Mathematical modeling of blood coagulation processes in intensive blood flow conditions

1. Mathematical modeling of blood coagulation processes in intensive blood flow conditions A.S. Rukhlenko1 K.E. Zlobina2 • G.Th. Guria1,2 • 1 –Moscow Institute of Physics and Technology 2 – National ResearchCenter for Hematology Research Moscow 2014

2. Intravascular thrombus formation General view to the interplay of processes Hemodynamics and rheology Change of rheological properties of the media Convection and diffusion Chemical kinetics of cascade reactions and processes of polymerization Wall shear stress and transmural pressure Vessel wall elasticity Change of vessel wall permeability and of concentration of activators Vessel wall and surrounding tissue

3. Intravascular clot growth at low Reynolds number [A.L. Chulichkov et al., 2000]

4. Intravascular clot growth in slow blood blow (Re<<1) Results of numerical simulations [A.L. Chulichkov et al., 2000]

5. Intravascular clot growth in slow blood flow (Re<<1) Results of numerical calculations [A.P. Guzevatikh et al., 2000]

6. Structure of diagram of liquid state of blood stability(Re<5*10-3) • Retarding of blood flow promotes activation of blood coagulation • The bigger stenosis the more likely thrombosis to occur Thrombus formation starts "II" "I" Liquid state of blood is stable Vessel with 50% stenosis, where μ denotes the rate of activator substances infiltration into the vessel [A.P. Guzevatykh et al., 2000]

7. Formation of polydisperse microthrombi • It was shown previously in our lab [Uzlova et al., 2008], that in intensive flow (Re ~ 100) a number of stages of microthrombi formation and growth precedes formation of large thrombi • Clouds of microthrombi may be detected by ultrasound techniques • Theoretical investigation of formation of friable clots without sharp phase border is the subject of present work [Uzlova S., Guria K., Guria G., 2008; Uzlova S.G., Guria K.G., Shevelev A.A. et al., 2008]

8. Influence of blood flow on vessel wall permeability • Experimental in vitro [Warboys et al., 2010, Jo et al., 1991, McIntire et al., 1995, Sill et al., 1995], and in vivo (а также ex vivo) [Kim et al., 2005, Lever et al., 1992, Williams, 1999, 2003] investigations give evidence for the fact that rise of wall shear stress may lead to reversible growth of endothelium permeability up to orders of magnitudefor some substances [McIntire et al., 1995]. • Rise of wall shear stressmay also lead to irreversible permeability growth e.g. due to the rupture of fibrous cap of atherosclerotic plaque. Comparative researches show [Gertz and Roberts, 1990, Fukumoto et al., 2008, Lovett and Rothwell, 2003, Slager et al., 2005] that atherosclerotic plaque rupturehappens predominantly in the high wall shear stress zones.

9. Pattern of thrombus formation processes development in intensive flows qualitatively differs from that in slow flows Re << 10-2 • Only solid thrombi are formed • Blood flow doesn’t alter vessel wall permeability • As a rule flow topology is trivial Re >> 10 • Microthrombi and friable clots are also formed • Blood flow may drastically alter vessel wall permeability • As a rule recirculation zones are formed behind stenoses

10. Blood coagulation in high-Reynolds flows • Wall shear stress in intensive flow in stenosed vessel may drastically change the permeability of the vessel • This situation refers to one of the most dangerous disease – atherosclerosis and subsequent intravascular thrombosis • We analyzed simplified case of 2D vessel geometry (lengthLx=7.5 cm, widthLy=1 cm)

11. Blood flow description • αp reflects the influence of the fibrin polymers network on blood flow (Darcy law)

12. Blood flow description • αp reflects the influence of the fibrin polymers network on blood flow (Darcy law)

13. Fibrin polymerization kinetics • Association and fragmentation processes: • Master equations: • Statistical moments: • Average number of monomers in polymer molecules: [Guria, Herrero, Zlobina, 2009]

14. Fibrin polymerization kinetics • Kinetics of polymerization in terms of moments: • Cutoff assumption: [Guria, Herrero, Zlobina, 2009]

15. Blood coagulation kinetics description [Guria, Herrero, Zlobina, 2009; Guria, Herrero, Zlobina, 2010]

16. Blood coagulation kinetics description [Guria, Herrero, Zlobina, 2009; Guria, Herrero, Zlobina, 2010]

17. Blood coagulation kinetics description [Guria, Herrero, Zlobina, 2009; Guria, Herrero, Zlobina, 2010]

18. Infiltration of procoagulant substances into the blood flow • It is supposed that procoagulant substances are able to infiltrate into the blood flow from surrounding tissue (variable u) • It is supposed that vessel wall permeability for the procoagulant substances μ depends on wall shear stress γsh in picewise-linear manner

19. Coefficients of polymers mass transfer • Diffusion coefficient: • Motility coefficient: • where: • denotes an average number of fibrin monomers in average-weighted polymer clot • Condition refers to semi-diluted solution [de Gennes, 1979 // Scaling concepts in polymer physics]

20. Under-threshold activation • The primary activator is infiltrated into the blood flow from the surrounding vessel tissue through the zone of high wall shear stress • As a result of activation of blood coagulation system clouds of microthrombi are formed in the blood • Mainly they are accumulated in recirculation zone

21. Under-threshold activation • The primary activator is infiltrated into the blood flow from the surrounding vessel tissue through the zone of high wall shear stress • As a result of activation of blood coagulation system clouds of microthrombi are formed in the blood • Mainly they are accumulated in recirculation zone

22. Presumable centres of initiation of clot growth • An areas with local maximum of microthrombi concentrations seem to be the most probable places for initiation of clot growth processes

23. Results of numerical simulation Fibrin filament growth Scenario №1 (Re = 130, )

24. Results of numerical simulation Two centers of fibrin filament growth Scenario №2 (Re = 130, )

25. Results of numerical simulation Stationary flattering fibrin filament formation Scenario №3 (Re = 200, )

26. Fluttering fibrin filament Results of experiments Uzlova S., Guria K., Guria G. Acoustic determination of early stages of intravascular blood coagulation // Philos Trans R Soc A. — 2008. — Vol. 366. — P. 3649–3661

27. Fibrin fibres

28. Flotating friable fibrin structures

29. Fibrin clots

30. Parametric plane relevant to activationof clot growth (Re > 10)

31. Parametric plane relevant to activationof clot growth (Re<5*10-3) Thrombus formation starts "II" "I" Liquid state of blood is stable [Guzevatykh et al., 2000]

32. Parametric plane relevant to activationof clot growth Thrombus formation starts "II" "I" Liquid state of blood is stable

33. The influence of stenosis shape

34. The influence of stenosis shape on activation threshold

35. Scaling power law № 1 • It was found that at the fixed Reynolds number the lag time of clot growth depends on the value of parameter in following way: • where () corresponds to the distance of representative point at the parametric plane to the border of liquid state stability

36. Scaling power law № 2 • It was found that at the fixed values of ( ) the lag time of clot growth depends on the value of Reynolds number in a following way: • where (Re - Recrit) corresponds to the distance of representative point at the parametric plane to the border of liquid state stability

37. Probable biomedical significance of the presented results Activation of blood coagulationmay happen both due to blood flow intensification (e.g. as a result of blood pressure rise) and due to its retarding (e.g. due to blood pressure drop). The most thrombogeneous are medium sized atherosclerotic plaques Detection of fibre-like structures in medical practice (e.g. by means of ultrasound techniques) has to be considered as early predictorof subsequent thrombosis.

38. Relevant publications • A.S. Rukhlenko, K.E. Zlobina, G.Th. Guria. Hydrodynamical activation of blood coagulation in stenosed vessels. Theoretical analysis // Computer Research and Modeling, 2012, vol. 4, no. 1, pp.155–184 • A.S. Rukhlenko, O.A. Dudchenko, K.E. Zlobina, G.Th. Guria. Threshold activation of blood coagulation as a result of elevated wall shear stress // Proceedings of MIPT. — 2012. — V. 4, N 2. • Rukhlenko A. S., Zlobina K. E., Guria G. T. Threshold activation of bloodcoagulation cascade in intensive flow and formation of fibre-like fibrin poly­mer networks // Proceedings of the International Conference “Instabilitiesand Control of Excitable Networks: From macro- to nano-systems”. Moscow:MAKS Press, 2012. Pp. 113–125. • Г.Т. Гурия. Как теоретическая физика трактует свертывание крови? // Наука, 2011, № 9, с. 51-57

39. Authors are grateful to following persons • Academician A.I. Vorobjev • O.A. Dudchenko • S.G. Uzlova, K.G. Guria • I.A. Romanets, A.R. Gagarina, D.A. Ivlev, O.A. Starikovskaya • The work was partially supported by ISTC grant #3744

40. Many thanks!

41. Blood Coagulation Cascade Graph [Guria G.Th., 2002; Uzlova et al. 2008 // Phil Trans Royal Soc A]

42. Coagulation cascade – threshold • Blood coagulation cascade is activated in thresholdmanner • Activation of the BCC could be achieved parametrically or dynamically • Blood is metastable under normal physiological conditions [Г.Т. Гурия. 2011 // Наука., № 9, с. 51-57]

43. Problems of «pretending to completeness» of mathematical description • There is a number of recently developed models operating with large number of variables and parameters (i.e. much more than 10): [Anand et al., 2003, 2005, 2008, Ataullakhanov and Panteleev, 2005, Shibeko et al., 2010, Jones and Mann; Leiderman and Fogelson, 2011, Hockin et al., 2002, Butenas et al., 2004; etc...] • The weak point of this approach is a large amount of uncertainty in constant rates values • It was shown in [Wagenvoord, Hemker, Hemker, 2006; Hemker, Kerdelo, Kremers, 2012] that up-do-date level of experiment-based data does not let to construct verifiable mathematical models of coagulation with large number of variables and parameters • That’s way in present work we have limited ourselves to the use of qualitative (i.e. phenomenological) mathematical models of blood coagulation cascade

44. Gelation criterion • It was assumed that fibrin gel was formed when the fibrin solution became half-diluted [de Gennes, 1979] • This means that neighboring polymer chains start to interweave each other • We assume that formation of polymer chains happens on chemical impurities (presumably phospholipids) with concentration n • Assuming that fibrin polymer chains behave as ideal chains (Rchain ~ l0(Nw K)0.5) we obtain the «gelation» criterion:

45. Filtration resistance of fibrin gel • It was assumed that if Nw < Nwpol fibrin polymer chains do not alter blood flow • The filtration resistance of porous media is known to depend on the mesh size: • The mesh size in half-diluted solution was estimated as (by [de Gennes, 1979]):

46. Diffusion and convection of polymer chains • To describe fibrin chains diffusion in the system the following asymptotic expression was used: • It gives well-known dependencies of diffusion on Nw when: • To take into account decrease of convective mass transfer due to chains interweaving following expression was used:

47. Computational mesh example

48. Zone 1 – located nearby the proximal end of separation line

49. Zone 2 – center of recirculation zone

50. Zone 3 – located nearby the distal end of separation line The local maximum of microthrombi concentration in zone 3 was not resolved by numerical calculations