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A continuum mechanics approach to determining the cellular velocity field within a wool follicle

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Abstract

A model, based on the principles of continuum mechanics, is presented for the analysis of cell-velocity fields within wool follicles. The model requires specification of three follicle characteristics in the form of spatially varying fields: viscosity, cell density and cell production rate. The viscosity is introduced as an attempt to model both complex intercellular interactions and individual cell deformation as the cells move. It is demonstrated that the distribution of cell production is more important than axial variation in viscosity in determining the overall flow pattern.

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References

  • Acheson, D. J. (1990). Elementary Fluid Dynamics. Oxford Applied Mathematics and Computing Science Series, Oxford: Oxford University Press.

    Google Scholar 

  • Batchelor, G. K. (1967). An Introduction to Fluid Dynamics, Cambridge, U.K.: Cambridge University Press.

    Google Scholar 

  • Black, J. L. (1987). Mechanisms controlling the rate of growth, composition and morphology of wool, in Merino Improvement Programs in Australia B. J. McGuirk (Ed.), Melbourne: Australian Wool Corporation, pp. 457–480.

    Google Scholar 

  • Black, J. L. and B. N. Nagorcka (1993). Wool growth, in Quantitative Aspects of Ruminant Digestion and Metabolism, J. M. Forbes and J. France (Eds), Cambridge, U.K.: Cambridge University Press, pp. 453–477.

    Google Scholar 

  • Bird, R. Byron, W. E. Stewart and Lightfoot E. N. (1987). Dynamics of polymeric liquids, in Fluid Mechanics 2nd edn vol. 1, New York: Wiley.

    Google Scholar 

  • Chapman, R. E., A. M. Downes and P. A. Wilson (1980). Migration and keratinization of cells in wool follicles. Aust. J. Biol. Sci. 33, 587–603.

    Google Scholar 

  • Chapman, R. E. and K. A. Ward (1979). Histological and biochemical features of the wool fibre and follicle, in Physiological and Environmental Limitations to Wool Growth J. L. Black and P. J. Reis (Eds), Armidale, New South Wales: University of New England Publishing Unit, pp. 193–208.

    Google Scholar 

  • Fox, L. (1950). The numerical solution of elliptic differential equations when the boundary conditions involve a derivative. Phil. Trans. A242, 345–378.

    Google Scholar 

  • Gandar, P. W., K. E. Kelly, P. M. Harris and D. W. Dellow (1990). Analysis of growth and cell proliferation in wool follicles, in Modelling Digestion and Metabolism in Farm Animals, A. B. Robson, and D. P. Poppi (Eds), Canterbury, New Zealand: Lincoln University, pp. 189–203.

    Google Scholar 

  • Holle, S. A. and M. J. Birtles (1990). An immunocytochemical method for studying patterns of cell proliferation in the wool follicle. N.Z. Vet. J. 38, 89–93.

    Google Scholar 

  • Hynd, P. I. (1989). Factors influencing cellular events in the wool follicle, in The Biology of Wool and Hair, G. E. Rogers, P. J. Reis, K. A. Ward and R. C. Marshall (Eds), New York: Chapman and Hall, pp. 169–184.

    Google Scholar 

  • Lighthill, M. J. (1975). Mathematical Biofluiddynamics. Philadelphia: Society for Industrial and Applied Mathematics.

    Google Scholar 

  • Murray, J. D. and G. F. Oster (1984). Cell traction models for generating pattern and form in morphogenesis. J. Math. Bio. 19, 265–279.

    MathSciNet  Google Scholar 

  • Nagorcka, B. N. and J. R. Mooney (1982). The role of a reaction-diffusion system in the formation of hair fibres. J. Theo. Bio. 98, 575–607.

    Article  MathSciNet  Google Scholar 

  • Orwin, D. F. G. and J. L. Woods (1982). Number changes and developmental potential of wool follicle cells in early stages of fibre differentiation. J. Ultrastructural Res. 80, 312–322.

    Article  Google Scholar 

  • Orwin, D. F. G., J. L. Woods and S. L. Ranford (1984). Cortical cell types and their distribution in wool fibres. Austral. J. Bio. Sci. 37, 237–255.

    Google Scholar 

  • Oster, G. F., J. D. Murray and P. K Maini (1985). A model for chondrogenic condensations in the developing limb: the role of extracellular matrix and cell tractions. J. Embryol. Exp. Morphol. 89, 93–112.

    Google Scholar 

  • Reis, P. J., B. N. Nagorcka, D. A. Tunks and S. G. Munro (1990). Variation of length growth rate and diameter of Merino wool fibres, in Proc. 8th International Wool Textile Research Conference, G. H. Crawshaw, (Ed.), Christchurch: Wool Research Organisation of New Zealand, pp. 580–589.

    Google Scholar 

  • Scott Blair, G. W. (1974). An Introduction to Biorheology, Amsterdam: Elsevier Scientific Publishing Company.

    Google Scholar 

  • Smith, G. D. (1989). Numerical Solution of Partial Differential Equations, 3rd edn, Oxford: Clarendon Press.

    Google Scholar 

  • Tanner, R. I. (1988). Engineering Rheology, Oxford: Clarendon Press.

    Google Scholar 

  • Viswanathan, R. V. (1957). Solution of Poisson’s equation by relaxation methods—normal gradient specified on curved boundaries. Math. Tab., Wash. 11, 67–78.

    MATH  MathSciNet  Google Scholar 

  • Wilson, P. A. and B. F. Short (1979). Cell proliferation and cortical cell production in relation to wool growth. Aust. J. Bio. Sci. 32, 317–327.

    Google Scholar 

  • Zlamál, M. (1967). Discretization and error estimates for elliptic boundary value problems of the fourth order. SIAM J. Numer. Anal. 4, 626–639.

    Article  MATH  MathSciNet  Google Scholar 

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Correspondence to K. Louie.

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Louie, K., Hall, A.J. & Gandar, P.W. A continuum mechanics approach to determining the cellular velocity field within a wool follicle. Bull. Math. Biol. 60, 79–100 (1998). https://doi.org/10.1006/bulm.1997.0025

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  • DOI: https://doi.org/10.1006/bulm.1997.0025

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