Reaction-diffusion waves of blood coagulation

Dynamic adaptive moving mesh finite-volume method for the blood flow and coagulation modeling

In this work, we develop numerical methods for the solution of blood flow and coagulation on dynamic adaptive moving meshes. We consider the blood flow as a flow of incompressible Newtonian fluid governed by the Navier-Stokes equations. The blood coagulation is introduced through the additional Darcy term, with a permeability coefficient dependent on reactions. To this end, we introduce moving mesh collocated finite-volume methods for the Navier-Stokes equations, advection-diffusion equations, and a method for the stiff cascade of reactions. A monolithic nonlinear system is solved to advance the solution in time. The finite volume method for the Navier-Stokes equations features collocated arrangement of pressure and velocity unknowns and a coupled momentum and mass flux. The method is conservative and inf-sup stable despite the saddle point nature of the system. It is verified on a series of analytical problems and applied to the blood flow problem in the deforming domain of the right ventricle, reconstructed from a time series of computed tomography scans. At last, we demonstrate the ability to model the coagulation process in deforming microfluidic capillaries.

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

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.

Clustering of Thrombin Generation Test Data Using a Reduced Mathematical Model of Blood Coagulation

Correct interpretation of the data from integral laboratory tests, including Thrombin Generation Test (TGT), requires biochemistry-based mathematical models of blood coagulation. The purpose of this study is to describe the experimental TGT data from healthy donors and hemophilia A (HA) and B (HB) patients. We derive a simplified ODE model and apply it to analyze the TGT data from healthy donors and HA/HB patients with in vitro added tissue factor pathway inhibitor (TFPI) antibody. This model allows the characterization of hemophilia patients in the space of three most important model parameters. The proposed approach may provide a new quantitative tool for the analysis of experimental TGT. Also, it gives a reduced model of coagulation verified against clinical data to be used in future theoretical large-scale modeling of thrombosis in flow.

Effects of hydrophobicity, tethering and size on flow-induced activation of von Willebrand factor multimers

Von Willebrand factor (VWF) is a multimeric protein of blood plasma that mediates platelet adhesion to injury under strong hemodynamic flows in arterias and microvasvulature. We present a 3D coarse-grained computer model of VWF multimers in flowing viscous fluid that explicitely grasps the dynamics, the conformational changes and the hydrodynamics-induced activation of adhesivity of these protein concatamers to GPIb receptor of blood platelets. The model is based on the fluctuating Lattice Boltzmann method for modelling the hydrodynamics in the simulation box and the Lagrangian particle dynamics coupled to the fluid by a viscous drag force. The model has been validated by the comparison with the experimental data found in literature. We studied the effect of hydrophobic interactions on the conformational dynamics of VWF multimers. The simulations suggest that the contour length is an important parameter that controls the functionality of VWF multimers in blood. We also demonstrate that tethering to the surface of a vessel wall promoted the flow-induced activation of VWF, while those multimers that remain untethered and move freely in the blood plasma require a stronger shearing to get activated.

Reduced model to predict thrombin and fibrin during thrombosis on collagen/tissue factor under venous flow: Roles of γ'-fibrin and factor XIa

During thrombosis, thrombin generates fibrin, however fibrin reversibly binds thrombin with low affinity E-domain sites (K_D = 2.8 μM) and high affinity γ’-fibrin sites (K_D = 0.1 μM). For blood clotting on collagen/tissue factor (1 TF-molecule/μm²) at 200 s^-1 wall shear rate, high μM-levels of intraclot thrombin suggest robust prothrombin penetration into clots. Setting intraclot zymogen concentrations to plasma levels (and neglecting cofactor rate limitations) allowed the linearization of 7 Michaelis-Menton reactions between 6 species to simulate intraclot generation of: Factors FXa (via TF/VIIa or FIXa), FIXa (via TF/FVIIa or FXIa), thrombin, fibrin, and FXIa. This reduced model [7 rates, 2 K_D’s, enzyme half-lives~1 min] predicted the measured clot elution rate of thrombin-antithrombin (TAT) and fragment F1.2 in the presence and absence of the fibrin inhibitor Gly-Pro-Arg-Pro. To predict intraclot fibrin reaching 30 mg/mL by 15 min, the model required fibrinogen penetration into the clot to be strongly diffusion-limited (actual rate/ideal rate = 0.05). The model required free thrombin in the clot (~100 nM) to have an elution half-life of ~2 sec, consistent with measured albumin elution, with most thrombin (>99%) being fibrin-bound. Thrombin-feedback activation of FXIa became prominent and reached 5 pM FXIa at >500 sec in the simulation, consistent with anti-FXIa experiments. In predicting intrathrombus thrombin and fibrin during 15-min microfluidic experiments, the model revealed “cascade amplification” from 30 pM levels of intrinsic tenase to 15 nM prothrombinase to 15 μM thrombin to 90 μM fibrin. Especially useful for multiscale simulation, this reduced model predicts thrombin and fibrin co-regulation during thrombosis under flow.

During blood clotting events, a complex series of reaction are involved. Simulation gives insights to the concentration of different enzymes which are at too low of concentration to be detected. However, the models are often large and difficult to solve for clotting under flow conditions. With a thin film approximation, we were able to simplify clotting under flow with parameters from literature, with only 3 adjusted in order to fit the experimental data. This model gave insights into the dynamics of the species involved, and the roles of γ’-fibrin and thrombin feedback activation. This reduced model may be useful in further multiscale simulations.

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.