Computational models of hemostasis: Degrees of complexity

The history of studies on blood clotting goes back to the emergence of civilized society itself. The foundations of the modern scientific study of hemostasis are based on the discovery of erythrocytes in blood in 1674 and, later, that of platelets in 1842. The causes of thrombosis are encapsulated in the Virchow Triad (dated to 1856), which refers, in modern terms, to hypercoagulability, alterations of hemodynamics (stasis), and endothelial injury. The understanding of coagulation, the network of reactions that underlies hemostasis and thrombosis, has evolved from a cascade (in 1964) into spatially distinct sets of reactions dependent on co-factors occurring on different cells in different tissues and linked together by diffusion and flow (as of 2015). Correspondingly, mathematical/computational models for hemostasis and thrombosis (which involve coagulation along with platelet aggregation in the presence of flow) have evolved in design complexity from Continuum temporal (or “homogeneous”) models to Continuum spatio-temporal models (with or without the flow) and lately into Discrete-Continuum spatio-temporal models with the flow. After a brief listing of the discoveries and historical personae that contributed to the understanding of hemostasis up to the present, the development of mathematical/computational models is traced from the late 1980s when they started gaining importance. Influential models are then highlighted. The models are reviewed in increasing order of design complexity (one of four possible methods of classification). The physiological significance of each and the insights they offer into hemostasis regulation are explained. © 2022 The Author

Traumatic vessel injuries initiating hemostasis generate high shear conditions

Blood flow is a major regulator of hemostasis and arterial thrombosis. The current view is that low and intermediate flows occur in intact healthy vessels, whereas high shear levels (>2000 s−1) are reached in stenosed arteries, notably during thrombosis. To date, the shear rates occurring at the edge of a lesion in an otherwise healthy vessel are nevertheless unknown. The aim of this work was to measure the shear rates prevailing in wounds in a context relevant to hemostasis. Three models of vessel puncture and transection were developed and characterized for a study that was implemented in mice and humans. Doppler probe measurements supplemented by a computational model revealed that shear rates at the edge of a wound reached high values, with medians of 22 000 s−1, 25 000 s−1, and 7000 s−1 after puncture of the murine carotid artery, aorta, or saphenous vein, respectively. Similar shear levels were observed after transection of the mouse spermatic artery. These results were confirmed in a human venous puncture model, where shear rates in a catheter implanted in the cubital vein reached 2000 to 27 000 s−1. In all models, the high shear conditions were accompanied by elevated levels of elongational flow exceeding 1000 s−1. In the puncture model, the shear rates decreased steeply with increasing injury size. This phenomenon could be explained by the low hydrodynamic resistance of the injuries as compared with that of the downstream vessel network. These findings show that high shear rates (>3000 s−1) are relevant to hemostasis and not exclusive to arterial thrombosis

Computational modeling of Quiescent Platelet Energy Metabolism in the Context of Whole-body Glucose Turnover

Platelets are anucleate blood cells circulating in the bloodstream for up to 9 days in quiescent state. Upon vessel wall injury, platelets become activated, change their shape and adhere to the vessel wall and each other, thus forming a thrombus and preventing the blood loss. To get energy for these processes, they can use oxidative phosphorylation and glycolysis utilizing blood glucose, stored glycogen or fatty acids as fuel. Yet, there is no agreement in experimental data on platelet functioning in quiescent and activated states. This study is a systematic analysis of the energy abilities of quiescent platelets through mathematical modeling of their energy metabolism by Flux Balance Analysis (FBA). As a result of the FBA analysis we concluded that a platelet even in quiescent state utilizes blood glucose at high rate (0.1 mM/s), producing lactate from 99% of it and about 0.2 mM/s ATP from glycolysis and respiration. Such high fluxes of glucose are not always available due to platelet’s glucose transporter (GLUT3)kinetic limitations. We positioned a “FBA” platelet in human glucose/insulin/glucagon PBPK/PD model to theoretically investigate platelet metabolism in close-to-real conditions. The main result of our study is that the stored glycogen could be daily used and resynthesized during platelet lifetime

Redistribution of TPA Fluxes in the Presence of PAI-1 Regulates Spatial Thrombolysis

The fibrin clot is gelatinous matter formed upon injury to stop blood loss and is later destroyed by fibrinolysis, an enzymatic cascade with feedback. Pharmacological fibrinolysis stimulation is also used to destroy pathological, life-threatening clots and thrombi (thrombolysis). The regulation of the nonlinear spatially nonuniform fibrinolytic process in thrombolysis is not currently well understood. We developed a reaction-diffusion-advection model of thrombolysis by tissue plasminogen activator (TPA) in an occluded vessel with a pressure gradient. Sensitivity-analysis-based model reduction was used to reveal the critical processes controlling different steps of thrombolysis. The propagation of thrombolysis in the system without flow was predominantly controlled by TPA diffusion, whereas transport of other active components was rendered nonessential either by their high fibrin-binding parameters and short lifetimes or their initial uniform distribution. The concentration of the main TPA inhibitor plasminogen activator inhibitor 1 (PAI-1) controlled both the extent of lysis propagation and the shape of fibrin spatial distribution during lysis. Interestingly, PAI-1 remained important even when its concentration was an order of magnitude below that of TPA because of its role at the edge of the diffusing TPA front. The system was robust to reaction rate constant perturbations. Using these data, a reduced model of thrombolysis was proposed. In the presence of flow, convection of TPA was the critical controlling process; although the role of PAI-1 concentration was much less in the presence of flow, its influence became greater in the presence of collateral bypassing vessels, which sufficiently reduced TPA flux through the thrombus. Flow bypass through the collateral vessel caused a decrease in TPA flux in the clotted vessel, which increased the PAI-1/TPA ratio, thus making PAI-1-induced inhibition relevant for the regulation of spatial lysis up to its arrest

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. © 2018 Elsevier B.V

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Modeling Thrombus Shell: Linking Adhesion Receptor Properties and Macroscopic Dynamics

Damage to arterial vessel walls leads to the formation of platelet aggregate, which acts as a physical obstacle for bleeding. An arterial thrombus is heterogeneous; it has a dense inner part (core) and an unstable outer part (shell). The thrombus shell is very dynamic, being composed of loosely connected discoid platelets. The mechanisms underlying the observed mobility of the shell and its (patho)physiological implications are unclear. To investigate arterial thrombus mechanics, we developed a novel, to our knowledge, two-dimensional particle-based computational model of microvessel thrombosis. The model considers two types of interplatelet interactions: primary reversible (glycoprotein Ib (GPIb)-mediated) and stronger integrin-mediated interaction, which intensifies with platelet activation. At high shear rates, the former interaction leads to adhesion, and the latter is primarily responsible for stable platelet aggregation. Using a stochastic model of GPIb-mediated interaction, we initially reproduced experimental curves that characterize individual platelet interactions with a von Willebrand factor-coated surface. The addition of the second stabilizing interaction results in thrombus formation. The comparison of thrombus dynamics with experimental data allowed us to estimate the magnitude of critical interplatelet forces in the thrombus shell and the characteristic time of platelet activation. The model predicts moderate dependence of maximal thrombus height on the injury size in the absence of thrombin activity. We demonstrate that the developed stochastic model reproduces the observed highly dynamic behavior of the thrombus shell. The presence of primary stochastic interaction between platelets leads to the properties of thrombus consistent with in vivo findings; it does not grow upstream of the injury site and covers the whole injury from the first seconds of the formation. А simplified model, in which GPIb-mediated interaction is deterministic, does not reproduce these features. Thus, the stochasticity of platelet interactions is critical for thrombus plasticity, suggesting that interaction via a small number of bonds drives the dynamics of arterial thrombus shell. © 2020 Biophysical Societ

In vitro flow-based assay: From simple toward more sophisticated models for mimicking hemostasis and thrombosis

In vitro flow-based assays are widely used to investigate the role of platelets and coagulation in hemostasis and thrombosis. Their main advantage over other assays relies on the fact that they integrate blood flow that regulates many aspects of platelet function, including adhesion, activation, and aggregation. Blood flow is also central in the regulation of coagulation through its ability to modulate the local concentrations of coagulation factors within and around thrombi. The most broadly used assay to study thrombus formation consists in perfusing whole blood over immobilized fibrillar collagen through a single channel, which helps to reproduce thrombus formation as it occurs in vivo after vascular injury, with platelets adhering, becoming activated, and forming a mural thrombus. This process can also be studied under conditions of thrombin generation, notably by recalcifying blood collected in sodium citrate. In this manuscript, we briefly discuss the advantages and limits of this broadly used "in vitro thrombus formation model." The main emphasis is on the description of the most recent developments regarding design of new flow models and new techniques, and how these may advance the landscape of in vitro studies into the formation of physiological or pathophysiological thrombi. Challenges linked to mimicking the formation of a hemostatic plug in a healthy vessel or a thrombus in diseased arteries and the complexity of reproducing the various aspects of venous thrombosis are discussed. Future directions are proposed to improve the physiological or pathophysiological relevance of current flow-based assays

Wall shear rates in human and mouse arteries: Standardization of hemodynamics for in vitro blood flow assays: Communication from the ISTH SSC subcommittee on biorheology

Hemodynamics play a central role in hemostasis and thrombosis by affecting all aspects linked to platelet functions and coagulation. In vitro flow devices are extensively used in basic research, pharmacological studies, antiplatelet agent screening, and development of diagnostic tools. Because hemodynamic conditions vary tremendously throughout the vascular tree and among different (patho)physiological processes, it is important to use flow conditions based on relevant biorheological reference ranges. Surprisingly, it is particularly difficult to find a concise overview of relevant hemodynamic parameters in various human and mouse vessels. To our knowledge, this is the first time an inventory of flow conditions in healthy, non-diseased, human and mouse vessels has been created. The objective of providing such a repertoire is to aid researchers in the field of hemostasis and thrombosis in choosing rheological conditions relevant in in vitro flow experiments and to promote harmonization of flow-based assays to facilitate comparative evaluations between studies. With reference to the human, we discuss relevant similarities and discrepancies in wall shear rates in the mouse, which are typically one order of magnitude greater in agreement with allometric scaling laws between species. Importantly, we bring the attention of the researchers to the fact that the relevant range of average wall shear rates in human arteries where clinically relevant arterial thrombosis occurs may fall as low as 100 to 200 s−1, thus significantly overlapping with what are considered “venous” shear rates. The same range for the murine arteries used for arterial thrombosis models may significantly exceed 1000 s−1 reaching values considered to be “pathological.”