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Journal of Medical and Biological Engineering

, Volume 39, Issue 4, pp 605–621 | Cite as

A Predictive Model of Thrombus Growth in Stenosed Vessels with Dynamic Geometries

  • Hamid Hosseinzadegan
  • Danesh K. TaftiEmail author
Original Article
  • 70 Downloads

Abstract

This paper introduces a dynamic model of platelet-rich thrombus growth in stenosed vessels using computational fluid dynamics methods. Platelet adhesion, aggregation and activation kinetics are modeled by solving mass transport equations for blood components involved in thrombosis. Arbitrary Lagrangian–Eulerian formulation is used to model the growing thrombi with variable geometry. The wall boundaries are discretely moved based on the amount of platelet deposition that occurs on vessel wall. To emulate the dynamic behavior of platelet adhesion kinetics during thrombus growth, a validated model for platelet adhesion, which calculates platelet-surface adhesion rates as a function of stenosis severity and Reynolds number, is applied to the model. Results of the present model for vessel occlusion times and platelet deposition in stenosed region are compared to ex vivo and in vitro experimental data. The model successfully predicts the nonlinear growth of thrombi in the stenosed area. These simulations provide a useful guideline to understand the effect of growing thrombus on thrombus growth rate, platelet activation kinetics and recurrence of embolism in highly stenosed arteries.

Keywords

Numerical modeling Platelet adhesion Platelet activation Shear stress Embolism 

List of Symbols

\(A_{ROI}\)

Region of interest area (mm2)

\(C_{i}\)

Concentration of species \(i\) (PLT ml−1, μΜ, U ml−1)

\(D_{i}\)

Total mass diffusivity of species \(i\) (m2 s−1)

\(D_{b,i}\)

Brownian diffusivity of species \(i\) (m2 s−1)

\(D_{s,i}\)

Enhanced diffusivity of species \(i\) (m2 s−1)

\([H]\)

Heparin concentration (U ml−1)

\(k_{{1,TxA_{2} }}\)

Reaction rate constant for inhibition of TxA2 (s−1)

\(k_{rs}\)

Resting platelet-surface adhesion rate (cm s−1)

\(k_{as}\)

Activated platelet-surface adhesion rate (cm s−1)

\(k_{aa}\)

Activated platelet–platelet adhesion rate (cm s−1)

\(L\)

Characteristic length (m)

\(M_{\infty }\)

Capacity of surface for first platelet layer (PLT cm−2)

\(M_{as}\)

Surface coverage due to resting platelets (PLT cm−2)

\(M_{as}\)

Surface coverage due to activated platelets (PLT cm−1)

\(M_{at}\)

Deposition due to activated platelets (PLT cm−1)

\(R_{RBC}\)

Radius of red blood cell (m)

\(Re\)

Reynolds number at inlet

\(Re_{apex}\)

Reynolds number at the apex

\(s_{{p,TxA_{2} }}\)

Rate of synthesis of TxA2 by platelet (nmol PLT−1 s−1)

\(S\)

Available free surface

\(t_{act}\)

Platelet activation time (s)

\(t_{qss}\)

Simulation time corresponding to quasi-steady state (s)

Greek

\(\beta\)

Conversion factor to convert thrombin concentration from \({\text{U}}\;{\text{ml}}^{ - 1}\) to μM (nmol U−1)

\(\dot{\gamma }\)

Local fluid shear rate (s−1)

\(\dot{\gamma }_{w}\)

Wall shear rate (s−1)

\(\varGamma\)

Griffith’s template model for the kinetics of the heparin-catalyzed inactivation of thrombin by antithrombin

\(\theta\)

Fraction of resting platelets that activate upon surface contact

\(\lambda_{ADP}\)

Amount of ADP per activated platelet (nmol PLT−1)

\(\mu\)

Plasma viscosity (Pa s)

\(\phi_{rt}\)

Rate of thrombin generation from prothrombin at the surface of resting platelets

\(\phi_{at}\)

Rate of thrombin generation from prothrombin at the surface of activated platelets

\(\rho\)

Plasma density (kg m−3)

Notes

Acknowledgements

This study was funded by the National Science Foundation (Grant CBET-1235790). The authors acknowledge the computational support and resources provided by Advanced Research Computing (ARC) at Virginia Tech. Authors declare that they have no conflict of interest. This article does not contain any studies with animals or human participants performed by any of the authors.

References

  1. 1.
    Colace, T. V., Muthard, R. W., & Diamond, S. L. (2012). Thrombus growth and embolism on tissue factor-bearing collagen surfaces under flow role of thrombin with and without fibrin. Arteriosclerosis, Thrombosis, and Vascular Biology, 32(6), 1466–1476.Google Scholar
  2. 2.
    Wu, W.-T., Jamiolkowski, M. A., Wagner, W. R., Aubry, N., Massoudi, M., & Antaki, J. F. (2017). Multi-constituent simulation of thrombus deposition. Scientific Reports, 7, 42720.Google Scholar
  3. 3.
    Xu, Z., Chen, N., Kamocka, M. M., Rosen, E. D., & Alber, M. (2008). A multiscale model of thrombus development. Journal of the Royal Society, Interface, 5(24), 705–722.Google Scholar
  4. 4.
    Xu, Z., Lioi, J., Mu, J., Kamocka, M. M., Liu, X., Chen, D. Z., et al. (2010). A multiscale model of venous thrombus formation with surface-mediated control of blood coagulation cascade. Biophysical Journal, 98(9), 1723–1732.Google Scholar
  5. 5.
    Hosseinzadegan, H., & Tafti, D. K. (2017). Prediction of thrombus growth: Effect of stenosis and reynolds number. Cardiovascular Engineering and Technology.  https://doi.org/10.1007/s13239-017-0304-3.Google Scholar
  6. 6.
    Jackson, S. P. (2007). The growing complexity of platelet aggregation. Blood, 109(12), 5087–5095.Google Scholar
  7. 7.
    Ruggeri, Z. M. (2002). Platelets in atherothrombosis. Nature Medicine, 8(11), 1227–1234.Google Scholar
  8. 8.
    Sixma, J. J., & Groot, P. G. (1994). Regulation of platelet adhesion to the vessel wall. Annals of the New York Academy of Sciences, 714(1), 190–199.Google Scholar
  9. 9.
    Skarja, G. A., Kinlough Rathbone, R. L., Perry, D. W., Rubens, F. D., & Brash, J. L. (1997). A cone and plate device for the investigation of platelet biomaterial interactions. Journal of Biomedical Materials Research, 34(4), 427–438.Google Scholar
  10. 10.
    Tschopp, T. B., Weiss, H. J., & Baumgartner, H. R. (1974). Decreased adhesion of platelets to subendothelium in von Willebrand’s disease. The Journal of Laboratory and Clinical Medicine, 83(2), 296–300.Google Scholar
  11. 11.
    Weiss, H. J. (1995). Flow-related platelet deposition on subendothelium. Thrombosis and Haemostasis, 74(1), 117–122.Google Scholar
  12. 12.
    Hosseinzadegan, H., & Tafti, D. K. (2017). Mechanisms of platelet activation, adhesion, and aggregation. Thrombosis and Haemostasis: Research, 1(2), 1–6.Google Scholar
  13. 13.
    Aarts, P. A., Bolhuis, P. A., Sakariassen, K. S., Heethaar, R. M., & Sixma, J. J. (1983). Red blood cell size is important for adherence of blood platelets to artery. Blood, 62(1), 212–214.Google Scholar
  14. 14.
    Cadroy, Y., & Hanson, S. R. (1990). Effects of red blood cell concentration on hemostasis and thrombus formation in a primate model. Blood, 75(11), 2185–2193.Google Scholar
  15. 15.
    Reasor, D. A., Jr., Mehrabadi, M., Ku, D. N., & Aidun, C. K. (2013). Determination of critical parameters in platelet margination. Annals of Biomedical Engineering, 41(2), 238–249.Google Scholar
  16. 16.
    Hosseinzadegan, H., & Tafti, D. K. (2017). Modeling thrombus formation and growth. Biotechnology and Bioengineering.  https://doi.org/10.1002/bit.26343.Google Scholar
  17. 17.
    Alenitsyn, A., Kondratyev, A., Mikhailova, I., & Siddique, I. (2010). Mathematical modeling of thrombus growth in mesenteric vessels. Mathematical Biosciences, 224(1), 29–34.MathSciNetzbMATHGoogle Scholar
  18. 18.
    Bark, D. L., & Ku, D. N. (2010). Wall shear over high degree stenoses pertinent to atherothrombosis. Journal of Biomechanics, 43(15), 2970–2977.Google Scholar
  19. 19.
    Mehrabadi, M., Casa, L. D. C., Aidun, C. K., & Ku, D. N. (2016). A predictive model of high shear thrombus growth. Annals of Biomedical Engineering, 44(8), 2339–2350.Google Scholar
  20. 20.
    Seo, J. H., Abd, T., George, R. T., & Mittal, R. (2016). A coupled chemo-fluidic computational model for thrombogenesis in infarcted left ventricles. American Journal of Physiology-Heart and Circulatory Physiology, 310(11), H1567–H1582.Google Scholar
  21. 21.
    Sorensen, E. N., Burgreen, G. W., Wagner, W. R., & Antaki, J. F. (1999). Computational simulation of platelet deposition and activation: I. Model development and properties. Annals of Biomedical Engineering, 27(4), 436–448.Google Scholar
  22. 22.
    Merrill, E. W. (1969). Rheology of blood. Physiological Reviews, 49(4), 863–888.Google Scholar
  23. 23.
    Levich, V. G. (1962). Physicochemical hydrodynamics. Englewood Cliffs: Prentice Hall.Google Scholar
  24. 24.
    Hosseinzadegan, H., & Tafti, D. K. (2016). Validation of a time dependent physio-chemical model for thrombus formation and growth. In ASME 2016 Fluids Engineering Division Summer Meeting collocated with the ASME 2016 Heat Transfer Summer Conference and the ASME 2016 14th International Conference on Nanochannels, Microchannels, and Minichannels (pp. V01AT04A007–V01AT04A007). New York: American Society of Mechanical Engineers.Google Scholar
  25. 25.
    Wootton, D. M., Markou, C. P., Hanson, S. R., & Ku, D. N. (2001). A mechanistic model of acute platelet accumulation in thrombogenic stenoses. Annals of Biomedical Engineering, 29(4), 321–329.Google Scholar
  26. 26.
    Fogelson, A. L. (1992). Continuum models of platelet aggregation: Formulation and mechanical properties. SIAM Journal on Applied Mathematics, 52(4), 1089–1110.MathSciNetzbMATHGoogle Scholar
  27. 27.
    Goodman, P. D., Barlow, E. T., Crapo, P. M., Mohammad, S. F., & Solen, K. A. (2005). Computational model of device-induced thrombosis and thromboembolism. Annals of Biomedical Engineering, 33(6), 780–797.Google Scholar
  28. 28.
    David, T., Thomas, S., & Walker, P. G. (2001). Platelet deposition in stagnation point flow: An analytical and computational simulation. Medical Engineering & Physics, 23(5), 299–312.Google Scholar
  29. 29.
    Soares, J. S., Sheriff, J., & Bluestein, D. (2013). A novel mathematical model of activation and sensitization of platelets subjected to dynamic stress histories. Biomechanics and Modeling in Mechanobiology, 12(6), 1127–1141.Google Scholar
  30. 30.
    Frojmovic, M. M., Mooney, R. F., & Wong, T. (1994). Dynamics of platelet glycoprotein IIb-IIIa receptor expression and fibrinogen binding. I. Quantal activation of platelet subpopulations varies with adenosine diphosphate concentration. Biophysical Journal, 67(5), 2060.Google Scholar
  31. 31.
    Weiss, H. J. (1982). Platelets: Pathophysiology and antiplatelet drug therapy. New York: AR Liss.Google Scholar
  32. 32.
    Jones, R. L., Wilson, N. H., & Marr, C. G. (2013). Thromboxane-like activity of prostanoids with aromatic substituents at C16 and C17. In Chemistry, Biochemistry, and Pharmacological Activity of Prostanoids: Including the Proceedings of a Symposium on the Chemistry and Biochemistry of Prostanoids Held at The University of Salford, England, 10–14 July 1978 (p. 210). New York: Elsevier.Google Scholar
  33. 33.
    Sorensen, E. N. (2002). Computational simulation of platelet transport, activation, and deposition. Pittsburgh: University of Pittsburgh.Google Scholar
  34. 34.
    Waluga, C., & Behbahani, M. (2008). Numerical simulation of platelet adhesion, activation and aggregation: application to Taylor-Couette systems (No. RWTH-CONV-112312). Fakultät für Mathematik, Informatik und Naturwissenschaften.Google Scholar
  35. 35.
    Neeves, K. B., Illing, D. A. R., & Diamond, S. L. (2010). Thrombin flux and wall shear rate regulate fibrin fiber deposition state during polymerization under flow. Biophysical Journal, 98(7), 1344–1352.Google Scholar
  36. 36.
    David, T. (2001). Platelet deposition in stagnation point flow: an analytical andcomputational simulation. Medical Engineering & Physics, 23, 299–312.Google Scholar
  37. 37.
    Sorensen, E. N., Burgreen, G. W., Wagner, W. R., & Antaki, J. F. (1999). Computational simulation of platelet deposition and activation: II. Results for Poiseuille flow over collagen. Annals of Biomedical Engineering, 27(4), 449–458.Google Scholar
  38. 38.
    Keller, K. H. (1971). Effect of fluid shear on mass transport in flowing blood. In Federation Proceedings, 30(5), 1591–1599.Google Scholar
  39. 39.
    Sorensen, E. N., Burgreen, G. W., Wagner, W. R., & Antaki, J. F. (1999). Computational simulation of platelet deposition and activation: I. Model development and properties. Annals of Biomedical Engineering, 27(4), 436–448.Google Scholar
  40. 40.
    Lévêque, A. (1928). Les Lois de la transmission de chaleur par convection, par André Lévêque. Paris: Dunod.Google Scholar
  41. 41.
    Hosseinzadegan, H., Pierron, O. N., & Hosseinian, E. (2014). Accurate modeling of air shear damping of a silicon lateral rotary micro-resonator for mems environmental monitoring applications. Sensors and Actuators, A: Physical, 216, 342–348.Google Scholar
  42. 42.
    Grunkemeier, J. M., Tsai, W. B., & Horbett, T. A. (1998). Hemocompatibility of treated polystyrene substrates: Contact activation, platelet adhesion, and procoagulant activity of adherent platelets. Journal of Biomedical Materials Research: An Official Journal of The Society for Biomaterials, The Japanese Society for Biomaterials, and the Australian Society for Biomaterials, 41(4), 657–670.Google Scholar
  43. 43.
    Rosing, J., Van Rijn, J. L., Bevers, E. M., van Dieijen, G., Comfurius, P., & Zwaal, R. F. (1985). The role of activated human platelets in prothrombin and factor X activation. Blood, 65(2), 319–332.Google Scholar
  44. 44.
    Varga-Szabo, D., Pleines, I., & Nieswandt, B. (2008). Cell adhesion mechanisms in platelets. Arteriosclerosis, Thrombosis, and Vascular Biology, 28(3), 403–412.Google Scholar
  45. 45.
    Adams, G. A., & Feuerstein, I. A. (1983). Maximum fluid concentrations of materials released from platelets at a surface. American Journal of Physiology-Heart and Circulatory Physiology, 244(1), H109–H114.Google Scholar
  46. 46.
    Griffith, M. J. (1982). Kinetics of the heparin-enhanced antithrombin III/thrombin reaction. Evidence for a template model for the mechanism of action of heparin. Journal of Biological Chemistry, 257(13), 7360–7365.Google Scholar
  47. 47.
    Folie, B. J., & Mcintire, L. V. (1989). Mathematical analysis of mural thrombogenesis. Concentration profiles of platelet-activating agents and effects of viscous shear flow. Biophysical Journal, 56(6), 1121–1141.Google Scholar
  48. 48.
    Govindarajan, V., Rakesh, V., Reifman, J., & Mitrophanov, A. Y. (2016). Computational study of thrombus formation and clotting factor effects under venous flow conditions. Biophysical Journal, 110(8), 1869–1885.Google Scholar
  49. 49.
    Gopalakrishnan, P., & Tafti, D. (2009). A parallel multiblock boundary fitted dynamic mesh solver for simulating flows with complex boundary movement. In 38th Fluid Dynamics Conference and Exhibit (p. 4142).Google Scholar
  50. 50.
    Bark, D. L., Para, A. N., & Ku, D. N. (2012). Correlation of thrombosis growth rate to pathological wall shear rate during platelet accumulation. Biotechnology and Bioengineering, 109(10), 2642–2650.Google Scholar
  51. 51.
    Casa, L. D. C., & Ku, D. N. (2014). High shear thrombus formation under pulsatile and steady flow. Cardiovascular Engineering and Technology, 5(2), 154–163.Google Scholar
  52. 52.
    Oliver, J. A., Monroe, D. M., Roberts, H. R., & Hoffman, M. (1999). Thrombin activates factor XI on activated platelets in the absence of factor XII. Arteriosclerosis, Thrombosis, and Vascular Biology, 19(1), 170–177.Google Scholar
  53. 53.
    Monroe, D. M., Roberts, H. R., & Hoffman, M. (1994). Platelet procoagulant complex assembly in a tissue factor initiated system. British Journal of Haematology, 88(2), 364–371.Google Scholar
  54. 54.
    Markou, C. P., Hanson, S. R., Siegel, J. M., & Ku, D. N. (1993). The role of high wall shear rate on thrombus formation in stenoses. ASME-PUBLICATIONS-BED, 26, 555.Google Scholar
  55. 55.
    Wellings, P. J., & Ku, D. N. (2012). Mechanisms of platelet capture under very high shear. Cardiovascular Engineering and Technology, 3(2), 161–170.Google Scholar

Copyright information

© Taiwanese Society of Biomedical Engineering 2018

Authors and Affiliations

  1. 1.Mechanical Engineering DepartmentVirginia Polytechnic Institute and State UniversityBlacksburgUSA

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