Advertisement

Cellular and Molecular Bioengineering

, Volume 3, Issue 1, pp 3–19 | Cite as

Application of Population Dynamics to Study Heterotypic Cell Aggregations in the Near-Wall Region of a Shear Flow

  • Yanping Ma
  • Jiakou Wang
  • Shile Liang
  • Cheng DongEmail author
  • Qiang Du
Article

Abstract

Our research focused on the polymorphonuclear neutrophils (PMNs) tethering to the vascular endothelial cells (EC) and the subsequent melanoma cell emboli formation in a shear flow, an important process of tumor cell extravasation from the circulation during metastasis. We applied population balance model based on Smoluchowski coagulation equation to study the heterotypic aggregation between PMNs and melanoma cells in the near-wall region of an in vitro parallel-plate flow chamber, which simulates in vivo cell-substrate adhesion from the vasculatures by combining mathematical modeling and numerical simulations with experimental observations. To the best of our knowledge, a multiscale near-wall aggregation model was developed, for the first time, which incorporated the effects of both cell deformation and general ratios of heterotypic cells on the cell aggregation process. Quantitative agreement was found between numerical predictions and in vitro experiments. The effects of factors, including: intrinsic binding molecule properties, near-wall heterotypic cell concentrations, and cell deformations on the coagulation process, are discussed. Several parameter identification approaches are proposed and validated which, in turn, demonstrate the importance of the reaction coefficient and the critical bond number on the aggregation process.

Keywords

Melanoma cells Leukocytes Endothelium Collisions Aggregates Probabilities Cell adhesion Shear conditions 

Notations

A

The reaction constant as a combination of A c, N r , N l , K on and K off (s−1)

Ac

The contact area between a PMN and a tumor cell (μm2)

Ca, We, Re

The Capillary number, the Weber number and the Reynolds number

Colln

The collision number between two kinds of cells (events)

CP, CT, C

The concentration of PMNs, tumor cells and bulk cell concentration, respectively, in the near-wall region (cells mL−1)

d

The diameter of an undeformed PMN cell (μm)

g

Acceleration due to gravity (m s−2)

H, Hdep, h

The height of a deformed PMN cell attached to the substrate, the height defined by the depth of field of the microscope objective used in the experimental observations and the distance from the substrate to the center of the cell, which represented the position of the cell (μm)

Iini, Itf

The increasing multiple of initial condition or tethering frequency

J

The incoming flux to collision region near PMN per concentration (kg−1 m3)

Kon, Kon(n)

The forward reaction rate per unit density for bond formation (s−1 μm−2)

Koff, Koff(n)

The backward reaction rate per cell for bond breakage (s−1)

n

Outward unit normal (vector) of collision region

n*

The approximation of the smallest number of bonds required for firm adhesion (bonds)

Nr, Nl

The concentration of receptor and ligand on cells (μm−2)

NPT, NP

The number of tethered PMN–tumor cell doublets and tethered PMN monomers on the substrate (cells)

Pn(t), Pn

The probability of having n bonds.

rp, rt

The radius of an undeformed PMN cell and a tumor cell (μm)

Tf

The tethering frequency of cells: including firmly adhered cell and rolling cells (events s−1 per view)

vavg, vrel

The average settling velocity for cells above height H dep and the relative velocity (difference between velocities) of two kinds of cells (μm s−1)

vs0, vs, vc

The free settling velocity, settling velocity and convection velocity of cells in the parabolic flow profile (μm s−1)

α

The aggregation percentage

βPT, β(i,j;i′,j′)

The coagulation kernel of tumor cells in the near-wall region to the tethered PMNs (μm3 s−1)

\( \hat{\beta }_{\text{PT}} ,\hat{\beta } \)

The collision rate between tethered PMNs and tumor cell monomers (μm3 s−1)

\( \dot{\gamma } \)

The shear rate in the close-wall region (s−1)

єPT

The adhesion efficiency between tethered PMNs and tumor cell monomers

μ

The viscosity of the fluid flow (Poise)

ρm, ρt

The fluid density, the tumor cell density (kg m−3)

σ

Membrane tension (N m−1)

Ψ

The concentration of cells within the region of height H dep

Ω

The collision region

Notes

Acknowledgments

The authors thank Dr. Meghan Hoskins for providing simulation data on hydrodynamic force. This work was supported by the National Institutes of Health grant CA-125707 and National Science Foundation grant CBET-0729091.

References

  1. 1.
    Abkarian, M., and A. Viallat. Dynamics of vesicles in a wall-bounded shear flow. Biophys. J. 89:1055–1066, 2005.CrossRefGoogle Scholar
  2. 2.
    Aldous, D. J. Deterministic and stochastic models for coalescence (aggregation and coagulation): a review of the Mean-Field Theory for Probabilities. Bernoulli 5:3–48, 1999.zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Belval, T., and J. Hellums. Analysis of shear-induced platelet aggregation with population balance mathematics. Biophys. J. 50:479–487, 1986.CrossRefGoogle Scholar
  4. 4.
    Caputo, K. E., and D. A. Hammer. Effect of microvillus deformability on leukocyte adhesion explored using adhesive dynamics simulations. Biophys. J. 89:187–200, 2005.CrossRefGoogle Scholar
  5. 5.
    Chang, K. C., D. F. Tees, and D. A. Hammer. The state diagram for cell adhesion under flow: leukocyte rolling and firm adhesion. Proc. Natl Acad. Sci. USA 97:11262–11267, 2000.CrossRefGoogle Scholar
  6. 6.
    Chesla, S. E., P. Selvaraj, and C. Zhu. Measuring two-dimensional receptor–ligand binding kinetics with micropipette. Biophys. J. 75:1553–1572, 1998.CrossRefGoogle Scholar
  7. 7.
    Davis, J. M., and J. C. Giddings. Influence of wall-retarded transport of retention and plate eight in field-flow fractionation. Sci. Technnol. 20:699–724, 1985.Google Scholar
  8. 8.
    Dong, C., J. Cao, E. J. Struble, and H. Lipowsky. Mechanics of leukocyte deformation and adhesion to endothelium in shear flow. Ann. Biomed. Eng. 27:298–312, 1999.CrossRefGoogle Scholar
  9. 9.
    Dong, C., and X. Lei. Biomechanics of cell rolling: shear flow, cell-surface adhesion, and cell deformability. J. Biomech. 33:35–43, 2000.CrossRefGoogle Scholar
  10. 10.
    Goldman, A. J., R. G. Cox, and H. Brenner. Slow viscous motion of a sphere parallel to a plane wall. II. Couette flow. Chem. Eng. Sci. 20:653–660, 1967.Google Scholar
  11. 11.
    Hammer, D. A., and S. M. Apte. Simulation of cell rolling and adhesion on surface in shear flow: general results and analysis of selectin-mediated neutrophil adhesion. Biophs. J. 63:35–57, 1992.CrossRefGoogle Scholar
  12. 12.
    Hentzen, E. R., S. Neelamegham, G. S. Kansas, J. A. Benanti, L. V. Smith, C. W. McIntire, and S. I. Simon. Sequential binding of CD11a/CD18 and CD11b/CD18 defines neutrophil capture and stable adhesion to intercellular adhesion molecule-1. Blood 95:911–920, 2000.Google Scholar
  13. 13.
    Hinds, M. T., Y. J. Park, S. A. Jones, D. P. Giddens, and B. R. Alevriadou. Local hemodynamics affect monocytic cell adhesion to a three-dimensional flow model coated with E-selectin. J. Biomech. 34:95–103, 2001.CrossRefGoogle Scholar
  14. 14.
    Hoskins, M., R. F. Kunz, J. Bistline, and C. Dong. Coupled flow–structure–biochemistry simulations of dynamic systems of blood cells using an adaptive surface tracking method. J. Fluids Struct. 25:936–953, 2009.CrossRefGoogle Scholar
  15. 15.
    Huang, P., and J. Hellums. Aggregation and disaggregation kinetics of human blood platelets: Part I. Development and validation of a population balance method. Biophys. J. 65:334–343, 1993.CrossRefGoogle Scholar
  16. 16.
    Huang, P., and J. Hellums. Aggregation and disaggregation kinetics of human blood platelets: Part II. Shear induced platelet aggregation. Biophys. J. 65:344–353, 1993.CrossRefGoogle Scholar
  17. 17.
    Khismatullin, D., and G. Truskey. A 3D numerical study of the effect of channel height on leukocyte deformation and adhesion in parallel-plate flow chambers. Microvasc. Res. 68:188–202, 2004.CrossRefGoogle Scholar
  18. 18.
    Khismatullin, D., and G. Truskey. 3D numerical simulation of receptor-mediated leukocyte adhesion to surfaces: effects of cell deformability and viscoelasticity. Phys. Fluids 17:53–73, 2005.CrossRefGoogle Scholar
  19. 19.
    Kolodko, A., K. Sabelfeld, and W. Wagner. A stochastic method for solving Smoluchowskis coagulation equation. Math. Comput. Simul. 49:57–79, 1999.zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Kwong, D., D. F. J. Tees, and H. L. Goldsmith. Kinetics and locus of failure of receptor ligand-mediated adhesion between latex spheres. II. Protein–protein bond. Biophys. J. 71:1115–1122, 1996.CrossRefGoogle Scholar
  21. 21.
    Laurenzi, I. J., and S. L. Diamond. Monte Carlo simulation of the heterotypic aggregation kinetics of platelets and neutrophils. Biophys. J. 77:1733–1746, 1999.CrossRefGoogle Scholar
  22. 22.
    Lei, X., M. B. Lawrence, and C. Dong. Influence of cell deformation on leukocyte rolling adhesion in shear flow. J. Biomech. Eng. 121:636–643, 1991.CrossRefGoogle Scholar
  23. 23.
    Liang, S., C. Fu, D. Wagner, H. Guo, D. Zhan, C. Dong, and M. Long. 2D kinetics of β2 integrin-ICAM-1 bindings between neutrophils and melanoma cells. Am. J. Physiol. 294:743–753, 2008.CrossRefGoogle Scholar
  24. 24.
    Liang, S., M. Hoskins, and C. Dong. Tumor cell extravasation mediated by leukocyte adhesion is shear rate-dependent on IL-8 signaling. Mol. Cell. Biomech. 7:77–91, 2009.Google Scholar
  25. 25.
    Liang, S., M. Hoskins, P. Khanna, R. F. Kunz, and C. Dong. Effects of the tumor-leukocyte microenvironment on melanoma–neutrophil adhesion to the endothelium in a shear flow. Cell. Mol. Bioeng. 1:189–200, 2008.CrossRefGoogle Scholar
  26. 26.
    Liang, S., M. Slattery, and C. Dong. Shear stress and shear rate differentially affect the multi-step process of leukocyte-facilitated melanoma adhesion. Exp. Cell Res. 310:282–292, 2005.CrossRefGoogle Scholar
  27. 27.
    Liang, S., M. Slattery, D. Wagner, S. I. Simon, and C. Dong. Hydrodynamic shear rate regulates melanoma-leukocyte aggregations, melanoma adhesion to the endothelium and subsequent extravasation. Ann. Biomed. Eng. 36:661–671, 2008.CrossRefGoogle Scholar
  28. 28.
    Long, M., H. L. Goldsmith, D. F. Tees, and C. Zhu. Probabilistic modeling of shear-induced formation and breakage of doublets cross-linked by receptor–ligand bonds. Biophys. J. 76:1112–1128, 1999.CrossRefGoogle Scholar
  29. 29.
    Lyczkowski, R. W., B. R. Alevriadou, M. Horner, C. B. Panchal, and S. G. Shroff. Application of multiphase computational fluid dynamics to analyze monocyte adhesion. Ann. Biomed. Eng. 37:1516–1533, 2009.CrossRefGoogle Scholar
  30. 30.
    McQuarrie, D. A. Kinetics of small systems. I. J. Phys. Chem. 38:433–436, 1963.CrossRefGoogle Scholar
  31. 31.
    Melder, R. J., L. L. Munn, S. Yamada, C. Ohkubo, and R. K. Jain. Selectin- and integrin mediated T-lymphocyte rolling and arrest on TNF-activated endothelium: augmentation by erythrocytes. Biophys. J. 69:2131–2138, 1995.CrossRefGoogle Scholar
  32. 32.
    Munn, L. L., R. J. Melder, and R. K. Jain. Analysis of cell flux in the parallel plate flow chamber: implications for cell capture studies. Biophys. J. 67:889–895, 1994.CrossRefGoogle Scholar
  33. 33.
    Munn, L. L., R. J. Melder, and R. K. Jain. Role of erythrocytes inleukocyte-endothelial interactions: mathematical model and experimental validation. Biophys. J. 71:466–478, 1996.CrossRefGoogle Scholar
  34. 34.
    Piper, J. W., R. A. Swerlick, and C. Zhu. Determining force dependence of two-dimensional receptor–ligand binding affinity by centrifugation. Biophys. J. 74:492–513, 1998.CrossRefGoogle Scholar
  35. 35.
    Rinker, K. D., V. Prabhakar, and G. A. Truskey. Effect of contact time and force on monocyte adhesion to vascular endothelium. Biophys. J. 80:1722–1732, 2001.CrossRefGoogle Scholar
  36. 36.
    Sabelfeld, K., and A. Kolodko. Stochastic Lagrangian models and algorithms for spatially inhomogeneous Smoluchowski equation. Math. Comput. Simul. 61:115–137, 2003.zbMATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    Shankaran, H., and S. Neelamegham. Nonlinear flow affects hydrodynamic forces and neutrophil adhesion rates in cone-plate viscometers. Biophys. J. 80:2631–2648, 2001.CrossRefGoogle Scholar
  38. 38.
    Shankaran, H., and S. Neelamegham. Effect of secondary flow on biological experiments in the cone-plate viscometer: Methods for estimating collision frequency, wall shear stress and inter-particle interactions in non-linear flow. Biorheology 38:275–304, 2001.Google Scholar
  39. 39.
    Slattery, M., and C. Dong. Neutrophils influence melanoma adhesion and migration under flow conditions. Int. J. Cancer 106:713–722, 2003.CrossRefGoogle Scholar
  40. 40.
    Slattery, M., S. Liang, and C. Dong. Distinct role of hydrodynamic shear in PMN-facilitated melanoma cell extravasation. Am. J. Physiol. 288(4):C831–C839, 2005.CrossRefGoogle Scholar
  41. 41.
    Smoluchowski, M. Mathematical theory of the kinetics of the coagulation of colloidal solutions. Z. Phys. Chem. 92:129, 1917.Google Scholar
  42. 42.
    Starkey, J. R., H. D. Liggitt, W. Jones, and H. L. Hosick. Influence of migratory blood cells on the attachment of tumor cells to vascular endothelium. Int. J. Cancer 34:535–543, 1984.CrossRefGoogle Scholar
  43. 43.
    Tees, D. F. J., O. Coenen, and H. L. Goldsmith. Interaction forces between red cells agglutinated by antibody. IV. Time and force dependence of break-up. Biophys. J. 65:1318–1334, 1993.CrossRefGoogle Scholar
  44. 44.
    Tees, D. F. J., and H. L. Goldsmith. Kinetics and locus of failure of receptor–ligand mediated adhesion between latex spheres. I. Proteincarbohydrate bond. Biophys. J. 71:1102–1114, 1996.CrossRefGoogle Scholar
  45. 45.
    Wang, J., M. Slattery, M. Hoskins, S. Liang, C. Dong, and Q. Du. Monte Carlo simulation of heterotypic cell aggregation in nonlinear shear flow. Math. Biosci. Eng. 3:683–696, 2006.zbMATHMathSciNetGoogle Scholar
  46. 46.
    Welch, D. R., D. J. Schissel, R. P. Howrey, and P. A. Aeed. Tumor-elicited polymorphonuclear cells, in contrast to “normal” circulating polymorphonuclear cells, stimulate invasive and metastatic potentials of rat mammary adnocardinoma cells. Proc. Natl Acad. Sci. USA 86:5859–5863, 1989.CrossRefGoogle Scholar
  47. 47.
    Wu, Q. D., J. H. Wang, C. Condron, D. Bouchier-Hayer, and H. P. Redmond. Human neutrophils facilitate tumor cell transendothelial migration. Am. J. Phsyiol. Cell Physiol. 280:814–822, 2001.Google Scholar
  48. 48.
    Zhang, Y., and S. Neelamegham. Estimating the efficiency of cell capture and arrest in flow chambers: study of neutrophil binding via E-selectin and ICAM-1. Biophys. J. 83:1934–1952, 2002.CrossRefGoogle Scholar
  49. 49.
    Zhang, Y., and S. Neelamegham. An analysis tool to quantify the efficiency of cell tethering and firm-adhesion in the parallel-plate flow chamber. J. Immunol. Methods 278:305–317, 2003.CrossRefGoogle Scholar
  50. 50.
    Zhu, C., G. Bao, and N. Wang. Cell mechanics: mechanical response, cell adhesion, and molecular deformation. Annu. Rev. Biomed. Eng. 2:189–226, 2000.CrossRefGoogle Scholar
  51. 51.
    Zhu, C., and S. E. Chesla. Dissociation of individual molecular bonds under force. In: Advances in Bioengineering, Vol. 36, edited by B. B. Simon. New York: ASME, 1997, pp. 177–178.Google Scholar
  52. 52.
    Zhu, C., and R. McEver. Catch bonds: physical models and biological functions. Mol. Cell. Biomech. 2(3):91–104, 2005.Google Scholar

Copyright information

© Biomedical Engineering Society 2010

Authors and Affiliations

  • Yanping Ma
    • 1
  • Jiakou Wang
    • 1
  • Shile Liang
    • 2
  • Cheng Dong
    • 2
    Email author
  • Qiang Du
    • 1
  1. 1.Department of MathematicsThe Pennsylvania State UniversityUniversity ParkUSA
  2. 2.Department of BioengineeringThe Pennsylvania State UniversityUniversity ParkUSA

Personalised recommendations