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2011, 8(2): 425-443. doi: 10.3934/mbe.2011.8.425

Blood coagulation dynamics: mathematical modeling and stability results

1. 

Department of Mathematics and CEMAT/IST, Instituto Superior Técnico, Technical University of Lisbon, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal

2. 

Department of Mathematics and CEMAT/IST, Faculty of Sciences and Technology, University of Algarve, Campus de Gambelas 8005-139 Faro, Portugal

3. 

Department of Technical Mathematics, Faculty of Mechanical Engineering, Czech Technical University, Náměstí 13, 121 35 Prague 2, Czech Republic

Received  March 2010 Revised  September 2010 Published  April 2011

The hemostatic system is a highly complex multicomponent biosystem that under normal physiologic conditions maintains the fluidity of blood. Coagulation is initiated in response to endothelial surface vascular injury or certain biochemical stimuli, by the exposure of plasma to Tissue Factor (TF), that activates platelets and the coagulation cascade, inducing clot formation, growth and lysis. In recent years considerable advances have contributed to understand this highly complex process and some mathematical and numerical models have been developed. However, mathematical models that are both rigorous and comprehensive in terms of meaningful experimental data, are not available yet. In this paper a mathematical model of coagulation and fibrinolysis in flowing blood that integrates biochemical, physiologic and rheological factors, is revisited. Three-dimensional numerical simulations are performed in an idealized stenosed blood vessel where clot formation and growth are initialized through appropriate boundary conditions on a prescribed region of the vessel wall. Stability results are obtained for a simplified version of the clot model in quiescent plasma, involving some of the most relevant enzymatic reactions that follow Michaelis-Menten kinetics, and having a continuum of equilibria.
Citation: Adélia Sequeira, Rafael F. Santos, Tomáš Bodnár. Blood coagulation dynamics: mathematical modeling and stability results. Mathematical Biosciences & Engineering, 2011, 8 (2) : 425-443. doi: 10.3934/mbe.2011.8.425
References:
[1]

M. Anand, K. Rajagopal and K. R. Rajagopal, A model incorporating some of the mechanical and biochemical factors underlying clot formation and dissolution in flowing blood, J. of Theoretical Medicine, 5 (2003), 183-218. doi: 10.1080/10273660412331317415.

[2]

M. Anand, K. Rajagopal and K. R. Rajagopal, 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. of Theoretical Biology, 253 (2008), 725-738. doi: 10.1016/j.jtbi.2008.04.015.

[3]

F. I. Ataullakhanov and M. A. Panteleev, Mathematical modeling and computer simulation in blood coagulation, Pathophysiol. Haemost. Thromb., 34 (2005), 60-70. doi: 10.1159/000089927.

[4]

S. P. Bhat and D. S. Bernstein, Nontangency-based Lyapunov tests for convergence and stability in systems having a continuum of equilibria, SIAM J. Control Optim., 42 (2003), 1745-1775. doi: 10.1137/S0363012902407119.

[5]

T. Bodnár and A. Sequeira, Numerical simulation of the coagulation dynamics of blood, Comp. Math. Methods in Medicine, 9 (2008), 83-104. doi: 10.1080/17486700701852784.

[6]

I. Borsi, A. Farina, A. Fasano and K. R. Rajagopal, Modelling the combined chemical and mechanical action for blood clotting, In "Nonlinear Phenomena with Energy Dissipation," volume 29 of GAKUTO Internat. Ser. Math. Sci. Appl., pages 53-72. Gakkōtosho, Tokyo, 2008.

[7]

S. L. Campbell and N. J. Rose, Singular perturbations of autonomous linear systems, SIAM Journal Math. Anal., 10 (1979), 542-551. doi: 10.1137/0510051.

[8]

M. H. Kroll, J. D. Hellums, L. V. McIntire, A. I. Schafer and J. L. Moake, Platelets and shear stress, Blood, 88 (1996), 1525-1541.

[9]

A. L. Kuharsky and A. L. Fogelson, Surface-mediated control of blood coagulation: The role of binding site densities and platelet deposition, Biophys. J., 80 (2001), 1050-1074. doi: 10.1016/S0006-3495(01)76085-7.

[10]

A. Leuprecht and K. Perktold, Computer simulation of non-Newtonian effects of blood flow in large arteries, Computer Methods in Biomechanics and Biomech. Eng., 4 (2001), 149-163.

[11]

L. Michaelis and M. L. Menten, Die kinetik der invertinwirkung, Biochem Z., 49 (1913), 333-369.

[12]

Y. H. Qiao, J. L. Liu and Y. J. Zeng, A kinetic model for simulation of blood coagulation and inhibition in the intrinsic path, J. of Medical Eng. and Technology, 29 (2005), 70-74. doi: 10.1080/03091900410001709079.

[13]

A. M. Robertson, A. Sequeira and M. V. Kameneva, Hemorheology, In "Hemodynamical Flows: Modeling, Analysis and Simulation (Oberwolfach Seminars)," volume 37, G.P. Galdi, R. Rannacher, A. M. Robertson, and S. Turek (Eds.), Birkhäuser Verlag, 2008, 63-120.

[14]

M. Schenone, B. C. Furie and B. Furie, The blood coagulation cascade, Curr. Opin. Hematol., 11 (2004), 272-277. doi: 10.1097/01.moh.0000130308.37353.d4.

[15]

L. A. Segel and M. Slemrod, The quasy-steady-state assumption: A case study in perturbation, SIAM Review, 32 (1989), 446-477. doi: 10.1137/1031091.

[16]

N. T. Wang and A. L. Fogelson, Computational methods for continuum models of platelet aggregation, J. Comput. Phys., 151 (1999), 649-675. doi: 10.1006/jcph.1999.6212.

show all references

References:
[1]

M. Anand, K. Rajagopal and K. R. Rajagopal, A model incorporating some of the mechanical and biochemical factors underlying clot formation and dissolution in flowing blood, J. of Theoretical Medicine, 5 (2003), 183-218. doi: 10.1080/10273660412331317415.

[2]

M. Anand, K. Rajagopal and K. R. Rajagopal, 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. of Theoretical Biology, 253 (2008), 725-738. doi: 10.1016/j.jtbi.2008.04.015.

[3]

F. I. Ataullakhanov and M. A. Panteleev, Mathematical modeling and computer simulation in blood coagulation, Pathophysiol. Haemost. Thromb., 34 (2005), 60-70. doi: 10.1159/000089927.

[4]

S. P. Bhat and D. S. Bernstein, Nontangency-based Lyapunov tests for convergence and stability in systems having a continuum of equilibria, SIAM J. Control Optim., 42 (2003), 1745-1775. doi: 10.1137/S0363012902407119.

[5]

T. Bodnár and A. Sequeira, Numerical simulation of the coagulation dynamics of blood, Comp. Math. Methods in Medicine, 9 (2008), 83-104. doi: 10.1080/17486700701852784.

[6]

I. Borsi, A. Farina, A. Fasano and K. R. Rajagopal, Modelling the combined chemical and mechanical action for blood clotting, In "Nonlinear Phenomena with Energy Dissipation," volume 29 of GAKUTO Internat. Ser. Math. Sci. Appl., pages 53-72. Gakkōtosho, Tokyo, 2008.

[7]

S. L. Campbell and N. J. Rose, Singular perturbations of autonomous linear systems, SIAM Journal Math. Anal., 10 (1979), 542-551. doi: 10.1137/0510051.

[8]

M. H. Kroll, J. D. Hellums, L. V. McIntire, A. I. Schafer and J. L. Moake, Platelets and shear stress, Blood, 88 (1996), 1525-1541.

[9]

A. L. Kuharsky and A. L. Fogelson, Surface-mediated control of blood coagulation: The role of binding site densities and platelet deposition, Biophys. J., 80 (2001), 1050-1074. doi: 10.1016/S0006-3495(01)76085-7.

[10]

A. Leuprecht and K. Perktold, Computer simulation of non-Newtonian effects of blood flow in large arteries, Computer Methods in Biomechanics and Biomech. Eng., 4 (2001), 149-163.

[11]

L. Michaelis and M. L. Menten, Die kinetik der invertinwirkung, Biochem Z., 49 (1913), 333-369.

[12]

Y. H. Qiao, J. L. Liu and Y. J. Zeng, A kinetic model for simulation of blood coagulation and inhibition in the intrinsic path, J. of Medical Eng. and Technology, 29 (2005), 70-74. doi: 10.1080/03091900410001709079.

[13]

A. M. Robertson, A. Sequeira and M. V. Kameneva, Hemorheology, In "Hemodynamical Flows: Modeling, Analysis and Simulation (Oberwolfach Seminars)," volume 37, G.P. Galdi, R. Rannacher, A. M. Robertson, and S. Turek (Eds.), Birkhäuser Verlag, 2008, 63-120.

[14]

M. Schenone, B. C. Furie and B. Furie, The blood coagulation cascade, Curr. Opin. Hematol., 11 (2004), 272-277. doi: 10.1097/01.moh.0000130308.37353.d4.

[15]

L. A. Segel and M. Slemrod, The quasy-steady-state assumption: A case study in perturbation, SIAM Review, 32 (1989), 446-477. doi: 10.1137/1031091.

[16]

N. T. Wang and A. L. Fogelson, Computational methods for continuum models of platelet aggregation, J. Comput. Phys., 151 (1999), 649-675. doi: 10.1006/jcph.1999.6212.

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