A computational investigation of occlusive arterial thrombosis

The generation of occlusive thrombi in stenotic arteries involves the rapid deposition of millions of circulating platelets under high shear flow. The process is mediated by the formation of molecular bonds of several distinct types between platelets; the bonds capture the moving platelets and stabilize the growing thrombi under flow. We investigated the mechanisms behind occlusive thrombosis in arteries with a two-phase continuum model. The model explicitly tracks the formation and rupture of the two types of interplatelet bonds, the rates of which are coupled with the local flow conditions. The motion of platelets in the thrombi results from competition between the viscoelastic forces generated by the interplatelet bonds and the fluid drag. Our simulation results indicate that stable occlusive thrombi form only under specific combinations for the ranges of model parameters such as rates of bond formation and rupture, platelet activation time, and number of bonds required for platelet attachment.

A model of fluid-structure and biochemical interactions for applications to subclinical leaflet thrombosis

Subclinical leaflet thrombosis (SLT) is a potentially serious complication of aortic valve replacement with a bioprosthetic valve in which blood clots form on the replacement valve. SLT is associated with increased risk of transient ischemic attacks and strokes and can progress to clinical leaflet thrombosis. SLT following aortic valve replacement also may be related to subsequent structural valve deterioration, which can impair the durability of the valve replacement. Because of the difficulty in clinical imaging of SLT, models are needed to determine the mechanisms of SLT and could eventually predict which patients will develop SLT. To this end, we develop methods to simulate leaflet thrombosis that combine fluid-structure interaction and a simplified thrombosis model that allows for deposition along the moving leaflets. Additionally, this model can be adapted to model deposition or absorption along other moving boundaries. We present convergence results and quantify the model's ability to realize changes in valve opening and pressures. These new approaches are an important advancement in our tools for modeling thrombosis because they incorporate both adhesion to the surface of the moving leaflets and feedback to the fluid-structure interaction.

Towards understanding the effect of fibrinogen interactions on fibrin gel structure

Fibrin gelation involves the enzymatic conversion of the plasma protein fibrinogen to fibrin monomers which then polymerize to form the gel that is a major structural component of a blood clot. Because fibrinogen provides the material from which fibrin is made, it is generally regarded as promoting the gelation process. However, fibrinogen can bind to a site on a fibrin oligomer, preventing another fibrin oligomer from binding there, thus slowing the polymerization process. "Soluble fibrin oligomers," which are mixtures of fibrin and fibrinogen, are found in the blood plasma and serve as biomarkers for various clotting disorders, so understanding the interplay between fibrin and fibrinogen during fibrin polymerization may have medical importance. We present a kinetic gelation model of fibrin polymerization which accounts for the dual and antagonistic roles of fibrinogen. It builds on our earlier model of fibrin polymerization that proposed a novel mechanism for branch formation, which is a necessary component of gelation. This previous model captured salient experimental observations regarding the determinants of the structure of the gel, but did not include fibrinogen binding. Here, we add to that model reactions between fibrinogen and fibrin, so oligomers are now mixtures of fibrin and fibrinogen, and characterizing their dynamics leads to equations of substantially greater complexity than previously. Using a moment generating function approach, we derive a closed system of moment equations and we track their dynamics until the finite time blow-up of specific second moments indicates that a gel has formed. In simulations begun with an initial mixture of fibrin and fibrinogen monomers, a sufficiently high relative concentration of fibrinogen prevents gelation; the critical concentration increases with the branch formation rate. In simulations begun with only fibrinogen monomers that are converted to fibrin at a specified rate, the rates of conversion, fibrinogen binding to oligomers, and branch formation together determine whether a gel forms, how long it takes to form, and the structural properties of the gel that results.

Inhibition of platelet-surface-bound proteins during coagulation under flow II: Antithrombin and heparin

Blood coagulation is a self-repair process regulated by activated platelet surfaces, clotting factors, and inhibitors. Antithrombin (AT) is one such inhibitor that impedes coagulation by targeting and inactivating several key coagulation enzymes. The effect of AT is greatly enhanced in the presence of heparin, a common anticoagulant drug. When heparin binds to AT, it either bridges with the target enzyme or induces allosteric changes in AT leading to more favorable binding with the target enzyme. AT inhibition of fluid-phase enzymes caused little suppression of thrombin generation in our previous mathematical models of blood coagulation under flow. This is because in that model, flow itself was a greater inhibitor of the fluid-phase enzymes than AT. From clinical observations, it is clear that AT and heparin should have strong inhibitory effects on thrombin generation, and thus we hypothesized that AT could be inhibiting enzymes bound to activated platelet surfaces that are not subject to being washed away by flow. We extended our mathematical model to include the relevant reactions of AT inhibition at the activated platelet surfaces as well as those for unfractionated heparin and a low molecular weight heparin. Our results show that AT alone is only an effective inhibitor at low tissue factor densities, but in the presence of heparin, it can greatly alter, and in some cases shut down, thrombin generation. Additionally, we studied each target enzyme separately and found that inactivation of no single enzyme could substantially suppress thrombin generation.

Inhibition of platelet-surface-bound proteins during coagulation under flow I: TFPI

Blood coagulation is a self-repair process regulated by activated platelet surfaces, clotting factors, and inhibitors. Tissue factor pathway inhibitor (TFPI) is one such inhibitor, well known for its inhibitory action on the active enzyme complex comprising tissue factor (TF) and activated clotting factor VII. This complex forms when TF embedded in the blood vessel wall is exposed by injury and initiates coagulation. A different role for TFPI, independent of TF:VIIa, has recently been discovered whereby TFPI binds a partially cleaved form of clotting factor V (FV-h) and impedes thrombin generation on activated platelet surfaces. We hypothesized that this TF-independent inhibitory mechanism on platelet surfaces would be a more effective platform for TFPI than the TF-dependent one. We examined the effects of this mechanism on thrombin generation by including the relevant biochemical reactions into our previously validated mathematical model. Additionally, we included the ability of TFPI to bind directly to and inhibit platelet-bound FXa. The new model was sensitive to TFPI levels and, under some conditions, TFPI could completely shut down thrombin generation. This sensitivity was due entirely to the surface-mediated inhibitory reactions. The addition of the new TFPI reactions increased the threshold level of TF needed to elicit a strong thrombin response under flow, but the concentration of thrombin achieved, if there was a response, was unchanged. Interestingly, we found that direct binding of TFPI to platelet-bound FXa had a greater anticoagulant effect than did TFPI binding to FV-h alone, but that the greatest effects occurred if both reactions were at play. The model includes activated platelets' release of FV species, and we explored the impact of varying the FV/FV-h composition of the releasate. We found that reducing the zymogen FV fraction of this pool, and thus increasing the fraction that is FV-h, led to acceleration of thrombin generation.

DEVELOPMENT OF FIBRIN BRANCH STRUCTURE BEFORE AND AFTER GELATION

In [Fogelson and Keener, Phys. Rev. E, 81 (2010), 051922], we introduced a kinetic model of fibrin polymerization during blood clotting that captured salient experimental observations about how the gel branching structure depends on the conditions under which the polymerization occurs. Our analysis there used a moment-based approach that is valid only before the finite time blow-up that indicates formation of a gel. Here, we extend our analyses of the model to include both pre-gel and post-gel dynamics using the PDE-based framework we introduced in [Fogelson and Keener, SIAM J. Appl. Math., 75 (2015), pp. 1346-1368]. We also extend the model to include spatial heterogeneity and spatial transport processes. Studies of the behavior of the model reveal different spatial-temporal dynamics as the time scales of the key processes of branch formation, monomer introduction, and diffusion are varied.

A Hybrid Semi-Lagrangian Cut Cell Method for Advection-Diffusion Problems with Robin Boundary Conditions in Moving Domains

We present a new discretization approach to advection-diffusion problems with Robin boundary conditions on complex, time-dependent domains. The method is based on second order cut cell finite volume methods introduced by Bochkov et al. [8] to discretize the Laplace operator and Robin boundary condition. To overcome the small cell problem, we use a splitting scheme along with a semi-Lagrangian method to treat advection. We demonstrate second order accuracy in the L ¹, L ², and L ^∞ norms for both analytic test problems and numerical convergence studies. We also demonstrate the ability of the scheme to convert one chemical species to another across a moving boundary.

Modeling and Simulation of the Ion-Binding-Mediated Swelling Dynamics of Mucin-like Polyelectrolyte Gels

Volume phase transitions in polyeletrolyte gels play important roles in many biophysical processes such as DNA packaging, nerve excitation, and cellular secretion. The swelling and deswelling of these charged polymer gels depend strongly on their ionic environment. In this paper, we present an extension to our previous two-fluid model for ion-binding-mediated gel swelling. The extended model eliminates the assumptions about the size similarity between the network and solvent particles, which makes it suitable for investigating of a large family of biologically relevant problems. The model treats the polyeletrolyte gel as a mixture of two materials, the network and the solvent. The dynamics of gel swelling is governed by the balance between the mechanical and chemical forces on each of these two materials. Simulations based on the model illustrate that the chemical forces are significantly influenced by the binding/unbinding reactions between the ions and the network, as well as the resulting distribution of charges within the gel. The dependence of the swelling rate on ionic bath concentrations is analyzed and this analysis highlights the importance of the electromigration of ions and the induced electric field in regulating gel swelling.

An Efficient High-Order Meshless Method for Advection-Diffusion Equations on Time-Varying Irregular Domains

We present a high-order radial basis function finite difference (RBF-FD) framework for the solution of advection-diffusion equations on time-varying domains. Our framework is based on a generalization of the recently developed Overlapped RBF-FD method that utilizes a novel automatic procedure for computing RBF-FD weights on stencils in variable-sized regions around stencil centers. This procedure eliminates the overlap parameter δ, thereby enabling tuning-free assembly of RBF-FD differentiation matrices on moving domains. In addition, our framework utilizes a simple and efficient procedure for updating differentiation matrices on moving domains tiled by node sets of time-varying cardinality. Finally, advection-diffusion in time-varying domains is handled through a combination of rapid node set modification, a new high-order semi-Lagrangian method that utilizes the new tuning-free overlapped RBF-FD method, and a high-order time-integration method. The resulting framework has no tuning parameters and has O(N logN) time complexity. We demonstrate high-orders of convergence for advection-diffusion equations on time-varying 2D and 3D domains for both small and large Peclet numbers. We also present timings that verify our complexity estimates. Finally, we utilize our method to solve a coupled 3D problem motivated by models of platelet aggregation and coagulation, once again demonstrating high-order convergence rates on a moving domain.

Pump efficacy in a two-dimensional, fluid-structure interaction model of a chain of contracting lymphangions

The transport of lymph through the lymphatic vasculature is the mechanism for returning excess interstitial fluid to the circulatory system, and it is essential for fluid homeostasis. Collecting lymphatic vessels comprise a significant portion of the lymphatic vasculature and are divided by valves into contractile segments known as lymphangions. Despite its importance, lymphatic transport in collecting vessels is not well understood. We present a computational model to study lymph flow through chains of valved, contracting lymphangions. We used the Navier-Stokes equations to model the fluid flow and the immersed boundary method to handle the two-way, fluid-structure interaction in 2D, non-axisymmetric simulations. We used our model to evaluate the effects of chain length, contraction style, and adverse axial pressure difference (AAPD) on cycle-mean flow rates (CMFRs). In the model, longer lymphangion chains generally yield larger CMFRs, and they fail to generate positive CMFRs at higher AAPDs than shorter chains. Simultaneously contracting pumps generate the largest CMFRs at nearly every AAPD and for every chain length. Due to the contraction timing and valve dynamics, non-simultaneous pumps generate lower CMFRs than the simultaneous pumps; the discrepancy diminishes as the AAPD increases. Valve dynamics vary with the contraction style and exhibit hysteretic opening and closing behaviors. Our model provides insight into how contraction propagation affects flow rates and transport through a lymphangion chain.

The Art and Science of Building a Computational Model to Understand Hemostasis

Computational models of various facets of hemostasis and thrombosis have increased substantially in the last decade. These models have the potential to make predictions that can uncover new mechanisms within the complex dynamics of thrombus formation. However, these predictions are only as good as the data and assumptions they are built upon, and therefore model building requires intimate coupling with experiments. The objective of this article is to guide the reader through how a computational model is built and how it can inform and be refined by experiments. This is accomplished by answering six questions facing the model builder: (1) Why make a model? (2) What kind of model should be built? (3) How is the model built? (4) Is the model a “good” model? (5) Do we believe the model? (6) Is the model useful? These questions are answered in the context of a model of thrombus formation that has been successfully applied to understanding the interplay between blood flow, platelet deposition, and coagulation and in identifying potential modifiers of thrombin generation in hemophilia A.

A MATHEMATICAL MODEL OF PLATELET AGGREGATION IN AN EXTRAVASCULAR INJURY UNDER FLOW

We present the first mathematical model of flow-mediated primary hemostasis in an extravascular injury which can track the process from initial deposition to occlusion. The model consists of a system of ordinary differential equations (ODEs) that describe platelet aggregation (adhesion and cohesion), soluble-agonist-dependent platelet activation, and the flow of blood through the injury. The formation of platelet aggregates increases resistance to flow through the injury, which is modeled using the Stokes-Brinkman equations. Data from analogous experimental (microfluidic flow) and partial differential equation models informed parameter values used in the ODE model description of platelet adhesion, cohesion, and activation. This model predicts injury occlusion under a range of flow and platelet activation conditions. Simulations testing the effects of shear and activation rates resulted in delayed occlusion and aggregate heterogeneity. These results validate our hypothesis that flow-mediated dilution of activating chemical adenosine diphosphate hinders aggregate development. This novel modeling framework can be extended to include more mechanisms of platelet activation as well as the addition of the biochemical reactions of coagulation, resulting in a computationally efficient high throughput screening tool of primary and secondary hemostasis.

Computationally Driven Discovery in Coagulation

Bleeding frequency and severity within clinical categories of hemophilia A are highly variable and the origin of this variation is unknown. Solving this mystery in coagulation requires the generation and analysis of large data sets comprised of experimental outputs or patient samples, both of which are subject to limited availability. In this review, we describe how a computationally driven approach bypasses such limitations by generating large synthetic patient data sets. These data sets were created with a mechanistic mathematical model, by varying the model inputs, clotting factor, and inhibitor concentrations, within normal physiological ranges. Specific mathematical metrics were chosen from the model output, used as a surrogate measure for bleeding severity, and statistically analyzed for further exploration and hypothesis generation. We highlight results from our recent study that employed this computationally driven approach to identify FV (factor V) as a key modifier of thrombin generation in mild to moderate hemophilia A, which was confirmed with complementary experimental assays. The mathematical model was used further to propose a potential mechanism for these observations whereby thrombin generation is rescued in FVIII-deficient plasma due to reduced substrate competition between FV and FVIII for FXa (activated factor X).

Clot Permeability, Agonist Transport, and Platelet Binding Kinetics in Arterial Thrombosis

The formation of wall-adherent platelet aggregates is a critical process in arterial thrombosis. A growing aggregate experiences frictional drag forces exerted on it by fluid moving over or through the aggregate. The magnitude of these forces is strongly influenced by the permeability of the developing aggregate; the permeability depends on the aggregate's porosity. Aggregation is mediated by formation of ensembles of molecular bonds; each bond involves a plasma protein bridging the gap between specific receptors on the surfaces of two different platelets. The ability of the bonds existing at any time to sustain the drag forces on the aggregate determines whether it remains intact or sheds individual platelets or larger fragments (emboli). We investigate platelet aggregation in coronary-sized arteries using both computational simulations and in vitro experiments. The computational model tracks the formation and breaking of bonds between platelets and treats the thrombus as an evolving porous, viscoelastic material, which moves differently from the background fluid. This relative motion generates drag forces which the fluid and thrombus exert on one another. These forces are computed from a permeability-porosity relation parameterized from experimental measurements. Basing this relation on measurements from occlusive thrombi formed in our flow chamber experiments, along with other physiological parameter values, the model produced stable dense thrombi on a similar timescale to the experiments. When we parameterized the permeability-porosity relation using lower permeabilities reported by others, bond formation was insufficient to balance drag forces on an early thrombus and keep it intact. Under high shear flow, soluble agonist released by platelets was limited to the thrombus and a boundary layer downstream, thus restricting thrombus growth into the vessel lumen. Adding to the model binding and activation of unactivated platelets through von Willebrand-factor-mediated processes allowed greater growth and made agonist-induced activation more effective.

A mathematical model of coagulation under flow identifies factor V as a modifier of thrombin generation in hemophilia A

BACKGROUND

The variability in bleeding patterns among individuals with hemophilia A, who have similar factor VIII (FVIII) levels, is significant and the origins are unknown.

OBJECTIVE

To use a previously validated mathematical model of flow-mediated coagulation as a screening tool to identify parameters that are most likely to enhance thrombin generation in the context of FVIII deficiency.

METHODS

We performed a global sensitivity analysis (GSA) on our mathematical model to identify potential modifiers of thrombin generation. Candidates from the GSA were confirmed by calibrated automated thrombography (CAT) and flow assays on collagen-tissue factor (TF) surfaces at a shear rate of 100 per second.

RESULTS

Simulations identified low-normal factor V (FV) (50%) as the strongest modifier, with additional thrombin enhancement when combined with high-normal prothrombin (150%). Low-normal FV levels or partial FV inhibition (60% activity) augmented thrombin generation in FVIII-inhibited or FVIII-deficient plasma in CAT. Partial FV inhibition (60%) boosted fibrin deposition in flow assays performed with whole blood from individuals with mild and moderate FVIII deficiencies. These effects were amplified by high-normal prothrombin levels in both experimental models.

CONCLUSIONS

These results show that low-normal FV levels can enhance thrombin generation in hemophilia A. Further explorations with the mathematical model suggest a potential mechanism: lowering FV reduces competition between FV and FVIII for factor Xa (FXa) on activated platelet surfaces (APS), which enhances FVIII activation and rescues thrombin generation in FVIII-deficient blood.

Kinetic model of two-monomer polymerization

We propose a kinetic gelation model of polymer growth with two monomeric types that have distinct functionalities (reaction sites), and can polymerize using different reaction types. The heterotypic aggregation of two monomer types is modeled using a moment generating function approach by tracking the temporal evolution of a closed system of moment equations up until gelation. We investigate several scenarios of polymerization with two distinct monomers that differ in the types of reactions that can occur. We determine numerical and analytical conditions for finite time blow-up (the emergence of an oligomer of infinite size) that depend on initial conditions, reaction rates, and number of reaction sites per monomer.

Hyperviscosity-Based Stabilization for Radial Basis Function-Finite Difference (RBF-FD) Discretizations of Advection-Diffusion Equations

We present a novel hyperviscosity formulation for stabilizing RBF-FD discretizations of the advectiondiffusion equation. The amount of hyperviscosity is determined quasi-analytically for commonly-used explicit, implicit, and implicit-explicit (IMEX) time integrators by using a simple 1D semi-discrete Von Neumann analysis. The analysis is applied to an analytical model of spurious growth in RBF-FD solutions that uses auxiliary differential operators mimicking the undesirable properties of RBF-FD differentiation matrices. The resulting hyperviscosity formulation is a generalization of existing ones in the literature, but is free of any tuning parameters and can be computed efficiently. To further improve robustness, we introduce a simple new scaling law for polynomial-augmented RBF-FD that relates the degree of polyharmonic spline (PHS) RBFs to the degree of the appended polynomial. When used in a novel ghost node formulation in conjunction with the recently-developed overlapped RBF-FD method, the resulting method is robust and free of stagnation errors. We validate the high-order convergence rates of our method on 2D and 3D test cases over a wide range of Peclet numbers (1-1000). We then use our method to solve a 3D coupled problem motivated by models of platelet aggregation and coagulation, again demonstrating high-order convergence rates.

Electrodiffusion-Mediated Swelling of a Two-Phase Gel Model of Gastric Mucus

Gastric mucus gel is known to exhibit dramatic and unique swelling behaviors in response to the ionic composition of the hydrating solution. This swelling behavior is important in the maintenance of the mucus layer lining the stomach wall, as the layer is constantly digested by enzymes in the lumen, and must be replenished by new mucus that swells as it is secreted from the gastric wall. One hypothesis suggests that the condensed state of mucus at secretion is maintained by transient bonds with calcium that form crosslinks. These crosslinks are lost as monovalent cations from the environment displace divalent crosslinkers, leading to a dramatic change in the energy of the gel and inducing the swelling behavior. Previous modeling work has characterized the equilibrium behavior of polyelectrolyte gels that respond to calcium crosslinking. Here, we present an investigation of the dynamic swelling behavior of a polyelectrolytic gel model of mucus. In particular, we quantified the rate at which a globule of initially crosslinked gel swells when exposed to an ionic bath. The dependence of this swelling rate on several parameters was characterized. We observed that swelling rate has a non-monotone dependence on the molarity of the bath solution, with moderate concentrations of available sodium inducing the fastest swelling.

A local and global sensitivity analysis of a mathematical model of coagulation and platelet deposition under flow

The hemostatic response involves blood coagulation and platelet aggregation to stop blood loss from an injured blood vessel. The complexity of these processes make it difficult to intuit the overall hemostatic response without quantitative methods. Mathematical models aim to address this challenge but are often accompanied by numerous parameters choices and thus need to be analyzed for sensitivity to such choices. Here we use local and global sensitivity analyses to study a model of coagulation and platelet deposition under flow. To relate with clinical assays, we measured the sensitivity of three specific thrombin metrics: lag time, maximum relative rate of generation, and final concentration after 20 minutes. In addition, we varied parameters of three different classes: plasma protein levels, kinetic rate constants, and platelet characteristics. In terms of an overall ranking of the model’s sensitivities, we found that the local and global methods provided similar information. Our local analysis, in agreement with previous findings, shows that varying parameters within 50-150% of baseline values, in a one-at-a-time (OAT) fashion, always leads to significant thrombin generation in 20 minutes. Our global analysis gave a different and novel result highlighting groups of parameters, still varying within the normal 50-150%, that produced little or no thrombin in 20 minutes. Variations in either plasma levels or platelet characteristics, using either OAT or simultaneous variations, always led to strong thrombin production and overall, relatively low output variance. Simultaneous variation in kinetics rate constants or in a subset of all three parameter classes led to the highest overall output variance, incorporating instances with little to no thrombin production. The global analysis revealed multiple parameter interactions in the lag time and final concentration leading to relatively high variance; high variance was also observed in the thrombin generation rate, but parameters attributed to that variance acted independently and additively.

A physics-based model for maintenance of the pH gradient in the gastric mucus layer

It is generally accepted that the gastric mucus layer provides a protective barrier between the lumen and the mucosa, shielding the mucosa from acid and digestive enzymes and preventing autodigestion of the stomach epithelium. However, the precise mechanisms that contribute to this protective function are still up for debate. In particular, it is not clear what physical processes are responsible for transporting hydrogen protons, secreted within the gastric pits, across the mucus layer to the lumen without acidifying the environment adjacent to the epithelium. One hypothesis is that hydrogen may be bound to the mucin polymers themselves as they are convected away from the mucosal surface and eventually degraded in the stomach lumen. It is also not clear what mechanisms prevent hydrogen from diffusing back toward the mucosal surface, thereby lowering the local pH. In this work we investigate a physics-based model of ion transport within the mucosal layer based on a Nernst-Planck-like equation. Analysis of this model shows that the mechanism of transporting protons bound to the mucus gel is capable of reproducing the trans-mucus pH gradients reported in the literature. Furthermore, when coupled with ion exchange at the epithelial surface, our analysis shows that bicarbonate secretion alone is capable of neutralizing the epithelial pH, even in the face of enormous diffusive gradients of hydrogen. Maintenance of the pH gradient is found to be robust to a wide array of perturbations in both physiological and phenomenological model parameters, suggesting a robust physiological control mechanism.NEW & NOTEWORTHY This work combines modeling techniques based on physical principles, as well as novel numerical simulations to test the plausibility of one hypothesized mechanism for proton transport across the gastric mucus layer. Results show that this mechanism is able to maintain the extreme pH gradient seen in in vivo experiments and suggests a highly robust regulation mechanism to maintain this gradient in the face of dynamic lumen composition.

Molecular and Physical Mechanisms of Fibrinolysis and Thrombolysis from Mathematical Modeling and Experiments

Despite the common use of thrombolytic drugs, especially in stroke treatment, there are many conflicting studies on factors affecting fibrinolysis. Because of the complexity of the fibrinolytic system, mathematical models closely tied with experiments can be used to understand relationships within the system. When tPA is introduced at the clot or thrombus edge, lysis proceeds as a front. We developed a multiscale model of fibrinolysis that includes the main chemical reactions: the microscale model represents a single fiber cross-section; the macroscale model represents a three-dimensional fibrin clot. The model successfully simulates the spatial and temporal locations of all components and elucidates how lysis rates are determined by the interplay between the number of tPA molecules in the system and clot structure. We used the model to identify kinetic conditions necessary for fibrinolysis to proceed as a front. We found that plasmin regulates the local concentration of tPA through forced unbinding via degradation of fibrin and tPA release. The mechanism of action of tPA is affected by the number of molecules present with respect to fibrin fibers. The physical mechanism of plasmin action (crawling) and avoidance of inhibition is defined. Many of these new findings have significant implications for thrombolytic treatment.

A Mathematical Model of Venous Thrombosis Initiation

We present a mathematical model for the initiation of venous thrombosis (VT) due to slow flow and the consequent activation of the endothelial cells (ECs) lining the vein, in the absence of overt mechanical disruption of the EC layer. It includes all reactions of the tissue factor (TF) pathway of coagulation through fibrin formation, incorporates the accumulation of blood cells on activated ECs, accounts for the flow-mediated delivery and removal of coagulation proteins and blood cells from the locus of the reactions, and accounts for the activity of major inhibitors including heparan-sulfate-accelerated antithrombin and activated protein C. The model reveals that the occurrence of robust thrombin generation (a thrombin burst) depends in a threshold manner on the density of TF on the activated ECs and on the concentration of thrombomodulin and the degree of heparan-sulfate accelerated antithrombin activity on those cells. Small changes in any of these in appropriate narrow ranges switches the response between "no burst" and "burst." The model predicts synergies among the inhibitors, both in terms of each inhibitor's multiple targets, and in terms of interactions between the different inhibitors. The model strongly suggests that the rate and extent of accumulation of activated monocytes, platelets, and MPs that can support the coagulation reactions has a powerful influence on whether a thrombin burst occurs and the thrombin response when it does. The slow rate of accumulation of cells supporting coagulation is one reason that the progress of VT is so much slower than that of arterial thrombosis initiated by subendothelial exposure.

Synergy Between Tissue Factor and Exogenous Factor XIa in Initiating Coagulation

OBJECTIVE

Recent evidence suggests involvement of coagulation factor XIa (FXIa) in thrombotic event development. This study was conducted to explore possible synergies between tissue factor (TF) and exogenous FXIa (E-FXIa) in thrombin generation.

APPROACH AND RESULTS

In thrombin generation assays, for increasing concentrations of E-FXIa with low, but not with high TF concentrations, peak thrombin significantly increased whereas lag time and time to peak significantly decreased. Similar dependencies of lag times and rates of thrombin generation were found in mathematical model simulations. In both in vitro and in silico experiments that included E-FXIa, thrombin bursts were seen for TF levels much lower than those required without E-FXIa. For in silico thrombin bursts initiated by the synergistic action of TF and E-FXIa, the mechanisms leading to the burst differed substantially from those for bursts initiated by high TF alone. For the synergistic case, sustained activation of platelet-bound FIX by E-FXIa, along with the feedback-enhanced activation of platelet-bound FVIIIa and FXa, was needed to elicit a thrombin burst. Furthermore, the initiation of thrombin bursts by high TF levels relied on different platelet FIX/FIXa binding sites than those involved in bursts initiated by low TF levels with E-FXIa.

CONCLUSIONS

Low concentrations of TF and exogenous FXIa, each too low to elicit a burst in thrombin production alone, act synergistically when in combination to cause substantial thrombin production. The observation about FIX/FIXa binding sites may have therapeutic implications.

A microfluidic model of hemostasis sensitive to platelet function and coagulation

Hemostasis is the process of sealing a vascular injury with a thrombus to arrest bleeding. The type of thrombus that forms depends on the nature of the injury and hemodynamics. There are many models of intravascular thrombus formation whereby blood is exposed to prothrombotic molecules on a solid substrate. However, there are few models of extravascular thrombus formation whereby blood escapes into the extravascular space through a hole in the vessel wall. Here, we describe a microfluidic model of hemostasis that includes vascular, vessel wall, and extravascular compartments. Type I collagen and tissue factor, which support platelet adhesion and initiate coagulation, respectively, were adsorbed to the wall of the injury channel and act synergistically to yield a stable thrombus that stops blood loss into the extravascular compartment in ~7.5 min. Inhibiting factor VIII to mimic hemophilia A results in an unstable thrombus that was unable to close the injury. Treatment with a P2Y12 antagonist to reduce platelet activation prolonged the closure time two-fold compared to controls. Taken together, these data demonstrate a hemostatic model that is sensitive to both coagulation and platelet function and can be used to study coagulopathies and platelet dysfunction that result in excessive blood loss.

Functional assay of antiplatelet drugs based on margination of platelets in flowing blood

A novel functional assay of antiplatelet drug efficacy was designed by utilizing the phenomena of platelet margination in flowing blood and transient platelet contacts with surface-immobilized platelet agonists. Flow margination enhances transient contacts of platelets with the walls of flow chambers covered with surface-immobilized proteins. Depending on the type and the surface density of the immobilized agonists, such transient interactions could "prime" the marginated platelet subpopulation for enhanced activation and adhesion downstream. By creating an upstream surface patch with an immobilized platelet agonist, platelet flow margination was used to test how effective antiplatelet drugs are in suppressing downstream platelet activation and adhesion. The platelet adhesion downstream was measured by a so-called "capture" patch region close to the distal end of the flow chamber. Platelet adhesion downstream was found to be dose-dependent on the upstream surface coverage of the "priming" patch, with immobilized fibrinogen acting as a platelet agonist. Several antiplatelet agents (acetylsalicylic acid, eptifibatide, and tirofiban) were evaluated for their efficacy in attenuating downstream adhesion after upstream platelet priming. The activation of the platelet population was found to be dependent on both the extent of the upstream agonist stimulus and the antiplatelet drug concentration. Such a relationship provides an opportunity to measure the efficacy of specific antiplatelet agents against the type and concentration of upstream platelet agonists.

A Radial Basis Function (RBF)-Finite Difference (FD) Method for Diffusion and Reaction-Diffusion Equations on Surfaces

In this paper, we present a method based on Radial Basis Function (RBF)-generated Finite Differences (FD) for numerically solving diffusion and reaction-diffusion equations (PDEs) on closed surfaces embedded in ℝ d . Our method uses a method-of-lines formulation, in which surface derivatives that appear in the PDEs are approximated locally using RBF interpolation. The method requires only scattered nodes representing the surface and normal vectors at those scattered nodes. All computations use only extrinsic coordinates, thereby avoiding coordinate distortions and singularities. We also present an optimization procedure that allows for the stabilization of the discrete differential operators generated by our RBF-FD method by selecting shape parameters for each stencil that correspond to a global target condition number. We show the convergence of our method on two surfaces for different stencil sizes, and present applications to nonlinear PDEs simulated both on implicit/parametric surfaces and more general surfaces represented by point clouds.

A Framework for Exploring the Post-gelation Behavior of Ziff and Stell's Polymerization Models

Ziff and Stell (Kinetics of polymer gelation, J. Chem. Phys., 73 (1980), 3492-3499.) pioneered the study of kinetic models of polymer growth and gelation which involve differential equations that describe the temporal evolution of oligomer concentrations and in which gelation is manifest as a finite-time singularity. Here we present a systematic framework for studying post-gelation behavior of these and related models that allows inclusion of the effects of diffusion and other transport mechanisms as well as those of sources and sinks, and which enables determination of, among other things, the final structure of the gel under a variety of conditions.

Fluid Mechanics of Blood Clot Formation

Intravascular blood clots form in an environment in which hydrodynamic forces dominate and in which fluid-mediated transport is the primary means of moving material. The clotting system has evolved to exploit fluid dynamic mechanisms and to overcome fluid dynamic challenges to ensure that clots that preserve vascular integrity can form over the wide range of flow conditions found in the circulation. Fluid-mediated interactions between the many large deformable red blood cells and the few small rigid platelets lead to high platelet concentrations near vessel walls where platelets contribute to clotting. Receptor-ligand pairs with diverse kinetic and mechanical characteristics work synergistically to arrest rapidly flowing cells on an injured vessel. Variations in hydrodynamic stresses switch on and off the function of key clotting polymers. Protein transport to, from, and within a developing clot determines whether and how fast it grows. We review ongoing experimental and modeling research to understand these and related phenomena.

An Immersed Boundary Method for Two-fluid Mixtures

We present an Immersed Boundary method for interactions between elastic boundaries and mixtures of two fluids. Each fluid has its own velocity field and volume-fraction. A penalty method is used to enforce the condition that both fluids' velocities agree with that of the elastic boundaries. The method is applied to several problems: Taylor's swimming sheet problem for a mixture of two viscous fluids, peristaltic pumping of a mixture of two viscous fluids, with and without immersed particles, and peristaltic pumping of a mixture of a viscous fluid and a viscoelastic fluid. The swimming sheet and peristalsis problems have received much attention recently in the context of a single viscoelastic fluid. Numerical results demonstrate that the method converges and show its capability to handle a number of flow problems of substantial current interest. They illustrate that for each of these problems, the relative motion between the two fluids changes the observed behaviors profoundly compared to the single fluid case.

The effect of factor VIII deficiencies and replacement and bypass therapies on thrombus formation under venous flow conditions in microfluidic and computational models

Clinical evidence suggests that individuals with factor VIII (FVIII) deficiency (hemophilia A) are protected against venous thrombosis, but treatment with recombinant proteins can increase their risk for thrombosis. In this study we examined the dynamics of thrombus formation in individuals with hemophilia A and their response to replacement and bypass therapies under venous flow conditions. Fibrin and platelet accumulation were measured in microfluidic flow assays on a TF-rich surface at a shear rate of 100 s⁻¹. Thrombin generation was calculated with a computational spatial-temporal model of thrombus formation. Mild FVIII deficiencies (5–30% normal levels) could support fibrin fiber formation, while severe (<1%) and moderate (1–5%) deficiencies could not. Based on these experimental observations, computational calculations estimate an average thrombin concentration of ∼10 nM is necessary to support fibrin formation under flow. There was no difference in fibrin formation between severe and moderate deficiencies, but platelet aggregate size was significantly larger for moderate deficiencies. Computational calculations estimate that the local thrombin concentration in moderate deficiencies is high enough to induce platelet activation (>1 nM), but too low to support fibrin formation (<10 nM). In the absence of platelets, fibrin formation was not supported even at normal FVIII levels, suggesting platelet adhesion is necessary for fibrin formation. Individuals treated by replacement therapy, recombinant FVIII, showed normalized fibrin formation. Individuals treated with bypass therapy, recombinant FVIIa, had a reduced lag time in fibrin formation, as well as elevated fibrin accumulation compared to healthy controls. Treatment of rFVIIa, but not rFVIII, resulted in significant changes in fibrin dynamics that could lead to a prothrombotic state.

Platelet motion near a vessel wall or thrombus surface in two-dimensional whole blood simulations

Computational simulations using a two-dimensional lattice-Boltzmann immersed boundary method were conducted to investigate the motion of platelets near a vessel wall and close to an intravascular thrombus. Physiological volume fractions of deformable red blood cells and rigid platelet-size elliptic particles were studied under arteriolar flow conditions. Tumbling of platelets in the red-blood-cell depleted zone near the vessel walls was strongly influenced by nearby red blood cells. The thickness of the red-blood-cell depleted zone was greatly reduced near a thrombus, and platelets in this zone were pushed close to the surface of the thrombus to distances that would facilitate their cohesion to it. The distance, nature, and duration of close platelet-thrombus encounters were influenced by the porosity of the thrombus. The strong influence on platelet-thrombus encounters of red-blood-cell motion and thrombus porosity must be taken into account to understand the dynamics of platelet attachment to a growing thrombus.

The effect of divalent vs. monovalent ions on the swelling of mucin-like polyelectrolyte gels: governing equations and equilibrium analysis

We introduce a comprehensive model of a mucin-like polyelectrolyte gel swelling-deswelling which includes the ion-mediated crosslinking of polymer strands and the exchange of divalent and monovalent ions in the gel. The gel is modeled as a multi-phase mixture which accounts for the polymer and solvent volume fractions and velocities as well as ionic species concentrations. Motion is determined by force balances involving viscous, drag, and chemical forces. The chemical forces are derived from a free energy which includes entropic contributions as well as the chemical and electrostatic interactions among the crosslinked polymer, uncrosslinked polymer, and the ionic solvent. The unified derivation produces all the classical effects (van't Hoff osmotic pressure, Donnan equilibrium potential, Nernst-Planck motion of ions) as well as expressions for Flory interaction parameter and the standard free energy parameters that explicitly depend on the gel chemistry and crosslink structure. For this model, we show how the interplay between ionic bath concentrations, ionic binding, and transient divalent crosslinking leads to a variety of swelled and deswelled phases/phase transitions. In particular, we show how the absorption of divalent ions can lead to a massive deswelling of the gel. We conclude that the unique properties of mucin-like gels can be explained by their ionic binding affinities and transient divalent crosslinking.

A study of different modeling choices for simulating platelets within the immersed boundary method

The Immersed Boundary (IB) method is a widely-used numerical methodology for the simulation of fluid-structure interaction problems. The IB method utilizes an Eulerian discretization for the fluid equations of motion while maintaining a Lagrangian representation of structural objects. Operators are defined for transmitting information (forces and velocities) between these two representations. Most IB simulations represent their structures with piecewise linear approximations and utilize Hookean spring models to approximate structural forces. Our specific motivation is the modeling of platelets in hemodynamic flows. In this paper, we study two alternative representations - radial basis functions (RBFs) and Fourier-based (trigonometric polynomials and spherical harmonics) representations - for the modeling of platelets in two and three dimensions within the IB framework, and compare our results with the traditional piecewise linear approximation methodology. For different representative shapes, we examine the geometric modeling errors (position and normal vectors), force computation errors, and computational cost and provide an engineering trade-off strategy for when and why one might select to employ these different representations.

Modelling fibrinolysis: 1D continuum models

Fibrinolysis is the enzymatic degradation of the fibrin mesh that stabilizes blood clots. Experiments have shown that coarse clots made of thick fibres sometimes lyse more quickly than fine clots made of thin fibres, despite the fact that individual thick fibres lyse more slowly than individual thin fibres. This paper aims at using a 1D continuum reaction-diffusion model of fibrinolysis to elucidate the mechanism by which coarse clots lyse more quickly than fine clots. Reaction-diffusion models have been the standard tool for investigating fibrinolysis, and have been successful in capturing the wave-like behaviour of lysis seen in experiments. These previous models treat the distribution of fibrin within a clot as homogeneous, and therefore cannot be used directly to study the lysis of fine and coarse clots. In our model, we include a spatially heterogeneous fibrin concentration, as well as a more accurate description of the role of fibrin as a cofactor in the activation of the lytic enzyme. Our model predicts spatio-temporal protein distributions in reasonable quantitative agreement with experimental data. The model also predicts observed behaviour such as a front of lysis moving through the clot with an accumulation of lytic proteins at the front. In spite of the model improvements, however, we find that 1D continuum models are unable to accurately describe the observed differences in lysis behaviour between fine and coarse clots. Features of the problems that lead to the inaccuracy of 1D continuum models are discussed. We conclude that higher-dimensional, multiscale models are required to investigate the effect of clot structure on lysis behaviour.

Modelling fibrinolysis: a 3D stochastic multiscale model

Fibrinolysis, the proteolytic degradation of the fibrin fibres that stabilize blood clots, is initiated when tissue-type plasminogen activator (tPA) activates plasminogen to plasmin, the main fibrinolytic enzyme. Many experiments have shown that coarse clots made of thick fibres lyse more quickly than fine clots made of thin fibres, despite the fact that individual thick fibres lyse more slowly than individual thin fibres. The generally accepted explanation for this is that a coarse clot with fewer fibres to transect will be degraded faster than a fine clot with a higher fibre density. Other experiments show the opposite result. The standard mathematical tool for investigating fibrinolysis has been deterministic reaction-diffusion models, but due to low tPA concentrations, stochastic models may be more appropriate. We develop a 3D stochastic multiscale model of fibrinolysis. A microscale model representing a fibre cross section and containing detailed biochemical reactions provides information about single fibre lysis times, the number of plasmin molecules that can be activated by a single tPA molecule and the length of time tPA stays bound to a given fibre cross section. Data from the microscale model are used in a macroscale model of the full fibrin clot, from which we obtain lysis front velocities and tPA distributions. We find that the fibre number impacts lysis speed, but so does the number of tPA molecules relative to the surface area of the clot exposed to those molecules. Depending on the values of these two quantities (tPA number and surface area), for given kinetic parameters, the model predicts coarse clots lyse faster or slower than fine clots, thus providing a possible explanation for the divergent experimental observations.

The influence of hindered transport on the development of platelet thrombi under flow

Vascular injury triggers two intertwined processes, platelet deposition and coagulation, and can lead to the formation of an intravascular clot (thrombus) that may grow to occlude the vessel. Formation of the thrombus involves complex biochemical, biophysical, and biomechanical interactions that are also dynamic and spatially-distributed, and occur on multiple spatial and temporal scales. We previously developed a spatial-temporal mathematical model of these interactions and looked at the interplay between physical factors (flow, transport to the clot, platelet distribution within the blood) and biochemical ones in determining the growth of the clot. Here, we extend this model to include reduction of the advection and diffusion of the coagulation proteins in regions of the clot with high platelet number density. The effect of this reduction, in conjunction with limitations on fluid and platelet transport through dense regions of the clot can be profound. We found that hindered transport leads to the formation of smaller and denser clots compared to the case with no protein hindrance. The limitation on protein transport confines the important activating complexes to small regions in the interior of the thrombus and greatly reduces the supply of substrates to these complexes. Ultimately, this decreases the rate and amount of thrombin production and leads to greatly slowed growth and smaller thrombus size. Our results suggest a possible physical mechanism for limiting thrombus growth.

Low-Reynolds-number swimming in viscous two-phase fluids

The fluid media surrounding many microorganisms are often mixtures of multiple materials with very different physical properties. The composition and rheology of the mixture may strongly affect the related locomotive behaviors. We study the classical Taylor's swimming sheet problem within a two-fluid model, which consists of two intermixed viscous fluids with different viscosities, with both numerical experiments and analysis. Our results indicate that both the swimming speed and efficiency may be decreased substantially relative to those for a single-phase fluid.

Blood clot formation under flow: the importance of factor XI depends strongly on platelet count

A previously validated mathematical model of intravascular platelet deposition and tissue factor (TF)-initiated coagulation under flow is extended and used to assess the influence on thrombin production of the activation of factor XI (fXI) by thrombin and of the activation of factor IX (fIX) by fXIa. It is found that the importance of the thrombin-fXIa-fIXa feedback loop to robust thrombin production depends on the concentration of platelets in the blood near the injury. At a near-wall platelet concentration of ~250,000/μL, typical in vessels in which the shear rate is <200 s(-1), thrombin activation of fXI makes a significant difference only at low densities of exposed TF. If the near-wall platelet concentration is significantly higher than this, either because of a higher systemic platelet count or because of the redistribution of platelets toward the vessel walls at high shear rates, then thrombin activation of fXI makes a major difference even for relatively high densities of exposed TF. The model predicts that the effect of a severe fXI deficiency depends on the platelet count, and that fXI becomes more important at high platelet counts.

Influence of the standard free energy on swelling kinetics of gels

Classical theories of gel swelling employ the mixing free energy, thereby ignoring any effects of the free energy of the pure phases,i.e., the polymer standard free energy. In this paper we present a model for the swelling kinetics of gels that incorporates the free energy, including the polymer standard free energy. We provide a complete analysis of how the swelling kinetics and stable states and sizes of the swelled gel depends on the free energy parameters and show that theories that use only the mixing free energy cannot correctly describe equilibrium states or the swelling kinetics.

Grow with the flow: a spatial-temporal model of platelet deposition and blood coagulation under flow

The body's response to vascular injury involves two intertwined processes: platelet aggregation and coagulation. Platelet aggregation is a predominantly physical process, whereby platelets clump together, and coagulation is a cascade of biochemical enzyme reactions. Thrombin, the major product of coagulation, directly couples the biochemical system to platelet aggregation by activating platelets and by cleaving fibrinogen into fibrin monomers that polymerize to form a mesh that stabilizes platelet aggregates. Together, the fibrin mesh and the platelet aggregates comprise a thrombus that can grow to occlusive diameters. Transport of coagulation proteins and platelets to and from an injury is controlled largely by the dynamics of the blood flow. To explore how blood flow affects the growth of thrombi and how the growing masses, in turn, feed back and affect the flow, we have developed the first spatial-temporal mathematical model of platelet aggregation and blood coagulation under flow that includes detailed descriptions of coagulation biochemistry, chemical activation and deposition of blood platelets, as well as the two-way interaction between the fluid dynamics and the growing platelet mass. We present this model and use it to explain what underlies the threshold behaviour of the coagulation system's production of thrombin and to show how wall shear rate and near-wall enhanced platelet concentrations affect the development of growing thrombi. By accounting for the porous nature of the thrombus, we also demonstrate how advective and diffusive transport to and within the thrombus affects its growth at different stages and spatial locations.

Toward an understanding of fibrin branching structure

The blood clotting enzyme thrombin converts fibrinogen molecules into fibrin monomers which polymerize to form a fibrous three-dimensional gel. The concentration of thrombin affects the architecture of the resulting gel, in particular, a higher concentration of thrombin produces a gel with more branch points per unit volume and with shorter fiber segments between branch points. We propose a mechanism by which fibrin branching can occur and show that this mechanism can lead to dependence of the gel's structure (at the time of gelation) on the rate at which monomer is supplied. A higher rate of monomer supply leads to a gel with a higher branch concentration and with shorter fiber segments between branch points. The origin of this dependence is explained.

Computational model of whole blood exhibiting lateral platelet motion induced by red blood cells

An Immersed Boundary method is developed in which the fluid's motion is calculated using the lattice Boltzmann method. The method is applied to explore the experimentally-observed lateral redistribution of platelets and platelet-sized particles in concentrated suspensions of red blood cells undergoing channel flow. Simulations capture red-blood-cell-induced lateral platelet motion and the consequent development of a platelet concentration profile that includes an enhanced concentration within a few microns of the channel walls. In the simulations, the near-wall enhanced concentration develops within approximately 400 msec starting from a random distribution of red blood cells and a uniform distribution of platelet-sized particles.

The effects of spatial inhomogeneities on flow through the endothelial surface layer

Flow through the endothelial surface layer (the glycocalyx and adsorbed plasma proteins) plays an important but poorly understood role in cell signaling through a process known as mechanotransduction. Characterizing the flow rates and shear stresses throughout this layer is critical for understanding how flow-induced ionic currents, deformations of transmembrane proteins, and the convection of extracellular molecules signal biochemical events within the cell, including cytoskeletal rearrangements, gene activation, and the release of vasodilators. Previous mathematical models of flow through the endothelial surface layer are based upon the assumptions that the layer is of constant hydraulic permeability and constant height. These models also assume that the layer is continuous across the endothelium and that the layer extends into only a small portion of the vessel lumen. Results of these models predict that fluid shear stress is dissipated through the surface layer and is thus negligible near endothelial cell membranes. In this paper, such assumptions are removed, and the resultant flow rates and shear stresses through the layer are described. The endothelial surface layer is modeled as clumps of a Brinkman medium immersed in a Newtonian fluid. The width and spacing of each clump, hydraulic permeability, and fraction of the vessel lumen occupied by the layer are varied. The two-dimensional Navier-Stokes equations with an additional Brinkman resistance term are solved using a projection method. Several fluid shear stress transitions in which the stress at the membrane shifts from low to high values are described. These transitions could be significant to cell signaling since the endothelial surface layer is likely dynamic in its composition, density, and height.

Fibrin gel formation in a shear flow

Blood clots are made up of platelets and fibrin gel, and the relative amount of fibrin is strongly influenced by the shear rate. In order to explore this phenomenon, this paper presents a model of fibrin gel formation over the surface of an injured blood vessel in a shear flow. A condition for gelation including source and sink terms of polymer is derived. A simplified model of coagulation, involving activation and inhibition of the enzyme thrombin and thrombin-mediated production of fibrin monomer, is combined with the model of gelation to explore how the shear rate and other parameters control the formation of fibrin gel. The results show that the thrombin inhibition rate, the gel permeability and the shear rate are key parameters in regulating the height of the fibrin gel.

Coagulation under flow: the influence of flow-mediated transport on the initiation and inhibition of coagulation

A mathematical model of intravascular coagulation is presented; it encompasses the biochemistry of the tissue factor pathway, platelet activation and deposition on the subendothelium, and flow- and diffusion-mediated transport of coagulation proteins and platelets. Simulation experiments carried out with the model indicate the predominant role played by the physical processes of platelet deposition and flow-mediated removal of enzymes in inhibiting coagulation in the vicinity of vascular injury. Sufficiently rapid production of factors IXa and Xa by the TF:VIIa complex can overcome this inhibition and lead to formation of significant amounts of the tenase complex on the surface of activated platelets and, as a consequence, to substantial thrombin production. Chemical inhibitors are seen to play almost no (TFPI) or little (AT-III and APC) role in determining whether substantial thrombin production will occur. The role of APC is limited by the necessity for diffusion of thrombin from the site of injury to nearby endothelial cells to form the thrombomodulin-thrombin complex and for diffusion in the reverse direction of the APC made by this complex. TFPI plays an insignificant part in inhibiting the TF:VIIa complex under the conditions studied whether its action involves sequential binding of TFPI to Xa and then TFPI:Xa to TF:VIIa, or direct binding of TFPI to Xa already bound to the TF:VIIa complex.

Platelet-wall interactions in continuum models of platelet thrombosis: formulation and numerical solution

A model is developed to describe the formation of platelet thrombi in coronary-artery-sized blood vessels. It involves interactions among a viscous, incompressible fluid; populations of non-activated and activated platelets; activating chemicals; and the vessel walls. Adhesion of platelets to the injured wall and cohesion between activated platelets is modelled using distributions of elastic links which generate stresses that can influence the fluid motion. The first version of the model presented involves two spatial scales: the microscale of the platelets and the macroscale of the vessel. A closure approximation is introduced that allows essential microscale behaviour to be computed while eliminating the necessity to explicitly track events on this scale. Computational methods are presented that meet the diverse challenges posed by the coupled nonlinear partial differential equations of the model and by the complex geometry of the constricted vessels in which the thrombosis simulations are carried out. Simulation results demonstrate that the model can produce thrombi that grow to occlude the vessel, that shear-stress exerted by the fluid on the thrombi can modify their subsequent growth and cause remodelling of their shape through small-scale local changes or large-scale structural breakup.

A mathematical study of volume shifts and ionic concentration changes during ischemia and hypoxia

The response of tissue to ischemia (cessation of blood flow and deprivation of oxygen) includes swelling of the intracellular space, shrinkage of the extracellular space, and an increase in the extracellular potassium concentration. The responses of cardiac and brain tissue to ischemia are qualitatively different in that cardiac tissue shows a rise in extracellular potassium over several minutes from about 5 to 10-12 mM followed by a plateau, while brain tissue shows a similar initial rise followed by a very rapid increase in extracellular potassium to levels of 50-80 mM. During hypoxia the flow of blood (or perfusate) is maintained and, while there is a substantial efflux of potassium from cells, there is little accumulation of potassium in the interstitium. A mathematical model is proposed and studied to try to elucidate the mechanism(s) underlying the increase in extracellular potassium, and the different time courses seen in neural and cardiac tissue. The model involves a Hodgkin-Huxley-type description of transmembrane ion currents, allows for ion concentrations as well as volumes to change for both the intracellular and extracellular space, and includes coupling of damaged tissue to nearby healthy tissue. The model produces a response to ischemia much like that seen in neural tissue, and the mechanism underlying this response in the model is determined. The same mechanism is not present in cardiac ion models, and this may explain the qualitative difference in response shown in cardiac tissue.

Probabilistic modeling of platelet aggregation: effects of activation time and receptor occupancy

A mathematical model is constructed to predict the probability that a collision between two activated platelets results in doublet formation mediated by fibrinogen cross-bridges. The model is used to explore the effect of time from activation, looking at both simultaneous and non-simultaneous activation times. Also considered are the impact of blood fibrinogen concentration and various shear rates. The idea of hydrodynamic efficiency [Tandon & Diamond (1997) Biophys. J.73, 2819-2835] is extended by varying the separation distance which is considered to be a collision. From fitting the model to data [Xia & Frojmovic (1994) Biophys. J.66, 2190-2201], it is found that the hydrodynamic efficiency corresponds to short interaction distances ( approximately 14 nm). The model predicts that the probability of forming a doublet increases quickly after activation, remains near its maximum for a significant time interval, and then declines. This may contribute to the regulation of the time and location of platelet aggregation, by ensuring that platelets are more likely to aggregate near an injury, rather than downstream in the vascular system. A newly activated platelet has a high probability of cross-bridging with an already activated platelet. Fibrinogen concentration strongly affects the time course and the equilibrium values of the aggregation probability. These results indicate the importance of considering the progression of the reaction between solution fibrinogen and surface receptors in determining a platelet's ability to aggregate.

Surface-mediated control of blood coagulation: the role of binding site densities and platelet deposition

A mathematical model of the extrinsic or tissue factor (TF) pathway of blood coagulation is formulated and results from a computational study of its behavior are presented. The model takes into account plasma-phase and surface-bound enzymes and zymogens, coagulation inhibitors, and activated and unactivated platelets. It includes both plasma-phase and membrane-phase reactions, and accounts for chemical and cellular transport by flow and diffusion, albeit in a simplified manner by assuming the existence of a thin, well-mixed fluid layer, near the surface, whose thickness depends on flow. There are three main conclusions from these studies. (i) The model system responds in a threshold manner to changes in the availability of particular surface binding sites; an increase in TF binding sites, as would occur with vascular injury, changes the system's production of thrombin dramatically. (ii) The model suggests that platelets adhering to and covering the subendothelium, rather than chemical inhibitors, may play the dominant role in blocking the activity of the TF:VIIa enzyme complex. This, in turn, suggests that a role of the IXa-tenase pathway for activating factor X to Xa is to continue factor Xa production after platelets have covered the TF:VIIa complexes on the subendothelium. (iii) The model gives a kinetic explanation of the reduced thrombin production in hemophilias A and B.

Membrane binding-site density can modulate activation thresholds in enzyme systems

The kinetic equations are analysed for a model system which is motivated by the reactions of blood coagulation, and which involves two zymogen-enzyme pairs each of which can exist in solution phase or bound to a membrane. The enzyme of each pair activates the zymogen of the other pair, and each enzyme is subject to first-order inactivation both in solution and when bound to the membrane. If enzyme activation happens exclusively or predominantly in the membrane phase, then the system displays a threshold response which can be modulated by varying the density of membrane binding sites for the zymogens and enzymes. For low densities of membrane binding sites, the system's response when challenged by a dose of enzyme quickly decays away. For high enough densities of membrane binding sites, the system responds with substantial and sustained enzyme production. Thus variations in surface-binding site densities can serve as a "switch", drastically altering the responsiveness of the system. Such a binding-site-mediated switching mechanism could have profound importance to the regulation of enzyme systems, in particular, the blood coagulation system.