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

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.

Connecting Mechanism for Artificial Blood Vessels with High Biocompatibility

This paper proposes a connecting mechanism for artificial vessels, which can be attached/detached with ease and does not deteriorate the biocompatibility of the vessels at the joint. The inner surface of the artificial vessels is designed to have high biocompatibility. In order to make the best of the property, the proposed connecting mechanism contacts and fixes the two artificial vessels whose contacting edges are turned inside out. In this manner, blood flowing inside the vessels only has contact with the biocompatible surface. The biocompatibility, or biofouling at the joint was investigated after in vitro blood circulation tests for 72 h with scanning electron microscopy. Blood coagulation for a short term (120 min) was evaluated by activated partial thromboplastin time (APTT). A decrease of APTT was observed, although it was too small to conclude that the connector augmented the blood coagulation. The micro dialysis device which our group has developed as the artificial kidney was inserted into the blood circulation system with the connector. Decrease of APTT was similarly small. These experiments verified that the proposed connector can be readily applicable for implantable medical devices.

Model predictions of deformation, embolization and permeability of partially obstructive blood clots under variable shear flow

Thromboembolism, one of the leading causes of morbidity and mortality worldwide, is characterized by formation of obstructive intravascular clots (thrombi) and their mechanical breakage (embolization). A novel two-dimensional multi-phase computational model is introduced that describes active interactions between the main components of the clot, including platelets and fibrin, to study the impact of various physiologically relevant blood shear flow conditions on deformation and embolization of a partially obstructive clot with variable permeability. Simulations provide new insights into mechanisms underlying clot stability and embolization that cannot be studied experimentally at this time. In particular, model simulations, calibrated using experimental intravital imaging of an established arteriolar clot, show that flow-induced changes in size, shape and internal structure of the clot are largely determined by two shear-dependent mechanisms: reversible attachment of platelets to the exterior of the clot and removal of large clot pieces. Model simulations predict that blood clots with higher permeability are more prone to embolization with enhanced disintegration under increasing shear rate. In contrast, less permeable clots are more resistant to rupture due to shear rate-dependent clot stiffening originating from enhanced platelet adhesion and aggregation. These results can be used in future to predict risk of thromboembolism based on the data about composition, permeability and deformability of a clot under specific local haemodynamic conditions.

Multi-Constituent Simulation of Thrombus Deposition

In this paper, we present a spatio-temporal mathematical model for simulating the formation and growth of a thrombus. Blood is treated as a multi-constituent mixture comprised of a linear fluid phase and a thrombus (solid) phase. The transport and reactions of 10 chemical and biological species are incorporated using a system of coupled convection-reaction-diffusion (CRD) equations to represent three processes in thrombus formation: initiation, propagation and stabilization. Computational fluid dynamic (CFD) simulations using the libraries of OpenFOAM were performed for two illustrative benchmark problems: in vivo thrombus growth in an injured blood vessel and in vitro thrombus deposition in micro-channels (1.5 mm × 1.6 mm × 0.1 mm) with small crevices (125 μm × 75 μm and 125 μm × 137 μm). For both problems, the simulated thrombus deposition agreed very well with experimental observations, both spatially and temporally. Based on the success with these two benchmark problems, which have very different flow conditions and biological environments, we believe that the current model will provide useful insight into the genesis of thrombosis in blood-wetted devices, and provide a tool for the design of less thrombogenic devices.

Threshold of microvascular occlusion: injury size defines the thrombosis scenario

Damage to the blood vessel triggers formation of a hemostatic plug, which is meant to prevent bleeding, yet the same phenomenon may result in a total blockade of a blood vessel by a thrombus, causing severe medical conditions. Here, we show that the physical interplay between platelet adhesion and hemodynamics in a microchannel manifests in a critical threshold behavior of a growing thrombus. Depending on the size of injury, two distinct dynamic pathways of thrombosis were found: the formation of a nonocclusive plug, if injury length does not exceed the critical value, and the total occlusion of the vessel by the thrombus otherwise. We develop a mathematical model that demonstrates that switching between these regimes occurs as a result of a saddle-node bifurcation. Our study reveals the mechanism of self-regulation of thrombosis in blood microvessels and explains experimentally observed distinctions between thrombi of different physical etiology. This also can be useful for the design of platelet-aggregation-inspired engineering solutions.

Recent advances in computational methodology for simulation of mechanical circulatory assist devices

Ventricular assist devices (VADs) provide mechanical circulatory support to offload the work of one or both ventricles during heart failure. They are used in the clinical setting as destination therapy, as bridge to transplant, or more recently as bridge to recovery to allow for myocardial remodeling. Recent developments in computational simulation allow for detailed assessment of VAD hemodynamics for device design and optimization for both children and adults. Here, we provide a focused review of the recent literature on finite element methods and optimization for VAD simulations. As VAD designs typically fall into two categories, pulsatile and continuous flow devices, we separately address computational challenges of both types of designs, and the interaction with the circulatory system with three representative case studies. In particular, we focus on recent advancements in finite element methodology that have increased the fidelity of VAD simulations. We outline key challenges, which extend to the incorporation of biological response such as thrombosis and hemolysis, as well as shape optimization methods and challenges in computational methodology.

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.

Simulation of platelet, thrombus and erythrocyte hydrodynamic interactions in a 3D arteriole with in vivo comparison

Cylindrical blood vessels, ellipsoid platelets and biconcave-shaped deformable erythrocytes (RBCs) are important participants in hemostasis and thrombosis. However, due to the challenge of combining these components in simulation tools, few simulation studies have included all of them in realistic three-dimensional models. In the present study, we apply a recently developed simulation model to incorporate these components and analyze the flow in a thrombotic tubular arteriole, particularly the detailed hydrodynamic interactions between the thrombus shape, RBCs and platelets. It was found that at certain azimuth positions, the velocity drops in the proximity of both the upstream and downstream edge of the thrombus, which is accompanied by a rapid velocity increase in the narrowed region. The RBCs alter the flow profiles significantly from the typical low Reynolds (Re) number flow, and also enhance the deposition of free flowing platelets onto the thrombus. By evaluating the platelet-thrombus interaction and platelet-RBC interaction together, several mechanisms of platelet deposition augmentation are identified. With in vivo data comparison, our model illustrates the potential of future thrombosis studies that incorporate detailed receptor-ligand adhesion modules.

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.

Modelling platelet-blood flow interaction using the subcellular element Langevin method

In this paper, a new three-dimensional modelling approach is described for studying fluid-viscoelastic cell interaction, the subcellular element Langevin (SCEL) method, with cells modelled by subcellular elements (SCEs) and SCE cells coupled with fluid flow and substrate models by using the Langevin equation. It is demonstrated that: (i) the new method is computationally efficient, scaling as (N) for N SCEs; (ii) cell geometry, stiffness and adhesivity can be modelled by directly relating parameters to experimentally measured values; (iii) modelling the fluid-platelet interface as a surface leads to a very good correlation with experimentally observed platelet flow interactions. Using this method, the three-dimensional motion of a viscoelastic platelet in a shear blood flow was simulated and compared with experiments on tracking platelets in a blood chamber. It is shown that the complex platelet-flipping dynamics under linear shear flows can be accurately recovered with the SCEL model when compared with the experiments. All experimental details and electronic supplementary material are archived at http://biomath.math.nd.edu/scelsupplementaryinformation/.

The use of computational fluid dynamics in the development of ventricular assist devices

Progress in the field of prosthetic cardiovascular devices has significantly contributed to the rapid advancements in cardiac therapy during the last four decades. The concept of mechanical circulatory assistance was established with the first successful clinical use of heart-lung machines for cardiopulmonary bypass. Since then a variety of devices have been developed to replace or assist diseased components of the cardiovascular system. Ventricular assist devices (VADs) are basically mechanical pumps designed to augment or replace the function of one or more chambers of the failing heart. Computational Fluid Dynamics (CFD) is an attractive tool in the development process of VADs, allowing numerous different designs to be characterized for their functional performance virtually, for a wide range of operating conditions, without the physical device being fabricated. However, VADs operate in a flow regime which is traditionally difficult to simulate; the transitional region at the boundary of laminar and turbulent flow. Hence different methods have been used and the best approach is debatable. In addition to these fundamental fluid dynamic issues, blood consists of biological cells. Device-induced biological complications are a serious consequence of VAD use. The complications include blood damage (haemolysis, blood cell activation), thrombosis and emboli. Patients are required to take anticoagulation medication constantly which may cause bleeding. Despite many efforts blood damage models have still not been implemented satisfactorily into numerical analysis of VADs, which severely undermines the full potential of CFD. This paper reviews the current state of the art CFD for analysis of blood pumps, including a practical critical review of the studies to date, which should help device designers choose the most appropriate methods; a summary of blood damage models and the difficulties in implementing them into CFD; and current gaps in knowledge and areas for future work.

A relation between near-wall particle-hemodynamics and onset of thrombus formation in abdominal aortic aneurysms

A novel computational particle-hemodynamics analysis of key criteria for the onset of an intraluminal thrombus (ILT) in a patient-specific abdominal aortic aneurysm (AAA) is presented. The focus is on enhanced platelet and white blood cell residence times as well as their elevated surface-shear loads in near-wall regions of the AAA sac. The generalized results support the hypothesis that a patient's AAA geometry and associated particle-hemodynamics have the potential to entrap activated blood particles, which will play a role in the onset of ILT. Although the ILT history of only a single patient was considered, the modeling and simulation methodology provided allow for the development of an efficient computational tool to predict the onset of ILT formation in complex patient-specific cases.

A multiscale model of venous thrombus formation with surface-mediated control of blood coagulation cascade

A combination of the extended multiscale model, new image processing algorithms, and biological experiments is used for studying the role of Factor VII (FVII) in venous thrombus formation. A detailed submodel of the tissue factor pathway of blood coagulation is introduced within the framework of the multiscale model to provide a detailed description of coagulation cascade. Surface reactions of the extrinsic coagulation pathway on membranes of platelets are studied under different flow conditions. It is shown that low levels of FVII in blood result in a significant delay in thrombin production, demonstrating that FVII plays an active role in promoting thrombus development at an early stage.

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.

Role of Computational Simulations in Heart Valve Dynamics and Design of Valvular Prostheses

Computational simulations are playing an increasingly important role in enhancing our understanding of the normal human physiological function, etiology of diseased states, surgical and interventional planning, and in the design and evaluation of artificial implants. Researchers are taking advantage of computational simulations to speed up the initial design of implantable devices before a prototype is developed and hence able to reduce animal experimentation for the functional evaluation of the devices under development. A review of the reported studies to date relevant to the simulation of the native and prosthetic heart valve dynamics is the subject of the present paper. Potential future directions toward multi-scale simulation studies for our further understanding of the physiology and pathophysiology of heart valve dynamics and valvular implants are also discussed.

An extended convection diffusion model for red blood cell-enhanced transport of thrombocytes and leukocytes

Transport phenomena of platelets and white blood cells (WBCs) are fundamental to the processes of vascular disease and thrombosis. Unfortunately, the dilute volume occupied by these cells is not amenable to fluid-continuum modeling, and yet the cell count is large enough that modeling each individual cell is impractical for most applications. The most feasible option is to treat them as dilute species governed by convection and diffusion; however, this is further complicated by the role of the red blood cell (RBC) phase on the transport of these cells. We therefore propose an extended convection-diffusion (ECD) model based on the diffusive balance of a fictitious field potential, Psi, that accounts for the gradients of both the dilute phase and the local hematocrit. The ECD model was applied to the flow of blood in a tube and between parallel plates in which a profile for the RBC concentration field was imposed and the resulting platelet concentration field predicted. Compared to prevailing enhanced-diffusion models that dispersed the platelet concentration field, the ECD model was able to simulate a near-wall platelet excess, as observed experimentally. The extension of the ECD model depends only on the ability to prescribe the hematocrit distribution, and therefore may be applied to a wide variety of geometries to investigate platelet-mediated vascular disease and device-related thrombosis.

A multiscale model of thrombus development

A two-dimensional multiscale model is introduced for studying formation of a thrombus (clot) in a blood vessel. It involves components for modelling viscous, incompressible blood plasma; non-activated and activated platelets; blood cells; activating chemicals; fibrinogen; and vessel walls and their interactions. The macroscale dynamics of the blood flow is described by the continuum Navier-Stokes equations. The microscale interactions between the activated platelets, the platelets and fibrinogen and the platelets and vessel wall are described through an extended stochastic discrete cellular Potts model. The model is tested for robustness with respect to fluctuations of basic parameters. Simulation results demonstrate the development of an inhomogeneous internal structure of the thrombus, which is confirmed by the preliminary experimental data. We also make predictions about different stages in thrombus development, which can be tested experimentally and suggest specific experiments. Lastly, we demonstrate that the dependence of the thrombus size on the blood flow rate in simulations is close to the one observed experimentally.