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Mathematical Modeling of Thrombin Generation and Wave Propagation: From Simple to Complex Models and Backwards

  • Alexey Tokarev
  • Nicolas Ratto
  • Vitaly VolpertEmail author
Chapter

Abstract

Blood coagulation is an extremely complex biochemical system consisting of more than twenty proteins involved in more than a hundred chemical reactions. Correct functioning of this system is indispensable for normal hemostasis, thus its malfunctions lead to life threatening bleedings and thromboses. Huge efforts are directed to understand how this system is organized and controlled, how its response can be predicted, and what experimental methods should be used in patient diagnostics and treatment. Here, we briefly review mathematical modeling approaches existing in this field. We pay special attention to the transitions from simple to complex models and to the reduction of complex models back to simple ones, as such reduction actually provides possibility to understand the fundamental mechanisms of functioning of complex biological systems besides coagulation.

Keywords

Blood coagulation Blood diseases Active media Autowaves Mathematical modeling Detailed models Phenomenological models Reduced models 

Notes

Acknowledgements

This work was partially supported by the “RUDN University Program 5-100” to A.T. and V.V. and by the Dynasty Foundation Fellowship to A.T.

References

  1. 1.
    H.C. Hemker, S. Béguin, Thrombin generation in plasma: its assessment via the endogenous thrombin potential. Thromb. Haemost. 74, 134–138 (1995)Google Scholar
  2. 2.
    H.C. Hemker, P. Giesen, R. AlDieri, V. Regnault, E. De Smed, R. Wagenvoord, T. Lecompte, S. Béguin, The calibrated automated thrombogram (CAT): a universal routine test for hyper- and hypocoagulability. Pathophysiol. Haemost. Thromb. 32, 249–253 (2002)CrossRefGoogle Scholar
  3. 3.
    N.M.M. Dashkevich, M.V. V. Ovanesov, A.N.N. Balandina, S.S.S. Karamzin, P.I.I. Shestakov, N.P.P. Soshitova, A.A.A. Tokarev, M.A.A. Panteleev, F.I.I. Ataullakhanov, Thrombin activity propagates in space during blood coagulation as an excitation wave. Biophys. J. 103, 2233–2240 (2012)CrossRefGoogle Scholar
  4. 4.
    A.S. Zhalyalov, M.A. Panteleev, M.A. Gracheva, F.I. Ataullakhanov, A.M. Shibeko, Co-ordinated spatial propagation of blood plasma clotting and fibrinolytic fronts. PLoS One 12 (2017)Google Scholar
  5. 5.
    A.N. Balandina, I.I. Serebriyskiy, A.V. Poletaev, D.M. Polokhov, M.A. Gracheva, E.M. Koltsova, D.M. Vardanyan, I.A. Taranenko, A.Y. Krylov, E.S. Urnova, K.V. Lobastov, A.V. Chernyakov, E.M. Shulutko, A.P. Momot, A.M. Shulutko, F.I. Ataullakhanov, Thrombodynamics—a new global hemostasis assay for heparin monitoring in patients under the anticoagulant treatment. PLoS One 13, 1–18 (2018)CrossRefGoogle Scholar
  6. 6.
    A.A. Onasoga-Jarvis, T.J. Puls, S.K. O’Brien, L. Kuang, H.J. Liang, K.B. Neeves, Thrombin generation and fibrin formation under flow on biomimetic tissue factor-rich surfaces. J. Thromb. Haemost. 12, 373–382 (2014)CrossRefGoogle Scholar
  7. 7.
    K.B. Neeves, D.A.R. Illing, S.L. Diamond, Thrombin flux and wall shear rate regulate fibrin fiber deposition state during polymerization under flow. Biophys. J. 98, 1344–1352 (2010)CrossRefGoogle Scholar
  8. 8.
    K.E. Brummel-Ziedins, S.J. Everse, K.G. Mann, T. Orfeo, Modeling thrombin generation: plasma composition based approach. J. Thromb. Thrombolysis 37, 32–44 (2014)CrossRefGoogle Scholar
  9. 9.
    K.B. Neeves, K. Leiderman, Mathematical models of hemostasis, in Trauma Induced Coagulopathy, ed. by A.l. EG et al. (Springer, New York, 2016), pp. 567–584Google Scholar
  10. 10.
    K.G. Mann, Is there value in kinetic modeling of thrombin generation? Yes. J. Thromb. Haemost. 10, 1463–1469 (2012)CrossRefGoogle Scholar
  11. 11.
    H.C. Hemker, S. Kerdelo, R.M.W. Kremers, Is there value in kinetic modeling of thrombin generation? No (unless …). J. Thromb. Haemost. 10, 1470–1477 (2012)CrossRefGoogle Scholar
  12. 12.
    K.E. Brummel-Ziedins, A.S. Wolberg, Global assays of hemostasis. Curr. Opin. Hematol. 21, 395–403 (2014)CrossRefGoogle Scholar
  13. 13.
    K.E. Brummel-Ziedins, Models for thrombin generation and risk of disease. J. Thromb. Haemost. 11(Suppl), 212–223 (2013)CrossRefGoogle Scholar
  14. 14.
    K. Leiderman, A. Fogelson, An overview of mathematical modeling of thrombus formation under flow. Thromb. Res. 133, S12–S14 (2014)CrossRefGoogle Scholar
  15. 15.
    E. Tsiklidis, C. Sims, T. Sinno, S.L. Diamond, Multiscale systems biology of trauma-induced coagulopathy. Wiley Interdiscip. Rev. Syst. Biol. Med. 10, 1–10 (2018)CrossRefGoogle Scholar
  16. 16.
    A.V. Belyaev, J.L. Dunster, J.M. Gibbins, M.A. Panteleev, V. Volpert, Modeling thrombosis in silico: frontiers, challenges, unresolved problems and milestones. Phys. Life Rev. 1, 1–39 (2018)Google Scholar
  17. 17.
    S.L. Diamond, Systems analysis of thrombus formation. Circ. Res. 118, 1348–1362 (2016)CrossRefGoogle Scholar
  18. 18.
    A.A. Tokarev, Y.V. Krasotkina, M.V. Ovanesov, M.A. Panteleev, M.A. Azhigirova, V.A. Volpert, F.I. Ataullakhanov, A.A. Butilin, Spatial dynamics of contact-activated fibrin clot formation in vitro and in silico in haemophilia B: effects of severity and Ahemphil B treatment. Math. Model. Nat. Phenom. 1, 124–137 (2006)MathSciNetCrossRefGoogle Scholar
  19. 19.
    K.G. Mann, S. Butenas, K. Brummel, The dynamics of thrombin formation. Arterioscler. Thromb. Vasc. Biol. 23, 17–25 (2003)CrossRefGoogle Scholar
  20. 20.
    M. Hoffman, Z.H. Meng, H.R. Roberts, D.M. Monroe, Rethinking the Coagulation Cascade. Jpn. J. Thromb. Hemost. 16, 70–81 (2005)CrossRefGoogle Scholar
  21. 21.
    A.Y. Kondratovich, A.V. Pokhilko, F.I. Ataullakhanov, Spatiotemporal dynamics of contact activation factors of blood coagulation. Biochim. Biophys. Acta. Gen. Subj. 1569, 86–104 (2002)CrossRefGoogle Scholar
  22. 22.
    M.A. Panteleev, V.I. Zarnitsina, F.I. Ataullakhanov, Tissue factor pathway inhibitor: a possible mechanism of action. Eur. J. Biochem. 269, 2016–2031 (2002)CrossRefGoogle Scholar
  23. 23.
    L. Smith, J. Engelbreth-Holm, W. Dameshek, R.S. Schwartz, E. Undritz, H. Braunsteiner, F. Pakesch, R.J.V. Pulvertaft, J.G. Humble, L.J. Witts, H.A. Magnus, B. von Bonsdorff, W. Nyberg, R. Gräsbeck, L. Heilmeyer, M. Bessis, L. Hallberg, K.N. von Kaulla, E. Deutsch, A. Videbaek, N.A. Feodorov, S.V. Skurkovich, A. Gajano, M. Seligmann, R. Di Guglielmo, J.R. Squire, E. Neumark, I.M. Nilsson, M. Blombäck, B. Blombäck, R. Biggs, C. Hougie, The nomenclature of blood coagulation factors. Acta Haematol. 24, 151–162 (1960)Google Scholar
  24. 24.
    R.G. Macfarlane, An enzyme cascade in the blood clotting mechanism and its function as a biochemical amplifier. Nature 202 498–499 (1964)CrossRefGoogle Scholar
  25. 25.
    S.N. Levine, Enzyme amplifier kinetics. Science(80) 152, 651–653 (1966)Google Scholar
  26. 26.
    H.C. Hemker, P.W. Hemker, General kinetics of enzyme cascades. Proc. R. Soc. Lond. Ser. B Biol. Sci. 173, 411–20 (1969)CrossRefGoogle Scholar
  27. 27.
    F. Martorana, A. Moro, On the kinetics of enzyme amplifier systems with negative feedback. Math. Biosci. 21, 77–84 (1974)CrossRefGoogle Scholar
  28. 28.
    V.V. Semenov, M.A. Khanin, Nonlinear effects in kinetics of blood coagulation. Biofizika [in Russia] 35, 139–141 (1990)Google Scholar
  29. 29.
    M.A. Khanin, V.V. Semenov, A mathematical model of the kinetics of blood coagulation. J. Theor. Biol. 136, 127–134 (1989)MathSciNetCrossRefGoogle Scholar
  30. 30.
    K.A. Bauer, B.L. Kass, H. ten Cate, J.J. Hawiger, R.D. Rosenberg, Factor IX is activated in vivo by the tissue factor mechanism. Blood 76, 731–736 (1990)Google Scholar
  31. 31.
    K.A. Bauer, B.L. Kass, H. ten Cate, M.A. Bednarek, J.J. Hawiger, R.D. Rosenberg, Detection of factor X activation in humans. Blood 74, 2007–2015 (1989)Google Scholar
  32. 32.
    J. Jesty, E. Beltrami, Positive feedbacks of coagulation: their role in threshold regulation. Arterioscler. Thromb. Vasc. Biol. 25, 2463–2469 (2005)CrossRefGoogle Scholar
  33. 33.
    H.L. Nossel, I. Yudelman, R.E. Canfield, V.P. Butler, Jr., K. Spanondis, G.D. Wildner, G.D. Qureshi, Measurement of fibrinopeptide A in human blood. J. Clin. Invest. 54, 43–53 (1974)CrossRefGoogle Scholar
  34. 34.
    S. Butenas, T. Orfeo, M.T. Gissel, K.E. Brummel, K.G. Mann, The significance of circulating factor IXa in blood. J. Biol. Chem. 279, 22875–22882 (2004)CrossRefGoogle Scholar
  35. 35.
    V.I. Zarnitsina, Study of mechanisms for arresting thrombus growth. PhD Thesis (1997)Google Scholar
  36. 36.
    J. Jesty, E. Beltrami, G. Willems, Mathematical analysis of a proteolytic positive-feedback loop: dependence of lag time and enzyme yields on the initial conditions and kinetic parameters. Biochemistry 32, 6266–6274 (1993)CrossRefGoogle Scholar
  37. 37.
    E. Beltrami, J. Jesty, Mathematical analysis of activation thresholds in enzyme-catalyzed positive feedbacks: application to the feedbacks of blood coagulation. Proc. Natl. Acad. Sci. U. S. A. 92, 8744–8748 (1995)CrossRefGoogle Scholar
  38. 38.
    F.I. Ataullakhanov, G.T. Guriia, Spatial aspects of the dynamics of blood coagulation. I. Hypothesis. Biophysics (Oxf) 39(1), 89–96 (1994)Google Scholar
  39. 39.
    F.I. Ataullakhanov, G.T. Guria, A.Y. Safroshkina, Spatial aspects of the dynamics of blood coagulation. II. Phenomenological model. Biophysics (Oxf) 39, 99–108 (1994)Google Scholar
  40. 40.
    G.M. Willems, T. Lindhout, W.T. Hermens, H.C. Hemker, Simulation model for thrombin generation in plasma. Haemostasis 21, 197–207 (1991)Google Scholar
  41. 41.
    R.J. Wagenvoord, P.W. Hemker, H.C. Hemker, The limits of simulation of the clotting system. J. Thromb. Haemost. 4, 1331–1338 (2006)CrossRefGoogle Scholar
  42. 42.
    R.M.W. Kremers, B. De Laat, R.J. Wagenvoord, H.C. Hemker, Computational modelling of clot development in patient-specific cerebral aneurysm cases: comment. J. Thromb. Haemost. 15, 395–396 (2017)CrossRefGoogle Scholar
  43. 43.
    F.I. Ataullakhanov, D.A. Molchanova, A.V. Pokhilko, A simulated mathematical model of the blood coagulation system intrinsic pathway. Biofiz. Russ. 40, 434–42 (1995)Google Scholar
  44. 44.
    A.V. Pokhilko, Threshold properties of the blood coagulation system, PhD Thesis (1994)Google Scholar
  45. 45.
    F.I. Ataullakhanov, A.V. Pohilko, E.I. Sinauridze, R.I. Volkova, Calcium threshold in human plasma clotting kinetics. Thromb. Res. 75, 383–394 (1994)CrossRefGoogle Scholar
  46. 46.
    M.A. Khanin, D.V. Rakov, A.E. Kogan, Mathematical model for the blood coagulation prothrombin time test. Thromb. Res. 89, 227–232 (1998)CrossRefGoogle Scholar
  47. 47.
    A.E. Kogan, D.V. Kardakov, M.A. Khanin, Analysis of the activated partial thromboplastin time test using mathematical modeling. Thromb. Res. 101, 299–310 (2001)CrossRefGoogle Scholar
  48. 48.
    J.H. Lawson, M. Kalafatis, S. Stram, K.G. Mann, A model for the tissue factor pathway to thrombin. I. An empirical study. J. Biol. Chem. 269, 23357–23366 (1994)Google Scholar
  49. 49.
    K.C. Jones, K.G. Mann, A model for the tissue factor pathway to thrombin. II. A mathematical simulation. J. Biol. Chem. 269, 23367–23373 (1994)Google Scholar
  50. 50.
    R.J. Leipold, T.A. Bozarth, A.L. Racanelli, I.B. Dicker, Mathematical model of serine protease inhibition in the tissue factor pathway to thrombin. J. Biol. Chem. 270, 25383–25387 (1995)CrossRefGoogle Scholar
  51. 51.
    M.F. Hockin, K.C. Jones, S.J. Everse, K.G. Mann, A model for the stoichiometric regulation of blood coagulation. J. Biol. Chem. 277, 18322–18333 (2002)CrossRefGoogle Scholar
  52. 52.
    T. Orfeo, S. Butenas, K.E. Brummel-Ziedins, K.G. Mann, The tissue factor requirement in blood coagulation. J. Biol. Chem. 280, 42887–42896 (2005)CrossRefGoogle Scholar
  53. 53.
    C.M. Danforth, T. Orfeo, K.G. Mann, K.E. Brummel-Ziedins, S.J. Everse, The impact of uncertainty in a blood coagulation model. Math. Med. Biol. 26, 323–336 (2009)MathSciNetCrossRefGoogle Scholar
  54. 54.
    C.M. Danforth, T. Orfeo, S.J. Everse, K.G. Mann, K.E. Brummel-Ziedins, Defining the boundaries of normal thrombin generation: investigations into hemostasis. PLoS One 7, e30385 (2012)CrossRefGoogle Scholar
  55. 55.
    K.E. Brummel-Ziedins, T. Orfeo, P.W. Callas, M. Gissel, K.G. Mann, E.G. Bovill, The prothrombotic phenotypes in familial protein C deficiency are differentiated by computational modeling of thrombin generation. PLoS One 7, e44378 (2012)CrossRefGoogle Scholar
  56. 56.
    M.C. Bravo, T. Orfeo, K.G. Mann, S.J. Everse, Modeling of human factor Va inactivation by activated protein C. BMC Syst. Biol. 6, 45 (2012)CrossRefGoogle Scholar
  57. 57.
    A.Y. Mitrophanov, F.R. Rosendaal, J. Reifman, Computational analysis of the effects of reduced temperature on thrombin generation: the contributions of hypothermia to coagulopathy. Anesth. Analg. 117, 565–574 (2013)CrossRefGoogle Scholar
  58. 58.
    A.Y. Mitrophanov, A.S. Wolberg, J. Reifman, Kinetic model facilitates analysis of fibrin generation and its modulation by clotting factors: implications for hemostasis-enhancing therapies. Mol. Biosyst. 10, 2347 (2014)CrossRefGoogle Scholar
  59. 59.
    V.I. Zarnitsina, A.V. Pokhilko, F.I. Ataullakhanov, A mathematical model for the spatio-temporal dynamics of intrinsic pathway of blood coagulation. I. The model description. Thromb. Res. 84, 225–236 (1996)CrossRefGoogle Scholar
  60. 60.
    V.I. Zarnitsina, A.V. Pokhilko, F.I. Ataullakhanov, A mathematical model for the spatio-temporal dynamics of intrinsic pathway of blood coagulation. II. Results. Thromb. Res. 84, 333–344 (1996)CrossRefGoogle Scholar
  61. 61.
    A.V. Pokhilko, F.I. Ataullakhanov, Spatial dynamics of blood coagulation. A mathematical model. Biol. Membr. 3, 250–263 (2002)Google Scholar
  62. 62.
    M.A. Panteleev, M. V. Ovanesov, D.A. Kireev, A.M. Shibeko, E.I. Sinauridze, N.M. Ananyeva, A.A. Butylin, E.L. Saenko, F.I. Ataullakhanov, Spatial propagation and localization of blood coagulation are regulated by intrinsic and protein C pathways, respectively. Biophys. J. 90, 1489–1500 (2006)CrossRefGoogle Scholar
  63. 63.
    A.M. Shibeko, E.S. Lobanova, M.A. Panteleev, F.I. Ataullakhanov, Blood flow controls coagulation onset via the positive feedback of factor VII activation by factor Xa. BMC Syst. Biol. 4 (2010)Google Scholar
  64. 64.
    A.N. Balandina, A.M. Shibeko, D.A. Kireev, A.A. Novikova, I.I. Shmirev, M.A. Panteleev, F.I. Ataullakhanov, Positive feedback loops for factor V and factor VII activation supply sensitivity to local surface tissue factor density during blood coagulation. Biophys. J. 101, 1816–1824 (2011)CrossRefGoogle Scholar
  65. 65.
    M.A. Panteleev, A.N. Balandina, E.N. Lipets, M. V. Ovanesov, F.I. Ataullakhanov, Task-Oriented modular decomposition of biological networks: trigger mechanism in blood coagulation. Biophys. J. 98, 1751–1761 (2010)CrossRefGoogle Scholar
  66. 66.
    A.L. Fogelson, Continuum models of platelet aggregation: formulation and mechanical properties. SIAM J. Appl. Math. 52, 1089–1110 (1992)MathSciNetCrossRefGoogle Scholar
  67. 67.
    A.L. Fogelson, A.L. Kuharsky, Membrane binding-site density can modulate activation thresholds in enzyme systems. J. Theor. Biol. 193, 1–18 (1998)CrossRefGoogle Scholar
  68. 68.
    A.L. Kuharsky, A.L. Fogelson, Surface-mediated control of blood coagulation: the role of binding site densities and platelet deposition. Biophys. J. 80, 1050–1074 (2001)CrossRefGoogle Scholar
  69. 69.
    K. Leiderman, A.L. Fogelson, Grow with the flow: a spatial-temporal model of platelet deposition and blood coagulation under flow. Math. Med. Biol. 28, 47–84 (2011)MathSciNetCrossRefGoogle Scholar
  70. 70.
    A.L. Fogelson, Y.H. Hussain, K. Leiderman, Blood clot formation under flow: the importance of factor XI depends strongly on platelet count. Biophys. J. 102, 10–18 (2012)CrossRefGoogle Scholar
  71. 71.
    K.G. Link, M.T. Stobb, J. Di Paola, K.B. Neeves, A.L. Fogelson, S.S. Sindi, K. Leiderman, A local and global sensitivity analysis of a mathematical model of coagulation and platelet deposition under flow. PLoS One 13, e0200917 (2018)CrossRefGoogle Scholar
  72. 72.
    A.A. Onasoga-Jarvis, K. Leiderman, A.L. Fogelson, M. Wang, M.J. Manco-Johnson, J.A. Di Paola, K.B. Neeves, The effect of factor VIII deficiencies and replacement and bypass therapies on thrombus formation under venous flow conditions in microfluidic and computational models. PLoS One 8(11), e78732 (2013)Google Scholar
  73. 73.
    V. Govindarajan, V. Rakesh, J. Reifman, A.Y. Mitrophanov, Computational study of thrombus formation and clotting factor effects under venous flow conditions. Biophys. J. 110, 1869–1885 (2016)CrossRefGoogle Scholar
  74. 74.
    S.W. Jordan, E.L. Chaikof, Simulated surface-induced thrombin generation in a flow field. Biophys. J. 101, 276–286 (2011)CrossRefGoogle Scholar
  75. 75.
    D.A. Molchanova, A.A. Butylin, F.I. Ataullakhanov, Investigation of dynamic blood coagulation regimes using a mathematical model. Biol. Membr. 21, 420–432 (2004)Google Scholar
  76. 76.
    V.I. Zarnitsina, F.I. Ataullakhanov, A.I. Lobanov, O.L. Morozova, Dynamics of spatially nonuniform patterning in the model of blood coagulation. Chaos 11, 57–70 (2001)CrossRefGoogle Scholar
  77. 77.
    E.S. Lobanova, F.I. Ataullakhanov, Unstable trigger waves induce various intricate dynamic regimes in a reaction-diffusion system of blood clotting. Phys. Rev. Lett. 91, 1–4 (2003)CrossRefGoogle Scholar
  78. 78.
    E.S. Lobanova, F.I. Ataullakhanov, Running pulses of complex shape in a reaction-diffusion model. Phys. Rev. Lett. 93, 1–4 (2004)CrossRefGoogle Scholar
  79. 79.
    E.S. Lobanova, E.E. Shnol, F.I. Ataullakhanov, Complex dynamics of the formation of spatially localized standing structures in the vicinity of saddle-node bifurcations of waves in the reaction-diffusion model of blood clotting. Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top. 70, 4 (2004)MathSciNetGoogle Scholar
  80. 80.
    E.A. Ermakova, M.A. Panteleev, E.E. Shnol, Blood coagulation and propagation of autowaves in flow. Pathophysiol. Haemost. Thromb. 34, 135–142 (2005)CrossRefGoogle Scholar
  81. 81.
    The Systems Biology Markup Language (SBML). http://sbml.org

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Alexey Tokarev
    • 1
    • 2
  • Nicolas Ratto
    • 1
    • 3
    • 4
  • Vitaly Volpert
    • 1
    • 3
    • 4
    Email author
  1. 1.Peoples’ Friendship University of Russia (RUDN University)MoscowRussian Federation
  2. 2.Dmitry Rogachev National Research Center of Pediatric HematologyOncology and ImmunologyMoscowRussian Federation
  3. 3.Institute Camille Jordan, UMR 5208 CNRSUniversity Lyon 1VilleurbanneFrance
  4. 4.INRIAUniversit de Lyon, Universit Lyon 1, Institute Camille JordanVilleurbanneFrance

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