Hostname: page-component-8448b6f56d-mp689 Total loading time: 0 Render date: 2024-04-23T21:56:21.370Z Has data issue: false hasContentIssue false
  • English
  • Français

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

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

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: michael.chen@adelaide.edu.au; laura.kimpton@maths.ox.ac.uk; colin.please@maths.ox.ac.uk; sarah.waters@maths.ox.ac.uk; helen.byrne@maths.ox.ac.uk 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: michael.chen@adelaide.edu.au; laura.kimpton@maths.ox.ac.uk; colin.please@maths.ox.ac.uk; sarah.waters@maths.ox.ac.uk; helen.byrne@maths.ox.ac.uk
J. P. WHITELEY
Affiliation:
Department of Computer Science, University of Oxford, Wolfson Building, Parks Road, Oxford OX1 3QD, UK e-mail: Jonathan.Whiteley@cs.ox.ac.uk
M. CASTILHO
Affiliation:
Department of Orthopaedics, University Medical Center Utrecht, Utrecht University, Utrecht, The Netherlands e-mail: M.DiasCastilho@umcutrecht.nl
J. MALDA
Affiliation:
Department of Orthopaedics, University Medical Center Utrecht, Utrecht University, Utrecht, The Netherlands e-mail: M.DiasCastilho@umcutrecht.nl Department of Equine Sciences, Faculty of Veterinary Medicine, Utrecht University, Utrecht, The Netherlands e-mail: j.malda@umcutrecht.nl
C. P. PLEASE
Affiliation:
Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK e-mails: michael.chen@adelaide.edu.au; laura.kimpton@maths.ox.ac.uk; colin.please@maths.ox.ac.uk; sarah.waters@maths.ox.ac.uk; helen.byrne@maths.ox.ac.uk
S. L. WATERS
Affiliation:
Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK e-mails: michael.chen@adelaide.edu.au; laura.kimpton@maths.ox.ac.uk; colin.please@maths.ox.ac.uk; sarah.waters@maths.ox.ac.uk; helen.byrne@maths.ox.ac.uk
H. M. BYRNE
Affiliation:
Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK e-mails: michael.chen@adelaide.edu.au; laura.kimpton@maths.ox.ac.uk; colin.please@maths.ox.ac.uk; sarah.waters@maths.ox.ac.uk; helen.byrne@maths.ox.ac.uk
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

References

Arthritis Research UK (2013) Osteoarthritis in general practice: data and perspectives.Google Scholar
Ateshian, G. (2007) On the theory of reactive mixtures for modeling biological growth. Biomech. Model. Mechanobiol. 6(6), 423–445.CrossRefGoogle ScholarPubMed
Auriault, J.-L. & Sanchez-Palencia, E. (1977) Etude du comportement macroscopique d’un milieu poreux saturé déformable. J. Méc. 16(4), 575–603.Google Scholar
Badia, S., Quaini, A. & Quarteroni, A. (2009) Coupling Biot and Navier-Stokes equations for modelling fluid-poroelastic media interaction. J. Comput. Phys. 228, 7986–8014.CrossRefGoogle Scholar
Baker, S. R., Banerjee, S., Bonin, K. & Guthold, M. (2016) Determining the mechanical properties of electrospun poly-ɛ-caprolactone (PCL) nanofibers using AFM and a novel fiber anchoring technique. Mater. Sci. Eng. C 59, 203–212.CrossRefGoogle Scholar
Bas, O., De-Juan-Pardo, E. M., Meinert, C., D’Angella, D., Baldwin, J. G., Bray, L. J., Wellard, R. M., Kollmannsberger, S., Rank, E., Werner, C., Klein, T. J., Catelas, I. & Hutmacher, D. W. (2017) Biofabricated soft network composites for cartilage tissue engineering. Biofabrication 7, 025014.CrossRefGoogle Scholar
Biot, M. A. (1962) Generalized theory of acoustic propagation in porous dissipative media. J. Acoust. Soc. Am. 34(5), 1254–1264.CrossRefGoogle Scholar
Biot, M. A. (1972) Theory of finite deformations of porous solids. Indiana Univ. Math. J. 21, 597–620.CrossRefGoogle Scholar
Bruna, M. & Chapman, S. J. (2015) Diffusion in spatially varying porous media. SIAM J. Appl. Math. 75(4), 1648–1674.CrossRefGoogle Scholar
Bukac, M., Yotov, I., Zakerzadeh, R. & Zunino, P. (2015) Partitioning strategies for the interaction of a fluid with a poroelastic material based on a Nitsche’s coupling approach. Comput. Methods Appl. Mech. Eng. 292, 138–170.CrossRefGoogle Scholar
Castilho, M., Feyen, D., Flandes-Iparraguirre, M., Hochleitner, G., Groll, J., Doevendans, P., Vermonden, T., Ito, K., Sluijter, J. & Malda, J. (2017) Melt electrospinning writing of poly-hydroxymethylglycolide-co-ɛ-caprolactone-based scaffolds for cardiac tissue engineering. Adv. Healthcare Mater. 6(18), 1700311.CrossRefGoogle ScholarPubMed
Castilho, M., Hochleitner, G., Wilson, W., van Rietbergen, B., Dalton, D. P., Groll, J. & Malda, J. (2018) Mechanical behavior of a soft hydrogel reinforced with three-dimensional printed microfibre scaffolds. Sci. Rep. 8, 1245.CrossRefGoogle ScholarPubMed
Collis, J., Brown, D. L., Hubbard, M. E. & O’Dea, R. D. (2017) Effective equations governing an active poroelastic medium. Proc. R. Soc. A 473, 20160755.CrossRefGoogle ScholarPubMed
Davit, Y., Bell, C. G., Byrne, H. M., Chapman, L. A., Kimpton, L. S., Lang, G. E., Leonard, K. H., Oliver, J. M., Pearson, N. C., Shipley, R. J., Waters, S. L., Whiteley, J. P., Wood, B. D. & Quintard, M. (2013) Homogenization via formal multiscale asymptotics and volume averaging: how do the two techniques compare? Adv. Water Resour. 62, 178–206.CrossRefGoogle Scholar
Detournay, E. & Cheng, A. H. D. (1993) Fundamentals of poroelasticity, Reprint of Chapter 5. In: Comprehensive Rock Engineering: Principles, Practice and Projects. Analysis and Design Method, Vol. II. Pergamon Press, Oxford, UK.Google Scholar
Dunlop, J. W. C. & Fratzl, P. (2015) Bioinspired composites: making a tooth mimic. Nat. Mater. 14, 1082–1083.CrossRefGoogle ScholarPubMed
Eschbach, F. O. & Huang, S. J. (1994) Hydrophilic-hydrophobic binary systems of poly (2-hydroxyethyl methacrylate) and polycaprolactone. Part I: Synthesis and characterization. J. Bioact. Compatible Polym. 9, 29–54.CrossRefGoogle Scholar
Eshraghi, S. & Das, S. (2010) Mechanical and microstructural properties of polycaprolactone scaffolds with one-dimensional, two-dimensional, and three-dimensional orthogonally oriented porous architectures produced by selective laser sintering. Acta Biomater. 6, 2467–2476.CrossRefGoogle ScholarPubMed
Gladman, A. S., Matsumoto, E. A., Nuzzo, R. G., Mahadevan, L. & Lewis, J. A. (2016) Biomimetic 4D printing. Nat. Mater. 15, 413–418.CrossRefGoogle ScholarPubMed
Goriely, A. & Amar, M. B. (2007) On the definition and modeling of incremental, cumulative, and continuous growth laws in morphoelasticity. Biomech. Model. Mechanobiol. 6(5), 289–296.CrossRefGoogle ScholarPubMed
Groll, J., Boland, T., Blunk, T., Burdick, J. A., Cho, D.-W., Dalton, P. D., Derby, B., Forgacs, G., Li, Q., Mironov, V. A., Moroni, L., Nakamura, M., Shu, W., Takeuchi, S., Vozzi, G., Woodfield, T. B. F., Xu, T., Yoo, J. J. & Malda, J. (2016) Biofabrication: reappraising the definition of an evolving field. Biofabrication 8, 013001.CrossRefGoogle ScholarPubMed
Hang, S. (2015) TetGen, a Delaunay-based quality tetrahedral mesh generator. ACM Trans. Math. Softw. 41(2), Article 11.Google Scholar
Holzapfel, G. A. (2000) Nonlinear Solid Mechanics: A Continuum Approach for Engineering. Wiley, Chichester.Google Scholar
Howell, P., Kozyreff, G. & Ockendon, J. (2009) Applied Solid Mechanics. Cambridge University Press, Cambridge.Google Scholar
HydroZONES, Last accessed 1 October 2017. http://hydrozones.eu.Google Scholar
Klika, V., Gaffney, E. A., Chen, Y.-C. & Brown, C. P. (2016) An overview of multiphase cartilage mechanical modelling and its role in understanding function and pathology. J. Mech. Behav. Biomed. Mater. 62, 139–157.CrossRefGoogle ScholarPubMed
Lemon, G., King, J. R., Byrne, H. M., Jensen, O. E. & Shakesheff, K. M. (2006) Mathematical modelling of engineered tissue growth using a multiphase porous flow mixture theory. J. Math. Bio 52(5), 571–594.CrossRefGoogle ScholarPubMed
Li, Z., Kupcsik, L., Yao, S.-J., Alini, M. & Stoddart, M. J. (2010) Mechanical load modulates chondrogenesis of human mesenchymal stem cells through the TGF-β pathway. J. Cell. Mol. Med. 14(6A), 1338–1346.CrossRefGoogle ScholarPubMed
Mikelic, A. & Wheeler, M. F. (2012) On the interface law between a deformable porous medium containing a viscous fluid and an elastic body. Math. Models Methods Appl. Sci. 22, 1250031.CrossRefGoogle Scholar
Mow, V., Kuei, S., Lai, W. & Armstrong, C. (1980) Biphasic creep and stress relaxation of articular cartilage in compression: theory and experiments. J. Biomech. Eng. 102(1), 73–84.CrossRefGoogle ScholarPubMed
Murad, M. A. & Loula, A. F. (1994) On stability and convergence of finite element approximations of Biot’s consolidation problem. Int. J. Numer. Methods Eng. 37, 645–667.CrossRefGoogle Scholar
O’Dea, R. D., Waters, S. L. & Byrne, H. M. (2010) A multiphase model for tissue construct growth in a perfusion bioreactor. Math. Med. Biol. 27(2), 95–127.CrossRefGoogle Scholar
Parnell, W. & Abrahams, I. (2008) Homogenization for wave propagation in periodic fibre-reinforced media with complex microstructure. i—theory. J. Mech. Phys. Solids 67, 2521–2540.CrossRefGoogle Scholar
Penta, R., Ambrosi, D. & Shipley, R. J. (2014) Effective governing equations for poroelastic growing media. Q. J. Mech. Appl. Math. 67, 69–91.CrossRefGoogle Scholar
Peter, M. A. (2009) Coupled reaction-diffusion processes inducing an evolution of the microstructure: analysis and homogenization. Nonlinear Anal. Theory Methods Appl. 70, 806–821.CrossRefGoogle Scholar
Piatnitski, A. & Ptashnyk, M. (2017) Homogenization of biomechanical models for plant tissues. Multiscale Model. Simul. 15(1), 339–387.CrossRefGoogle Scholar
Ptashnyk, M. & Seguin, B. (2016) The impact of microfibril orientations on the biomechanics of plant cell walls and tissues. Bull. Math. Biol. 78, 2135–2164.CrossRefGoogle ScholarPubMed
Shipley, R., Jones, G., Dyson, R., Sengers, B., Bailey, C., Catt, C., Please, C. & Malda, J. (2009) Design criteria for a printed tissue engineering construct: a mathematical homogenization approach. J. Theor. Biol. 259(3), 489–502.CrossRefGoogle ScholarPubMed
Shipley, R. J. & Chapman, S. J. (2010) Multiscale modelling of fluid and drug transport in vascular tumours. Bull. Math. Biol. 72, 1464–1491.CrossRefGoogle ScholarPubMed
Sunkara, V. & von Kleist, M. (2016) Coupling cellular phenotype and mechanics to understand extracellular matrix formation and homeostasis in osteoarthritis. IFAC-PapersOnLine 49(26), 038–043.CrossRefGoogle Scholar
Tan, E., Ng, S. & Lim, C. (2005) Tensile testing of a single ultrafine polymeric fiber. Biomaterials 26, 1453–1456.CrossRefGoogle ScholarPubMed
Visser, J., Melchels, F. P. W., Jeon, J. E., van Bussel, E. M., Kimpton, L. S., Byrne, H. M., Dhert, W. J. A., Dalton, P. D., Hutmacher, D. W. & Malda, J. (2015) Reinforcement of hydrogels using three-dimensionally printed microfibres. Nat. Commun. 6, 6933.CrossRefGoogle ScholarPubMed
Wegst, U. G. K., Ba, H., Saiz, E., Tomsia, A. P. & Ritchie, R. O. (2015) Bioinspired structural materials. Nat. Mater. 14, 23–26.CrossRefGoogle ScholarPubMed