CMM-2011 – Computer Methods in Mechanics 9–12 May 2011, Warsaw, Poland Blood coagulation: a puzzle for biologists, a maze for mathematicians Antonio Fasano 1 , Rafael F. Santos 2 and Adelia Sequeira 3 1 Università degli Studi di Firenze, Dipartimento di Matematica ”‘U. Dini”’, Viale Morgagni 67/a, 50134 Firenze, Italy, e-mail: fasano@math.unifi.it 2 Faculty of Sciences and Technology, University of Algarve, Department of Mathematics and CEMAT/ IST, Campus de Gambelas, 8005-139 Faro, Portugal e-mail: rsantos@ualg.pt 3 Instituto Superior Técnico, Technical University of Lisbon, Department of Mathematics and CEMAT/ IST, Av. Rovisco Pais, 1049-001 Lisboa, Portugal e-mail: adelia.sequeira@math.ist.utl.p Abstract Blood coagulation is a process of extreme complexity and of fundamental importance that has been studied since antiquity. In this talk we shall go briefly through the historical stages of the studies on blood coagulation, and we will devote most of the time to illustrating the modern view (the so-called cell based model), in contrast to the theories that have dominated till recent times (∼2005). A discussion on the complex mechanisms of blood clotting cannot be disjoint from an analysis (necessarily superficial) of bleeding disorders and thrombophilia. In a very recent survey paper [1] we have summarized the main features of the process and we have illustrated both the modern approach, and some history, presenting also a concise review of mathematical models. The material collected there will be the subject of this lecture, which consists of two presentations: the fist talk (presented by prof. Antonio Fasano) will deal with the biological introductory part, while the second talk ( delivered by prof. Adelia Sequeira) will be devoted to mathematical models. Keywords: blood coagulation, bleeding disorders, models of hemodynamics, blood rheology, biochemical and mechanical factors 1. Part I: Biological mechanisms and models of blood coagulation. The circulatory system is equipped with an incredibly complex chemical and mechanical machinery ready to repair lesions which may occur to blood vessels by sealing them with a clot (or thrombus). A clot is a gel like structure consisting of a polymer (fibrin) network entrapping various blood components. The ingredients necessary to lead to the clot formation are either present in the blood, or reside in the endothelium of the blood vessels. Actually, the ones contained in the blood vessel walls become available immediately after a lesion of the tunica intima, so even in the presence of a tiny internal lesion. Indeed the clotting process (accompanied by the clot dissolution process) is taking place continuously in the organism on a small scale, so to keep the circulatory system in good shape. Of course, it also occurs on a larger scale when needed. It is absolutely crucial that the clotting mechanism is set in motion only when it is really necessary, remaining silent in normal conditions, and at the same time that the coagulation process is terminated before it occludes the vessel, allowing blood to flow normally. Indeed, after the process of hemostasis is completed, the thrombus will be gradually removed by means of another process known as fibrinolysis. To be more precise, it is not correct to look at these processes in a sequential way. The hemostasis and the fibrinolysis machines are both active at the same time and whether a thrombus is growing or retreating is the result of an unbalance between the two processes. The clotting process takes place in two steps: primary and secondary hemostasis. Primary hemostasis occurs very rapidly: platelets form a compact network with the help of a multimer available at the injury site, known as von Willebrand Factor. Both platelets and von Willebrand Factor have multiple roles also in the next phase. Secondary hemostasis goes through a chain of reactions with a highly positive feedback, in which many other “Factors” intervene (the very first one being the Tissue Factor, present in the blood vessels wall). Such factors (numbered from I to XIII) come in two forms: activated and non-activated. Some of them form complexes that perform specific actions. For instance, the combination of the activated form of Factors V and X gives rise to an enzyme (prothrombinase), which triggers the final stage of coagulation. The reaction cascade has the aim of producing fibrin, making the skeleton of the thrombus, which is further consolidated by the production of cross links in the network. Fibrinolysis is likewise a chemical process with positive feedback, eventually producing the enzyme that can break the fibrin structure. Many drugs can be effectively used to inhibit coagulation, interfering with the action of specific elements in the chain. This is another big chapter in blood coagulation. Since we are addressing a composite audience, we will not enter the intricate world of the chemical structure of the numerous proteins and enzymes and on the details of the reactions, but we will try to describe the basic mechanisms in the simplest possible way. Moreover, it is very important to point out that hemostasis (both primary and secondary) is not just of chemical nature, but some crucial steps (the activation of platelets, as well as the action of von Willebrand Factor) require the intervention of shear stress beyond some threshold. Thus chemistry and fluid dynamics are strongly coupled, clearly making the problem extremely complicated. Besides the physiological process of hemostasis following an injury, it is well known that unwanted coagulation can take place due to temporary reduced blood flow rate or stasis, leading to deep vein thrombosis (DVT), and stimulating puzzling questions concerning the absence of the triggering mechanical stress. A lot can be learned from what may go wrong in the processes of hemostasis and fibrinolysis. Many different kinds of bleeding disorders have been studied, progressively identifying their cause in the deficiency or dysfunction of one of the many elements having some role in the formation or in the dissolution