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A MULTIPHASE MULTISCALE MODEL FOR NUTRIENT-LIMITED TISSUE GROWTH, PART II: A SIMPLIFIED DESCRIPTION

Part of:Mathematical biology in general

Published online by Cambridge University Press:  18 September 2019

E. C. HOLDENOpen the ORCID record for E. C. HOLDEN[Opens in a new window],
S. J. CHAPMANOpen the ORCID record for S. J. CHAPMAN[Opens in a new window],
B. S. BROOKOpen the ORCID record for B. S. BROOK[Opens in a new window] and
R. D. O’DEAOpen the ORCID record for R. D. O’DEA[Opens in a new window]
E. C. HOLDEN
Affiliation:
Centre for Mathematical Medicine and Biology, School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK email pmxeh5@exmail.nottingham.ac.uk, bindi.brook@nottingham.ac.uk, reuben.odea@nottingham.ac.uk
S. J. CHAPMAN
Affiliation:
Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK email chapman@maths.ox.ac.uk
B. S. BROOK
Affiliation:
Centre for Mathematical Medicine and Biology, School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK email pmxeh5@exmail.nottingham.ac.uk, bindi.brook@nottingham.ac.uk, reuben.odea@nottingham.ac.uk
R. D. O’DEA*
Affiliation:
Centre for Mathematical Medicine and Biology, School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK email pmxeh5@exmail.nottingham.ac.uk, bindi.brook@nottingham.ac.uk, reuben.odea@nottingham.ac.uk
*
✉reuben.odea@nottingham.ac.uk
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Abstract

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In this paper, we revisit our previous work in which we derive an effective macroscale description suitable to describe the growth of biological tissue within a porous tissue-engineering scaffold. The underlying tissue dynamics is described as a multiphase mixture, thereby naturally accommodating features such as interstitial growth and active cell motion. Via a linearization of the underlying multiphase model (whose nonlinearity poses a significant challenge for such analyses), we obtain, by means of multiple-scale homogenization, a simplified macroscale model that nevertheless retains explicit dependence on both the microscale scaffold structure and the tissue dynamics, via so-called unit-cell problems that provide permeability tensors to parameterize the macroscale description. In our previous work, the cell problems retain macroscale dependence, posing significant challenges for computational implementation of the eventual macroscopic model; here, we obtain a decoupled system whereby the quasi-steady cell problems may be solved separately from the macroscale description. Moreover, we indicate how the formulation is influenced by a set of alternative microscale boundary conditions.

Keywords

multiscale homogenizationmixture theorytissue growthtissue engineeringporous flow

MSC classification

Primary:92B05: General biology and biomathematics

Information

Type
Research Article
Information
The ANZIAM Journal , Volume 61 , Issue 4 , October 2019, pp. 368 - 381
Copyright
© 2019 Australian Mathematical Society

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