Antonio Fasano
Blood coagulation is an extremely complex process which is the result of the action of platelets and of a large number of chemicals going through a chemical cascade. Its aim is the formation of a clot, sealing a wound The clot evolution leads to a free boundary problem. It goes in parallel with the process of clot dissolution (fibrinolysis), taking place with a slower time scale. Due to its complexity, the process may fail in various ways because of pathological conditions, leading to thrombosis or bleeding disorders of various types, that have also been the subject of mathematical models. The classical 3-pathway cascade model for blood coagulation, that was formulated in 1964, has been questioned after forty years. Though it is now ascertained to be wrong, its influence has been so strong that many new publications still refer to it. During the last few years a new model has been proposed in the medical literature (the so called cell-based model) and new mathematical papers have been written accordingly. Recently two opposite trends have been observed in mathematical models: on one side a tendency towards "completeness" with an incredible number of pde's describing the biochemistry in great detail (but sometimes ignoring platelets!); on the other side a tendency to focus just on the role of platelets. Those ways of approaching the problem have their own advantages and drawbacks. The "complete models" fail in any case to consider elements of great importance, that, very surprisingly, have been systematically ignored in the huge literature on the subject. The models considering just platelets can be used only for some very early stage of the process. A basic feature of any realistic coagulation model is the coupling between the biochemistry, the evolution of platelets population, and the flow of blood (in turn influenced by the growing clot). Thus blood rheology has a basic role. Blood rheology is known to be a very complicated subject and many different options have been offered. Nevertheless, the main point here is not which rheological model is preferable for blood, but the boundary conditions for blood flow. All models on blood coagulation use a no-slip condition. We prove that even a modest slip can have a dominant influence, depending on the geometry of the growing clot. We will also make a general discussion on the strategy to approach the problem (How many ingredients should be included? How to simplify the description of the chemistry? What targets can be considered realistic? etc.). New perspectives should also account for the most recent discoveries, suggesting that the cell-based model too may need some revision.
