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Numerical Modelling of the Atherosclerotic Plaques Effect in Damaged Vessels on Coagulation Dynamics

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The presence of atherosclerotic plaques in the vessels leads to serious diseases of the cardiovascular system. In this paper, the problems of the dynamics of coagulation processes in stenotic vessels are studied. The system of Navier-Stokes equations for an incompressible fluid, supplemented by equations of changing the concentration of metabolites (inhibitor, activator and fibrinogen) is considered. A segment of a vessel with a damaged wall with atherosclerotic plaques is considered as computational domain. The effects of the size and position of these plaques on the dynamics of clot propagation are studied. Blood is considered as a Newtonian fluid as well as non-Newtonian fluid. In the case when the atherosclerotic plaque is located in front of the affected area of the vessel, numerical analysis has showed that due to an obstacle in the vessel, a sharp flow of fluid through the area with atherosclerotic plaque has been detected, then the flow slows down, thereby slowing down the spreading rate of the fibrin clot. With an increase in the size of the plaque, a slowdown in the growth rate of fibrinogen has been observed. In the case when blood is considered as a non-Newtonian fluid, numerical calculations lead to a slowdown in the spread of fibrinogen in space compared to the case of a Newtonian fluid.
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Blood Coagulation; Concentration of Metabolites; Non-Newtonian Fluid; Navier-Stokes Equations; Reaction-Diffusion System

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J.W. Weisel, The mechanical properties of fibrin for basic scientists and clinicians, Biophysical Chemistry 112 (2004) 267 - 276.

A. Fogelson, H.Yu, A. Kuharsky, Computational Modeling of Blood Clotting: Coagulation and Three-dimensional Platelet Aggregation. Polymer and Cell Dynamics, 145-154, 2003.

J.W. Weisel, C. Nagaswami, Computer modeling of fibrin polymerization kinetics correlated with electron microscope and turbidity observations: clot structure and assembly are kinetically controlled, Biophysical Journal, Volume 63, Issue 1, 1992, Pages 111-128.

M. Anand, K Rajagopal, A Model for the Formation and Lysis of Blood Clots. Pathophysiol Haemos Thromb 2005;34:109-120.

S. Butenas, K. Mann, Blood coagulation, Biochem. (Moscow) 67 (31), 3-12, 2002.

T. Orfeo, S. Butenas, K. Brummel-Ziedins, K. Mann, The tissue factor requirement in blood coagulation, J. Biol. Chem. 280 (52), 42887-42896, 2005.

H. Hemker, Thrombin generation, an essential step in haemostasis and thrombosis, Haemost. Thromb. 3, 477-491, 1993.

H. Hemker, S. Beguin, Thrombin generation in plasma: its assessment via the endogenous thrombin potential, Thromb. Haemost. 74 (1), 134-138, 1995.

T. Orfeo, K. Brummel-Ziedins, M. Gissel, S. Butenas, K. Mann, The nature of the stable blood clot procoagulant activities, J. Biol. Chem., 283 (15), 9776-9786, 2008.

D. Gailani, G. Broze, Factor xi activation in a revised model of blood coagulation, Science 253 (5022), 909-912, 1991.

S. Bungay, P. Gentry, R. Gentry, A mathematical model of lipid-mediated thrombin generation, Mathematical Medicine and Biology 20 (1),105-129, 2003.

K. C. Jones, K. G. Mann, A model for the tissue factor pathway to thrombin, Journal of Biological Chemistry 269 (37), 23367-73, 1994.

M. Hockin, A model for the stoichiometric regulation of blood coagulation, Journal of Biological Chemistry, 277 (21), 18322-33, 2002.

M. Khanin, D. Rakov, A. Kogan, Mathematical model for the blood coagulation prothrombin time test, Thrombosis research, 89 (5), 227-32,1998.

C. Xu, X. H. Xu, Y. Zeng, Y. W. Chen, Simulation of a mathematical model of the role of the TFPI in the extrinsic pathway of coagulation, Computers in Biology and Medicine 35 (5), 435-45, 2005.

Henri H. Versteeg, Johan W. M. Heemskerk, Marcel Levi, and Pieter H. Reitsma, New Fundamentals in Hemostasis, Physiological Reviews 2013 93:1, 327-358.

C. Longstaff, K. Kolev, Basic mechanisms and regulation of fibrinolysis, J. Thromb Haemost., 13(Suppl. 1): 98-105, 2015

A. Fogelson, A mathematical model and numerical method for studying platelet adhesion and aggregation during blood clotting, Journal of Computational Physics, Volume 56, Issue 1, 1984, Pages 111-134, ISSN 0021-9991.

V. Govindarajan, V. Rakesh, J. Reifman, A.Y. Mitrophanov, Computational Study of Thrombus Formation and Clotting Factor Effects under Venous Flow Conditions, Biophysical Journal, Volume 110, Issue 8, 2016, Pages 1869-1885, ISSN0006-3495,

C. Menichini, X.Y. Xu, Mathematical modeling of thrombus formation in idealized models of aortic dissection: initial findings and potential applications. J. Math. Biol. 73, 1205-1226, 2016.

J. Biasetti, An integrated fluid-chemical model toward modeling the formation of intra-luminal thrombus in abdominal aortic aneurysms, Frontiers in physiology, Т. 3, 266, 2016.

A.S. Bedekar, K. Pant, Y. Ventikos, S. Sundaram,A Computational Model Combining Vascular Biology and Haemodynamics for Thrombosis Prediction in Anatomically Accurate Cerebral Aneurysms, Food and Bioproducts Processing, Volume 83, Issue 2, 2005, Pages 118-126, ISSN 0960-3085.

A. Yazdani, H. Li, JD Humphrey, GE Karniadakis, A General Shear-Dependent Model for Thrombus Formation, PLoS Comput Biol 13(1): e1005291, 2017.

S. Cito, M. Mazzeo, L. Badimon, A Review of Macroscopic Thrombus Modeling Methods, Thrombosis Research, Volume 131, Issue 2, 2013, Pages 116-124, ISSN 0049-3848.

R. Guy, A. Fogelson, J. Keener, Fibrin gel formation in a shear flow, Math. Med. Biol. 24, 111-130, 2007.

L. M. Haynes, Y. Dubief, K. Mann, Membrane binding events in the initiation and propagation phases of tissue factor-initiated zymogen activation under flow, Journal of Biological Chemistry 287 (8) 5225-5234, 2012.

A. I. Lobanov, T. K. Starozhilova, The effect of convective flows on blood coagulation processes, Pathophysiology of haemostasis and thrombosis, 34 (2-3), 121-134, 2005.

A. L. Fogelson, N. Tania, Coagulation under flow: the influence of flow mediated transport on the initiation and inhibition of coagulation, Pathophysiology of haemostasis and thrombosis, 34 (2-3), 91-108,2005.

S. W. Jordan, E. L. Chaikof, Simulated surface-induced thrombin generation in a flow field, Biophysical journal 101 (2), 276-286, 2011.

S. Gir, S. Slack, V. Turito, A numerical analysis of factor x activation in the presence of tissue factor-factor viia complex in a ow reactor, Annals of Biomedical Engineering, 24 (8), 394-399,1996.

A. Sequeira, R. Santos, T. Bodnar, Blood coagulation dynamics: mathematical modeling and stability results, Mathematical biosciences and engineering, 8 (2), 425-443, 2011.

T. Bodnar, A. Sequeira, Numerical simulation of the coagulation dynamics of blood, Computational and Mathematical Methods in Medicine 9 (2), 83-104, 2008.

A. Sequeira, T. Bodnar, Blood coagulation simulations using a viscoelastic model, Mathematical Modelling of Natural Phenomena, 9 (6), 34-45, 2014.

E. S. Babushkina, N. M. Bessonov, F. I. Ataullakhanov, M. A. Panteleev, Continuous modeling of arterial platelet thrombus formation using a spatial adsorption equation, PLoS ONE 10 (10), e0141068, 2015.

A. Tokarev, I. Sirakov, G. Panasenko, V. Volpert, E. Shnol, A. Butylin, F. Ataullakhanov, Continuous mathematical model of platelet thrombus formation in blood ow, Russian Journal of Numerical Analysis and Mathematical Modelling, 27 (2), 192-212, 2012.

A. L. Fogelson, Y. H. Hussain, K. Leiderman, Blood clot formation under ow: The importance of factor xi depends strongly on platelet count, Biophys. J. 102, 10-18, 2012.

J.W. Bauer, L. Xu, E.A. Vogler, C. A. Siedlecki, Surface dependent contact activation of factor XII and blood plasma coagulation induced by mixed thiol surfaces, 2017, Biointerphases, V 12, N 2, 10.1116/1.4983634.

F. Gaertner, S. Massberg, Blood coagulation in immunothrombosis-At the frontline of intravascular immunity, Seminars in Immunology, Volume 28, Issue 6, 2016, Pages 561-569, ISSN 1044-5323.

R. D. Averett, B. Menn, E. H. Lee, C. C. Helms, T. Barker, M. Guthold, A Modular Fibrinogen Model that Captures the Stress-Strain Behavior of Fibrin Fibers, Biophysical Journal, Volume 103, Issue 7, 2012, Pages 1537-1544, ISSN 0006-3495.

S. Yesudasan, X.Wang, Averett, R.D. Fibrin polymerization simulation using a reactive dissipative particle dynamics method, Biomechanics and Modeling in Mechanobiology, 17, 1389-1403, 2018.

M. Anand, K.R. Rajagopal, A Short Review of Advances in the Modelling of Blood Rheology and Clot Formation. Fluids, 2, 35, 1-9, 2017.

T. Yamaguchi, T. Ishikawa, Y. Imai et al. Particle-Based Methods for Multiscale Modeling of Blood Flow in the Circulation and in Devices: Challenges and Future Directions. Ann Biomed Eng 38, 1225-1235, 2010.

S. Maussumbekova, A. Beketaeva, Application of Immersed Boundary Method in Modeling of Thrombosis in the Blood Flow, Mathematical Modeling of Technological Processes. Springer, 2015, pp. 108-117.

C.S.Peskin, Flow patterns around heart valves: A numerical method, Journal of Computational Physics, Issue 10, 1972, pp. 252- 271.

I. Lee, H. Choi, A discrete-forcing immersed boundary method for the fluid-structure interaction of an elastic slender body, Journal of Computational Physics, Volume 280, 2015, Pages 529-546, ISSN 0021-9991.

G.C. Bourantas, D.S. Lampropoulos, B.F. Zwick, V.C. Loukopoulos, A. Wittek, K. Miller, Immersed boundary finite element method for blood flow simulation, Computers & Fluids, Volume 230, 2021, 105162, ISSN 0045-7930.

F.I. Ataullakhanov., M.A Panteleev. Mathematical modeling and computer simulation in blood coagulation, Pathophysiol Haemost Thromb; 34: 60-70, 2005.

A.J. Chorin Numerical solution of the Navier-Stokes equations. Math Comput, 22(104):745-762, 1968.

J. Kim, P. Moin, Application of a fractional-step method to incompressible Navier-Stokes equations, J. of Computational Physics, 59, N 2. 308-323, 1985.

K. Anupindi, Y.Delorme, DA Shetty, Frankel SH. A novel multiblock immersed boundary method for large eddy simulation of complex arterial hemodynamics. J Comput Phys, 254:200-18, 2013.

D. de Zélicourt, L. Ge, C. Wang, F. Sotiropoulos, A. Gilmanov, A.Yoganathan, Flow simulations in arbitrarily complex cardiovascular anatomies-An unstructured Cartesian grid approach. Comput & Fluids 2009;38(9):1749-62.

E.A Fadlun, R. Verzicco, P. Orlandi, J. Mohd-Yusof, Combined immersed-boundary finite-difference methods for three-dimensional complex flow simulations, J. of Computational Physics. 2000. 161, N 1. 35-60.

M. Pourquie, W.P. Breugem, B.J. Boersma, Some issues related to the use of immersed boundary methods to represent square obstacles, Int. J. for Multiscale Computational Engineering, 2009. 7, N 6. 509-522.

F. Domenichini, On the consistency of the direct forcing method in the fractional step solution of the Navier-Stokes equations, J. of Computational Physics, 227, N 12. 6372-6384, 2008.

R.D.Guy, D.A. Hartenstine, On the accuracy of direct forcing immersed boundary methods with projection methods, J. of Computational Physics,. 229, N 7. 2479-2496, 2010.

K. Taira, T.Colonius, The immersed boundary method: a projection approach, J. of Computational Physics, 225, N 2. 2118-2137, 2007.

Y. Mori, C.S. Peskin, Implicit second-order immersed boundary methods with boundary-mass, Computer Methods in Applied Mechanics and Engineering, 197, N 25-28. 2049-2067, . 2008.

MO Khan, K. Valen-Sendstad, D. A. Steinman, Direct numerical simulation of laminar-turbulent transition in a non-axisymmetric stenosis model for Newtonian vs. Shear-thinning non-Newtonian rheology, Flow Turbul Combust, 102(1):43-72, 2019.

G. C. Bourantas, B. L.Cheeseman, R. Ramaswamy, I. F. Sbalzarini, Using DC PSE operator discretization in Eulerian meshless collocation methods improves their robustness in complex geometries, Comput & Fluids, 136:285-300, 2016.

J. Mikhal, B. Geurts, Development and application of a volume penalization immersed boundary method for the computation of blood flow and shear stresses in cerebral vessels and aneurysms. J Math Biol, 67(6-7):1847-75, 2013.

T. Gao, Y.-H Tseng., X.-Y. Lu, An improved hybrid Cartesian/immersed boundary method for fluid-solid flows, Int. J. for Numerical Methods in Fluids. 2007. 55, N 12. 1189-1211.

S. Su, M.-C. Lai, C.-An Lin, An immersed boundary technique for simulating complex flows with rigid boundary, Computers & Fluids, V. 36, Issue 2, 313-324, 2007, ISSN 0045-7930.

G. Valencia, J. Núñez, and J. Duarte, Multiobjective optimization of a plate heat exchanger in a waste heat recovery organic rankine cycle system for natural gas engines, Entropy, vol. 21, no. 7, 2019.

H. S Udaykumar, R. Mittal, P. Rampunggoon, A. Khanna, A sharp interface Cartesian grid method for simulating flows with complex moving boundaries, J. of Computational Physics, 174, N 1. 345-380, 2001.

M. Braza, P.Chassaing, H.H. Minh, Numerical study and physical analysis of the pressure and velocity fields in the near wake of a circular cylinder, J. of Fluid Mechanics, 165. 79-130, 1986.

C. Liu, X. Zheng, C.H. Sung, Preconditioned multigrid methods for unsteady incompressible flows, J. of Computational Physics, 139, N 1. 35-57, 1998.

S. Maussumbekova, A. Beketaeva, Numerical modeling of the dynamics of blood flow through thrombosis, International Journal of Biology and Biomedical Engineering, 12, стр. 59-65, 2018.

M.M. Cross, Rheology of non-Newtonian fluids: a new flow equation for pseudoplastic systems, Journal of Colloid Science 20(5), 417-437, (1965).


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