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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. doi: 10.1080/10273660412331317415. Google Scholar

[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. doi: 10.1016/j.jtbi.2008.04.015. Google Scholar

[3]

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

[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. doi: 10.1137/S0363012902407119. Google Scholar

[5]

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

[6]

I. Borsi, A. Farina, A. Fasano and K. R. Rajagopal, Modelling the combined chemical and mechanical action for blood clotting,, In, 29 (2008), 53. Google Scholar

[7]

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

[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. Google Scholar

[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. doi: 10.1016/S0006-3495(01)76085-7. Google Scholar

[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. Google Scholar

[11]

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

[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. doi: 10.1080/03091900410001709079. Google Scholar

[13]

A. M. Robertson, A. Sequeira and M. V. Kameneva, Hemorheology,, In, (2008), 63. Google Scholar

[14]

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

[15]

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

[16]

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

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. doi: 10.1080/10273660412331317415. Google Scholar

[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. doi: 10.1016/j.jtbi.2008.04.015. Google Scholar

[3]

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

[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. doi: 10.1137/S0363012902407119. Google Scholar

[5]

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

[6]

I. Borsi, A. Farina, A. Fasano and K. R. Rajagopal, Modelling the combined chemical and mechanical action for blood clotting,, In, 29 (2008), 53. Google Scholar

[7]

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

[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. Google Scholar

[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. doi: 10.1016/S0006-3495(01)76085-7. Google Scholar

[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. Google Scholar

[11]

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

[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. doi: 10.1080/03091900410001709079. Google Scholar

[13]

A. M. Robertson, A. Sequeira and M. V. Kameneva, Hemorheology,, In, (2008), 63. Google Scholar

[14]

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

[15]

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

[16]

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

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