# A Short Review of Advances in the Modelling of Blood Rheology and Clot Formation

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{®}Living Heart Human Model). Prediction of clot/plaque formation and growth in regions of flow recirculation and stagnation in such devices requires reliable models that can be integrated in computational simulations of blood flow. In an earlier review article [2], the salient features and microstructural underpinnings of the rheology of blood and clots, the prominent features and biochemical reactions underlying the formation, growth and lysis of clots, and the various pathologies that are a consequence of clotting, were summarized. We also posited an integrated framework for the study of blood flow with clot formation. In this review article, we summarize the constitutive models in both the field of blood rheology and that of clot formation and growth, and bring the reader abreast of the developments since the appearance of [2].

## 2. Advances in Modelling

#### 2.1. Constitutive Models for Blood

#### 2.1.1. Model of Anand et al. (2013)

#### 2.2. Models for Clot Formation and Lysis

#### 2.2.1. Single-Scale Models

#### 2.2.2. Multi-Scale Models

## 3. Discussion

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Baillargeon, B.; Rebelo, N.; Fox, D.D.; Taylor, R.L.; Kuhl, E. The Living Heart Project: A robust and integrative simulator for human heart function. Eur. J. Mech. A Solids
**2014**, 48, 38–47. [Google Scholar] [CrossRef] [PubMed] - Anand, M.; Rajagopal, K.; Rajagopal, K.R. A model incorporating some of the mechanical and biochemical factors underlying clot formation and dissolution in flowing blood. J. Theor. Med.
**2003**, 5, 183–218. [Google Scholar] [CrossRef] - Charm, S.E.; Kurland, G.S. Viscometry of human blood for shear rate of 0–100,000 sec
^{−1}. Nature**1965**, 206, 617–618. [Google Scholar] [CrossRef] [PubMed] - Thurston, G.B. Viscoelasticity of human blood. Biophys. J.
**1972**, 12, 1205–1217. [Google Scholar] [CrossRef] - Thurston, G.B. Frequency and shear rate dependence of viscoelasticity of blood. Biorheology
**1973**, 10, 375–381. [Google Scholar] [PubMed] - Thurston, G.B. Rheological Parameters for the viscosity, viscoelasticity and thixotropy of Blood. Biorheology
**1979**, 16, 149–162. [Google Scholar] [PubMed] - McMillan, D.E.; Utterback, N.G.; Nasrinasrabadi, M.; Lee, M.M. An Instrument to evaluate the time dependent flow properties of blood at moderate shear rates. Biorheology
**1986**, 23, 63–74. [Google Scholar] [PubMed] - Womersley, J.R. Method for the calculation of velocity, rate of flow and viscous drag in arteries when the pressure gradient is known. J. Physiol.
**1955**, 127, 553–563. [Google Scholar] [CrossRef] [PubMed] - Casson, N. A flow equation for pigment-oil suspensions of the printing ink type. In Rheology of Disperse Systems; Mill, C.C., Ed.; Pergamon Press: Oxford, UK, 1959; pp. 84–104. [Google Scholar]
- Liepsch, D.; Moravec, S. Pulsatile flow of Non-Newtonian fluids in distensible models of human arteries. Biorheology
**1984**, 21, 571–586. [Google Scholar] [PubMed] - Cho, Y.I.; Kensey, K.R. Effects of the non-newtonian viscosity of blood on flows in a diseased arterial vessel. Part I: Steady flows. Biorheology
**1991**, 28, 241–262. [Google Scholar] [PubMed] - Quemada, D. A non-linear Maxwell model of biofluids: Application to normal blood. Biorheology
**1993**, 30, 253–265. [Google Scholar] [PubMed] - Sharp, M.K.; Thurston, G.B.; Moore, J.E. The effect of blood viscoelasticity on pulastile flow in stationary and axially moving tubes. Biorheology
**1996**, 33, 185–206. [Google Scholar] [CrossRef] - Phillips, W.M.; Deutsch, S. Toward a constitutive equation for blood. Biorheology
**1975**, 12, 383–389. [Google Scholar] [PubMed] - Yeleswarapu, K.K. Evaluation of Continuum Models for Characterizing the Constitutive Behavior of Blood. Ph.D. Thesis, University of Pittsburgh, Pittsburgh, PA, USA, 1996. [Google Scholar]
- Sun, N.; De Kee, D. Simple shear, hysteresis and yield stress in biofluids. Can. J. Chem. Eng.
**2001**, 79, 36–41. [Google Scholar] - Anand, M.; Rajagopal, K.R. A shear-thinning viscoelastic fluid model for describing the flow of blood. Int. J. Cardiovasc. Med. Sci.
**2004**, 4, 59–68. [Google Scholar] - Owens, R.G. A new micro-structure based constitutive model for blood. J. Non-Newton. Fluid Mech.
**2006**, 40, 57–70. [Google Scholar] [CrossRef] - Capek, M. A Non-Newtonian Model of Blood Capturing Segregation of Erythrocytes. Unpublished, Necas Center for Mathematical Modeling. 2014. [Google Scholar]
- Massoudi, M.; Phuoc, T.X. Pulsatile flow of blood using a modified second grade fluid model. Comput. Math. Appl.
**2008**, 56, 199–211. [Google Scholar] [CrossRef] - Wu, W.T.; Aubry, N.; Massoudi, M.; Antaki, J.F. Transport of platelets induced by red blood cells based on mixture theory. Int. J. Eng. Sci.
**2017**, 118, 16–27. [Google Scholar] [CrossRef] - Bodnar, T.; Rajagopal, K.R.; Sequeira, A. Simulation of the three-dimensional flow of blood using a shear-thinning viscoelastic fluid model. Math. Model. Nat. Phenom.
**2011**, 6, 1–24. [Google Scholar] [CrossRef] - Anand, M.; Kwack, J.H.; Masud, A. A new generalized Oldroyd-B model for blood flow in complex geometries. Int. J. Eng. Sci.
**2013**, 72, 78–88. [Google Scholar] [CrossRef] - Rajagopal, K.R. Multiple Natural Configurations in Continuum Mechanics; Technical Report “Report 6”; Institute of Computational and Applied Mechanics, University of Pittsburgh: Pittsburgh, PA, USA, 1995. [Google Scholar]
- Rajagopal, K.R.; Srinivasa, A.R. A thermodynamic framework for rate-type fluids. J. Non-Newton. Fluid Mech.
**2000**, 88, 207–227. [Google Scholar] [CrossRef] - Rajagopal, K.R. On implicit constitutive theories. Appl. Math.
**2013**, 48, 279–319. [Google Scholar] [CrossRef] - Mann, K.G. Thrombin formation. Chest
**2003**, 124, 4S–10S. [Google Scholar] [CrossRef] [PubMed] - Orfeo, T.; Butenas, S.; Brummel-Ziedins, K.E.; Mann, K.G. The tissue factor requirement in blood coagulation. J. Biol. Chem.
**2005**, 280, 42887–42896. [Google Scholar] [CrossRef] [PubMed] - Panteleev, M.A.; Dashkevich, N.M.; Ataullakhanov, F.I. Hemostasis and thrombosis beyond biochemistry: Roles of geometry, flow and diffusion. Thromb. Res.
**2015**, 136, 699–711. [Google Scholar] [CrossRef] [PubMed] - Hockin, M.F.; Jones, K.C.; Everse, S.J.; Mann, K.G. A Model for the stoichiometric regulation of blood coagulation. J. Biol. Chem.
**2002**, 277, 18322–18333. [Google Scholar] [CrossRef] [PubMed] - Diamond, S.L. Systems biology of coagulation. J. Thromb. Haemost.
**2013**, 11, 224–232. [Google Scholar] [CrossRef] [PubMed] - Kuharsky, A.L.; Fogelson, A.L. Surface-mediated control of blood coagulation: The role of binding site densities and platelet deposition. Biophys. J.
**1997**, 80, 1050–1074. [Google Scholar] [CrossRef] - Panteleev, M.A.; Ovanesov, M.V.; Kireev, D.A.; Shibeko, A.M.; Sinauridze, E.I.; Ananyeva, N.M.; Butylin, A.A.; Saenko, E.L.; Ataullakhanov, F.I. Spatial propagation and localization of blood coagulation are regulated by intrinsic and protein C Pathways, respectively. Biophys. J.
**2006**, 90, 1489–1500. [Google Scholar] [CrossRef] [PubMed] - Anand, M.; Rajagopal, K.; Rajagopal, K.R. 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. Theor. Biol.
**2008**, 253, 725–738. [Google Scholar] [CrossRef] [PubMed] - Luan, D.; Zai, M.; Varner, J.D. Computationally derived points of fragility of a human cascade are consistent with current therapeutic strategies. PLoS Comput. Biol.
**2007**, 3, e142. [Google Scholar] [CrossRef] [PubMed] - Chatterjee, M.S.; Denney, W.S.; Jing, H.; Diamond, S.L. Systems biology of coagulation initiation: Kinetics of thrombin generation in resting and activated human blood. PLoS Comput. Biol.
**2010**, 6, e1000950. [Google Scholar] [CrossRef] [PubMed] - Susree, M.; Anand, M. A mathematical model for in vitro coagulation of blood: Role of platelet count and inhibition. Sadhana
**2017**, 42, 291–305. [Google Scholar] - Shibeko, A.M.; Panteleev, M.A. Untangling the complexity of blood coagulation network: Use of computational modelling in pharmacology and diagnostics. Brief. Bioinform.
**2016**, 17, 429–439. [Google Scholar] [CrossRef] [PubMed] - Bodnar, T.; Sequeira, A. Numerical simulation of the coagulation dynamics of blood. Comput. Math. Methods Med.
**2008**, 9, 83–104. [Google Scholar] [CrossRef] - Sequeira, A.; Bodnar, T. Blood coagulation simulations using a viscoelastic model. Math. Model. Nat. Phenom.
**2014**, 9, 34–45. [Google Scholar] [CrossRef] - Leiderman, K.; Fogelson, A.L. Grow with the flow: A spatial-temporal model of platelet deposition and blood coagulation under flow. Math. Med. Biol.
**2011**, 28, 47–84. [Google Scholar] [CrossRef] [PubMed] - Leiderman, K.; Fogelson, A.L. The influence of hindered transport on the development of platelet thrombi under flow. Bull. Math. Biol.
**2013**, 75, 1255–1283. [Google Scholar] [CrossRef] [PubMed] - Leiderman, K.; Fogelson, A.L. An overview of mathematical modeling of thrombus formation under flow. Thromb. Res.
**2014**, 133, S12–S14. [Google Scholar] [CrossRef] [PubMed] - Fogelson, A.L.; Neeves, K.B. Fluid mechanics of blood clot formation. Annu. Rev. Fluid Mech.
**2015**, 47, 377–403. [Google Scholar] [CrossRef] [PubMed] - Wu, W.T.; Jamiolkowski, M.A.; Wagner, W.R.; Aubry, N.; Massoudi, M.; Antaki, J.F. Multi-Constituent Simulation of Thrombus Deposition. Sci. Rep.
**2017**, 7, 42720. [Google Scholar] [CrossRef] [PubMed] - Xu, Z.; Chen, N.; Kamocka, M.; Rosen, E.D.; Alber, M.S. A multiscale model of thrombus development. J. R. Soc. Interface
**2008**, 5, 705–722. [Google Scholar] [CrossRef] [PubMed] - Jones, K.C.; Mann, K.G. A model for the tissue factor pathway to thrombin. II. A mathematical simulation. J. Biol. Chem.
**1994**, 269, 23367–23373. [Google Scholar] [PubMed] - Xu, Z.; Lioi, J.; Mu, J.; Kamocka, M.M.; Liu, X.; Chen, D.Z.; Rosen, E.D.; Alber, M.S. A multiscale model of venous thrombus formation with surface-mediated control of blood coagulation cascade. Biophys. J.
**2010**, 98, 1723–1732. [Google Scholar] [CrossRef] [PubMed] - Xu, Z.; Kim, O.; Kamocka, M.M.; Rosen, E.D.; Alber, M.S. Multiscale models of thrombogenesis. Wiley Interdiscip. Rev. Syst. Biol. Med.
**2012**, 4, 237–246. [Google Scholar] [CrossRef] [PubMed] - Bessonov, N.; Sequeira, A.; Simakov, S.; Vassilevskii, Y.; Volpert, V. Methods of blood flow modelling. Math. Model. Nat. Phenom.
**2016**, 11, 1–25. [Google Scholar] [CrossRef] - Mann, K.G. Is there value in kinetic modeling of thrombin generation? Yes. J. Thromb. Haemost.
**2012**, 10, 1463–1469. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**Configurations of a viscoelastic fluid body with instantaneous elasticity. Reproduced with permission from Anand, M., et al., International Journal of Engineering Science; published by Elsevier, 2013 [23].

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Anand, M.; Rajagopal, K.R.
A Short Review of Advances in the Modelling of Blood Rheology and Clot Formation. *Fluids* **2017**, *2*, 35.
https://doi.org/10.3390/fluids2030035

**AMA Style**

Anand M, Rajagopal KR.
A Short Review of Advances in the Modelling of Blood Rheology and Clot Formation. *Fluids*. 2017; 2(3):35.
https://doi.org/10.3390/fluids2030035

**Chicago/Turabian Style**

Anand, Mohan, and Kumbakonam Ramamani Rajagopal.
2017. "A Short Review of Advances in the Modelling of Blood Rheology and Clot Formation" *Fluids* 2, no. 3: 35.
https://doi.org/10.3390/fluids2030035