# Modelling of blood thrombosis at microscopic and mesoscopic scales

### Abstract

Blood coagulation at the place of the complete severing of a vessel or puncturing of a vessel sidewall is usually a beneficial reaction, as it protects the body from bleeding and maintains hemostasis, while the formation of a blood clot inside the blood vessel is a pathological phenomenon, which is highly dangerous, and sometimes leads to serious complications. In this paper, two scales of modelling blood thrombosis will be introduced using numerical methods and fluid dynamics. The meso-scale model of the flow is described by Navier-Stokes equations and the blood thrombosis model is based on equations of transport and diffusion. The equations describing levels of concentrations of factors responsible for blood coagulation can be implemented into a solver solving Navier-Stokes equations, what will enable simulation of blood flow and estimation of the risk of thrombus formation related to flow conditions. The proposed micro-scale model is using molecular dynamics to simulate interactions between blood cells and vascular walls. An effective combination of both models is possible thanks to the introduction of the multiple-time stepping algorithm, which enables a full visualization of blood flow, coupling molecular interaction with the fluid mechanics equation. The goal of the paper is to present the latest literature review on the possibilities of blood coagulation modelling in two scales and the main achievements in blood thrombosis research: the key role of transport and experimental background.

### Keywords

multi-scale model, molecular dynamics, fluid dynamics, blood rheology, blood thrombosis,### References

[1] M.M. Aleman, B.L. Walton, J.R. Byrnes, A.S. Wolberg. Fibrinogen and red blood cells in venous thrombosis. Thrombosis Research, 133: S38–S40, 2014.[2] T. Almomani, H.S. Udaykumar, J.S. Marshall, K.B. Chandran. Micro-scale dynamic simulation of erythrocyteplatelet interaction in blood flow. Annals of Biomedical Engineering, 36(6): 905–920, 2008.

[3] M. Anand, K. Rajagopal, K.R. Rajagopal. A model for the formation and lysis of blood clots. Pathophysiology of Haemostasis and Thrombosis, 34: 109–120, 2005.

[4] A.C. Ashwood, S.J. Vanden Hogen, M.A. Rodarte, C.R. Kopplin, D.J. Rodr´ıguez, E.R. Hurlburt, T.A. Shedd. A multiphase, micro-scale PIV measurement technique for liquid film velocity measurements in annular twophase flow. International Journal Multiphase Flow, 68: 27–39, 2015.

[5] F.I. Ataullakhanov, M.A. Panteleev. Mathematical modeling and computer simulation in blood coagulation. Pathophysiology of Haemostasis and Thrombosis, 34(2–3): 60–70, 2006.

[6] L. Badimon, J. Badimon, A. Galvez, J. Chesebro, V. Fuster. Influence of arterial damage and wall shear rate on platelet deposition. Ex vivo study in a swine model. Arteriosclerosis, 6: 312–320, 1986.

[7] P. Bagcji. Mesoscale simulation of blood flow in small vessels. Biophysical Journal, 92: 1858–1877, 2007.

[8] F. Bajd, I. Sersa. Mathematical modeling of blood clot fragmentation during flow-mediated thrombolysis. Biophysical Journal, 104: 1181–1190, 2013.

[9] D. Basmadjian. The effect of flow and mass transport in thrombogenesis. Annals of Biomedical Engineering, 18: 685–709, 1990.

[10] E. Beltrami, J. Jesty. Mathematical analysis of activation thresholds in enzyme-catalyzed positive feedbacks: application to the feedbacks of blood coagulation. Proceedings of National Academy of Sciences of the United States of America, 92: 8744–8748, 1995.

[11] E. Beltrami, J. Jesty. The role of membrane patch size and flow in regulating a proteolytic feedback threshold on a membrane: possible application in blood coagulation. Mathematical Biosciences, 172: 1–13, 2001.

[12] B. Bebenek. Flows in the Circulatory System [in Polish: Przepływy w układzie krwionosnym]. Politechnika Krakowska, Kraków, Polska, 1999.

[13] T. Bodn´ar, A. Sequeira. Numerical simulation of the coagulation dynamics of blood. Computational and Mathematical Methods in Medicine, 9: 83–104, 2008.

[14] T. Bodn´ar, A. Sequeira, M. Prosi. On the shear-thinning and viscoelastic effects of blood flow under various flow rates. Applied Mathematics and Computation, 217: 5055–5067, 2011.

[15] T. Bodn´ar, G.P. Galdi, S. Necasova. Fluid-Structure Interaction and Biomedical Applications. Springer, Basel, Switzerland, 2014.

[16] A. Bouchnita, T. Galochkina, P. Kurbatova, P. Nony, V. Volpert. Conditions of microvessel occlusion for blood coagulation in flow. International Journal of Numerical Methods in Biomedical Engineering, 33: 1–15, 2017.

[17] A. Bouchnita, A. Tosenberger, V. Volpert. On the regimes of blood coagulation. Applied Mathematics Letters, 51: 74–79, 2016.

[18] L.F. Brass, S.L. Diamond. Transport physics and biorheology in the setting of hemostasis and thrombosis. Journal of Thrombosis and Haemostasis, 14: 906–917, 2016.

[19] S.E. Charm, W. McComis, G. Kurland. Rheology and structure of blood suspension. Journal of Applied Physiology, 19: 127–133, 1964.

[20] R. Chebbi. Dynamic of blood flow: modeling of the F°ahræus and Lindqvist effect. Journal of Biological Physics, 41: 313–326, 2015.

[21] J. Chen, X. Lu, W. Wang. Non-Newtonian effects of blood flow on hemodynamics in distal vascular graft anastomoses. Journal of Biomechanics, 38: 1983–1995, 2006.

[22] I. Cimrak, I. Jancigova. Computational Blood Cell Mechanics: Road Towards Models and Biomedical Applications. Taylor & Francis Ltd., 2018.

[23] S. Cito, M.D. Mazzeo, L. Badimon. A review of macroscopic thrombus modeling methods. Thrombosis Research, 131: 116–124, 2013.

[24] J.C.A. Cluitmans, V. Chokkalingam, A.M. Janssen, R. Brock, W.T.S. Huck, G.J.C.G.M. Bosman. Alterations in red blood cell deformability during storage: a microfluidic approach. BioMed Research International, 764268, 9 pages, 2014.

[25] G. Cokelet, H. Goldsmith. Decreased hydrodynamic resistance in the two-phase flow of blood through small vertical tubes at low flow rates.Circulation Research, 68: 1–17, 1991.

[26] Z. Dabrowski. Blood Physiology. PWN, Warszawa, Poland, 2000.

[27] M.C.H. De Visser, L.A. Sandkuijl, R.P.M. Lensen, H.L. Vos, F.R. Rosendaal, R.M. Bertina. Linkage analysis of factor VIII and von Willebrand factor loci as quantitative trait loci. Journal of Thrombosis and Haemostasis, 1: 1771–1776, 2003.

[28] L. Diamond. Systems biology of coagulation. Journal of Thrombosis and Haemostasis, 11: 224–232, 2013.

[29] A. Fasano, R. Santos, A. Sequeira. Blood coagulation: a puzzle for biologists, a maze for mathematicians. In: D. Ambrosi, A. Quarteroni, G. Rozza [Eds.], Modeling of Physiological Flows. MS&A – Modeling, Simulation and Applications, pp. 41–75, Springer-Verlag, Italy, 2012.

[30] A. Fasano, A. Sequeira. Hemomath: The Mathematics of Blood. Springer, Cham, Switzerland, 2017.

[31] R. Feng, M. Xenos, M. Girdhar, W. Kang, J.W. Davenport, Y. Deng, D. Bluestein. Viscous flow simulation in a stenosis model using discrete particle dynamics: a comparison between DPD and CFD. Biomechanics and Modeling in Mechanobiology, 11: 119–129, 2012.

[32] N. Filipovic, M. Kojic, A. Tsuda. Modelling thrombosis using dissipative particle dynamics method. Philosophical Transactions of the Royal Society of London A, 366: 3265–3279, 2008.

[33] A. Fisher, J. Rossmann. Effect of non-Newtonian behavior on hemodynamics of cerebral aneurysms. Journal of Biomechanical Engineering, 131(9), 9 pages, 2009.

[34] M.H. Flamm, S.L. Diamond. Multiscale systems biology and physics of thrombosis under flow. Annals of Biomedical Engineering, 40: 2355–2364, 2012.

[35] A.L. Fogelson, K.B. Neeves. Fluid mechanics of blood clot formation. Annual Review of Fluid Mechanics, 47: 377–403, 2015.

[36] A.L. Fogelson. Continuum models of platelet aggregation: formulation and mechanical properties. SIAM Journal of Applied Mathematics, 52: 1089–1110, 1992.

[37] A.L. Fogelson. Continuum models of platelet aggregation: mechanical properties and chemically-induced phase transitions. Contemporary Mathematics, 141: 279–294, 1993.

[38] A.L. Fogelson, D. Robert. Platelet-wall interactions in continuum models of platelet thrombosis: formulation and numerical solution. Mathematical Medicine and Biology: A Journal of the IMA, 21: 293–334, 2004.

[39] S.P. Fu, Z. Peng, H. Yuan, R. Kfoury, Y.N. Young. Lennard-Jones type pair-potential method for coarse-grained lipid bilayer membrane simulations in LAMMPS. Computer Physics Communications, 210: 193–203, 2017.

[40] Y. Fu, J. Wu, J. Wu, R. Sun, Z. Ding, C. Dong. Micro-PIV measurements of the flow field around cells in flow chamber. Journal of Hydrodynamics, 27: 562–568, 2015.

[41] H. Gharahi, B. Zambrano, D. Zhu, J. DeMarco, S. Baek. Computational fluid dynamic simulation of human carotid artery bifurcation based on anatomy and volumetric blood flow rate measured with magnetic resonance imaging. International Journal of Advances in Engineering Sciences and Applied Mathematics, 8: 40–60, 2016.

[42] V. Govindarajan, V. Rakesh, J. Reifman, A.Y. Mitrophanov. Computational study of thrombus formation and clotting factor effects under venous flow conditions. Biophysical Journal, 108: 1869–1885, 2016.

[43] B. Guerciotti, C. Vergara. Computational Comparison Between Newtonian and Non-Newtonian Blood Rheologies in Stenotic Vessels. MOX-Report No. 19/2016, Department of Mathematics, Politecnico di Milano, Milano, Italy, 2016.

[44] L. Hook, D. Anderson, R. Langer, P. Williams, M. Davies, M. Alexander. High throughput methods applied in biomaterial development and discovery. Biomaterials, 31: 187–198, 2010.

[45] H. Hosseinzadegan, D.F. Tafti. Modeling thrombus formation and growth. Biotechnology and Bioengineering, 114: 2154–2172, 2017.

[46] A. Jain, A. Graveline, A. Waterhouse, A. Vernet, R. Flaumenhaft, D. Ingber. A shear gradient-activated microfluidic device for automated monitoring of whole blood haemostasis and platelet function. Nature Communications, 7, article number: 10176, 2016.

[47] A. Jonasova, J. Vimmr. Numerical simulation of non-Newtonian blood flow in bypass models. Applied Mathematics and Mechanics, 8: 10179–10180, 2008.

[48] H. Kamada, Y. Imai, M. Nakamura, T. Ishikawa, T. Yamaguchi. Computational analysis on the mechanical interaction between a thrombus and red blood cells: possible causes of membrane damage of red blood cells at microvessels. Medical Engineering and Physics, 34: 1411–1420, 2012.

[49] M. Kopernik, A. Milenin. Two-scale finite element model of multi-layer blood chamber of POLVAD EXT. Archives of Civil and Mechanical Engineering, 12: 178–185, 2012.

[50] M. Kopernik, J. Nowak. Numerical modelling of the opening process of the three-coating aortic valve. Archives of Mechanics, 61(3–4): 171–193, 2009.

[51] D. Ku. Blood flow in arteries. Annual Review of Fluid Mechanics, 29: 399–434, 1997.

[52] D. Leslie et al. A bioinspired omniphobic surface coating on medical devices prevents thrombosis and biofouling. Nature Biotechnology, 32: 1134–1140, 2014.

[53] X. Li, Z. Peng, H. Lei, M. Dao, G. Karniadakis. Probing red blood cell mechanics, rheology and dynamics with a two-component multi-scale model. Philosophical Transactions of the Royal Society A, 372, 1, 2014.

[54] K.Y. Lin, T.C. Shih, S.H. Chou, Z.Y. Chen, C.H. Hsu, C.Y. Ho. Computational fluid dynamics with application of different theoretical flow models for the evaluation of coronary artery stenosis on CT angiography: comparison with invasive fractional flow reserve. Biomedical Physics and Engineering Express, 2, 065011, 2016, doi: 10.1088/2057-1976/2/6/065011.

[55] Z. Liu, Y. Deng, Y. Wu. Topology Optimization Theory for Laminar Flow. Springer Nature Customer Service Center GmbH, 2017.

[56] A.I. Lobanov, T.K. Starozhilova. The effect of convective flows on blood coagulation processes. Pathophysiology of Haemostatis and Thrombosis, 34: 121–134, 2005.

[57] K. Lykov, X. Li, H. Lei, I.V. Pivkin, G.E. Karniadakis. Inflow/outflow boundary conditions for particle-based blood flow simulations: application to arterial bifurcations and trees. PLoS Computional Biology, 11(8): e1004410, 2015.

[58] S. Maussumbekova, A. Beketaeva. Application of immersed boundary method in modelling of thrombosis in the blood flow. In: N. Danaev, Y. Shokin, A.Z. Darkhan [Eds.], Mathematical Modeling of Technological Processes. CITech 2015. Communications in Computer and Information Science, vol. 549. Springer, Cham, 2015.

[59] S.K. Mitra, S. Chakraborty [Eds.]. Microfluidics and Nanofluidics Handbook. CRC Press, Boca Raton, USA, 2018.

[60] M. Nakamura, S. Wada. Mesoscopic blood flow simulation considering hematocrit-dependent viscosity. Journal of Biomechanical Science and Engineering, 5: 578–590, 2010.

[61] P. Neofytou, D. Drikakis. Non-Newtonian flow instability in a channel with a sudden expansion. Journal of Non-Newtonian Fluid Mechanics, 111: 127–150, 2003.

[62] P. Nilsson, K. Ekdahl, P. Magnusson, H. Qu, H. Iwata, D. Ricklin, J. Hong, J.D. Lambris, B. Nilsson, Y. Teramura. Autoregulation of thromboinflammation on biomaterial surfaces by a multicomponent therapeutic coating. Biomaterials, 34: 985–994, 2013.

[63] K. Papadopoulos, M. Gavaises, C. Atkin. A simplified mathematical model for thrombin generation. Medical Engineering and Physics, 36: 196–204, 2014.

[64] H. Park, E. Yeom, S.-J. Seo, J.-H. Lim, S.-J. Lee. Measurement of real pulsatile blood flow using X-ray PIV technique with CO2 microbubbles. Scientific Reports, 5, article number: 8840, 2015.

[65] J. Pavlova, A. Fasano, J. Janela, A. Sequeira. Numerical validation of a synthetic cell-based model of blood coagulation. Journal of Theoretical Biology, 380: 367–379, 2015.

[66] S.J. Plimpton. Fast parallel algorithms for short-range molecular dynamics. Journal of Computional Physics, 117: 1–19, 1995.

[67] A. Pries, D. Neuhaus, P. Gaehtgens. Blood viscosity in tube flow: dependence on diameter and hematocrit. American Journal of Physiology – Heart and Circulatory Physiology, 263: H1770–1778, 1992.

[68] A.M. Robertson, A. Sequeira, R.G. Owens. Rheological models for blood. In: L. Formaggia, A. Quarteroni, A. Veneziani [Eds.], Cardiovascular Mathematics. Modeling and Simulation of the Circulatory System, 211–241. Springer-Verlag, Italy, 2009.

[69] D.S. Sankar, A.K. Nagar, A.V. Kumar. Mathematical analysis of single and two-phase flow of blood in narrow arteries with multiple contrictions. Journal of Applied Fluid Mechanics, 8: 871–883, 2015.

[70] J. Soares, C. Gao, Y. Alemu, M. Slepian, D. Bluestein. Simulation of platelets suspension flowing through a stenosis model using a dissipative particle dynamics approach. Annals of Biomedical Engineering, 41: 2318–2333, 2013.

[71] E.N. Sorensen, G.W. Burgreen, W.R. Wagner, J.F. Antaki. Computational simulation of platelet deposition and activation: II. Results for Poiseuille flow over collagen. Annals of Biomedical Engineering, 27: 449–458, 1999.

[72] M. Thiriet, K. Parker. Cardiovascular Mathematics. Springer-Verlag, Milano, Italy, 2009.

[73] P. Tokarczyk, M. Kopernik. Development of Meso-Scale Model of Blood Thrombosis. Book of Abstracts. Wydawnictwo Instytutu Zrównowazonej Energetyki, K. Styszko et al. [Eds.], AGH ISC 2018, 10–12 October 2018, Krakow, Poland, 2018.

[74] M. Toloui, B. Firoozabadi, M. Saidi. A numerical study of the effects of blood rheology and vessel deformability on the hemodynamics of carotid bifurcation. Scienta Iranica, 19: 119–126, 2012.

[75] A. Tosenberger. Blood flow modelling and applications to blood coagulation and atherosclerosis. Ph.D. Dissertation, Lyon, 2014.

[76] A. Tosenberger, F. Ataullakhanov, N. Bessonov, M. Panteleev, A. Tokarev, V. Volpert. Modelling of thrombus growth in flow with a DPD-PDE method. Journal of Theoretical Biology, 337: 30–41, 2013.

[77] A. Tosenberger, F. Ataullakhanov, N. Bessonov, M. Panteleev, A. Tokarev, V. Volpert. Modelling of plateletfibrin clot formation in flow with a DPD–PDE method. Journal of Mathematical Biology, 72: 649–681, 2016.

[78] E. Tsiklidis, C. Sims, T. Sinno, S.L. Diamond. Multiscale systems biology of trauma-induced coagulopathy. WIREs Systems Biology and Medicine, 10(4): e1418, 2018, doi.org/10.1002/wsbm.1418.

[79] A. Valencia, A. Zarate, M. Galvez, L. Badilla. Non-Newtonian blood flow dynamics in a right internal carotid artery with a saccular aneurysm. International Journal for Numerical Methods in Fluids, 50: 751–764, 2006.

[80] P. Vennemann, J. Westerweel. Full-field blood velocity measurement techniques. Lecture Notes – ABIOMED, 91–108, 2005.

[81] S. Wada, M. Nakamura. Multiscale analysis of blood flow: Modeling and simulation of multiple red blood cell flow. Processings of the 12th Asian Congress of Fluid Mechanics. Daejeon, Korea, 2008.

[82] E. Wazna. Platelet-mediated regulation of immunity [in Polish: Płytki krwi jako regulatory procesów odpornosciowych]. Postepy Higieny i Medycyny Doswiadczalnej, 60: 265–277, 2006.

[83] J.O. Wilkes. Fluid Mechanics for Chemical Engineers. Prentice Hall, 2017.

[84] W.-T. Wu, N. Aubry, M. Massoudi, J.F. Antaki.Transport of platelets induced by red blood cells based on mixture theory. International Journal of Engineering Science, 118: 16–27, 2017.

[85] Z. Wu, Z. Xu, O. Kim, M. Alber. Three-dimensional multi-scale model of deformable platelets adhesion to vessel wall in blood flow. Philosophical Transactions of the Royal Society Of London Series A, 372 (2021), 2014.

[86] A.R. Wufsus, N.E. Macera, K.B. Neeves. The hydraulic permeability of blood clots as a function of fibrin and platelet density. Biophysical Journal, 104: 1812–1823, 2013.

[87] Z. Xu, N. Chen, M.M. Kamocka, E.D. Rosen, M. Alber. A multiscale model of thrombus development. Journal of the Royal Society Interface, 5: 705–722, 2008.

[88] Z. Xu, O. Kim, M.M. Kamocka, E.D. Rosen, M. Alber. Multiscale models of thrombogenesis. WIREs Systems Biology and Medicine, 4: 237–246, 2012.

[89] Z. Xu, J. Lioi, J. Mu, M.M. Kamocka, X. Liu, D.Z. Chen, E.D. Rosen, M. Alber. A multiscale model of venous thrombus formation with surface-mediated control of blood coagulation cascade. Journal of Biophysics, 98: 1723–1732, 2010.

[90] K. Yu, Y. Mei, N. Hadjesfandiari, J.N. Kizhakkedathu. Engineering biomaterials surfaces to modulate the host response. Colloids and Surfaces B: Biointefaces, 124: 69–79, 2014.

[91] P. Zhang, C. Gao, N. Zhang, M.J. Slepian, Y. Deng, D. Bluestein. Multiscale particle-based modeling of flowing platelets in blood plasma using dissipative particle dynamics and coarse grained molecular dynamics. Cellular and Molecular Bioengineering, 7: 552–574, 2014.

[92] J.-B. Zhang, Z.-B. Kuang. Study on blood constitutive parameters in different blood constitutive equations. Journal of Biomechanics, 33: 355–360, 2000.

[93] P. Zhang, N. Zhang, Y. Deng, D. Bluestein. A multiple time stepping algorithm for efficient multiscale modeling of platelets flowing in blood plasma. Journal of Computational Physics, 284: 668–686, 2015.

[94] S. Zhu, Y. Lu, T.R. Sinno, S.L. Diamond. Dynamics of thrombin generation and flux from clots during whole human blood flow over collagen/tissue factor surfaces. Biological Chemistry, 291: 23027–23035, 2016.

**Computer Assisted Methods in Engineering and Science**, [S.l.], v. 25, n. 1, p. 21-45, feb. 2019. ISSN 2299-3649. Available at: <https://cames.ippt.pan.pl/index.php/cames/article/view/227>. Date accessed: 26 jan. 2022. doi: http://dx.doi.org/10.24423/cames.227.