The effect of flow and mass transport in thrombogenesis

The paper presents a mathematical analysis of the contributions of flow and mass transport to a single reactive event at a blood vessel wall. The intent is to prepare the ground for a comprehensive study of the intertwining of these contributions with the reaction network of the coagulation cascade. We show that in all vessels with local mural activity, or in "large" vessels (d greater than 0.1 mm) with global reactivity, events at the tubular wall can be rigorously described by algebraic equations under steady conditions, or by ordinary differential forms (ODEs) during transient conditions. This opens up important ways for analyzing the combined roles of flow, transport, and coagulation reactions in thrombosis, a task hitherto considered to be completely intractable. We report extensively on the dependence of transport coefficient kL and mural coagulant concentration Cw on flow, vessel geometry, and reaction kinetics. It is shown that for protein transport, kL varies only weakly with shear rate gamma in large vessels, and not at all in the smaller tubes (d less than 10(-2) mm). For a typical protein, kL approximately 10(-3) cm s-1 within a factor of 3 in most geometries, irrespective of the mural reaction kinetics. Significant reductions in kL (1/10-1/1,000) leading to high-coagulant accumulation are seen mainly in stagnant zones vicinal to abrupt expansions and in small elliptical tubules. This is in accord with known physical observations. More unexpected are the dramatic increases in accumulation which can come about through the intervention of an autocatalytic reaction step, with Cw rising sharply toward infinity as the ratio of reaction to transport coefficient approaches unity. Such self-catalyzed reactions have the ability to act as powerful amplifiers of an otherwise modest influence of flow and transport on coagulant concentration. The paper considers as well the effect on mass transport of transient conditions occasioned by coagulation initiation or pulsatile flow. During initiation, instantaneous flux varies with diffusivity and bulk concentration, favouring the early adsorption/consumption of proteins with the highest abundance and mobility. This is akin to the 'Vroman effect' seen in narrow, stagnant spaces. The effect of flow pulsatility on kL has the potential, after prolonged cycling, of bringing about segregation or accumulation of proteins, with consequences for the coagulation process.