

1.1 Significance and Rationale for a Multiscale Model

Cardiovascular diseases account for over 31% of deaths globally and in the USA annually. Whether due to acute thrombosis associated with myocardial infarction or progressive intermittent atherothrombotic events, significant ventricular dysfunction may result, leading to heart failure. Presently over 6.5 million patients suffer from heart failure in USA, and their number is expected to grow by nearly 50% by 2030. Of those, a significant proportion will become candidates for mechanical circulatory support and prosthetic cardiovascular devices, also burdened with thromboembolic risk and complications, mandating antithrombotic medications for their recipients. Despite antithrombotic therapy, ventricular assist device patients have approximately a 12% stroke rate, of which 18% prove fatal. The alarming thrombosis rates, coupled with complex antithrombotic and anticoagulant management, necessitate a solution to save numerous lives and drastically reduce the ensuing healthcare costs.

Thrombosis in vascular disease is potentiated by the interaction of blood constituents with an injured vascular wall and the nonphysiologic flow patterns characterizing cardiovascular pathologies. Both initiate and enhance the hemostatic response through chronic platelet activation. Similarly, the thrombus and thromboemboli generated in devices are composed primarily of platelets. The elevated flow stresses they induce chronically activate platelets despite aggressive antithrombotic therapy. However, limitations in characterizing the mechanical stimuli, and the complex biochemical and morphological changes platelets undergo in response, still hamper our ability to successfully model such intricate phenomena. The coupling of the disparate spatiotemporal scales between molecular-level events and the macroscopic transport represents a major modeling and computational challenge, which requires a multidisciplinary integrated multiscale numerical approach. Continuum approaches are limited in their ability to cover the smaller molecular mechanisms such as filopodia formation during platelet activation. Utilizing molecular dynamics (MD) to cover the multiscales involved is computationally prohibitive. Innovative multiscale approaches are essential for elucidating this vexing problem (Bluestein et al. 2014; Zhang et al. 2014).

1.2 Flow-Induced Mechanisms of Platelet Activation, Aggregation, and Thrombosis

The “Virchow’s triad” of altered blood, surface, and flow establishes the latter as the most complex component of the triad – the local flow patterns role in platelet activation and aggregation, interaction with the vascular wall, and deposition onto foreign surfaces. It determines where a thrombus will form, its size, and composition and whether it will remain at its initiation nidus or embolize (Sorensen et al. 1999). Shear-induced platelet activation (SIPA) shows consistent “dose” and time response characteristics of biochemical agonists. It causes both aggregation and thrombin generation – the most potent platelet activator, with an established activation threshold (Kroll et al. 1996). Aggregation of platelets and deposition onto thrombogenic surfaces (Jackson 2007) have three distinct shear ranges:
  1. (i)

    Low-intermediate shear in veins and larger arteries (<103 s−1). Aggregation is predominantly mediated by αIIbβ3 on free-flowing platelets which engage fibrinogen (Fg) adsorbed on platelet monolayers that compose the initial thrombi.

     
  2. (ii)

    High shear in arterial microcirculation or moderate arterial stenosis (103 − 104 s−1). Aggregation is more vWF-dependent and involves the activity of both GPIba and αIIbβ3.

     
  3. (iii)

    Pathological or “hyper” shear in atherothrombosis and cardiovascular devices (>104 s−1). Aggregation is mediated exclusively by vWF-GPIb bonds, not requiring platelet activation or adhesion via αIIbβ3.

     

1.3 Aggregation, Adhesion, and Wall Interaction

Shear-induced flow mechanisms are enhanced by eddies, flow vortices, and flow separation (Nesbitt et al. 2009), with prothrombotic agonists trapped in regions of recirculating flow. Shear-induced aggregation has two characteristic phases: (i) reversible phase between two discoid platelets and the presence of unstable platelet aggregates and (ii) stable phase that involves platelet shape change and granular release (Maxwell et al. 2007; Nesbitt et al. 2009). Unstable aggregates are formed with vWF alone, while stable aggregates require both vWF and fibrinogen, allowing for a shorter formation time and increased recruitment of platelets to aggregates (Maxwell et al. 2007). Initial aggregates typically involve clustering of platelets around a single activated platelet via membrane tethers, not pseudopodia, whereas stable aggregates require platelet sphering and filopodial projections. Increasing the shear rate from 1,800 to 5,000 s−1 increases the number of discoid platelet aggregates converting to stable aggregates, and αIIbβ3 is required for membrane tether formation between platelets in reversible aggregates (Maxwell et al. 2007). In stenotic regions, initial platelet recruitment occurs at the stenosis apex, and aggregates form at the downstream expansion zone where the flow decelerates. Larger decelerations result in larger aggregates embedded in the growing thrombus.

1.4 Continuum Approaches for Blood Flow at the Microscale

Several continuum approaches try to couple the timescales between molecular and macroscopic transport. In those the cellular level is represented by field equations and moving boundaries to allow the cell to change its shape with interfaces that have different timescales and properties (N’Dri et al. 2003; Shyy et al. 2001). Models of, e.g., adhesion of rolling leukocytes to endothelium are based on either equilibrium concepts (Bell 1978; Berk and Evans 1991; Evans 1985a,b) or kinetics approach (Dembo et al. 1988; Hammer and Lauffenburger 1987; Dong et al. 1999). Bond association and dissociation occur according to the forward and reverse reaction rates (N’Dri et al. 2003; Shyy et al. 2001). Initial models (Skalak et al. 1981a, 1989, 1981b; Weiss 1990) neglected cell rheological properties and were limited in their predictive capability (Bell 1978; Dembo et al. 1988). Flow-induced stresses acting on a cell may weaken the adhesion bonds or extract receptor molecules from the cell surface (Alon et al. 1997, 1998) with progress toward understanding the receptor-mediated cell adhesion process (Lauffenburger et al. 1987; Linderman and Lauffenburger 1986, 1988). Detailed review on such simulations using probabilistic and Monte Carlo approaches was published (Zhu 2000). Flow-mediated transport and initiation and inhibition of coagulation were modeled extensively by the Fogelson group (Fogelson and Tania 2005).

1.5 Coarse-Grained Particle Methods at the Mesoscale

CFD simulations are limited to continuum systems (Dwinzel et al. 2002; Groot and Warren 1997), representing a 10 nm to 100 μm gap between microscopic biochemistry and macroscopic scales. DPD models possess important properties of mesoscopic systems: DPD can easily model heterogeneous fluids, allowing simulations of processes which are very difficult to model by continuum approaches. Initial work focused on multiphase nature of these flows (Espanol 1998; Boek et al. 1996) to address limitations of continuum approaches that represent them consistently (Albert et al. 1998; Kechagia et al. 2002; Davis 2002). Blood flow was modeled with DPD (Dzwinel et al. 2003) with plasma represented by dissipative fluid particles, and red blood cells (RBCs) and capillary walls by coarse-grained (CG) particles ensemble (Boryczko et al. 2003, 2004), demonstrating aggregation and deformation. A triple-decker MD-DPD model interfaced atomistic-mesoscopic-continuum flow regimes (Fedosov and Karniadakis 2009). A flow-cell interaction model (Sweet et al. 2011) used subcellular elements to simulate cell motion, with the cell coupled to the plasma flow by the Langevin equation (Skeel and Izaguirre 2002). DPD was initially used in simple geometries (Fedosov and Karniadakis 2009; Soddemann et al. 2003; Espanol and Warren 1995) for modeling viscous flows with no-slip boundary conditions (Koplik and Banavar 1995a,b; Revenga et al. 1999; Pivkin and Karniadakis 2005), expanding to higher Reynolds numbers (Qian and Wang 2005), and more recently in a stenosis model (Gao et al. 2017).

2 Hierarchical Multiscale Modeling of Platelet Thrombosis

Numerical modeling of thrombus formation and growth is a challenging problem due to multiscale and multiphysics nature of clotting process, which involves fluid mechanics, cell mechanics, and biochemistry. Diverse studies have addressed this problem on different scales such as cellular, meso, and continuum levels, whereas attempts have been made to bridge these different scales to model the process at the initial phase of platelet activation and aggregation (cf. Xu et al. 2010, Zhang et al. 2014, and Grinberg et al. 2013). These studies may be broadly put in three distinct modeling strategies: cellular/subcellular modeling of adhesive dynamics, platelet transport, and aggregation in whole blood; continuum-based modeling of blood flow treating platelets as Lagrangian particles; and continuum-based modeling of thrombus formation and growth using empirical correlations for platelet deposition rates.

A major challenge for multiscale modeling of thrombogenesis is the coupling of the disparate spatiotemporal scales characterizing flow effects on deformable platelets and the resulting adhesion/aggregation bond scales. A schematic flowchart depicts flow-induced processes leading to clotting and thrombosis and the feedback between the scales (Fig. 2). A successful multiscale model requires molecular-level characterization of mechano-transduction processes activating platelets, the ensuing clotting reactions, and their interactions with the flow. Departing from traditional continuum approaches, a multiscale model is proposed by combining DPD interfaced with coarse-grained molecular dynamics (CGMD) – this integrated “DPD-CGMD” multiscale model that can bridge the gap across the vast range of the spatiotemporal scales. Starting with flow patterns with high propensity to activate and aggregate platelets, macro-to-mesoscale flow scales are modeled using DPD and micro-to-nanoscales using CGMD. An interactive DPD-CGMD dynamic coupling interface scheme will reflect any changes in platelets and thrombus morphology and feed back to affect the hemodynamics and vice versa, resulting in a multiscale model to bridge the gap between the macroscopic flow scales and the cellular/molecular scales.
Fig. 2

Schematic flowchart of the multiscale flow-induced clotting, including mechanical and biochemical interactions

2.1 Molecular Dynamics Modeling of Receptor-Ligand Interactions

Understanding how force affects receptor and ligand binding and unbinding is a long-standing effort in mechanobiology. Bond dissociation rates typically increase under mechanical stress; however, bond stability can be enhanced through specialized mechanisms induced by force, including catch bonds and switching to a slip bond with a slower off-rate (flex bonds) (Kim et al. 2015). At sites of vascular injury, hydrodynamic force in the bloodstream acting on vWF is pivotal in regulating the binding of the vWF A1 domain to GPIba on platelets and commencing the cross-linking of platelets by vWF to form a platelet plug. vWF circulates in the form of long, disulfide bonded concatemers, with tens to hundreds of monomers, which mostly adopt a compact, irregularly coiled conformation during normal hemodynamics (see Fig. 3). At sites of hemorrhage, flow changes from shear to elongational. On the transition from low to high shear and from shear to elongational flow, irregularly coiled molecules extend to a threadlike shape, and elongational (tensile) force is exerted throughout their lengths. Molecular elongation exposes the multiple A1 binding sites in vWF concatemers for multivalent binding to GPIba (Springer 2014).
Fig. 3

Conformation of vWF under elongation and structural form of its A1 domain bound to GPIba. (a) Schematic organization of domains in vWF and head-to-head and tail-to-tail linkage of vWF monomers into concatemers. (b) The vWF A1-GPIba complex forms a super β-sheet at the interface between the A1 β3 and GPIba β14 strands. (Reprinted with permission from Springer 2014)

Atomic force microscope (AFM) experiments have been used extensively to probe the force required to rupture single-molecule complexes and have provided additional insights into their binding properties. By performing force spectroscopic measurements of binding/unbinding force distributions at different relaxation/pulling rates, these AFM experiments have been able to quantify the kinetics of ligand-receptor interactions.

To reveal the microscopic processes underlying the AFM observations, numerical simulations of binding/unbinding events for different complexes have been performed using steered molecular dynamics (SMD) technique. SMD allows us to explore these processes on timescales accessible to MD simulations. The basic idea behind any SMD simulation is to apply an external force to one or more atoms. In addition, you can keep another group of atoms fixed and study the behavior of the complex under various conditions. The only major difference between the AFM experiment and simulations concerns the value for the pulling velocity. Whereas the AFM experiment is carried out on a millisecond timescale, MD simulations are limited to nanoseconds; therefore, one has to consider thermal fluctuations and dissipation. Because of these nonequilibrium phenomena, the rupture force should vary systematically with rupture speed, and thus the computed rupture forces should be able to be extrapolated to the experimental timescale (Grubmüller et al. 1996).

2.2 An Integrated DPD-CGMD Modeling Approach

The multiscale approach (Fig. 4) incorporates (i) top/macro blood flow scale using DPD fluid, with length scales down to \(\mathcal {O}(\mathrm {m})\), where multiple platelets each composed of an ensemble of attractive particles are suspended in a discrete particle medium with viscous fluid properties and interact with the blood vessel walls and each other, and (ii) bottom/cellular scale employing CGMD, \(\mathcal {O}\,(\mathrm {nm})\) scales, in which platelets with multiple sub-components (actin filaments cytoskeleton, bilayer membrane, and cytoplasm) evolve during activation as pseudopodia grow and platelets lose their quiescent discoid shape. The fully interactive DPD-CGMD dynamic coupling interface passes the information across the scales as individual platelet activation evolves at the cellular bottom scale in response to viscous shear. This multiscale approach (Fig. 4) depicts the (a) scale exchange of shear from the DPD blood flow and (b) state exchange based on the coarse-graining of MD modeling of the individual platelets, and their shape change is transferred – interactively affecting the DPD fluid hemodynamics.
Fig. 4

Multiscale DPD-CGMD approach for modeling flow-induced platelet activation

2.2.1 Coarse-Grained Molecular Dynamics Model of Platelet Cellular Structure

The cellular and subcellular structure of a resting platelet can be modeled using a CGMD approach in Fig. 5 (Zhang et al. 2017). A resting platelet has a discoid shape with a 2 mum semimajor axis and 0.5 mum semiminor axis. The peripheral zone is modeled as a homogeneous elastic material bilayer constructed by 2D triangulation method. A shell of 300 AA thickness represents the phospholipid bilayer membrane (100 AA) and an exterior coat (150–200 Å). The membrane is allowed to deform under strain. The organelle zone, represented by the cytoplasm, is composed of homogeneous nonbonded particles filling the gap between the membrane and the cytoskeleton. At the fluid-platelet interface, the membrane prevents fluid particles from penetrating while maintaining the flowing platelet flipping dynamics.
Fig. 5

CGMD model of the intercellular constituents structure of human platelets. Detailed connections of the filamentous core with actin filaments and the membrane particles are shown in the upper left corner. The rigid filamentous core made by carbon-70 structure at the center of platelet. One end of the actin bundles is attached to the core, and the other end is attached to the membrane. The cytoplasm fills the space between the membrane and the cytoskeleton. (Reprinted with permission from Zhang et al. 2017)

The cytoplasm rheology is modeled using a Morse potential (Morse 1929). The cytoskeleton consists of two types of actin-based filaments: (a) a rigid filamentous core and (b) an assembly of radially spanning elastic actin filaments that mediates the contractility. A carbon-70 structure is used to generate the oval-shaped core. Each actin filament is individually extensible and is tethered to the core. An α-helical structure mimics the spring-loaded molecular mechanism, with its spiral conformation continuously stretchable under external force.

A CGMD potential for the elastic membrane and filamentous core along with a Morse potential for the cytoplasm and the MD force fields for the actin filaments are adopted in this platelet model. The CGMD describes the lowest scale of the platelet intracellular constituents, including the bilayer membrane, the cytoplasm, and a cytoskeleton composed of extensible actin filaments of the pseudopods at the micro- to nanoscales. CGMD potential is written as
$$\displaystyle \begin{aligned} V_{CGMD}(r) = \sum_{\textit{bonds}}{k_b(r-r_0)^2} + \sum_{LJ} { 4 \varepsilon_{ij} \big[\big(\frac{\sigma_{ij}}{r}\big)^{12} - \big(\frac{\sigma_{ij}}{r}\big)^6\big] } \ , \end{aligned} $$
(1)
and describes the deformability of the membrane (the first term) and the interaction between the membrane and intercellular particles (the second term). A MD potential without an electrostatic term for the actin-based filaments is
$$\displaystyle \begin{aligned} V_{MD} = \sum_{\textit{bonds}}{k_b(r-r_0)^2} + \sum_{\textit{angles}}{k_{\theta}(\theta - \theta_0)^2} + \sum_{\textit{torsion}}{k_{\phi}(1+\textit{cos}(n \phi - \sigma))} \\ + \sum_{LJ}{4\varepsilon_{ij}\big[\big(\frac{\sigma_{ij}}{r}\big)^{12} - \big(\frac{\sigma_{ij}}{r}\big)^6\big]} \ ,\end{aligned} $$
(2)
where kb, kθ, and kϕ are force constants for the α-helical structure, r0 and θ0 are the equilibrium distance and angle, and n is the symmetry of the rotor and δ is the phase. In the L-J term, εij is the well depth of the L-J potential, σij is the finite distance and r is the interparticle distance, and θ is the angle and ϕ is the torsional angle. Morse potential is used for the cytoplasm
$$\displaystyle \begin{aligned} V_{\textit{Morse}} (r) = \varepsilon [ e^{\alpha(1-\frac{r}{R})} - 2e^{\frac{\alpha}{2}(1-\frac{r}{R})} ] \ , \end{aligned} $$
(3)
describing a pairwise nonbonded interaction for viscous flows at coarse-grained scales. R is the distance of minimum energy ε, and α is a parameter that measures the curvature of the potential around R, where r is the interparticle distance.

2.2.2 Modeling Blood Plasma Using DPD Fluid Model

In the standard DPD method, the pairwise forces consist of three components: (i) conservative force, \(\mathbf {{f}}^C_{ij} = a_{c} (1 - r_{ij}/r_c) \hat {\mathbf {{r}}}_{ij}\); (ii) dissipative force, \(\mathbf {{f}}^D_{ij} = \gamma \omega _d(r_{ij}) (\hat {\mathbf {{r}}}_{ij} \cdot \hat {\mathbf {{v}}}_{ij} ) \hat {\mathbf {{r}}}_{ij}\); and (iii) random force, \(\mathbf {{f}}^R_{ij} = \sigma \omega _r (r_{ij}) (\zeta _{ij} / \sqrt {\varDelta t}) \hat {\mathbf {{r}}}_{ij}\). Hence, the total force on particle i is given by \(\mathbf {{f}}_i = \sum _{i \ne j} ({\mathbf {f}}_{ij}^C + {\mathbf {f}}_{ij}^D+ {\mathbf {f}}_{ij}^R)\), where the sum acts over all particles j within a cutoff radius rc, and ac, γ, and σ are the conservative, dissipative, and random coefficients, respectively, rij is the distance with the corresponding unit vector \(\hat {\mathbf {{r}}}_{ij}\), \(\hat {{\mathbf {v}}}_{ij}\) is the unit vector for the relative velocity, ζij is a Gaussian random number with zero mean and unit variance, and Δt is the simulation time step size. The parameters γ and σ and the weight functions are coupled through the fluctuation-dissipation theorem and are related by \(\omega _d = \omega _r^2\) and σ2 = 2γkBT, where kB is the Boltzmann constant and T is the temperature of the system. The weight function ωr(rij) is given by
$$\displaystyle \begin{aligned} \omega_r \left( {r_{ij} } \right) = \left\{ \begin{array}{l} \; ({1 - \frac{{r_{ij} }}{{r_c }}})^k \quad r_{ij} < r_c \\ \; 0 \qquad\qquad\\ r_{ij} \ge r_c \\ \end{array} \right. \, , \end{aligned} $$
(4)
with k = 1 in the original DPD form, whereas k < 1 is used to increase the plasma viscosity. A hybrid DPD-Morse model for blood flow simulations through severe stenotic microchannel is proposed by combining the Morse potential and classical DPD conservative force and is being used in order to reduce the fluid compressibility (Gao et al. 2017). The compression simulation presents a much smaller isothermal compressibility of the DPD-Morse fluid than that of the DPD fluid with \(\kappa ^{DM}_T/\kappa ^{DPD}_T \approx 0.3\), which shows DPD-Morse fluid is more resistant to compression under the same condition.

2.2.3 DPD-CGMD Spatial Interfacing

Hybrid force field is developed for describing the dynamic fluid-platelet interaction (Zhang et al. 2014) and is written as
$$\displaystyle \begin{aligned} d\mathbf{{v}}_i = \frac{1}{m_i} \sum_{j \neq i}^{N} ( \nabla U(r_{ij}) dt + \mathbf{{f}}^D_{ij} dt + \mathbf{{f}}^R_{ij} \sqrt[]{dt} ) \ , \end{aligned} $$
(5)
where U(rij) is the standard Lennard-Jones potential, with εp and σp the same as the energy and length characteristic parameters in CGMD, and \(\mathbf {{f}}^D_{ij}\) and \(\mathbf {{f}}^R_{ij}\) are the standard DPD dissipative and random forces, respectively. The L-J force term helps the cytoskeleton-confined shapes and the incompressibility of platelets against the applied shear stress of circumfluent plasma flow. The dissipative and random force terms maintain the flow local thermodynamic and mechanical properties and exchange momentum to express interactions between the platelet and the flow particles. A no-slip boundary condition was applied at the fluid-membrane surface interface. A dissipative or drag force was added to enforce the no-slip boundary condition at the fluid and membrane interface, so that the fluid particles are dragged by the dissipative forces of the membrane particles as they are getting closer to the membrane surface, mimicking boundary layer mechanism where one layer drags its adjacent layers. The hard-core L-J force simultaneously provides a bounce-back reflection of fluid particles on the membrane (to prevent fluid particles from penetrating through the platelet membrane) with the no-slip achieved by slowing down the fluid particles as the fluid particles are getting closer to the membrane surface.

Determination of the parameters for the spatial interface is conducted through numerical experiments for maintaining the Jefferys orbits of spheroids in a shear flow. The Jefferys orbits solution (Jeffery 1922) describes the rotation of isolated ellipsoidal particles immersed in viscous shear flow and is widely used as a benchmark analytical solution. Zhang et al. (2014) presented the comparison of this multiscale fluid-platelet simulation results with Jeffery’s orbit solution in Couette flows.

The shapes of the rigid and the deformable platelet models following the immersion are widely different: the rigid model maintains an ideal ellipsoidal shape, but the deformable model responds to the interactions between platelet and the hemodynamic flow stresses around it (Fig. 6(left)). The accumulation of shear stress-exposure time produced over time (Fig. 6(right)) indicated that the flow-induced stresses in the rigid platelet model are on average about 2.6 times larger than those of a deformable model. The hydrodynamic stress will be less on deformable objects, as the surface will move and deform absorbing part of the external force. Thus, the rigidity of the platelet model is not a negligible factor when estimating flow-induced stresses on the platelets. Neglecting the platelet deformability may overestimate the stress on the platelet membrane, which in turn may lead to erroneous predictions of the platelet activation under viscous shear flow conditions.
Fig. 6

DPD-CGMD spatial interfacing for a platelet cell. (left) Shapes of the rigid and the deformable platelet models following immersion in the plasma fluid. (right) Accumulation of shear stress-exposure time product (y-axis) on the rigid and deformable platelet models over time (x-axis in μs). (Reprinted with permission from Zhang et al. 2014)

2.2.4 Modeling the Highly Resolved Mechano-Transduction Process of Hemodynamic Stresses in Platelets

The DPD-CGMD multiscale model described above enables an accurate study for the mechano-transduction process of the hemodynamic stresses acting onto the platelet membrane and transmitted to such intracellular constituents as cytoplasm and cytoskeleton. The platelets constituents continuously deform in response to the flow-induced stresses. Figure 7 shows the mapping of hemodynamic stresses on the membrane, cytoplasm, and cytoskeletal structure of the flowing and flipping platelet under a shear flow of 45 dyne∕cm2. Comparatively, the cytoplasm experiences a slightly lower stress level of 51.46 dyne∕cm2, implying that the deformable membrane absorbs part of the extracellular hemodynamic forces. The spread of the stress distribution on cytoplasm has a much larger standard deviation of 34.72 than the membrane’s. This is expected, as the cytoplasm is more fluidic than the membrane, thereby affecting the spread of the stress distributions. The distribution of mechanical stresses on the actin filaments is 48.15 ± 11.33 dyne∕cm2. The actin-based filament structure experiences the smallest average stress level than the membrane and cytoplasm. The stress distribution on the actin filaments indicates that mostly the membrane-adjacent apical tips of the filaments that are attached to the membrane base experience higher stresses than the innermost parts. It is evident that simplifying the mechano-transduction process by using a completely rigid platelet model leads to erroneous estimation of the hemodynamic stresses acting on the platelet membrane and intracellular constituents.
Fig. 7

The shear flow-induced mechano-transduction process. Hemodynamic stress distributions on the membrane, cytoplasm, and cytoskeleton (actin filaments). (Reprinted with permission from Zhang et al. 2017)

2.2.5 Modeling the Activation of Platelets and Formation of Filopodia

Formation of filopodia observed during early-stage platelet activation is modeled in Pothapragada et al. (2015). Simulating the dynamics of varied filopodia formations is enabled by exploring the parameter space of this CGMD platelet model. The dynamic simulation of the filopodia formation is achieved by incrementing the length and thickness parameters of filopod. The rest of the parameters remain constant during the simulation to preserve the structural integrity of the platelet during the formation of filopodia. The change of the filopod length and thickness is determined by correlating with in vitro experiments. In experiments, the platelets are exposed to constant shear stresses over a defined duration. Figure 8 shows (a) the scanning electron microscopy (SEM) images of filopodia formations under varied combinations of shear stress and exposure time and (b) corresponding simulated formation of filopods on the deformable platelet model.
Fig. 8

Visual comparison of experimental and simulated filopodia formation. (a) SEM image after exposure to (i) 1 dyne∕cm2 for 4 min, (ii) 70 dyne∕cm2 for 4 min, and (iii) 70 dyne∕cm2 for 1 min. (b) Simulated filopodia formation. (Reprinted with permission from Pothapragada et al. 2015)

Flow-mediated platelet shape change can be induced by exposing the platelets to a range of physiologic and pathologic shear stresses encountered during blood flow. Platelets isolated by gel filtration are diluted in Hepes-buffered modified Tyrodes solution with 0.1% bovine serum albumin, recalcified with a physiological concentration of CaCl2, and exposed to shear stresses ranging from the physiologic 1 dyne∕cm2 to the pathologic 70 dyne∕cm2 for up to 4 mins in the hemodynamic shearing device (HSD), a programmable cone-plate-Couette viscometer that exposes cell suspensions to uniform dynamic shear stresses (Sheriff et al. 2013). During exposure, samples are withdrawn using a LabView-controlled syringe-capillary viscometer and immediately fixed in 1% paraformaldehyde to block additional shape change (Pothapragada et al. 2015). Fixed platelets are then placed on glass slides precoated with poly-L-lysine, dehydrated through a graded ethanol series, and sputter-coated with gold particles prior to imaging. Images are obtained at 30,000x magnification in a SEM and analyzed using a MATLAB image processing toolbox (Pothapragada et al. 2015). Platelet boundary parameters, such as the major and minor axis lengths, and circularity are measured by fitting an ellipse to the platelet perimeter, while filopod thickness is obtained by manually defining the intersection with the elliptical platelet body. Filopod lengths are then obtained by morphological erosion of the central filopod axis. These parameters are then recorded in a database and grouped by shear condition (i.e., shear stress and exposure time). The analyzed images are compared with snapshots of the simulation results in Fig. 8. The lengths of filopodia generated due to shear stresses in this study range from 0.24 to 2.74 μm and thicknesses range from 0.06 to 0.73 μm, and the simulations are able to achieve results within this range (Pothapragada et al. 2015).

2.2.6 Measuring Platelet Stiffness and Deformation with Dielectrophoresis-Mediated Electro-Deformation

A critical component of any biological model is validation through in vitro or in vivo experiments. Platelets behave as viscoelastic materials, storing energy as they are subject to shear deformation until their capacity to store and withstand this deformation is exceeded, resulting in shape change, fragmentation, and subsequent activation and thrombus formation. A variety of methods have been used to quantify single cell stiffness, including atomic force microscopy (AFM), cytoindenter, flow cytometry, magnetic tweezers, magnetic twisting cytometry, microfluidics, micropipette aspiration, microplate manipulation, molecular force spectroscopy, optical stretchers, and optical tweezers. Many of these methods require anchored, adhered, or shear-insensitive cells and are unsuitable for platelets, which are free-floating and sensitive to both shear forces and foreign surfaces. These limitations have been bypassed through the use of dielectrophoresis (DEP), in which neutral particles are exposed to a nonuniform electric field, leading to their translational motion. This method has been adapted as a means of cellular electro-deformation (EDF), in which mechanical properties are determined by trapping and deforming platelets exposed to an oscillating electric field without the need for surface contact, substratum attachment, or induction of membrane indentation or damage (Leung et al. 2015). This approach utilizes a microelectrode chip consisting of a patterned thin-film titanium-gold array on a glass substrate, covered by hollow silicone-coated polymer space for the fluid chamber and topped by a coverslip for microscopic observation. Isolated platelets placed in the fluid chamber are exposed to an electric current in a sine-wave pattern formed using a function generator coupled with a current amplifier, allowing modulation of the applied voltage. Platelet deformation is imaged via fluorescence microscopy, recorded with a CCD camera, and analyzed using NIH ImageJ software (Leung et al. 2015). Time-averaged electromagnetic force (F) on the platelet surface is calculated by integrating Maxwell’s stress tensor (T) over the cell body Ω, multiplied by the out-of-plane thickness d, with stress calculated over the platelet surface area (Chen et al. 2011):
$$\displaystyle \begin{gathered} F = d\int_{\partial\varOmega}{n \cdot T dS} \ , \end{gathered} $$
(6a)
$$\displaystyle \begin{gathered} \mathrm{\mathbf{T}} = \begin{bmatrix} \varepsilon_0 \varepsilon_r E_x^2 - \frac{1}{2} \varepsilon_0 \varepsilon_r (E_x^2 + E_y^2) & \varepsilon_0 \varepsilon_r E_x E_y \\ \varepsilon_0 \varepsilon_r E_x E_y & \varepsilon_0 \varepsilon_r E_y^2 - \frac{1}{2} \varepsilon_0 \varepsilon_r (E_x^2 + E_y^2) \end{bmatrix} \ , \end{gathered} $$
(6b)
where the electric field intensities in the x- and y-directions are given as Ex and Ey, respectively, and εr and ε0 are the respective medium and vacuum relative permittivity. An increase in voltage yields an increase in deformation that is stiffness-dependent, as verified by comparing with platelets fixed with paraformaldehyde. The magnitude of DEP forces increases by the square of the electric field strength (15–90 V), yielding a Young’s modulus of 3.5 ± 1.4 kPa for resting platelets, comparable to those measured using micropipette aspiration and AFM.

2.3 Seamless Multiscale Modeling of Thrombosis Using Dissipative Particle Dynamics

Dissipative particle dynamics, a mesoscopic particle-based hydrodynamics approach, has been effectively used to model plasma and suspending cells in blood in many applications. The advantage is the seamless integration of hydrodynamics and cell mechanics in a single framework. Further, a new submodel of DPD that is able to address chemical transport has been formulated (Li et al. 2015). The multiscale model of thrombosis includes four main processes: the hemodynamics, coagulation kinetics and transport of species, blood cell mechanics, and platelet adhesive dynamics. DPD can provide the correct hydrodynamic behavior of fluids at the mesoscale, and it has been successfully applied to study blood, where a coarse-grained representation of the cell membrane is used for red blood cells and platelets.

Transport of chemical species in the coagulation cascade is typically described by a set of continuum advection-diffusion-reaction (ADR) partial differential equations (cf. Eq. 12). In the context of Lagrangian particle-based methods such as DPD, these equations can be written as
$$\displaystyle \begin{aligned} \frac{d[C_i]}{dt}=\nabla(D_i\nabla [C_i])+Q_i^S \ , \end{aligned} $$
(7)
where [Ci] denotes the concentration of ith reactant and Di is the corresponding diffusion coefficient. This system can be modeled by transport DPD, which is developed as an extension of the classic DPD framework with extra variables for describing concentration fields. Therefore, equations of motion and concentration for each particle of mass mi are written by \(\left ( \mathrm {d}{\mathbf {r}}_{i} = {\mathbf {v}}_{i}\,\mathrm {dt} \,; \quad\mathrm {d}{\mathbf {v}}_{i} = {\mathbf {f}}_{i}/m_i\, \mathrm {dt} \,; \quad\mathrm {d}C_{i}=Q_i/m_i\,\mathrm {dt} \right )\) and integrated using a velocity-Verlet algorithm, where Ci represents the concentration of a specie per unit mass carried by a particle and \(Q_i= \sum _{i \ne j} (Q_{ij}^D+Q_{ij}^R)+Q_i^{S}\) is the corresponding concentration flux. We note that Ci can be a vector Ci containing N components, i.e., {C1, C2, …, CN}i when N chemical species are considered. In the transport DPD model, the total concentration flux accounts for the Fickian flux \(Q_{ij}^D\) and random flux \(Q_{ij}^R\), which are given by
$$\displaystyle \begin{gathered} Q_{ij}^D = -\kappa_{ij} \omega_{dc}(r_{ij}) \left(C_{i} - C_j \right) \ , \end{gathered} $$
(8a)
$$\displaystyle \begin{gathered} Q_{ij}^R = \varepsilon_{ij} \omega_{rc}(r_{ij}) \zeta _{ij} \varDelta t^{-1/2} \ , \end{gathered} $$
(8b)
where κij and εij determine the strength of the Fickian and random fluxes. \(Q_{i}^{S}\) represents the source term for the concentration due to chemical reactions. Further, the fluctuation-dissipation theorem is employed to relate the random terms to the dissipative terms, i.e., \(\omega _{dc} = \omega _{rc}^2\), and the contribution of the random flux \(Q_{ij}^R\) to the total flux is negligible.
To describe the cell mechanics, a coarse-grained representation of the cell membrane structure (lipid bilayer + cytoskeleton) can be adopted for RBCs and platelets. The membrane is defined as a set of Nv DPD particles with Cartesian coordinates xi, i ∈ 1, …, Nv in a two-dimensional triangulated network created by connecting the particles with wormlike chain (WLC) bonds. The free energy of the system is given by
$$\displaystyle \begin{aligned} V_t = V_s + V_b + V_a + V_v \, , \end{aligned} $$
(9)
where Vs is the stored elastic energy associated with the bonds, Vb is the bending energy, and Va and Vv are the energies due to cell surface area and volume constraints, respectively. RBCs are extremely deformable under shear, whereas platelets in their resting (i.e., unactivated) shape are nearly rigid. We parametrize the RBC membrane such that we can reproduce the optical tweezer stretching tests (see Fig. 9). In the case of platelets, we choose shear modulus and bending rigidity sufficiently large to ensure its rigid behavior. Platelets become more deformable upon activation through reordering the actin network in their membrane, which is accompanied by the release of their content to the plasma.
Fig. 9

Particle representation of blood cell structure and their mechanics and simulation of clotting in whole blood. (a) Coarse-grained DPD model of RBC membrane and the stretching response of stress-free RBC model for different coarse-graining levels compared with the optical tweezer stretching experiment. (b) Coarse-grained model of platelet membrane and its flipping dynamics when placed in a linear shear flow (theory of rigid particle dynamics in shear flow can be found in Jeffery 1922). (c) Seamless DPD simulation of whole blood flowing over a site of injury in a channel showing RBCs, platelets, and vWF ligands (green particles on the lower wall). Contours of thrombin concentration are plotted as the solution of the coupled ADR equations for the coagulation cascade

The platelet adhesive dynamics model describes the adhesive dynamics of receptors on the platelet membrane binding to their ligands. The cell adhesive dynamics model was first formulated in the work of Hammer and Apte (1992) and extended to platelets by others. This model utilizes a Monte Carlo method to determine each bond formation/dissociation event based on specific receptor-ligand binding kinetics. The probability of bond formation Pf and probability of bond rupture Pr are estimated using the following equations:
$$\displaystyle \begin{aligned} P_f &= 1 - \mathrm{exp}(-k_f \varDelta t) \, , \end{aligned} $$
(10a)
$$\displaystyle \begin{aligned} P_r &= 1 - \mathrm{exp}(-k_r \varDelta t) \, , \end{aligned} $$
(10b)
where kf and kr are the rates of formation and dissociation, respectively. Following the Bell model (cf. Bell 1978), the force-dependent kr is evaluated by
$$\displaystyle \begin{aligned} k_r = k_r^0 \ \mathrm{exp} \left( \frac{\sigma_r |F_b|}{k_BT} \right) \, , \end{aligned} $$
(11)
where \(k_r^0\) is the intrinsic dissociation rate and σr is the corresponding reactive compliance derived from atomic force microscope (AFM) experiments and Fb is the pulling force applied to the bond.

2.4 Continuum and Particle/Continuum Modeling of Thrombosis

At the continuum level, macroscopic numerical models have been developed by treating blood as incompressible Newtonian fluid (or non-Newtonian in small arterioles and capillaries), thus leading to continuum fields for blood velocity and pressure and the transport of enzymes, which can be resolved using an Eulerian description. To study thrombosis in both physiological and pathological conditions, some of these models treat platelets as concentration fields similar to chemical species that follow specific ADR transport equations. The governing equations for the incompressible blood flow coupled with the ADR equations related to the biochemistry of coagulation are written as
$$\displaystyle \begin{gathered} \rho (\frac{\partial \textbf{{u}}}{\partial t}+\textbf{{u}} \cdot \nabla \textbf{{u}}) = -\nabla p + \nabla\cdot{\boldsymbol{\tau}} \, , \end{gathered} $$
(12a)
$$\displaystyle \begin{gathered} \nabla \cdot \textbf{{u}} = 0 \, , \end{gathered} $$
(12b)
$$\displaystyle \begin{gathered} \frac{\partial c_i}{\partial t} + \textbf{{u}} \cdot \nabla c_i = D_i \nabla^2 c_i + S_i \ , \end{gathered} $$
(12c)
where u is the flow velocity, p is the pressure and τ is the deviatoric viscous stress tensor, ci and Di are the concentration and diffusion coefficient for each reactant, respectively, and Si represents the rate of production or destruction of that reactant. The shear stress tensor can be written as τ = 2μD where D is the strain rate tensor. The viscosity μ may be assumed constant for Newtonian blood or a function of the local shear rate using an empirical correlation for non-Newtonian blood. The system of equations (12) are usually solved using a finite element Galerkin formulation subject to proper Dirichlet and Neumann boundary conditions for velocity, pressure, and concentrations at the inlets/outlets and on the wall.
It is also possible to take individual platelets into account by treating them as Lagrangian particles using a method that is able to couple particle-particle particle-wall interactions with the background flow. Force coupling method (FCM) provides such platform for two-way coupling of platelets (treated as rigid spherical particles) with the background flow (cf. Yazdani et al. 2017). This numerical approach has the advantage of tracking thousands of platelets forming aggregates at the site of injury and effectively capturing the shape and extent of thrombus. As a result, the thrombus shape modeled by FCM is affected by the local hydrodynamics and fluid stresses. FCM can be easily incorporated in the Navier-Stokes equation by introducing a body-force term due to the particles
$$\displaystyle \begin{aligned} \mathbf{{f}}(\mathbf{{x}},t) = \sum_{n=1}^N \mathbf{{F}}^n \ \varDelta\left( \mathbf{{x}}-\mathbf{{Y}}\right) \ , \end{aligned} $$
(13)
where Fn is the force due to particle n. The effect of platelets is transferred to the flow field through the body force term f(x, t) on the right-hand side of the Navier-Stokes Equation 12a. Further, a phenomenological model based on a Morse potential VMorse may be used to describe the attractive/repulsive forces between platelets. The contribution of each platelet whose center of mass is located at Yn to the flow at position x is smoothed by a Gaussian distribution kernel \(\varDelta \left ( \mathbf {{x}}-\mathbf {{Y}}\right )\).

3 Concurrent Coupling Using a Domain Decomposition Approach

Most current work on multiscale modeling is in the setting of the so-called concurrent coupling methods, i.e., the microscale and the macroscale models are linked together on the fly as the computation goes on (Weinan et al. 2007). In most biological problems such as blood clotting, localized processes occur in isolated regions. For these problems, the microscopic model is only necessary near the local biologically active domains. Further away it is adequate to use the macroscopic model. In this case, the macro-micro coupling is localized, and domain decomposition can be used by either using overlapping interfaces or embedded subdomains. For example, in modeling of blood clotting in cerebral aneurysm shown in Fig. 10, an approach is adopted in which the flow field is solved in the full geometry using a continuum 3D solver, while small atomistic subdomains are embedded in isolated regions identified with higher platelet residence times and higher propensity of platelet deposition inside the sac of the aneurysm.
Fig. 10

Concurrent coupling of continuum-atomistic modeling of platelet aggregation at the wall of an aneurysm. (a) The patient-specific geometry was constructed from MRI imaging and contains three embedded atomistic domains to resolve the microrheology. Continuum blood flow is computed with \(\mathcal {O}(0.1\,\textit {mm})\) resolution and platelet aggregation with \(\mathcal {O}(1\,\mu {\textit {m}})\) resolution. (b) Asynchronous time-matching and synchronization of the continuum and atomistic (based on DPD) solvers. (c) Platelet aggregation at the wall of the aneurysm. Two atomistic domains ΩA1,A2 in which the platelet aggregation initiates are fully embedded in the continuum domain. Yellow dots correspond to active platelets and red dots to resting platelets (both modeled by hard spheres in DPD). Streamlines depict instantaneous velocity field in the (i) onset of clot formation and in (ii) clot formation as it progresses in time and space. (Reprinted with permission from Grinberg et al. 2013)

The bridging of scales is achieved through designing appropriate interface conditions between the heterogeneous solvers. The interface conditions are derived to respect the physical conservation constraints and the requirements of each solver. Specifically, mass conservation is of primary importance, while momentum and energy conservation are desirable but present several challenges, with the latter being nearly impossible in practice. Interface conditions are concurrently enforced by constructing appropriate projection, interpolation, denoising, and statistical moment-matching techniques that aim to transfer information across the different domains in a compatible way with the requirements and data format of each solver.

Coupling of atomistic and continuum solvers requires the calculation and communication of averaged properties, such as fluid velocity and density, across heterogeneous solver interfaces. For example, atomistic solvers require a local velocity flux to be imposed at each voxel of the atomistic domain. This is achieved through constructing appropriate interpolation and projection operators that are capable of mapping the continuum velocity field onto the atomistic domain. Specifically, to enforce mass conservation, the continuum solver computes the fluxes through the surface interfaces with the atomistic domains, and particles are inserted in the atomistic domain in such a way that these fluxes are preserved.

Accurate computing of averaged fields through processing of non-stationary atomistic data presents several challenges including geometrical complexity of the atomistic domains, thermal fluctuations, and flow unsteadiness. Therefore, effective averaging and filtering techniques are required before the data are transmitted to the continuum domains. In particular, to filter out the stochastic component (thermal fluctuations) at all interfaces with the continuum domain, a window proper orthogonal decomposition (WPOD) has been used and shown to be very effective (Grinberg et al. 2014).

4 Multiple Time-Stepping (MTS) Algorithm for Efficient Multiscale Modeling

The disparate spatial scales between the DPD flow and the CGMD platelet model are handled by a hybrid force field. However, the disparity in temporal scales between the two models represents a huge challenge that may become prohibitive in the practice of multiscale modeling. A multiple time-stepping (MTS) algorithm is the necessary solution for this kind of multiscale models. Conventional MTS algorithms manage to differentiate multi-stepping for up to one order of magnitude of scale (Han et al. 2007) and are being applied to classical MD simulations. In order to handle three to four orders of magnitude disparity in the temporal scales between DPD and CGMD, a novel improved MTS scheme hybridizing DPD and CGMD is developed through utilizing four different time-stepping sizes. In this MTS scheme, the DPD-based fluid system advances at the largest time step, the fluid-platelet interface at a middle time step size, and the non-bonded and bonded potentials for the CGMD-based platelet system at two smallest time step sizes (Zhang et al. 2015).

MTS for top-scale DPD uses a modified velocity-Verlet integrator that is derived from stochastic Trotter formula and is written as
$$\displaystyle \begin{gathered} r(\varDelta t) = r(0) + \varDelta t \cdot v(0) + \frac{\varDelta t^2}{2m} F[r(0), v(0)] \, , \end{gathered} $$
(14a)
$$\displaystyle \begin{gathered} \tilde{v} (\varDelta t) = v(0) + \lambda \cdot \frac{\varDelta t}{m} \cdot F[r(0), v(0)] \, , \end{gathered} $$
(14b)
$$\displaystyle \begin{gathered} \mathrm{compute} \quad F [ r(\varDelta t), \tilde{v} (\varDelta t) ] \, , \end{gathered} $$
(14c)
$$\displaystyle \begin{gathered} \tilde{v} (\varDelta t) = v(0) + \lambda \cdot \frac{\varDelta t}{m} \cdot \big( F[r(0), v(0)] + F[r(\varDelta t), \tilde{v}(\varDelta t)] \big) \ , \end{gathered} $$
(14d)
where the empirical factor λ accounts for the additive effects of stochastic interactions and λ = 0.5 restores the velocity-Verlet integrator with \(\mathcal {O}(\varDelta t^2)\). MTS for the bottom-scale CGMD subdivides the pairwise forces into short- and long-ranged forces with different time steps δt and nδt where n is a positive integer. It uses the velocity-Verlet integrator as follows:
$$\displaystyle \begin{gathered} v\big(\frac{\varDelta t}{2}\big) = v(0) + \frac{\varDelta t}{2m} \cdot F[r(0)] \, , \end{gathered} $$
(15a)
$$\displaystyle \begin{gathered} r(\varDelta t) = r(0) + \varDelta t \cdot v \big( \frac{\varDelta t}{2} \big) \, , \end{gathered} $$
(15b)
$$\displaystyle \begin{gathered} v(\varDelta t) = v \big( \frac{\varDelta t}{2} \big) + \frac{\varDelta t}{2m} \cdot F[r(\varDelta t)] \ . \end{gathered} $$
(15c)
DPD-CGMD temporal interfacing scheme decomposes the whole integrator process into four levels as shown in Fig. 11. The topmost two levels use the scheme in Eq. 14, referred to as DPD-MTS, because both of them employ the DPD thermostatting method. The bottommost two levels use the scheme in Eq. 15, referred to as CGMD-MTS, since both of them employ the conservative potentials. In each of DPD-MTS and CGMD-MTS, the integrator is subdivided into two timescales, one for the soft potential with a larger step size and the other for the hard potential with a smaller step size.
Fig. 11

Multiple time step sizes in MTS for multiscale DPD-CGMD model. (Reprinted with permission from Zhang et al. 2015)

MTS algorithm is a trade-off between speed and accuracy for multiscale modeling. Energy conservation and maintenance of adequate precisions of other measures must be verified while accelerating computations. We investigate the microscopic shape changes of platelets in response to macroscopic flow-induced stresses. Thus, in measuring numerical solutions, we focus on the accuracy of characterizing the hybrid system and the flowing platelets, as well as the accuracy of calculating the dynamic flow-induced stresses on the surface membrane. To compare, the standard time stepping (STS) that operates a single time step size is examined against MTS. The numerical results demonstrated that the microscopic measures for single platelets are more sensitive to the MTS parameters than the macroscopic measures for the hybrid system. The results reaffirm that a separation of temporal scales in MTS considerably improves the efficiency of utilizing parallel computing resources, as compared to conventional single-scale methods in which considerable time is wasted conducting massive unnecessary computations. For example, completing 1-ms multiscale simulation of a 10-million particle system is reduced from 2.6 years to 3.5 days only (Zhang et al. 2015, 2016).

5 Comments on Long-Term Modeling of Thrombus Formation

Any proposed multiscale numerical model for thrombus formation has to account for short-term platelet aggregation, midterm thrombus growth, and long-term clot stabilization and remodeling (see Fig. 12). The physiologic timescales of blood coagulation dynamics and thrombosis are in the order of minutes, hence making long-time CGMD or DPD computations of such systems at the cellular level impractical. In addition, resorting to coarse-scale simulations at the macroscale may not necessarily resolve this difficulty as acceleration in the chemical reactions in the coagulation model is needed. Despite the importance of temporal scale bridging in biological applications, there has been less attention than spatial scale coupling in the literature.
Fig. 12

Serial/concurrent coupling of numerical models of thrombus formation and growth/remodeling in a dissecting aneurysm. The platelet activation/aggregation model, based on FCM, is the most expensive subprocess in this framework and, thus, is only used to model the formation and propagation of the clot for several seconds after the injury. Once a stable thrombus is formed, we transform the platelets Lagrangian distribution in the aggregate clusters into continuum fields of clot volume fraction, which can be subsequently used in a phase-field numerical approach to model interaction of blood with thrombus and its further remodeling in a concurrent scheme

In contrast to the concurrent coupling, serial coupling schemes determine an effective macroscale model from the microscale model in a preprocessing step and use the resulted macroscale model subsequently. Such serial coupling methods are largely limited to parameter passing yet are very useful in addressing long-term biological processes provided there is a timescale separation between its relevant subprocesses. For example, in Fig. 12, a possible strategy to tackle the long-term simulation of clot formation and remodeling upon arterial wall injury that normally occurs over days is proposed. Here, the subprocesses are identified as (i) the acute formation of thrombus and platelet aggregation over the first tens of seconds, (ii) the midterm formation of fibrin network at the site of thrombus and stabilization of the clot, and (iii) the long-term remodeling of the clot, which occurs over days. As noted in the figure, due to the separation of timescales between phases (i) and (ii), a serial coupling between the fine-scale particle/continuum (based on FCM) and the coarse-scale multiphasic continuum models is possible. The challenge is the long-term simulation of particles actively coupled with flow; the complexities in geometry and flow conditions and the large size of injuries may require hundreds of thousands of FCM particles to represent platelets, which imposes a restrictively high computational cost for such simulations in large domains. To overcome this difficulty, it is possible to initially distribute particles in the regions of interest that contain thrombogenic surfaces. The timescales of phases (ii) and (iii) are not separated, which implicates a concurrent coupling scheme between these two subprocesses.

Notes

Acknowledgements

This work was supported by NIH grants U01HL116323 (Yazdani, A.), NHLBI R21HL096930-01, NIBIB Quantum U01EB012487, and NHLBI U01HL131052 (Bluestein, D.) and XSEDE grants DMS140019, DMS150011 (Zhang, P.), and DMS140007 (Yazdani, A.).

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Alireza Yazdani
    • 1
  • Peng Zhang
    • 2
    Email author
  • Jawaad Sheriff
    • 2
  • Marvin J. Slepian
    • 4
  • Yuefan Deng
    • 3
  • Danny Bluestein
    • 2
  1. 1.Division of Applied MathematicsBrown UniversityProvidenceUSA
  2. 2.Department of Biomedical EngineeringStony Brook UniversityStony BrookUSA
  3. 3.Department of Applied Mathematics and StatisticsStony Brook UniversityStony BrookUSA
  4. 4.Department of Medicine, Biomedical Engineering DepartmentUniversity of ArizonaTucsonUSA

Section editors and affiliations

  • Ming Dao
    • 1
  • George E Karniadakis
    • 2
  1. 1.Department of Materials Science and EngineeringMassachusetts Institute of TechnologyCambridgeUSA
  2. 2.Division of Applied MathematicsBrown UniversityProvidenceUSA

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