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A novel coupled system of non-local integro-differential equations modelling Young’s modulus evolution, nutrients’ supply and consumption during bone fracture healing

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Abstract

During fracture healing, a series of complex coupled biological and mechanical phenomena occurs. They include: (i) growth and remodelling of bone, whose Young’s modulus varies in space and time; (ii) nutrients’ diffusion and consumption by living cells. In this paper, we newly propose to model these evolution phenomena. The considered features include: (i) a new constitutive equation for growth simulation involving the number of sensor cells; (ii) an improved equation for nutrient concentration accounting for the switch between Michaelis–Menten kinetics and linear consumption regime; (iii) a new constitutive equation for Young’s modulus evolution accounting for its dependence on nutrient concentration and variable number of active cells. The effectiveness of the model and its predictive capability are qualitatively verified by numerical simulations (using COMSOL) describing the healing of bone in the presence of damaged tissue between fractured parts.

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Authors and Affiliations

  1. Warsaw University of Technology, Warsaw, Poland

    Yanfei Lu & Tomasz Lekszycki

  2. Medical University of Warsaw, Warsaw, Poland

    Tomasz Lekszycki

  3. International Research Center MeMOCS “Mathematics and Mechanics of Complex Systems”, L’Aquila, Italy

    Tomasz Lekszycki

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  1. Yanfei Lu
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Lu, Y., Lekszycki, T. A novel coupled system of non-local integro-differential equations modelling Young’s modulus evolution, nutrients’ supply and consumption during bone fracture healing. Z. Angew. Math. Phys. 67, 111 (2016). https://doi.org/10.1007/s00033-016-0708-1

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