Skip to main content
SpringerLink
Log in

A free boundary problem modeling thrombus growth

Model development and numerical simulation using the level set method

  • Published:
Journal of Mathematical Biology Aims and scope Submit manuscript

Abstract

Recently, neglect of either shear stress, surface saturation, or thrombus growth in mathematical models of platelet deposition has been identified as leading cause of inability to match experimental evidence. While the consideration of shear stress is necessary to obtain at least some qualitative agreement, purely shear-dependent approaches yield notable quantitative discrepancies. In a previous paper, the author demonstrated that surface saturation significantly improves model predictions. However, discrepancies still persist when thrombus growth is neglected. Therefore, the present work develops a free boundary problem which takes this into account. Numerical simulations are performed using the level set method. The results agree well with measurements in stagnation point flow and tubular expansions, which demonstrates the coupling of flow, platelet adhesion, and aggregate growth in primary hemostasis.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Affeld K, Reininger AJ, Gadischke J, Grunert K, Schmidt S, Thiele F (1995) Fluid mechanics of the stagnation point flow chamber and its platelet deposition. Artif Organs 19(7): 597–602

    Article  Google Scholar 

  • Babuska I (1969) Numerical solution of boundary value problems by the perturbated variational principle. Technical report, University of Maryland

  • Barstad RM, Kierulf P, Sakariassen KS (1996) Collagen induced thrombus formation at the apex of eccentric stenoses—a time course study with non-anticoagulated human blood. Thromb Haemostasis 75(4): 685–692

    Google Scholar 

  • Becker R, Braack M (2000) Multigrid techniques for finite elements on locally refined meshes. Numer Linear Algebra Appl 7(6): 363–379

    Article  MATH  MathSciNet  Google Scholar 

  • Becker R, Braack M (2004) A two-level stabilization scheme for the Navier-Stokes equations. In: Feistauer M et al (eds) ENUMATH 2003. Numerical Mathematics and Advanced Applications. Springer, Heidelberg, pp 123–130

    Google Scholar 

  • Becker R, Braack M, Dunne T, Meidner D, Richter T, Schmich M, Vexler B (2006) The finite element toolkit Gascoigne (2006). http://www.gascoigne.uni-hd.de

  • Becker R, Dunne T (2007) VisuSimple: an interactive open source visualization utility for scientific computing. In: Jäger W, Rannacher R, Warnatz J (eds) Reactive flows, diffusion and transport: from experiments via mathematical modeling to numerical simulation and optimization. Springer, New York. http://visusimple.uni-hd.de

  • Braack M, Burman E (2006) Local projection stabilization for the Oseen problem and its interpretation as a variational multiscale method. SIAM J Numer Anal 43(6): 2544–2566

    MATH  MathSciNet  Google Scholar 

  • Brash JL, Brophy JM, Feuerstein IA (1976) Adhesion of platelets to artificial surfaces: effect of red cells. J Biomed Mater Res 10: 429–443

    Article  Google Scholar 

  • Chen S, Merriman B, Osher S, Smereka P (1997) A simple level set method for solving Stefan problems. J Comput Phys 135: 8–29

    Article  MATH  MathSciNet  Google Scholar 

  • David T, Thomas S, Walker PG (2001) Platelet deposition in stagnation point flow: an analytical and computational simulation. Med Eng Phys 23: 299–312

    Article  Google Scholar 

  • Goldsmith HL, Turitto VT (1986) Rheological aspects of thrombosis and haemostasis: basic principles and applications. Thromb Haemostasis 55(3): 415–435

    Google Scholar 

  • Javierre E, Vuik C, Vermolen FJ, Segal A (2005) A level set method for particle dissolution in a binary alloy. Technical report, Delft University of Technology

  • Jordan A, David T, Homer-Vanniasinkam S, Graham A, Walker PG (2004) The effects of margination and red cell augmented platelet diffusivity on platelet adhesion in complex flow. Biorheology 41: 641–653

    Google Scholar 

  • Juric D, Tryggvason G (1996) A front-tracking method for dendritic solidification. J Comput Phys 123: 127–148

    Article  MATH  MathSciNet  Google Scholar 

  • Karino T, Goldsmith HL (1979) Adhesion of human platelets to collagen on the walls distal to a tubular expansion. Microvasc Res 17(3): 238–262

    Article  Google Scholar 

  • Kroll MH, Hellums JD, McIntire LV, Schafer AI, Moake JL (1996) Platelets and shear stress. Blood 88(5): 1525–1541

    Google Scholar 

  • Kulkarni S, Dopheide SM, Yap CL, Ravanat C, Freund M, Mangin P, Heel KA, Street A, Harper IS, Lanza F, Jackson SP (2000) A revised model of platelet aggregation. J Clin Invest 105(6): 783–791

    Article  Google Scholar 

  • Lee D, Chiu Y-L, Jen C-Y (1999) Wall stresses and platelet adhesion in a t-junction. Proc Natl Sci Counc ROC (A) 23(2): 303–310

    Google Scholar 

  • Nitsche J (1971) Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen die keinen Randbedingungen unterworfen sind. Abh Math Univ Hamburg 36: 9–15

    Article  MATH  MathSciNet  Google Scholar 

  • Osher S, Fedkiw RP (2003) Level set methods and dynamic implicit surfaces. Springer, Berlin

    MATH  Google Scholar 

  • Rannacher R (1984) Finite element solution of diffusion problems with irregular data. Numer Math 43: 309–327

    Article  MATH  MathSciNet  Google Scholar 

  • Saad Y, Schultz MH (1986) GMRES: a generalized minimal residual algorithm for solving non-symmetric linear systems. SIAM J Sci Stat Comput 7: 856–869

    Article  MATH  MathSciNet  Google Scholar 

  • Sakariassen KS, Baumgartner HR (1989) Axial dependence of platelet-collagen interactions in flowing blood: upstream thrombus growth impairs downstream platelet adhesion. Arteriosclerosis 9(1): 33–42

    Google Scholar 

  • Savage B, Saldivar E, Ruggeri ZM (1996) Initiation of platelet adhesion by arrest onto fibrinogen or translocation on von Willebrand factor. Cell 84(2): 289–297

    Article  Google Scholar 

  • Sorensen EN, Burgreen GW, Wagner WR, Antaki JF (1999) Computational simulation of platelet deposition and activation: I. Model development and properties. Ann Biomed Eng 27(4): 436–448

    Article  Google Scholar 

  • Sussman M, Smereka P, Osher S (1994) A level set approach for computing solutions to incompressible two-phase flow. J Comput Phys 114: 146–159

    Article  MATH  Google Scholar 

  • Tryggvason G, Bunner B, Esmaeeli A, Juric D, Al-Rawahi N, Tauber W, Han J, Nas S, Jan Y-J (2001) A front-tracking method for the computations of multiphase flow. J Comput Phys 169: 708–759

    Article  MATH  Google Scholar 

  • Weller FF (2008) Modeling, analysis, and simulation of thrombosis and hemostasis. PhD thesis, University of Heidelberg. http://www.ub.uni-heidelberg.de/archiv/8558

  • Weller FF (2008) Platelet deposition in non-parallel flow. Influence of shear stress and changes in surface reactivity. J Math Biol 57: 333–359

    Article  MATH  MathSciNet  Google Scholar 

  • Zydney AL, Colton CK (1988) Augmented solute transport in the shear flow of a concentrated suspension. Physicochem Hydrodyn 10: 77–96

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, University of Heidelberg, Im Neuenheimer Feld 293, 69120, Heidelberg, Germany

    Frédéric Frank Weller

Authors
  1. Frédéric Frank Weller
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Frédéric Frank Weller.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Weller, F.F. A free boundary problem modeling thrombus growth. J. Math. Biol. 61, 805–818 (2010). https://doi.org/10.1007/s00285-009-0324-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00285-009-0324-1

Keywords

Mathematics Subject Classification (2000)

Search

Navigation