A mathematical model of the kinetics of platelets and plasma hemostasis system interaction

Mathematical Modeling and Computer Simulation in Blood Coagulation

Over the last two decades, mathematical modeling has become a popular tool in study of blood coagulation. The in silico methods were able to yield interesting and significant results in the understanding of both individual reaction mechanisms and regulation of large sections of the coagulation cascade. The objective of this paper is to review the development of theoretical research in hemostasis and thrombosis, to summarize the main findings, and outline problems and possible prospects in the use of mathematical modeling and computer simulation approaches. This review is primarily focused on the studies dealing with: (1) the membrane-dependent reactions of coagulation; (2) regulation of the coagulation cascade, including effects of positive and negative feedback loops, diffusion of coagulation factors, and blood flow.

Mathematical modeling of material-induced blood plasma coagulation

Contact activation of the intrinsic pathway of the blood coagulation cascade is initiated when a procoagulant material interacts with coagulation factor XII, (FXII) yielding a proteolytic enzyme FXIIa. Procoagulant surface properties are thought to play an important role in activation. To study the mechanism of interaction between procoagulant materials and blood plasma, a mathematical model that is similar in form and in derivation to Michaelis-Menten enzyme kinetics was developed in order to yield tractable relationships between dose (surface area and energy) and response (coagulation time (CT)). The application of this model to experimental data suggests that CT is dependent on the FXIIa concentration and that the amount of FXIIa generated can be analyzed using a model that is linearly dependent on contact time. It is concluded from these experiments and modeling analysis that the primary mechanism for activation of coagulation involves autoactivation of FXII by the procoagulant surface or kallikrein-mediated reciprocal activation of FXII. FXIIa-induced self-amplification of FXII is insignificant.

A Model Incorporating Some of the Mechanical and Biochemical Factors Underlying Clot Formation and Dissolution in Flowing Blood

Multiple interacting mechanisms control the formation and dissolution of clots to maintain blood in a state of delicate balance. In addition to a myriad of biochemical reactions, rheological factors also play a crucial role in modulating the response of blood to external stimuli. To date, a comprehensive model for clot formation and dissolution, that takes into account the biochemical, medical and rheological factors, has not been put into place, the existing models emphasizing either one or the other of the factors. In this paper, after discussing the various biochemical, physiologic and rheological factors at some length, we develop a model for clot formation and dissolution that incorporates many of the relevant crucial factors that have a bearing on the problem. The model, though just a first step towards understanding a complex phenomenon, goes further than previous models in integrating the biochemical, physiologic and rheological factors that come into play.

Analysis of the activated partial thromboplastin time test using mathematical modeling

Activated partial thromboplastin time (APTT) is a laboratory test for the diagnosis of blood coagulation disorders. The test consists of two stages: The first one is the preincubation of a plasma sample with negatively charged materials (kaolin, ellagic acid etc.) to activate factors XII and XI; the second stage begins after the addition of calcium ions that triggers a chain of calcium-dependent enzymatic reactions resulting in fibrinogen clotting. Mathematical modeling was used for the analysis of the APTT test. The process of coagulation was described by a set of coupled differential equations that were solved by the numerical method. It was found that as little as 2.3 x 10(-9) microM of factor XIIa (1/10000 of its plasma concentration) is enough to cause the complete activation of factor XII and prekallikrein (PK) during the first 20 s of the preincubation phase. By the end of this phase, kallikrein (K) is completely inhibited, residual activity of factor XIIa is 54%, and factor XI is activated by 26%. Once a clot is formed, factor II is activated by 4%, factor X by 5%, factor IX by 90%, and factor XI by 39%. Calculated clotting time using protein concentrations found in the blood of healthy people was 40.5 s. The most pronounced prolongation of APTT is caused by a decrease in factor X concentration.

Mathematical model for the blood coagulation prothrombin time test

A mathematical model for the prothrombin time test is proposed. The time course of clotting factor activation during coagulation was calculated, and the sensitivity of the test to a decrease in the concentrations of coagulation proteins or their activities was studied. The model predicts that only severe coagulation disorders connected with a more than five-fold decrease in the concentrations or activities of the blood coagulation factors can be revealed by the test.

Mathematical model of serine protease inhibition in the tissue factor pathway to thrombin

A mathematical model has been developed to simulate the generation of thrombin by the tissue factor pathway. The model gives reasonable predictions of published experimental results without the adjustment of any parameter values. The model also accounts explicitly for the effects of serine protease inhibitors on thrombin generation. Simulations to define the optimum affinity profile of an inhibitor in this system indicate that for an inhibitor simultaneously potent against VIIa, IXa, and Xa, inhibition of thrombin generation decreases dramatically as the affinity for thrombin increases. Additional simulations show that the reason for this behavior is the sequestration of the inhibitor by small amounts of thrombin generated early in the reaction. This model is also useful for predicting the potency of compounds that inhibit thrombosis in rats. We believe that this is the first mathematical model of blood coagulation that considers the effects of exogenous inhibitors. Such a model, or extensions thereof, should be useful for evaluating targets for therapeutic intervention in the processes of blood coagulation.

Contact activation of the plasma coagulation cascade. I. Procoagulant surface chemistry and energy

Contact activation of the intrinsic pathway of porcine blood plasma coagulation is shown to be a steep exponential-like function of procoagulant surface energy, with low activation observed for poorly water-wettable surfaces and very high activation for fully water-wettable surfaces. Test procoagulants studied were a system of oxidized polystyrene films with varying wettability (surface energy) and glass discs bearing close-packed self-assembled silane monolayers (SAMs) with well-defined chemistry consisting of 12 different terminating chemical functionalities. A monotonic trend of increasing coagulation activation with increasing water wettability was observed for the oxidized polystyrene system whereas results with SAM procoagulants suggested a level of chemical specificity over and above the surface energy trend. In particular, it was noted that coagulation activation by SAMs terminated with--CO2H was much higher than anticipated based on surface wettability whereas--NH3(+)-terminated SAMs exhibited very low procoagulant activity. SAMs terminated in--(CH2)2(CF2)7CF3 behaved as anticipated based on surface energy with very low procoagulant activity and did not exhibit special properties sometimes attributed to perfluorinated compounds. Quantitative ranking of the inherent coagulation activation properties of procoagulant surfaces was obtained by application of a straightforward phenomenological model expressed in a closed-form mathematical equation relating coagulation time to procoagulant surface area. Fit of the model with a single adjustable parameter to experimental measurements of porcine platelet-poor plasma coagulation time was very good, implying that assertions and simplifications of the model adequately simulated reality. Two important propositions of the model were that (1) the number of putative "activating sites" scaled linearly with procoagulant surface area, and (2) contact activation of the plasma coagulation cascade was catalytic in the sense that these activating sites were not consumed or "poisoned" by irreversible or slowly reversible protein adsorption during coagulation. An extension of the coagulation model proposed that procoagulant activation properties scale exponentially with the surface density of polar (acid-base) sites, which, in turn, was related to procoagulant wettability.