Bulletin of Mathematical Biology

, Volume 69, Issue 2, pp 747–764 | Cite as

Simulations of Chemotaxis and Random Motility in 2D Random Porous Domains

Original Article


We discuss a generic computational model of eukariotic chemotaxis in 2D random porous domains. The model couples the fully time-dependent finite-difference solution of a reaction–diffusion equation for the concentration field of a chemoattractant to biased random walks representing individual chemotactic cells. We focus in particular on the influence of consumption of chemoattractant by the boundaries of obstacles with irregular shapes which are distributed randomly in the domain on the chemotactic response of the cells. Cells are stimulated to traverse a field of obstacles by a line source of chemoattractant. We find that the reactivity of the obstacle boundaries with respect to the chemoattractant strongly determines the transit time of cells through two primary mechanisms. The channeling effect arises because cells are effectively repelled from surfaces which consume chemoattractant, and opposing surfaces therefore act to keep cells in the middle of channels. This reduces traversal times relative to the case with unreactive boundaries, provided that the appropriate Péclet number relating the strength of reactivity to diffusion in governing chemoattractant transport is neither too low nor too high. The dead-zone effect arises due to a realistic threshold on the chemotactic response, which at steady state results in portions of the domain having no detectable gradient. Of these two, the channeling effect is responsible for 90% of the sensitivity of transit times to boundary reactivity. Based on these results, we speculate that it may be possible to tune the rates of cellular penetration into porous domains by engineering the reactivity of the internal surfaces to cytokines.


Chemotaxis Simulation Finite difference Porosity 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Alt, W., 1980. Biased random walk models for chemotaxis and related diffusion approximations. J. Math. Biol. 9, 147–177.MATHCrossRefMathSciNetGoogle Scholar
  2. Anderson, A.R.A., Chaplain, M.A.J., 1998. A mathematical model for capillary network formation in the absence of endothelial cell proliferation. Appl. Math. Lett. 11, 109–114.MATHCrossRefGoogle Scholar
  3. Chaplain, M.A.J., 2000. Mathematical modeling of angiogenesis. J. Neuro-Oncol. 50, 37–51.CrossRefGoogle Scholar
  4. Dunn, G.A., Brown, A.F., 1987. A unified approach to analyzing cell motility. J. Cell Sci. Suppl. 8, 81–102.Google Scholar
  5. Dziubla, T.D., 2002. Macroporous hydrogels as vascularizable soft tissue-implant interfaces: Material characterization, in vitro evaluation, computer simulations and applications in implantable drug delivery devices. Ph.D. thesis, Drexel University.Google Scholar
  6. Ferrara, N., 1999. Vascular endothelial growth factor: molecular and biological aspects. Curr. Top. Microbiol. Immunol. 237, 1–30.Google Scholar
  7. Francis, K., Palsson, B.O., 1997. Effective intercellular communication distances are determined by the relative time constant for cyto/chemokine secretion and diffusion. Proc. Nat. Acad. Sci. USA 94, 12258–12262.CrossRefGoogle Scholar
  8. Friedl, P., Zanker, K.S., Bröcker, E.B., 1998. Cell migration strategies in 3-d extracellular matrix: Differences in morphology, cell matrix interactions and integrin function. Micro. Res. Tech. 43, 369–378.CrossRefGoogle Scholar
  9. Golden, M.A., Hanson, S.R., Kirkman, T.R., Schneider, P.A., Clowes, A.W., 1990. Healing of polytetrafluoroethylene arterial grafts is influenced by graft porosity. J. Vasc. Surg. 11, 838–844.CrossRefGoogle Scholar
  10. Gordon, P., 1965. Nonsymmetric difference equations. J. Soc. Ind. Appl. Math. 13, 667.Google Scholar
  11. Gourlay, A.R., 1970. Hopscoth: A fast second-order partial differential equation solver. J. Inst. Math. Appl. 6, 375–390.MATHCrossRefMathSciNetGoogle Scholar
  12. Ito, H., Koefoed, M., Tiyapatanaputi, P., Gromov, K., Goater, J.J., Carmouche, J., Zhang, X., Rubery, P., Rabinowitz, J., Samulski, R.J., Nakamura, T., Soballe, K., O'Keefe, R.J., Boyce, B.F., Schwarz, E.M., 2005. Remodeling of cortical bone allografts mediated by adherent rAAV-RANKL and VEGF gene therapy. Nat. Med. 11, 291–297.CrossRefGoogle Scholar
  13. Jabbarzadeh, E., Abrams, C.F., 2005. Chemotaxis and random motility in unsteady chemoattractant fields: A computational study. J. Theor. Biol. 235, 221–232.CrossRefMathSciNetGoogle Scholar
  14. Keller, E.F., Segel, L.A., 1971. Model for chemotaxis. J. Theor. Biol. 30, 225–235.CrossRefGoogle Scholar
  15. Kidd, K.R., Nagle, R.B., Williams, S., 2002. Angiogenesis and neovascularization associated with extracellular matrix-modified porous implants. J. Biomed. Mater. Res. 59, 366–477.CrossRefGoogle Scholar
  16. Lauffenburger, D.A., Horwitz, A.F., 1996. Cell migration: A physically integrated progress. Cell 84, 359–369.CrossRefGoogle Scholar
  17. Liekens, S., Clercq, E.D., Neyts, J., 2001. Angiogenesis: Regulators and clinical applications. Biochem. Pharmacol. 61, 253–370.CrossRefGoogle Scholar
  18. Maheshwari, G., Lauffenburger, D.A., 1998. Deconstructing (and reconstructing) cell migration. Micro. Res. Tech. 43, 358–368.CrossRefGoogle Scholar
  19. Matsumoto, M., Nishimura, T., 1998. Mersenne twister: A 623-dimensionally equidistributed uniform pseudorandom number generator. ACM Trans. Model Comput. Simulation 8, 3–30.MATHCrossRefGoogle Scholar
  20. Othmer, H.G., Stevens, A., 1997. Aggregation, blowup, and collapse: The abc's of taxis in reinforced random walks. SIAM J. Appl. Math. 57, 1044–1081.MATHCrossRefMathSciNetGoogle Scholar
  21. Risau, W., 1997. Mechanisms of angiogenesis. Nature 386, 671–674.CrossRefGoogle Scholar
  22. Rivero, M.A., Tranquillo, R.T., Buettner, H.M., Lauffenburger, D.A., 1989. Transport models for chemotactic cell populations based on individual cell behavior. Chem. Eng. Sci. 44, 2881–2897.CrossRefGoogle Scholar
  23. Sanders, J.E., Malcolm, S.G., Bale, S.D., Wang, Y.N., Lamont, S., 2002. Prevascularization of a biomaterial using a chorioallontoic membrane. Microvasc. Res. 64, 174–178.CrossRefGoogle Scholar
  24. Serini, G., Ambrosi, D., Giraudo, E., Gamba, A., Preziosi, L., Bussolino, F., 2003. Modeling the early stages of vascular network assembly. EMBO J. 22, 1771–1779.CrossRefGoogle Scholar
  25. Sharkawy, A.A., Klitzman, B., Truskey, G.A., Reichert, W.M., 1998. Engineering the tissue which encapsulates subcutaneous implants. II. Plasma–tissue exchange properties. J. Biomed. Mater. Res. 40, 598–605.CrossRefGoogle Scholar
  26. Shenderov, A.D., Sheetz, M.P., 1997. Inversely correlated cycles in speed and turning in an ameba: An oscillatory model of cell locomotion. Biophys. J. 72, 2382–2389.CrossRefGoogle Scholar
  27. Sieminski, A.L., Gooch, K.J., 2000. Biomaterial–microvasculature interactions. Biomaterials 22, 2233–2241.CrossRefGoogle Scholar
  28. Sleeman, B.D., Wallace, I.P., 2002. Tumour induced angiogenesis as a reinforced random walk: Modelling capillary network formation without endothelial cell proliferation. Math. Comput. Model 36, 339–358.MATHCrossRefGoogle Scholar
  29. Stokes, C.L., Lauffenburger, D.A., 1991. Analysis of the roles of microvessel endothelial cell random motility and chemotaxis in angiogenesis. J. Theor. Biol. 3, 377–403.CrossRefGoogle Scholar
  30. Tong, S., Yuan, F., 2001. Numerical simulation of angiogenesis in the cornea. Microvas. Res. 61, 14–27.CrossRefGoogle Scholar
  31. Zhang, Z., Zou, W., Wang, J., Gu, J., Dang, Y., Li, B., Zhao, L., Qian, C., Qian, Q., Liu, X., 2005. Suppression of tumor growth by oncolytic adenvirus-meditaed delivery of antiangiogenic gene, souble Flt-1. Mol. Therapy 11, 553–562.CrossRefGoogle Scholar
  32. Zigmond, S.H., 1977. Ability of polymorphonuclear leukocytes to orient in gradients of chemotactic factors. J. Cell Biol. 75, 606–616.CrossRefGoogle Scholar
  33. Zygourakis, K., 1996. Quantification and regulation of cell migration. Tissue Eng. 2, 1–26.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  1. 1.Department of Chemical and Biological EngineeringDrexel UniversityPhiladelphiaUSA

Personalised recommendations