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Network Development in Biological Gels: Role in Lymphatic Vessel Development

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Abstract

In this paper, we present a model that explains the prepatterning of lymphatic vessel morphology in collagen gels. This model is derived using the theory of two phase rubber material due to Flory and coworkers and it consists of two coupled fourth order partial differential equations describing the evolution of the collagen volume fraction, and the evolution of the proton concentration in a collagen implant; as described in experiments of Boardman and Swartz (Circ. Res. 92, 801–808, 2003). Using linear stability analysis, we find that above a critical level of proton concentration, spatial patterns form due to small perturbations in the initially uniform steady state. Using a long wavelength reduction, we can reduce the two coupled partial differential equations to one fourth order equation that is very similar to the Cahn–Hilliard equation; however, it has more complex nonlinearities and degeneracies. We present the results of numerical simulations and discuss the biological implications of our model.

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Authors and Affiliations

  1. Centre for Industrial and Applied Mathematics and Centre for Mathematical Biology, Mathematical Institute, 24-29 St Giles’, Oxford, OX1 3LB, UK

    Tiina Roose

  2. Department of Mathematics and Statistics, University of Limerick, Limerick, Republic of Ireland

    Andrew C. Fowler

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  1. Tiina Roose
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  2. Andrew C. Fowler
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Correspondence to Tiina Roose.

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This work was supported by the Royal Society (London) by the award of a University Research Fellowship.

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Roose, T., Fowler, A.C. Network Development in Biological Gels: Role in Lymphatic Vessel Development. Bull. Math. Biol. 70, 1772–1789 (2008). https://doi.org/10.1007/s11538-008-9324-3

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  • DOI: https://doi.org/10.1007/s11538-008-9324-3

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