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Journal of Mathematical Biology

, Volume 63, Issue 4, pp 757–777 | Cite as

Dynamic formation of oriented patches in chondrocyte cell cultures

  • Marcus J. GroteEmail author
  • Viviana Palumberi
  • Barbara Wagner
  • Andrea Barbero
  • Ivan Martin
Article

Abstract

Growth factors have a significant impact not only on the growth dynamics but also on the phenotype of chondrocytes (Barbero et al. in J. Cell. Phys. 204:830–838, 2005). In particular, as chondrocytes approach confluence, the cells tend to align and form coherent patches. Starting from a mathematical model for fibroblast populations at equilibrium (Mogilner et al. in Physica D 89:346–367, 1996), a dynamic continuum model with logistic growth is developed. Both linear stability analysis and numerical solutions of the time-dependent nonlinear integro-partial differential equation are used to identify the key parameters that lead to pattern formation in the model. The numerical results are compared quantitatively to experimental data by extracting statistical information on orientation, density and patch size through Gabor filters.

Keywords

Cell alignment Pattern formation Stability Integro-partial differential equations Image processing 

Mathematics Subject Classification (2000)

92C17 92D25 92-08 65M06 

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References

  1. Abramowitz M, Stegun IA (1964) Handbook of mathematical functions with formulas, graphs, and mathematical tables. Dover Publications, New YorkzbMATHGoogle Scholar
  2. Barbero A, Grogan S, Schafer D, Heberer M, Mainil-Varlet P, Martin I (2004) Age related changes in human articular chondrocyte yield, proliferation and post-expansion chondrogenic capacity. Osteoarthr Cartil 12: 476–484CrossRefGoogle Scholar
  3. Barbero A, Palumberi V, Wagner B, Sader R, Grote MJ, Martin I (2005) Experimental and mathematical study of the influence of growth factors on the growth kinetics of adult human articular chondrocytes. J Cell Physiol 204: 830–838CrossRefGoogle Scholar
  4. Barbero A, Ploegert S, Heberer M, Martin I (2003) Plasticity of clonal populations of dedifferentiated adult human articular chondrocytes. Arthr Rheum 48: 1315–1325CrossRefGoogle Scholar
  5. Beattie G, Cirulli V, Lopez A, Hayek A (1997) Ex vivo expansion of human pancreatic endocrine cells. J Clin Endocrinol Metab 82: 1852–1856CrossRefGoogle Scholar
  6. Carpenter M, Cui X, Hu Z, Jackson J, Sherman S, Seiger A, Wahlberg L (1999) In vitro expansion of a multipotent population of human neural progenitor cells. Exp Neurol 158: 265–278CrossRefGoogle Scholar
  7. Chipot M, Edelstein-Keshet L (1983) A mathematical theory of size distribution in tissue culture. J Math Biol 16: 115–130MathSciNetzbMATHCrossRefGoogle Scholar
  8. Civelecoglu G, Edelstein-Keshet L (1994) Modelling the dynamics of f-actin in the cell. Bull Math Biol 56: 587–616Google Scholar
  9. Daugman J (1985) Uncertainty relations for resolution in space, spatial frequency, and orientation optimized by two-dimensional visual cortical filters. J Opt Soc Am A 2: 1160–1169CrossRefGoogle Scholar
  10. Deasy B, Qu-Peterson Z, Greenberger J, Huard J (2002) Mechanisms of muscle stem cell expansion with cytokines. Stem Cells 20: 50–60CrossRefGoogle Scholar
  11. Deenick E, Gett A, Hodgkin P (2003) Stochastic model of T dell proliferation: a calculus revealing il-2 regulation of precursor frequencies, cell cycle time, and survival. J Immunol 170: 4963–4972Google Scholar
  12. Edelstein-Keshet L, Ermentrout G (1990) Models for contact-mediated pattern formation: cells that form parallel arrays. J Math Biol 29: 3–58MathSciNetCrossRefGoogle Scholar
  13. Elsdale T (1973) The generation and maintenance of parallel arrays in cultures of diploid fibroblasts. In: Kulonen E, Pikkarainen J (eds) Biology of fibroblast. Academic Press, London, pp 41–58Google Scholar
  14. Elsdale T, Bard J (1972) Collagen substrata for studies on cell behaviours. J Cell Biol 54: 626–637CrossRefGoogle Scholar
  15. Erickson C (1978) Analysis of the formation of parallel arrays in bhk cells in vitro. Exp Cell Res 115: 303–315CrossRefGoogle Scholar
  16. Forsyth DA, Ponce J (2003) Computer vision, a modern approach. Prentice Hall, Englewood CliffsGoogle Scholar
  17. Franklin J (1959) Numerical stability in digital and anlogue computation for diffusion problems. J Math Phys 37: 305–315MathSciNetzbMATHGoogle Scholar
  18. Gabor D (1946) Theory of comminication. J IEE 93(26): 429–457Google Scholar
  19. Gouillou A, Lago B (1960) Domaine de stabilité associé aux formules d’intégration numérique d’équations différentielles, à pas séparés et à pas liés. Recherche de formules à grand rayon de stabilité. Ier Congr. Ass. Fran. Calcul. AFCAL pp 43–56Google Scholar
  20. Hairer EWG (2002) Solving ordinary differential equation II. Springer, BerlinGoogle Scholar
  21. Hairer E, Nørsett SPWG (1987) Solving ordinary differential equations. I: nonstiff problems. Springer, BerlinzbMATHGoogle Scholar
  22. Jakob M, Demarteau O, Schafer D, Hinterman B, Dick W, Heberer M, Martin I (2001) Specific growth factors during the expansion and redifferentiation of adult human articular chondrocytes enhance chondrogenesis and cartilaginous tissue formation in vitro. J Cell Biochem 81: 368–377CrossRefGoogle Scholar
  23. Kress R (1999) Linear integral equations. Springer, BerlinzbMATHCrossRefGoogle Scholar
  24. Langer R, Vacanti J (1993) Tissue engineering. Science 260: 920–926CrossRefGoogle Scholar
  25. Markoff CL (1916) Ueber Polynome, die in einem gegebenen Intervall möglichst wenig von Null abweichen. Math Ann 77: 213–258MathSciNetzbMATHCrossRefGoogle Scholar
  26. Mogilner A, Edelstein-Keshet L (1995) Selecting a common direction, how orientational order can arise from simple contact responses between interacting cells. J Math Biol 33: 619–660MathSciNetzbMATHCrossRefGoogle Scholar
  27. Mogilner A, Edelstein-Keshet L (1996) Spatio-angular order in populations of self-aligning objects: formation of oriented patches. Physica D 89: 346–367MathSciNetzbMATHCrossRefGoogle Scholar
  28. Mogilner A, Edelstein-Keshet L, Ermentrout GB (1996) Selecting a common direction. II. peak-like solutions representing total alignment of cell clusters. J Math Biol 34: 811–842MathSciNetzbMATHGoogle Scholar
  29. Sharma B, Elisseeff JH (2004) Engineering structurally organized cartilage and bone tissues. Ann Biomed Eng 32: 148–159CrossRefGoogle Scholar
  30. Stewart J, Masi T, Cumming A, Molnar G, Wentworth B, Sampath K, McPherson J, Yaeger P (2003) Characterization of proliferating human skeletal muscle-derived cells in vitro: differential modulation of myoblast markers by tgf-beta2. J Cell Physiol 196: 70–78CrossRefGoogle Scholar
  31. Trefethen L, Trefethen A, Reddy S, Driscoll D (1993) Hydrodynamics without eigenvalues. Science 261: 578MathSciNetCrossRefGoogle Scholar
  32. Trinkaus JP (1985) Further thoughts on directional cell-movement during morphogenesis. J Neurosci Res 13: 1–19CrossRefGoogle Scholar
  33. Yuan C (1958) Some difference schemes for the solution of the first boundary value problem for linear differential equations with partial derivatives. Master’s thesis, Moscow State UniversityGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Marcus J. Grote
    • 1
    Email author
  • Viviana Palumberi
    • 1
  • Barbara Wagner
    • 2
  • Andrea Barbero
    • 3
  • Ivan Martin
    • 3
  1. 1.Institute of MathematicsBaselSwitzerland
  2. 2.Weierstrass Institute for Applied Analysis and StochasticsBerlinGermany
  3. 3.Department of BiomedicineUniversity HospitalBaselSwitzerland

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