Dynamics of spatially nonuniform patterning in the model of blood coagulation

Efficient multi-fidelity computation of blood coagulation under flow

Clot formation is a crucial process that prevents bleeding, but can lead to severe disorders when imbalanced. This process is regulated by the coagulation cascade, a biochemical network that controls the enzyme thrombin, which converts soluble fibrinogen into the fibrin fibers that constitute clots. Coagulation cascade models are typically complex and involve dozens of partial differential equations (PDEs) representing various chemical species' transport, reaction kinetics, and diffusion. Solving these PDE systems computationally is challenging, due to their large size and multi-scale nature. We propose a multi-fidelity strategy to increase the efficiency of coagulation cascade simulations. Leveraging the slower dynamics of molecular diffusion, we transform the governing PDEs into ordinary differential equations (ODEs) representing the evolution of species concentrations versus blood residence time. We then Taylor-expand the ODE solution around the zero-diffusivity limit to obtain spatiotemporal maps of species concentrations in terms of the statistical moments of residence time, [Formula: see text], and provide the governing PDEs for [Formula: see text]. This strategy replaces a high-fidelity system of N PDEs representing the coagulation cascade of N chemical species by N ODEs and p PDEs governing the residence time statistical moments. The multi-fidelity order (p) allows balancing accuracy and computational cost providing a speedup of over N/p compared to high-fidelity models. Moreover, this cost becomes independent of the number of chemical species in the large computational meshes typical of the arterial and cardiac chamber simulations. Using a coagulation network with N = 9 and an idealized aneurysm geometry with a pulsatile flow as a benchmark, we demonstrate favorable accuracy for low-order models of p = 1 and p = 2. The thrombin concentration in these models departs from the high-fidelity solution by under 20% (p = 1) and 2% (p = 2) after 20 cardiac cycles. These multi-fidelity models could enable new coagulation analyses in complex flow scenarios and extensive reaction networks. Furthermore, it could be generalized to advance our understanding of other reacting systems affected by flow.

Efficient multi-fidelity computation of blood coagulation under flow

Clot formation is a crucial process that prevents bleeding, but can lead to severe disorders when imbalanced. This process is regulated by the coagulation cascade, a biochemical network that controls the enzyme thrombin, which converts soluble fibrinogen into the fibrin fibers that constitute clots. Coagulation cascade models are typically complex and involve dozens of partial differential equations (PDEs) representing various chemical species' transport, reaction kinetics, and diffusion. Solving these PDE systems computationally is challenging, due to their large size and multi-scale nature. We propose a multi-fidelity strategy to increase the efficiency of coagulation cascade simulations. Leveraging the slower dynamics of molecular diffusion, we transform the governing PDEs into ordinary differential equations (ODEs) representing the evolution of species concentrations versus blood residence time. We then Taylor-expand the ODE solution around the zero-diffusivity limit to obtain spatiotemporal maps of species concentrations in terms of the statistical moments of residence time, , and provide the governing PDEs for . This strategy replaces a high-fidelity system of N PDEs representing the coagulation cascade of N chemical species by N ODEs and p PDEs governing the residence time statistical moments. The multi-fidelity order( p ) allows balancing accuracy and computational cost, providing a speedup of over N/p compared to high-fidelity models. Using a simplified coagulation network and an idealized aneurysm geometry with a pulsatile flow as a benchmark, we demonstrate favorable accuracy for low-order models of p = 1 and p = 2. These models depart from the high-fidelity solution by under 16% ( p = 1) and 5% ( p = 2) after 20 cardiac cycles. The favorable accuracy and low computational cost of multi-fidelity models could enable unprecedented coagulation analyses in complex flow scenarios and extensive reaction networks. Furthermore, it can be generalized to advance our understanding of other systems biology networks affected by blood flow.

Patient-Specific Modelling of Blood Coagulation

Blood coagulation represents one of the most studied processes in biomedical modelling. However, clinical applications of this modelling remain limited because of the complexity of this process and because of large inter-patient variation of the concentrations of blood factors, kinetic constants and physiological conditions. Determination of some of these patients-specific parameters is experimentally possible, but it would be related to excessive time and material costs impossible in clinical practice. We propose in this work a methodological approach to patient-specific modelling of blood coagulation. It begins with conventional thrombin generation tests allowing the determination of parameters of a reduced kinetic model. Next, this model is used to study spatial distributions of blood factors and blood coagulation in flow, and to evaluate the results of medical treatment of blood coagulation disorders.

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.

Modeling thrombosis in silico: Frontiers, challenges, unresolved problems and milestones

Hemostasis is a complex physiological mechanism that functions to maintain vascular integrity under any conditions. Its primary components are blood platelets and a coagulation network that interact to form the hemostatic plug, a combination of cell aggregate and gelatinous fibrin clot that stops bleeding upon vascular injury. Disorders of hemostasis result in bleeding or thrombosis, and are the major immediate cause of mortality and morbidity in the world. Regulation of hemostasis and thrombosis is immensely complex, as it depends on blood cell adhesion and mechanics, hydrodynamics and mass transport of various species, huge signal transduction networks in platelets, as well as spatiotemporal regulation of the blood coagulation network. Mathematical and computational modeling has been increasingly used to gain insight into this complexity over the last 30 years, but the limitations of the existing models remain profound. Here we review state-of-the-art-methods for computational modeling of thrombosis with the specific focus on the analysis of unresolved challenges. They include: a) fundamental issues related to physics of platelet aggregates and fibrin gels; b) computational challenges and limitations for solution of the models that combine cell adhesion, hydrodynamics and chemistry; c) biological mysteries and unknown parameters of processes; d) biophysical complexities of the spatiotemporal networks' regulation. Both relatively classical approaches and innovative computational techniques for their solution are considered; the subjects discussed with relation to thrombosis modeling include coarse-graining, continuum versus particle-based modeling, multiscale models, hybrid models, parameter estimation and others. Fundamental understanding gained from theoretical models are highlighted and a description of future prospects in the field and the nearest possible aims are given.

Reaction-diffusion waves of blood coagulation

One of the main characteristics of blood coagulation is the speed of clot growth. In the current work we consider a mathematical model of the coagulation cascade and study existence, stability and speed of propagation of the reaction-diffusion waves of blood coagulation. We also develop a simplified one-equation model that reflects the main features of the thrombin wave propagation. For this equation we estimate the wave speed analytically. The resulting formulas provide a good approximation for the speed of wave propagation in a more complex model as well as for the experimental data.

Influence of Antithrombin on the Regimes of Blood Coagulation: Insights from the Mathematical Model

Blood coagulation is regulated through a complex network of biochemical reactions of blood factors. The main acting enzyme is thrombin whose propagation in blood plasma leads to fibrin clot formation. Spontaneous clot formation is normally controlled through the action of different plasma inhibitors, in particular, through the thrombin binding by antithrombin. In the current study we develop a mathematical model of clot formation both in quiescent plasma and in blood flow and determine the analytical conditions on the antithrombin concentration corresponding to different regimes of blood coagulation.

Oscillons localized inside breathing periodical structures in a two-variable model of a one-dimensional infinite excitable reaction-diffusion system

A two-variable model of a one-dimensional (1D), infinite, excitable, reaction-diffusion system describing oscillons localized inside an expanding breathing periodical structure emitting traveling impulses is presented. The model is based on two coupled catalytic (enzymatic) reactions.

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.

On propagation of excitation waves in moving media: the FitzHugh-Nagumo model

Background

Existence of flows and convection is an essential and integral feature of many excitable media with wave propagation modes, such as blood coagulation or bioreactors.

Methods/Results

Here, propagation of two-dimensional waves is studied in parabolic channel flow of excitable medium of the FitzHugh-Nagumo type. Even if the stream velocity is hundreds of times higher that the wave velocity in motionless medium (), steady propagation of an excitation wave is eventually established. At high stream velocities, the wave does not span the channel from wall to wall, forming isolated excited regions, which we called “restrictons”. They are especially easy to observe when the model parameters are close to critical ones, at which waves disappear in still medium. In the subcritical region of parameters, a sufficiently fast stream can result in the survival of excitation moving, as a rule, in the form of “restrictons”. For downstream excitation waves, the axial portion of the channel is the most important one in determining their behavior. For upstream waves, the most important region of the channel is the near-wall boundary layers. The roles of transversal diffusion, and of approximate similarity with respect to stream velocity are discussed.

Conclusions

These findings clarify mechanisms of wave propagation and survival in flow.

An allosteric mechanism for switching between parallel tracks in mammalian sulfur metabolism

Methionine (Met) is an essential amino acid that is needed for the synthesis of S-adenosylmethionine (AdoMet), the major biological methylating agent. Methionine used for AdoMet synthesis can be replenished via remethylation of homocysteine. Alternatively, homocysteine can be converted to cysteine via the transsulfuration pathway. Aberrations in methionine metabolism are associated with a number of complex diseases, including cancer, anemia, and neurodegenerative diseases. The concentration of methionine in blood and in organs is tightly regulated. Liver plays a key role in buffering blood methionine levels, and an interesting feature of its metabolism is that parallel tracks exist for the synthesis and utilization of AdoMet. To elucidate the molecular mechanism that controls metabolic fluxes in liver methionine metabolism, we have studied the dependencies of AdoMet concentration and methionine consumption rate on methionine concentration in native murine hepatocytes at physiologically relevant concentrations (40–400 µM). We find that both [AdoMet] and methionine consumption rates do not change gradually with an increase in [Met] but rise sharply (∼10-fold) in the narrow Met interval from 50 to 100 µM. Analysis of our experimental data using a mathematical model reveals that the sharp increase in [AdoMet] and the methionine consumption rate observed within the trigger zone are associated with metabolic switching from methionine conservation to disposal, regulated allosterically by switching between parallel pathways. This regulatory switch is triggered by [Met] and provides a mechanism for stabilization of methionine levels in blood over wide variations in dietary methionine intake.

Methionine is an essential amino acid that is highly toxic at elevated levels, and the liver is primarily responsible for buffering its concentration in circulation. Intracellularly, methionine is needed for the synthesis of S-adenosylmethionine (AdoMet), the major biological methylating agent. Methionine used for AdoMet synthesis can be replenished via remethylation of homocysteine. Alternatively, homocysteine can be converted to cysteine via the transsulfuration pathway. A specific feature of liver methionine metabolism is the existence of twin pathways for AdoMet synthesis and degradation. In this study, we analyzed the dependence of methionine metabolism on methionine concentration in liver cells using a combined experimental and theoretical approach. We find a sharp transition in rat hepatocyte metabolism from methionine conservation to a disposal mode with an increase in methionine concentration above its physiological range. Mathematical modeling reveals that this transition is afforded by an allosteric mechanism for switching between parallel metabolic pathways. This study demonstrates a novel mechanism of trigger behavior in biological systems by which the substrate for the metabolic network switches metabolic flux between parallel tracks for sustaining two different metabolic modes.

Localized patterns in reaction-diffusion systems

We discuss a variety of experimental and theoretical studies of localized stationary spots, oscillons, and localized oscillatory clusters, moving and breathing spots, and localized waves in reaction-diffusion systems. We also suggest some promising directions for future research in this area.

Steady state and (bi-) stability evaluation of simple protease signalling networks

Signal transduction networks are complex, as are their mathematical models. Gaining a deeper understanding requires a system analysis. Important aspects are the number, location and stability of steady states. In particular, bistability has been recognised as an important feature to achieve molecular switching. This paper compares different model structures and analysis methods particularly useful for bistability analysis. The biological applications include proteolytic cascades as, for example, encountered in the apoptotic signalling pathway or in the blood clotting system. We compare three model structures containing zero-order, inhibitor and cooperative ultrasensitive reactions, all known to achieve bistability. The combination of phase plane and bifurcation analysis provides an illustrative and comprehensive understanding of how bistability can be achieved and indicates how robust this behaviour is. Experimentally, some so-called "inactive" components were shown to have a residual activity. This has been mostly ignored in mathematical models. Our analysis reveals that bistability is only mildly affected in the case of zero-order or inhibitor ultrasensitivity. However, the case where bistability is achieved by cooperative ultrasensitivity is severely affected by this perturbation.

Dorsoventral axis formation in the Drosophila embryo--shaping and transducing a morphogen gradient

The graded nuclear location of the transcription factor Dorsal along the dorsoventral axis of the early Drosophila embryo provides positional information for the determination of different cell fates. Nuclear uptake of Dorsal depends on a complex signalling pathway comprising two parts: an extracellular proteolytic cascade transmits the dorsoventral polarity of the egg chamber to the early embryo and generates a gradient of active Spätzle protein, the ligand of the receptor Toll; an intracellular cascade downstream of Toll relays this graded signal to embryonic nuclei. The slope of the Dorsal gradient is not determined by diffusion of extracellular or intracellular components from a local source, but results from self-organised patterning, in which positive and negative feedback is essential to create and maintain the ratio of key factors at different levels, thereby establishing and stabilising the graded spatial information for Dorsal nuclear uptake.

Blood coagulation and propagation of autowaves in flow

This study analyses the effect of flow and boundary reactions on spatial propagation of waves of blood coagulation. A simple model of coagulation in plasma consisting of three differential reaction-diffusion equations was used for numerical simulations. The vessel was simulated as a two-dimensional channel of constant width, and the anticoagulant influence of thrombomodulin present on the undamaged vessel wall was taken into account. The results of the simulations showed that this inhibition could stop the coagulation process in the absence of flow in narrow channels. For the used mathematical model of coagulation this was the case if the width was below 0.2 mm. In wider vessels, the process could be stopped by the rapid blood flow. The required flow rate increased with the increase of the damage region size. For example, in a 0.5-mm wide channel with 1-mm long damage region, the propagation of coagulation may be terminated at the flow rate of more than 20 mm/min.

The Effect of Convective Flows on Blood Coagulation Processes

Two mathematical models of clot growth in the fluid flows have been considered. The first one is the model of embolus growth in a wall-adjacent flow. The effect of hydrodynamic flows on proceeding chemical reactions and the backward effect of the growing clot on the flow are taken into account. The growing thrombus is assumed to be porous and having low permeability, that is in good agreement with experimental data. The exact solutions determining the distribution of a fluid velocity close to the embolus have been used. Numerical analysis of these solutions have demonstrated that hydrodynamic flows can essentially affect the processes of blood coagulation, and consequently on the clot structure. Their presence might lead to the destruction of chemical fronts having a cylindrical symmetry and formation of the so-called chemical spots. The second model describes the initial stage of thrombus growing in the hemorrhage into a natural internal space. It permits accounting for vessel geometry and provides studying the effects of geometric parameters on fluid flows and coagulation processes. The process of thrombus growth is shown to depend on the ratio of typical values of blood velocity in the vessel and rate of chemical reactions.

Mathematical Modeling and Computer Simulation in Blood Coagulation

Over the last two decades, mathematical modeling has become a popular tool in study of blood coagulation. The in silico methods were able to yield interesting and significant results in the understanding of both individual reaction mechanisms and regulation of large sections of the coagulation cascade. The objective of this paper is to review the development of theoretical research in hemostasis and thrombosis, to summarize the main findings, and outline problems and possible prospects in the use of mathematical modeling and computer simulation approaches. This review is primarily focused on the studies dealing with: (1) the membrane-dependent reactions of coagulation; (2) regulation of the coagulation cascade, including effects of positive and negative feedback loops, diffusion of coagulation factors, and blood flow.

Mathematical modeling of material-induced blood plasma coagulation

Contact activation of the intrinsic pathway of the blood coagulation cascade is initiated when a procoagulant material interacts with coagulation factor XII, (FXII) yielding a proteolytic enzyme FXIIa. Procoagulant surface properties are thought to play an important role in activation. To study the mechanism of interaction between procoagulant materials and blood plasma, a mathematical model that is similar in form and in derivation to Michaelis-Menten enzyme kinetics was developed in order to yield tractable relationships between dose (surface area and energy) and response (coagulation time (CT)). The application of this model to experimental data suggests that CT is dependent on the FXIIa concentration and that the amount of FXIIa generated can be analyzed using a model that is linearly dependent on contact time. It is concluded from these experiments and modeling analysis that the primary mechanism for activation of coagulation involves autoactivation of FXII by the procoagulant surface or kallikrein-mediated reciprocal activation of FXII. FXIIa-induced self-amplification of FXII is insignificant.

Complex dynamics of the formation of spatially localized standing structures in the vicinity of saddle-node bifurcations of waves in the reaction-diffusion model of blood clotting

Local activation in a one-dimensional three-component reaction-diffusion model of blood clotting may lead to a formation of spatially localized standing structures (peaks) via several complex scenarios. In the first scenario, two concentration pulses first propagate from the site of activation, then stop and transform into peaks [Zarnitsina et al., Chaos 11, 57 (2001)]. Here, we examine this scenario, and also describe a different scenario of peak formation. In this scenario, two trigger waves propagate initially in opposite directions away from the site of activation. Then they stop and change direction of propagation toward each other to the activation site, where they interact and form a peak. Both of these scenarios of stable peak formation are observed in the vicinity of saddle-node bifurcation and may be viewed as a memory of the extinct wave modes.

Running pulses of complex shape in a reaction-diffusion model

In a one-dimensional reaction-diffusion model of an active medium, stable steady-state wave pulses of a new type are described. They are called multihumped because their waveforms contain several maxima of similar size. Presumably, the multihumped pulses arise via a bifurcation at which an unstable trigger wave disappears. The parameter governing this bifurcation is the diffusion coefficient for the model inhibitor. The model is analyzed by varying this parameter to determine the conditions for the emergence of multihumped pulses. The results of this analysis show how their waveform and dynamics of excitation depend on the inhibitor diffusion coefficient.

Pattern formation by vascular mesenchymal cells

In embryogenesis, immature mesenchymal cells aggregate and organize into patterned tissues. Later in life, a pathological recapitulation of this process takes place in atherosclerotic lesions, when vascular mesenchymal cells organize into trabecular bone tissue within the artery wall. Here we show that multipotential adult vascular mesenchymal cells self-organize in vitro into patterns that are predicted by a mathematical model based on molecular morphogens interacting in a reaction-diffusion process. We identify activator and inhibitor morphogens for stripe, spot, and labyrinthine patterns and confirm the model predictions in vitro. Thus, reaction-diffusion principles may play a significant role in morphogenetic processes in adult mesenchymal cells.

Unstable trigger waves induce various intricate dynamic regimes in a reaction-diffusion system of blood clotting

In this work we demonstrate that the unstable trigger waves, connecting stable and unstable spatially uniform steady states, can create intricate dynamic regimes in one-dimensional three-component reaction-diffusion model describing blood clotting. Among the most interesting regimes are the composite and replicating waves running at a constant velocity. The front part of the running composite wave remains constant, while its rear part oscillates in a complex manner. The rear part of the running replicating wave periodically gives rise to new daughter waves, which propagate in the direction opposite the parent wave. The domain of these intricate regimes in parameter space lies in the region of monostability near the region of bistability.

A Model Incorporating Some of the Mechanical and Biochemical Factors Underlying Clot Formation and Dissolution in Flowing Blood

Multiple interacting mechanisms control the formation and dissolution of clots to maintain blood in a state of delicate balance. In addition to a myriad of biochemical reactions, rheological factors also play a crucial role in modulating the response of blood to external stimuli. To date, a comprehensive model for clot formation and dissolution, that takes into account the biochemical, medical and rheological factors, has not been put into place, the existing models emphasizing either one or the other of the factors. In this paper, after discussing the various biochemical, physiologic and rheological factors at some length, we develop a model for clot formation and dissolution that incorporates many of the relevant crucial factors that have a bearing on the problem. The model, though just a first step towards understanding a complex phenomenon, goes further than previous models in integrating the biochemical, physiologic and rheological factors that come into play.

Inwardly rotating spiral waves in a reaction-diffusion system

Almost 30 years have passed since the discovery of concentric (target) and spiral waves in the spatially extended Belousov-Zhabotinsky (BZ) reaction. Since then, rotating spirals and target waves have been observed in a variety of physical, chemical, and biological reaction-diffusion systems. All of these waves propagate out from the spiral center or pacemaker. We report observations of inwardly rotating spirals found in the BZ system dispersed in water droplets of a water-in-oil microemulsion. These "antispirals" were also generated in computer simulations.