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Multiscale modelling and homogenisation of fibre-reinforced hydrogels for tissue engineering

Part of: Partial differential equations Homogenization, determination of effective properties Physiological, cellular and medical topics Flows in porous media; filtration; seepage Representations of solutions Generalities, axiomatics, foundations of continuum mechanics of solids

Published online by Cambridge University Press:  22 November 2018

M. J. CHEN Open the ORCID record for M. J. CHEN [Opens in a new window] ,
L. S. KIMPTON ,
J. P. WHITELEY ,
M. CASTILHO ,
J. MALDA ,
C. P. PLEASE ,
S. L. WATERS  and
H. M. BYRNE
M. J. CHEN*
Affiliation:
Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK e-mails: [email protected]; [email protected]; [email protected]; [email protected]; [email protected] School of Mathematical Sciences, The University of Adelaide, North Terrace, Adelaide SA 5005, Australia
L. S. KIMPTON
Affiliation:
Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK e-mails: [email protected]; [email protected]; [email protected]; [email protected]; [email protected]
J. P. WHITELEY
Affiliation:
Department of Computer Science, University of Oxford, Wolfson Building, Parks Road, Oxford OX1 3QD, UK e-mail: [email protected]
M. CASTILHO
Affiliation:
Department of Orthopaedics, University Medical Center Utrecht, Utrecht University, Utrecht, The Netherlands e-mail: [email protected]
J. MALDA
Affiliation:
Department of Orthopaedics, University Medical Center Utrecht, Utrecht University, Utrecht, The Netherlands e-mail: [email protected] Department of Equine Sciences, Faculty of Veterinary Medicine, Utrecht University, Utrecht, The Netherlands e-mail: [email protected]
C. P. PLEASE
Affiliation:
Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK e-mails: [email protected]; [email protected]; [email protected]; [email protected]; [email protected]
S. L. WATERS
Affiliation:
Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK e-mails: [email protected]; [email protected]; [email protected]; [email protected]; [email protected]
H. M. BYRNE
Affiliation:
Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK e-mails: [email protected]; [email protected]; [email protected]; [email protected]; [email protected]
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Abstract

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Tissue engineering aims to grow artificial tissues in vitro to replace those in the body that have been damaged through age, trauma or disease. A recent approach to engineer artificial cartilage involves seeding cells within a scaffold consisting of an interconnected 3D-printed lattice of polymer fibres combined with a cast or printed hydrogel, and subjecting the construct (cell-seeded scaffold) to an applied load in a bioreactor. A key question is to understand how the applied load is distributed throughout the construct. To address this, we employ homogenisation theory to derive equations governing the effective macroscale material properties of a periodic, elastic–poroelastic composite. We treat the fibres as a linear elastic material and the hydrogel as a poroelastic material, and exploit the disparate length scales (small inter-fibre spacing compared with construct dimensions) to derive macroscale equations governing the response of the composite to an applied load. This homogenised description reflects the orthotropic nature of the composite. To validate the model, solutions from finite element simulations of the macroscale, homogenised equations are compared to experimental data describing the unconfined compression of the fibre-reinforced hydrogels. The model is used to derive the bulk mechanical properties of a cylindrical construct of the composite material for a range of fibre spacings and to determine the local mechanical environment experienced by cells embedded within the construct.

Keywords

Homogenisationelasticityporoelasticity

MSC classification

Primary: 74Q15: Effective constitutive equations
Secondary: 92C50: Medical applications (general) 76S99: None of the above, but in this section 74A40: Random materials and composite materials 35C20: Asymptotic expansions 35-04: Explicit machine computation and programs (not the theory of computation or programming)
Type
Papers
Information
European Journal of Applied Mathematics , Volume 31 , Issue 1 , February 2020 , pp. 143 - 171
Copyright
© Cambridge University Press 2018 

Footnotes

*

Joint first authors

The research leading to these results has received funding from the European Union Seventh Framework Programme (FP7/2007-2013) under grant agreement no. 309962 (HydroZONES). The authors gratefully thank the Utrecht-Eindhoven strategic alliance and the European Research Council (consolidator grant 3D-JOINT, no. 647426) for the financial support.

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