Journal of Mathematical Biology

, Volume 72, Issue 3, pp 649–681 | Cite as

Modelling of platelet–fibrin clot formation in flow with a DPD–PDE method

  • A. Tosenberger
  • F. Ataullakhanov
  • N. Bessonov
  • M. Panteleev
  • A. Tokarev
  • V. Volpert
Article

Abstract

The paper is devoted to mathematical modelling of clot growth in blood flow. Great complexity of the hemostatic system dictates the need of usage of the mathematical models to understand its functioning in the normal and especially in pathological situations. In this work we investigate the interaction of blood flow, platelet aggregation and plasma coagulation. We develop a hybrid DPD–PDE model where dissipative particle dynamics (DPD) is used to model plasma flow and platelets, while the regulatory network of plasma coagulation is described by a system of partial differential equations. Modelling results confirm the potency of the scenario of clot growth where at the first stage of clot formation platelets form an aggregate due to weak inter-platelet connections and then due to their activation. This enables the formation of the fibrin net in the centre of the platelet aggregate where the flow velocity is significantly reduced. The fibrin net reinforces the clot and allows its further growth. When the clot becomes sufficiently large, it stops growing due to the narrowed vessel and the increase of flow shear rate at the surface of the clot. Its outer part is detached by the flow revealing the inner part covered by fibrin. This fibrin cap does not allow new platelets to attach at the high shear rate, and the clot stops growing. Dependence of the final clot size on wall shear rate and on other parameters is studied.

Keywords

Blood coagulation Clot growth Hybrid models Dissipative particle dynamics 

Mathematics Subject Classification

92C35 92C50 74F10 

Notes

Acknowledgments

The study was supported by the Russian Foundation for Basic Research Grants 14-04-00670, 15-04-99595, 15-51-15008, by Dynasty Foundation Fellowship and by the Russian Academy of Sciences Presidium Basic Research Programs “Molecular and Cellular Biology” and “Basic Research for Development of Biomedical Technologies”, by the Russian Federation President Grant for Young Doctors of Science MD-6347.2015.4, and by “Russian Federation Presidential Scholarship for Young Scientists and Graduate Students”. The work was also supported by the research Grant ANR “Bimod”.

References

  1. Abaeva AA, Canault M, Kotova YN, Obydennyy SI, Yakimenko AO, Podoplelova NA, Kolyadko VN, Chambost H, Mazurov AV, Ataullakhanov FI, Nurden AT, Alessi MC, Panteleev MA (2013) Procoagulant platelets form an -granule protein-covered ”cap” on their surface that promotes their attachment to aggregates. J Biol Chem 288(41):29621–29632CrossRefGoogle Scholar
  2. Alenitsyn AG, Kondratyev AS, Mikhailova I, Siddique I (2008) Mathematical modeling of thrombus growth in microvessels. J Prime Res Math 4:195–205MATHGoogle Scholar
  3. Allen MP, Tidesley DJ (1987) Computer simulation of liquids. Clarendon, OxfordMATHGoogle Scholar
  4. Anand M, Rajagopal K, Rajagopal KR (2005) A model for the formation and lysis of blood clots. Pathophysiol Haemost Thromb 34(2–3):109–120CrossRefGoogle Scholar
  5. Anand M, Rajagopal K, Rajagopal KR (2008) A model for the formation, growth, and lysis of clots in quiescent plasma. A comparison between the effects of antithrombin III deficiency and protein C deficiency. J Theor Biol 253(4):725–738 Epub 2008 Apr 25CrossRefGoogle Scholar
  6. Arya M, Kolomeisky AB, Romo GM, Cruz MA, López JA, Anvari B (2005) Dynamic force spectroscopy of glycoprotein Ib-IX and von Willebrand factor. Biophys J 88:4391–4401CrossRefGoogle Scholar
  7. Barynin IA, Starkov IA, Khanin MA (1999) Mathematical models in hemostasis physiology. Izv Akad Nauk Ser Biol 1:59–66 (in Russian)Google Scholar
  8. Begent N, Born GV (1970) Growth rate in vivo of platelet thrombi, produced by iontophoresis of ADP, as a function of mean blood flow velocity. Nature 227(5261):926–930CrossRefGoogle Scholar
  9. Bessonov N, Babushkina E, Golovashchenko FG, Tosenberger A, Ataullakhanov F, Panteleev M, Tokarev A, Volpert V (2013) Numerical simulations of blood flows with non-uniform distribution of erythrocytes and platelets. Russ J Numer Anal Math Model 28(5):443–458Google Scholar
  10. Bessonov N, Babushkina E, Golovashchenko FG, Tosenberger A, Ataullakhanov F, Panteleev M, Tokarev A, Volpert V (2014) Numerical modelling of cell distribution in blood flow. Math Model Nat Phenom 9(6):69–84Google Scholar
  11. Bodnar T, Sequeria A (2008) Numerical simulation of the coagulation dynamics of blood. Comput Math Methods Med 9(2):83–104MathSciNetCrossRefMATHGoogle Scholar
  12. Brown AEX, Litvinov RI, Discher DE, Purohit PK, Weisel JW (2009) Multiscale mechanics of fibrin polymer: gel stretching with protein unfolding and loss of water. Science 325:741CrossRefGoogle Scholar
  13. CDC, Centers for Disease Control and Prevention (2002) State-specific mortality from sudden cardiac death-United States, 1999. MMWR Morb Mortal Wkly Rep 51(6): 123–126Google Scholar
  14. Chiu YL, Chou YL, Jen CY (1988) Platelet deposition onto fibrin-coated surfaces under flow conditions. Blood Cells 13(3):437–450Google Scholar
  15. Coburn LA (2010) Studies of platelet GPIb-alpha and von Willebrand factor bond formation under flow. PhD thesis at Georgia Institute of Technology and Emory UniversityGoogle Scholar
  16. Dashkevich NM, Ovanesov MV, Balandina AN, Karamzin SS, Shestakov PI, Soshitova NP, Tokarev AA, Panteleev MA, Ataullakhanov FI (2012) Thrombin activity propagates in space during blood coagulation as an excitation wave. Biophys J 103(10):2233–2240CrossRefGoogle Scholar
  17. Falati S, Gross P, Merrill-Skoloff G, Furie BC, Furie B (2002) Real-time in vivo imaging of platelets, tissue factor and fibrin during arterial thrombus formation in the mouse. Nat Med 8:1175–1181CrossRefGoogle Scholar
  18. Fedosov DA (2010) Multiscale modeling of blood flow and soft matter. PhD dissertation at Brown UniversityGoogle Scholar
  19. Fedosov DA, Pivkin IV, Karniadakis GE (2008) Velocity limit in DPD simulations of wall-bounded flows. J Comput Phys 227:2540–2559MathSciNetCrossRefMATHGoogle Scholar
  20. Filipovic N, Kojic M, Tsuda A (2008) Modelling thrombosis using dissipative particle dynamics method. Philos Trans R Soc A 366:3265–3279MathSciNetCrossRefGoogle Scholar
  21. Fogelson AL (2007) Cell-based models of blood clotting. Single-cell-based models in biology and medicine. In: Anderson ARA, Chaplain MAJ, Rejniak KA (eds) Mathematics and biosciences in interaction. Birkhäuser, Basel, pp 243–269Google Scholar
  22. Fogelson AL, Guy RD (2008) Immersed-boundary-motivated models of intravascular platelet aggregation. Comput Methods Appl Mech Eng 197:2250–2264MathSciNetCrossRefMATHGoogle Scholar
  23. Frojmovic MM, Mooney RF, Wong T (1994) Dynamics of platelet glycoprotein IIb-IIIa receptor expression and fibrinogen binding. I. Quantal activation of platelet subpopulations varies with adenosine diphosphate concentration. Biophys J 67(5):2060–2068CrossRefGoogle Scholar
  24. Groot RD, Warren PB (1997) Dissipative particle dynamics: bridging the gap between atomistic and mesoscopic simulation. J Chem Phys 107(11):4423–4435CrossRefGoogle Scholar
  25. Guy RD, Fogelson AL, Keener JP (2007) Fibrin gel formation in a shear flow. Math Med Biol 24(1):111–130CrossRefMATHGoogle Scholar
  26. Jackson SP (2007) The growing complexity of platelet aggregation. Blood 109(12):5087–5095 Epub 2007 Feb 20CrossRefGoogle Scholar
  27. Jackson SP, Nesbitt WS, Westein E (2009) Dynamics of platelet thrombus formation. J Thromb Haemost 7(Suppl 1):17–20CrossRefGoogle Scholar
  28. Jen CJ, Hu SJ, Wu HJ, Lin TS, Mao CW (1990) Platelet–fibrin interaction in the suspension and under flow conditions. Adv Exp Med Biol 281:277–285CrossRefGoogle Scholar
  29. Jen CJ, Lin JS (1991) Direct observation of platelet adhesion to fibrinogen- and fibrin-coated surfaces. Am J Physiol 261(5 Pt 2):H1457–H1463Google Scholar
  30. Jen CJ, Tai YW (1992) Morphological study of platelet adhesion dynamics under whole blood flow condition. Platelets 3(3):145–153CrossRefGoogle Scholar
  31. Kamocka MM, Mu J, Liu X, Chen N, Zollman A, Sturonas-Brown B, Dunn K, Xu Z, Chen DZ, Alber MS, Rosen ED (2010) Two-photon intravital imaging of thrombus development. J Biomed Opt 15:016020CrossRefGoogle Scholar
  32. Karttunen M, Vattulainen I, Lukkarinen A (2004) A novel methods in soft matter simulations. Springer, BerlinCrossRefGoogle Scholar
  33. Krasotkina YV, Sinauridze EI, Ataullakhanov FI (2000) Spatiotemporal dynamics of fibrin formation and spreading of active thrombin entering non-recalcified plasma by diffusion. Biochim Biophys Acta 1474:337–345CrossRefGoogle Scholar
  34. Kulkarni S, Dopheide SM, Yap CL, Ravanat C, Freund M, Mangin P, Heel KA, Street A, Harper IS, Lanza F, Jackson SP (2000) A revised model of platelet aggregation. J Clin Invest 105(6):783–791CrossRefGoogle Scholar
  35. Leiderman K, Fogelson A (2014) An overview of mathematical modeling of thrombus formation under flow. Thromb Res 133(Suppl 1):S12–S14CrossRefGoogle Scholar
  36. Litvinov RI, Bennett JS, Weisel JW, Shumanz H (2005) Multi-step fibrinogen binding to the integrin \(\alpha \)IIb\(\beta \)3 detected using force spectroscopy. Biophys J 89:2824–2834CrossRefGoogle Scholar
  37. Pallister CJ, Watson MS (2010) Haematology. Scion Publishing LtdGoogle Scholar
  38. Panteleev MA, Ovanesov MV, Kireev DA, Shibeko AM, Sinauridze EI, Ananyeva NM, Butylin AA, Saenko EL, Ataullakhanov FI (2006) Spatial propagation and localization of blood coagulation are regulated by intrinsic and protein C pathways, respectively. Biophys J 90:1489–1500CrossRefGoogle Scholar
  39. Pivkin IV, Karniadakis GE (2005) A new method to impose no-slip boundary conditions in dissipative particle dynamics. J Comput Phys 207:114–128MathSciNetCrossRefMATHGoogle Scholar
  40. Pivkin IV, Richardson PD, Karniadakis G (2006) Blood flow velocity effects and role of activation delay time on growth and form of platelet thrombi. PNAS 103:17164–17169CrossRefGoogle Scholar
  41. Pivkin IV, Richardson PD, Karniadakis GE (2009) Effect of red blood cells on platelet aggregation. Eng Med Biol Mag IEEE 28(2):32–37CrossRefGoogle Scholar
  42. Schiller UD (2005) Dissipative particle dynamics. A study of the methodological background. Diploma thesis at Faculty of Physics University of BielefeldGoogle Scholar
  43. Schneider SW, Nuschele S, Wixforth A, Gorzelanny C, Alexander-Katz A, Netz RR, Schneider MF (2007) Shear-induced unfolding triggers adhesion of von Willebrand factor fibers. Proc Natl Acad Sci USA 104(19):7899–7903CrossRefGoogle Scholar
  44. Shankaran H, Alexandridis P, Neelamegham S (2003) Aspects of hydrodynamic shear regulating shear-induced platelet activation and self-association of von Willebrand factor in suspension. Blood 101(7):2637–2645CrossRefGoogle Scholar
  45. Shibeko AM, Lobanova ES, Panteleev MA, Ataullakhanov FI (2010) Blood flow controls coagulation onset via the positive feedback of factor VII activation by factor Xa. BMC Syst Biol 4:5CrossRefGoogle Scholar
  46. Sweet CR, Chatterjee S, Xu Z, Bisordi K, Rosen ED, Alber M (2011) Modelling platelet–blood flow interaction using the subcellular element Langevin method. J R Soc Interface 8:1760–1771CrossRefGoogle Scholar
  47. Tokarev A, Sirakov I, Panasenko G, Volpert V, Shnol E, Butylin A, Ataullakhanov F (2012) Continuous mathematical model of platelet thrombus formation in blood flow. Russ J Numer Anal Math Model 27(2):192–212MathSciNetCrossRefGoogle Scholar
  48. Tosenberger A, Salnikov V, Bessonov N, Babushkina E, Volpert V (2011) Particle dynamics methods of blood flow simulations. Math Model Nat Phenom 6(5):320–332MathSciNetCrossRefMATHGoogle Scholar
  49. Tosenberger A, Ataullakhanov F, Bessonov N, Panteleev M, Tokarev A, Volpert V (2012) Modelling of thrombus growth and growth stop in flow by the method of dissipative particle dynamics. Russ J Numer Anal Math Model 27(5):507–522MathSciNetCrossRefMATHGoogle Scholar
  50. Tosenberger A, Ataullakhanov F, Bessonov N, Panteleev M, Tokarev A, Volpert V (2013) Modelling of thrombus growth in flow with a DPD–PDE method. J Theor Biol 337:30–41MathSciNetCrossRefGoogle Scholar
  51. Turitto VT, Baumgartner HR (1979) Platelet interaction with subendothelium in flowing rabbit blood: effect of blood shear rate. Microvasc Res 17(1):38–54CrossRefGoogle Scholar
  52. Voronov RS, Stalker TJ, Brass LF, Diamond SL (2013) Simulation of intrathrombus fluid and solute transport using in vivo clot structures with single platelet resolution. Ann Biomed Eng 41(6):1297–1307CrossRefGoogle Scholar
  53. Wagner CL, Mascelli MA, Neblock DS, Weisman HF, Coller BS, Jordan RE (1996) Analysis of GPIIb/IIIa receptor number by quantification of 7E3 binding to human platelets. Blood 88(3):907–914Google Scholar
  54. Weisel JW (2008) Enigmas of blood clot elasticity. Science 320:456CrossRefGoogle Scholar
  55. Wellings PJ (2011) Mechanisms of platelet capture under very high shear. Master thesis at Georgia Institute of TechnologyGoogle Scholar
  56. Windberger U, Bartholovitsch A, Plasenzotti R, Korak KJ, Heinze G (2003) Whole blood viscosity, plasma viscosity and erythrocyte aggregation in nine mammalian species: reference values and comparison of data. Exp Physiol 88:431–440CrossRefGoogle Scholar
  57. Wooton DM, Ku DN (1999) Fluid mechanics of vascular systems and thrombus. Annu Rev Biomed Eng 01:299–329CrossRefGoogle Scholar
  58. Xu Z, Chen N, Kamocka MM, Rosen ED, Alber M (2008) A multiscale model of thrombus development. J R Soc Interface 5:705–722CrossRefGoogle Scholar
  59. Xu Z, Chen N, Shadden S, Marsden JE, Kamocka MM, Rosen ED, Alber M (2009) Study of blood flow impact on growth of thrombi using a multiscale model. Soft Matter 5:769–779CrossRefGoogle Scholar
  60. Xu Z, Lioi J, Mu J, Kamocka MM, Liu X, Chen DZ, Rosen ED, Alber M (2010) A multiscale model of venous thrombus formation with surface-mediated control of blood coagulation cascade. Biophys J 98:1723–1732CrossRefGoogle Scholar
  61. Xu Z, Christleyy S, Lioiz J, Kim O, Harveyx C, Sun W, Rosen ED, Alber M (2012) Multiscale model of fibrin accumulation on the blood clot surface and platelet dynamics. Methods Cell Biol 110:367–388Google Scholar
  62. Yakimenko AO, Verholomova FY, Kotova YN, Ataullakhanov FI, Panteleev MA (2012) Identification of different proaggregatory abilities of activated platelet subpopulations. Biophys J 102(10):2261–2269CrossRefGoogle Scholar
  63. Zaidi TN, McIntire LV, Farrell DH, Thiagarajan P (1996) Adhesion of platelets to surface-bound fibrinogen under flow. Blood 88(8):2967–2972Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • A. Tosenberger
    • 1
  • F. Ataullakhanov
    • 2
  • N. Bessonov
    • 3
  • M. Panteleev
    • 2
  • A. Tokarev
    • 2
  • V. Volpert
    • 4
  1. 1.Institut des Hautes Etudes ScientifiquesBures-sur-YvetteFrance
  2. 2.Federal Research and Clinical Centre of Pediatric Hematology, Oncology and ImmunologyMinistry of Healthcare of the Russian FederationMoscowRussian Federation
  3. 3.Institute of Mechanical Engineering ProblemsSaint PetersburgRussian Federation
  4. 4.Institut Camille Jordan, UMR 5208 CNRSLyonFrance

Personalised recommendations