title: A multiscale analysis of nutrient transport and biological tissue growth in vitro.
abstract: In this paper, we consider the derivation of macroscopic equations appropriate to describe the growth of biological tissue, employing a multiple-scale homogenization method to accommodate explicitly the influence of the underlying microscale structure of the material, and its evolution, on the macroscale dynamics. Such methods have been widely used to study porous and poroelastic materials; however, a distinguishing feature of biological tissue is its ability to remodel continuously in response to local environmental cues. Here, we present the derivation of a model broadly applicable to tissue engineering applications, characterized by cell proliferation and extracellular matrix deposition in porous scaffolds used within tissue culture systems, which we use to study coupling between fluid flow, nutrient transport, and microscale tissue growth. Attention is restricted to surface accretion within a rigid porous medium saturated with a Newtonian fluid; coupling between the various dynamics is achieved by specifying the rate of microscale growth to be dependent upon the uptake of a generic diffusible nutrient. The resulting macroscale model comprises a Darcy-type equation governing fluid flow, with flow characteristics dictated by the assumed periodic microstructure and surface growth rate of the porous medium, coupled to an advection-reaction equation specifying the nutrient concentration. Illustrative numerical simulations are presented to indicate the influence of microscale growth on macroscale dynamics, and to highlight the importance of including experimentally relevant microstructural information to correctly determine flow dynamics and nutrient delivery in tissue engineering applications.
authors.affiliation: School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, UK reuben.o'dea@nottingham.ac.uk.
authors.forename: R D
authors.initials: RD
authors.lastname: O'Dea
authors.affiliation: School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, UK and School of Science and Technology, Nottingham Trent University, Clifton Campus, Nottingham NG11 8NS, UK.
authors.forename: M R
authors.initials: MR
authors.lastname: Nelson
authors.affiliation: Institute for Science & Technology in Medicine, Keele University, Guy Hilton Research Centre, Stoke-on-Trent ST4 7QB, UK.
authors.forename: A J
authors.initials: AJ
authors.lastname: El Haj
authors.affiliation: Oxford Centre for Industrial and Applied Mathematics, Mathematical Institute, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK.
authors.forename: S L
authors.initials: SL
authors.lastname: Waters
authors.affiliation: Department of Computer Science, University of Oxford, Wolfson Building, Parks Road, Oxford OX1 3QD, UK and Oxford Centre for Collaborative Applied Mathematics, Mathematical Institute, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK.
authors.forename: H M
authors.initials: HM
authors.lastname: Byrne
mesh_terms: D001692:Biological Transport; D057446:Hydrodynamics; D008954:Models, Biological; D016062:Porosity; D023822:Tissue Engineering; D054457:Tissue Scaffolds
keywords: multiscale homogenization; porous flow; tissue engineering