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Mathematical analysis of mural thrombogenesis. Concentration profiles of platelet-activating agents and effects of viscous shear flow.

BJ Folie — 2008-04-17T18:24:40-00:00

Biophys. J., Vol. 56, No. 6. (1 December 1989), pp. 1121-1141.

The concentration profiles of adenosine diphosphate (ADP), thromboxane A2 (TxA2), thrombin, and von Willebrand factor (vWF) released extracellularly from the platelet granules or produced metabolically on the platelet membrane during thrombus growth, were estimated using finite element simulation of blood flow over model thrombi of various shapes and dimensions. The wall fluxes of these platelet-activating agents were estimated for each model thrombus at three different wall shear rates (100 s-1, 800 s-1, and 1,500 s-1), employing experimental data on thrombus growth rates and sizes. For that purpose, whole human blood was perfused in a parallel-plate flow chamber coated with type l fibrillar human collagen, and the kinetic data collected and analyzed by an EPl-fluorescence video microscopy system and a digital image processor. It was found that thrombin concentrations were large enough to cause irreversible platelet aggregation. Although heparin significantly accelerated thrombin inhibition by antithrombin lll, the remaining thrombin levels were still significantly above the minimum threshold required for irreversible platelet aggregation. While ADP concentrations were large enough to cause irreversible platelet aggregation at low shear rates and for small aggregate sizes, TxA2 concentrations were only sufficient to induce platelet shape change over the entire range of wall shear rates and thrombi dimensions studied. Our results also indicated that the local concentration of vWF multimers released from the platelet alpha-granules could be sufficient to modulate platelet aggregation at low and intermediate wall shear rates (less than 1,000 s-1). The sizes of standing vortices formed adjacent to a growing aggregate and the embolizing stresses and the torque, acting at the aggregate surface, were also estimated in this simulation. It was found that standing vortices developed on both sides of the thrombus even at low wall shear rates. Their sizes increased with thrombus size and wall shear rate, and were largely dependent upon thrombus geometry. The experimental observation that platelet aggregation occurred predominantly in the spaces between adjacent thrombi, confirmed the numerical prediction that those standing vortices are regions of reduced fluid velocities and high concentrations of platelet-activating substances, capable of trapping and stimulating platelets for aggregation. The average shear stress and normal stress, as well as the torque, acting to detach the thrombus, increased with increasing wall shear rate. Both stresses were found to be nearly independent of thrombus size and only weekly dependent upon thrombus geometry. Although both stresses had similar values at low wall shear rates, the average shear stress became the predominant embolizing stress at high wall shear rates.

Platelet active concentration profiles near growing thrombi. A mathematical consideration.

JA Hubbell — 2008-04-17T18:20:44-00:00

Biophys. J., Vol. 50, No. 5. (1 November 1986), pp. 937-945.

When blood contacts foreign material surfaces, platelets usually adhere and form aggregates on those surfaces, generating mural thrombi. The mechanism of mural thrombogenesis is not completely understood, but one hypothesis states that the local release of certain platelet-active substances from the platelets composing an initial small thrombus stimulates additional platelet recruitment to that thrombus, resulting in growth of the cell aggregate. The purpose of this paper is to investigate the feasibility of this hypothesis. Concentration profiles of adenosine diphosphate (ADP), thromboxane A2 (TxA2), and thrombin were computed in the vicinity of growing model thrombi 10 and 20 micron long. Wall shear rates of 100, 500, and 1,500 s-1 were considered for blood flowing through a thin rectangular slit 200 micron wide coated with collagen, a predominant subendothelial protein. The local concentrations of ADP and TxA2 were marginally large enough to stimulate platelet activation individually, while local thrombin levels can be much greater than required for stimulation. Antithrombin III, a natural thrombin inhibitor, did not significantly reduce the thrombin concentrations, but antithrombin III accelerated by heparin greatly reduced the local thrombin concentrations. The reduced thrombin levels may, however, still be large enough to activate platelets.

Computational Simulation of Platelet Deposition and Activation: II. Results for Poiseuille Flow over Collagen

Erik Sorensen — 2008-04-17T18:17:08-00:00

Annals of Biomedical Engineering, Vol. 27, No. 4. (1 July 1999), pp. 449-458.

We have previously described the development of a two-dimensional computational model of platelet deposition onto biomaterials from flowing blood (Sorensen et al., Ann. Biomed. Eng. 27:436–448, 1999). The model requires estimation of four parameters to fit it to experimental data: shear-dependent platelet diffusivity and three platelet-deposition-related reaction rate constants. These parameters are estimated for platelet deposition onto a collagen substrate for simple parallel-plate flow of whole blood in both the presence and absence of thrombin. One set of experimental results is used as a benchmark for model-fitting purposes. The “trained” model is then validated by applying it to additional test cases from the literature for parallel-plate Poiseuille flow over collagen at both higher and lower wall shear rates, and in the presence of various anticoagulants. The predicted values agree very well with the experimental results for the training cases, and good reproduction of deposition trends and magnitudes is obtained for the heparin, but not the citrate, validation cases. The model is formulated to be easily extended to synthetic biomaterials, as well as to more complex flows. © 1999 Biomedical Engineering Society.

Computational Simulation of Platelet Deposition and Activation: I. Model Development and Properties

Erik Sorensen — 2008-04-17T18:16:07-00:00

Annals of Biomedical Engineering, Vol. 27, No. 4. (1 July 1999), pp. 436-448.

To better understand the mechanisms leading to the formation and growth of mural thrombi on biomaterials, we have developed a two-dimensional computational model of platelet deposition and activation in flowing blood. The basic formulation is derived from prior work by others, with additional levels of complexity added where appropriate. It is comprised of a series of convection-diffusion-reaction equations which simulate platelet-surface and platelet-platelet adhesion, platelet activation by a weighted linear combination of agonist concentrations, agonist release and synthesis by activated platelets, platelet-phospholipid-dependent thrombin generation, and thrombin inhibition by heparin. The model requires estimation of four parameters to fit it to experimental data: shear-dependent platelet diffusivity and resting and activated platelet-surface and platelet-platelet reaction rate constants. The model is formulated to simulate a wide range of biomaterials and complex flows. In this article we present the basic model and its properties; in Part II (Sorensen et al., Ann. Biomed. Eng. 27:449–458, 1999) we apply the model to experimental results for platelet deposition onto collagen. © 1999 Biomedical Engineering Society.

Boundary-layer Type Solutions for Initial Platelet Activation and Deposition

T David — 2008-04-17T18:14:27-00:00

Journal of Theoretical Medicine (January 2002), pp. 95-108.

This paper presents, on the basis of high Peclet number, a mathematical model for the activation and initial adhesion of flowing platelets onto a surface. In contrast to past work, the model is applicable to general 2D and axi-symmetric flows where the wall shear stress is known a priori. Results indicate that for high activation reaction rates there exist two layers, one containing only activated platelets and the other both activated and non-activated platelets. Fundamental relationships are proposed between the adhesion rate of platelets to the surface and the characteristic parameters of Peclet number and Reynolds number. Activation in the bulk fluid (blood) is characterised by the Damkohler number, which is a function of activation rate and the free-stream velocity. It is shown that, as the free-stream velocity varies, there exists a maximum of activated platelet flux to the wall for particular values of the velocity. These values, at which the maximum occur, are themselves functions of the platelet activation rate. As the free-stream velocity increases the activation of platelets ceases altogether and adhesion is reduced to a very small value strengthening the hypothesis of the correlation between atherogenesis/thrombogenesis and areas of low shear.

Activation and extinction models for platelet adhesion

T David — 2008-04-17T18:06:10-00:00

Biorheology, Vol. 39, No. 3-4. (2002), pp. 293-298.

Adherent platelets are an important part of both thrombus formation and in certain stages of atherogenesis. Platelets can be activated by potent chemicals released from adherent platelets and adhere far more readily than unactivated ones. An analytical and numerical model is presented utilising high Peclet number for the activation and adhesion of platelets in shear flows. The model uses a similarity transformation, which characterises the relationship between convective, diffusive transport and the bulk platelet activating reaction mechanism. A first order surface reaction mechanism is used to model platelet adhesion at the wall (cell) surface. The reduced Damköhler number, Mscr, characterises the importance of the bulk reaction and includes both convective and diffusive terms. For a high rate of blood flow (Mscr→0) the activation of platelets can effectively be terminated. In contrast, for (Mscr→∞) an inner layer of activated platelets exists with an infinitesimally thin reaction sheet separating activated and non-activated platelets. This characterisation by the Damköhler number highlights results found clinically, in that thrombus forms in areas of low shear (high Mscr) and in some cases an increased blood flow (low Mscr) can inhibit the activation of platelets completely. The model shows the critical balance that exists between convection, diffusion and reaction.

Blood flow velocity effects and role of activation delay time on growth and form of platelet thrombi

Igor Pivkin — 2008-04-17T17:52:05-00:00

Proceedings of the National Academy of Sciences, Vol. 103, No. 46. (14 November 2006), pp. 17164-17169.

Mural thrombi are composed dominantly of platelets and develop under a blood flow. Portions can break off and are carried in the blood flow as emboli. Thrombus growth rates are affected by the velocity of the blood flow, but they do not simply increase with it, they exhibit a maximum, with subsequent decrease. Whereas this variation indicates an interaction of biochemical and physical processes, studies have concentrated widely on understanding only the biochemical processes. Here we show results of simulation of thrombus formation in 3D flows by accounting for the movements of individual platelets. Each platelet follows prescribed rules for interactions while the local flow around the thrombus continuously adjusts to the growing structure of the thrombus, also when embolization occurs. With an activation delay time assigned to each platelet we demonstrate the dependence of thrombus growth rate on blood velocity as found experimentally by Begent and Born [Begent N, Born GV (1970) Nature 227:926-930]. With activated platelets having mutual tensile action sustainable up to a prescribed distance we achieve thrombus growth faster than with shorter maximum distances that make a thrombus less porous; when the prescribed maximum distance is large enough the thrombus shape is not like a "hill" but like a "carpet." We find that thrombus growth rate is enhanced by modest pulsatility but less so when pulsations are amplified in part because of more embolization. 10.1073/pnas.0608546103

Activation waves in a model of platelet aggregation: existence of solutions and stability of travelling fronts

Victor Moll — 2008-04-17T17:32:25-00:00

Journal of Mathematical Biology, Vol. 31, No. 7. (1993), pp. 675-701.

Platelets cohere to one another to form platelet aggregates as part of the blood's clotting response. The ability of a platelet to participate in this process depends on its prior ‘activation’ by chemicals released into the blood plasma by other activated platelets. We study the piecewise-linear system of reaction-diffusion equations which, in one spatial dimension, describe the chemically-mediated spread of platelet activation. We establish the existence of classical solutions to this system of equations, and show that these solutions do not blow up in finite time. We also explicitly construct travelling front solutions and discuss their stability. Finally, we present numerical evidence which suggests that for a broad range of initial data with the correct limiting values at ± 8, the solution to the initial value problem rapidly evolves into the travelling front solution provided the front is linearly stable.

An estimated shape function for drift in a platelet-transport model.

C Yeh — 2008-04-17T17:20:05-00:00

Biophys. J., Vol. 67, No. 3. (1 September 1994), pp. 1252-1259.

Prior work has shown that concentration profiles of platelets in flowing whole blood and of platelet-sized beads in flowing blood suspensions can include near-wall excesses. A model to describe this phenomenon was built about a single-component convective diffusion equation. To incorporate redistribution to preferred sites by shear flows of red cell suspensions, the model used a drift shape function (in addition to the commonly used augmented diffusion coefficient). This paper reports experiments that provide an average concentration profile from which the shape function for that model is calculated; the experiments and shape function are for the particular conditions of 40% hematocrit, platelet-sized latex beads (2.5 microns diameter), tube ID of 217 microns, and a wall shear rate of 555 s-1. Less precise estimates of the shape function obtained from data of previous studies indicate that the shape function is similar for the hematocrit of 15%.

Coagulation under Flow: The Influence of Flow-Mediated Transport on the Initiation and Inhibition of Coagulation

AL Fogelson — 2008-04-10T19:53:08-00:00

Pathophysiology of Haemostasis and Thrombosis, Vol. 34 (2005), pp. 91-108.

A mathematical model of intravascular coagulation is presented; it encompasses the biochemistry of the tissue factor pathway, platelet activation and deposition on the subendothelium, and flow- and diffusion-mediated transport of coagulation proteins and platelets. Simulation experiments carried out with the model indicate the predominant role played by the physical processes of platelet deposition and flow-mediated removal of enzymes in inhibiting coagulation in the vicinity of vascular injury. Sufficiently rapid production of factors IXa and Xa by the TF:VIIa complex can overcome this inhibition and lead to formation of significant amounts of the tenase complex on the surface of activated platelets and, as a consequence, to substantial thrombin production. Chemical inhibitors are seen to play almost no (TFPI) or little (AT-III and APC) role in determining whether substantial thrombin production will occur. The role of APC is limited by the necessity for diffusion of thrombin from the site of injury to nearby endothelial cells to form the thrombomodulin-thrombin complex and for diffusion in the reverse direction of the APC made by this complex. TFPI plays an insignificant part in inhibiting the TF:VIIa complex under the conditions studied whether its action involves sequential binding of TFPI to Xa and then TFPI:Xa to TF:VIIa, or direct binding of TFPI to Xa already bound to the TF:VIIa complex.

Surface-Mediated Control of Blood Coagulation: The Role of Binding Site Densities and Platelet Deposition

Andrew Kuharsky — 2008-03-27T18:00:29-00:00

Biophys. J., Vol. 80, No. 3. (1 March 2001), pp. 1050-1074.

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.

Model of platelet transport in flowing blood with drift and diffusion terms.

EC Eckstein — 2008-03-27T17:46:50-00:00

Biophys. J., Vol. 60, No. 1. (1 July 1991), pp. 53-69.

A drift term is added to the convective diffusion equation for platelet transport so that situations with near-wall excesses of platelets can be described. The mathematical relationship between the drift and the fully developed, steady-state platelet concentration profile is shown and a functional form of the drift that leads to concentration profiles similar to experimentally determined profiles is provided. The transport equation is numerically integrated to determine concentration profiles in the developing region of a tube flow. With the approximate drift function and typical values of augmented diffusion constant, the calculated concentration profiles have near-wall excesses that mimic experimental results, thus implying the extended equation is a valid description of rheological events. Stochastic differential equations that are equivalent to the convective diffusion transport equation are shown, and simulations with them are used to illustrate the impact of the drift term on platelet concentration profiles during deposition in a tube flow.