Dynamics of spatially nonuniform patterning in the model of blood coagulation.

We propose a reaction-diffusion model that describes in detail the cascade of molecular events during blood coagulation. In a reduced form, this model contains three equations in three variables, two of which are self-accelerated. One of these variables, an activator, behaves in a threshold manner. An inhibitor is also produced autocatalytically, but there is no inhibitor threshold, because it is generated only in the presence of the activator. All model variables are set to have equal diffusion coefficients. The model has a stable stationary trivial state, which is spatially uniform and an excitation threshold. A pulse of excitation runs from the point where the excitation threshold has been exceeded. The regime of its propagation depends on the model parameters. In a one-dimensional problem, the pulse either stops running at a certain distance from the excitation point, or it reaches the boundaries as an autowave. However, there is a parameter range where the pulse does not disappear after stopping and exists stationarily. The resulting steady-state profiles of the model variables are symmetrical relative to the center of the structure formed. (c) 2001 American Institute of Physics.

A mathematical model for the spatio-temporal dynamics of intrinsic pathway of blood coagulation. II. Results.

This paper continues our study (see Part I) where we modeled the spatio-temporal dynamics of the intrinsic pathway of blood coagulation. Here, we analyzed this model and showed that it describes the threshold behavior of coagulation. When activation is subthreshold (which produces not more than 0.07 nM factor XIa at saturating free calcium concentrations of 2 mM or higher), the concentration of generated thrombin remains below 0.01 nM. At the abovethreshold activation corresponding to factor XIa exceeding 0.07 nM, the concentration of thrombin explosively increases and then abruptly decreases. The peak concentration of thrombin reaches hundreds nM. With respect to free calcium concentration, the system also behaves in a threshold manner. For activation corresponding to 0.3 nM factor XIa, the threshold concentration of free calcium where the outburst of explosive thrombin generation occur is equal to 0.21 mM. The model simulations are in a good agreement with the experimentally recorded kinetics of thrombin generation at different concentrations of free calcium (1). Analysis of the spatial dynamics of coagulation showed that if activation exceeded the threshold level at a certain point, the concentration wave of thrombin arises and propagates at a high speed from the activation zone. The parameters of this wave depends mainly on the efficiency of the feedback loops. The feedback loops through the backbone factors of the intrinsic pathway (autoactivation of factor X or activation of factor XI by thrombin) has a potential for the unlimited propagation of the thrombin wave. With increasing activity of activated protein C (the effect equivalent to that of thrombomodulin), oscillating regimes arise in the model. The first thrombin wave is followed by several secondary running waves. The amplitudes of secondary waves increases to the periphery of the clot consolidating its surface layer.

A mathematical model for the spatio-temporal dynamics of intrinsic pathway of blood coagulation. I. The model description.

We developed and analyzed the mathematical model of the intrinsic pathway based on the current biochemical data on the kinetics of blood coagulation individual stages. The model includes eight differential equations describing the spatio-temporal dynamics of activation of factors XI, IX, X, II, I, VIII, V, and protein C. The assembly of tenase and prothrombinase complexes is considered as a function of calcium concentration. The spatial dynamics of coagulation was analyzed for the one-dimensional case. We examined the formation of active factors, their spreading, and growth of the clot from the site of injury in the direction perpendicular to the vessel wall, into the blood thickness. We assumed that the site of injury (in the model one boundary of the space segment under examination) becomes a source of the continuous influx of factor XIa. In the first part, we described the model, selected the parameters, etc. In the second part, we compared the model with experimental data obtained in the homogeneous system and analyzed the spatial dynamics of the clot growth.