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      A mathematical model to quantify the effects of platelet count, shear rate, and injury size on the initiation of blood coagulation under venous flow conditions

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          Abstract

          Platelets upregulate the generation of thrombin and reinforce the fibrin clot which increases the incidence risk of venous thromboembolism (VTE). However, the role of platelets in the pathogenesis of venous cardiovascular diseases remains hard to quantify. An experimentally validated model of thrombin generation dynamics is formulated. The model predicts that a high platelet count increases the peak value of generated thrombin as well as the endogenous thrombin potential (ETP) as reported in experimental data. To investigate the effects of platelets density, shear rate, and wound size on the initiation of blood coagulation, we calibrate a previously developed model of venous thrombus formation and implement it in 3D using a novel cell-centered finite-volume solver. We conduct numerical simulations to reproduce in vitro experiments of blood coagulation in microfluidic capillaries. Then, we derive a reduced one-equation model of thrombin distribution from the previous model under simplifying hypotheses and we use it to determine the conditions of clotting initiation on the platelet count, the shear rate, and the plasma composition. The initiation of clotting also exhibits a threshold response to the size of the wounded region in good agreement with the reported experimental findings.

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          A model for the stoichiometric regulation of blood coagulation.

          We have developed a model of the extrinsic blood coagulation system that includes the stoichiometric anticoagulants. The model accounts for the formation, expression, and propagation of the vitamin K-dependent procoagulant complexes and extends our previous model by including: (a) the tissue factor pathway inhibitor (TFPI)-mediated inactivation of tissue factor (TF).VIIa and its product complexes; (b) the antithrombin-III (AT-III)-mediated inactivation of IIa, mIIa, factor VIIa, factor IXa, and factor Xa; (c) the initial activation of factor V and factor VIII by thrombin generated by factor Xa-membrane; (d) factor VIIIa dissociation/activity loss; (e) the binding competition and kinetic activation steps that exist between TF and factors VII and VIIa; and (f) the activation of factor VII by IIa, factor Xa, and factor IXa. These additions to our earlier model generate a model consisting of 34 differential equations with 42 rate constants that together describe the 27 independent equilibrium expressions, which describe the fates of 34 species. Simulations are initiated by "exposing" picomolar concentrations of TF to an electronic milieu consisting of factors II, IX, X, VII, VIIa, V, and VIIII, and the anticoagulants TFPI and AT-III at concentrations found in normal plasma or associated with coagulation pathology. The reaction followed in terms of thrombin generation, proceeds through phases that can be operationally defined as initiation, propagation, and termination. The generation of thrombin displays a nonlinear dependence upon TF, AT-III, and TFPI and the combination of these latter inhibitors displays kinetic thresholds. At subthreshold TF, thrombin production/expression is suppressed by the combination of TFPI and AT-III; for concentrations above the TF threshold, the bolus of thrombin produced is quantitatively equivalent. A comparison of the model with empirical laboratory data illustrates that most experimentally observable parameters are captured, and the pathology that results in enhanced or deficient thrombin generation is accurately described.
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            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.
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              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.
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                Author and article information

                Contributors
                Role: ConceptualizationRole: Data curationRole: InvestigationRole: MethodologyRole: ValidationRole: VisualizationRole: Writing – original draft
                Role: Formal analysisRole: Funding acquisitionRole: InvestigationRole: MethodologyRole: ResourcesRole: SoftwareRole: ValidationRole: VisualizationRole: Writing – original draft
                Role: ConceptualizationRole: Data curationRole: Formal analysisRole: MethodologyRole: SupervisionRole: ValidationRole: Writing – review & editing
                Role: ConceptualizationRole: Formal analysisRole: Funding acquisitionRole: InvestigationRole: MethodologyRole: ResourcesRole: SoftwareRole: SupervisionRole: ValidationRole: Writing – review & editing
                Role: ConceptualizationRole: Formal analysisRole: Funding acquisitionRole: InvestigationRole: Project administrationRole: SupervisionRole: ValidationRole: Writing – review & editing
                Role: Editor
                Journal
                PLoS One
                PLoS ONE
                plos
                plosone
                PLoS ONE
                Public Library of Science (San Francisco, CA USA )
                1932-6203
                2020
                29 July 2020
                : 15
                : 7
                : e0235392
                Affiliations
                [1 ] Ecole Centrale Casablanca, Casablanca, Morocco
                [2 ] Marchuk Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow, Russia
                [3 ] Services de Pharmacologie Clinique, Hospices Civils de Lyon, Lyon, France
                [4 ] Sechenov University, Moscow, Russia
                [5 ] Moscow Institute of Physics and Technology, Dolgoprudny, Russia
                [6 ] Institut Camille Jordan, Université Lyon 1, Villeurbanne, France
                [7 ] INRIA team Dracula, INRIA Lyon La Doua, Villeurbanne, France
                [8 ] Peoples’ Friendship University of Russia (RUDN University), Moscow, Russia
                University of New South Wales, AUSTRALIA
                Author notes

                Competing Interests: The authors have declared that no competing interests exist.

                Author information
                http://orcid.org/0000-0002-2729-6484
                Article
                PONE-D-19-27289
                10.1371/journal.pone.0235392
                7390270
                32726315
                54fda57e-6741-4743-b195-aa0f12c2f5d2
                © 2020 Bouchnita et al

                This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

                History
                : 16 October 2019
                : 16 June 2020
                Page count
                Figures: 8, Tables: 2, Pages: 23
                Funding
                Funded by: Russian Foundation for Basic Research
                Award ID: 18-31-20048
                Award Recipient :
                Funded by: RUDN University Program
                Award ID: 5-100
                Award Recipient :
                Funded by: Moscow Center for Fundamental and Applied Mathematics
                Award Recipient :
                K. Terekhov was supported by RFBR grant 18-31-20048. Yu. Vassilevski was supported by the world-class research center "Moscow Center for Fundamental and Applied Mathematics" (agreement with the Ministry of Education and Science of the Russian Federation No. 075-15-2019-1624). V. Volpert was supported by the “RUDN University Program 5-100.” The development of the 3D numerical model of thrombus growth under venous flow conditions was supported by Russian Science Foundation grant 14-31-00024. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
                Categories
                Research Article
                Biology and Life Sciences
                Anatomy
                Body Fluids
                Blood
                Platelets
                Medicine and Health Sciences
                Anatomy
                Body Fluids
                Blood
                Platelets
                Biology and Life Sciences
                Physiology
                Body Fluids
                Blood
                Platelets
                Biology and Life Sciences
                Cell Biology
                Cellular Types
                Animal Cells
                Blood Cells
                Platelets
                Biology and Life Sciences
                Biochemistry
                Proteins
                Thrombin
                Biology and Life Sciences
                Anatomy
                Body Fluids
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                Blood Plasma
                Medicine and Health Sciences
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                Biology and Life Sciences
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                Body Fluids
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                Biology and Life Sciences
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                Medicine and Health Sciences
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                Custom metadata
                The entire set of equations, boundary conditions and parameter values used is presented in the article. These data are sufficient to reproduce the results presented in the article.

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