Hostname: page-component-8448b6f56d-t5pn6 Total loading time: 0 Render date: 2024-04-24T06:47:27.831Z Has data issue: false hasContentIssue false
  • English
  • Français

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. HOLDEN Open the ORCID record for E. C. HOLDEN [Opens in a new window] ,
S. J. CHAPMAN Open the ORCID record for S. J. CHAPMAN [Opens in a new window] ,
B. S. BROOK Open the ORCID record for B. S. BROOK [Opens in a new window]  and
R. D. O’DEA Open 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
Get access Rights & Permissions [Opens in a new window]

Abstract

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alarcón, T., Owen, M. R., Byrne, H. M. and Maini, P. K., “Multiscale modelling of tumour growth and therapy: the influence of vessel normalisation on chemotherapy”, Comput. Math. Methods Med. 7 (2006) 85–119; doi:10.1080/10273660600968994.Google Scholar
Bensoussan, A., Lions, J.-L. and Papanicolaou, G., Asymptotic analysis for periodic structures, Volume 5 of Stud. in Math. Appl. (North-Holland, Amsterdam, 1978); ISBN: 9780080875262.Google Scholar
Bowen, R. M., “Incompressible porous media models by use of the theory of mixtures”, Internat. J. Engrg. Sci. 18 (1980) 1129–1148; doi:10.1016/0020-7225(80)90114-7.Google Scholar
Collis, J., Brown, D. L., Hubbard, M. E. and O’Dea, R. D., “Effective equations governing an active poroelastic medium”, Proc. R. Soc. Lond. Ser. A 473 (2017) 20160755; doi:10.1098/rspa.2016.0755.Google Scholar
Collis, J., Hubbard, M. E. and O’Dea, R. D., “Computational modelling of multiscale, multiphase fluid mixtures with application to tumour growth”, Comput. Methods Appl. Mech. Engrg. 309 (2016) 554–578; doi:10.1016/j.cma.2016.06.015.Google Scholar
Collis, J., Hubbard, M. E. and O’Dea, R. D., “A multi-scale analysis of drug transport and response for a multi-phase tumour model”, European J. Appl. Math. 28 (2017) 499–534; doi:10.1017/S0956792516000413.Google Scholar
Davit, Y. et al. , “Homogenization via formal multiscale asymptotics and volume averaging: how do the two techniques compare?”, Adv. Water Resour. 62 (2013) 178–206; doi:10.1016/j.advwatres.2013.09.006.Google Scholar
Drew, D. A. and Passman, S. L., Theory of multicomponent fluids, Volume 135 of Appl. Math. Sci. (Springer-Verlag, Berlin, Heidelberg, 2006) 1–46; doi:10.1007/b97678.Google Scholar
Holden, E. C., Collis, J., Brook, B. S. and O’Dea, R. D., “A multiphase multiscale model for nutrient limited tissue growth”, ANZIAM J. 59 (2018) 499–532; doi:10.1017/S1446181118000044.Google Scholar
Irons, L., Collis, J. and O’Dea, R. D., “Microstructural influences on growth and transport in biological tissue: a multiscale description”, in: Microscale transport modelling in biological processes (Academic Press, Elsevier, London, 2017) 311–334; doi:10.1016/B978-0-12-804595-4.00012-2.Google Scholar
Jakus, A. E., Secor, E. B., Rutz, A. L., Jordan, S. W., Hersam, M. C. and Shah, R. N., “Three-dimensional printing of high-content graphene scaffolds for electronic and biomedical applications”, ACS Nano 9 (2015) 4636–4648; doi:10.1021/acsnano.5b01179.Google Scholar
Keller, J. B., “Darcy’s law for flow in porous media and the two-space method”, in: Nonlinear partial differential equations in engineering and applied sciences, Volume 54 of Lect. Notes in Pure Appl. Math. (Marcel Dekker, New York, 1980) 429–443; doi:10.1201/9780203745465.Google Scholar
Lemon, G., King, J. R., Byrne, H. M., Jensen, O. and Shakesheff, K., “Mathematical modelling of engineered tissue growth using a multiphase porous flow mixture theory”, J. Math. Biol. 52 (2006) 571–594; doi:10.1007/s00285-005-0363-1.Google Scholar
Macklin, P., McDougall, S., Anderson, A. R. A., Chaplain, M. A. J., Cristini, V. and Lowengrub, J., “Multiscale modelling and nonlinear simulation of vascular tumour growth”, J. Math. Biol. 58 (2009) 765–798; doi:10.1007/s00285-008-0216-9.Google Scholar
Marle, C., “On macroscopic equations governing multiphase flow with diffusion and chemical reactions in porous media”, Internat. J. Engrg. Sci. 20 (1982) 643–662; doi:10.1016/0020-7225(82)90118-5.Google Scholar
Marle, C., “The effect of weak inertia on flow through a porous medium”, J. Fluid Mech. 222 (1991) 647–663; doi:10.1017/S0022112091001258.Google Scholar
Mei, C. C. and Vernescu, B., Homogenization methods for multiscale mechanics (World Scientific, Singapore, 2010); doi:10.1142/7427.Google Scholar
Miller, E. D., Fisher, G. W., Weiss, L. E., Walker, L. M. and Campbell, P. G., “Dose-dependent cell growth in response to concentration modulated patterns of FGF-2 printed on fibrin”, Biomaterials 27 (2006) 2213–2221; doi:10.1016/j.biomaterials.2005.10.021.Google Scholar
Miller, E. D., Li, K., Kanade, T., Weiss, L. E., Walker, L. M. and Campbell, P. G., “Spatially directed guidance of stem cell population migration by immobilized patterns of growth factors”, Biomaterials 32 (2011) 2775–2785; doi:10.1016/j.biomaterials.2010.12.005.Google Scholar
O’Dea, R. D., Nelson, M., El Haj, A., Waters, S. and Byrne, H. M., “A multiscale analysis of nutrient transport and biological tissue growth in vitro”, Math. Med. Biol. 32 (2015) 345–366; doi:10.1093/imammb/dqu015.Google Scholar
Osborne, J. M., Walter, A., Kershaw, S. K., Mirams, G. R., Pathmanathan, P., Gavaghan, D., Jensen, O. E., Maini, P. K. and Byrne, H. M., “A hybrid approach to multi-scale modelling of cancer”, Philos. Trans. R. Soc. Lond. Ser. A 368 (2010) 5013–5028; doi:10.1098/rsta.2010.0173.Google Scholar
Pavliotis, G. A. and Stuart, A., Multiscale methods: averaging and homogenization, Volume 53 of Texts in Appl. Math. (Springer-Verlag, New York, 2008); doi:10.1007/978-0-387-73829-1.Google Scholar
Penta, R., Ambrosi, D. and Shipley, R. J., “Effective governing equations for poroelastic growing media”, Quart. J. Mech. Appl. Math. 67 (2014) 69–91; doi:10.1093/qjmam/hbt024.Google Scholar
Shipley, R. J. and Chapman, S. J., “Multiscale modelling of fluid and drug transport in vascular tumours”, Bull. Math. Biol. 72 (2010) 1464–1491; doi:10.1007/s11538-010-9504-9.Google Scholar
Visser, J., Melchels, F. P., Jeon, J. E., van Bussel, E. M., Kimpton, L. S., Byrne, H. M., Dhert, W. J., Dalton, P. D., Hutmacher, D. W. and Malda, J., “Reinforcement of hydrogels using three-dimensionally printed microfibres”, Nat. Commun. 6 (2015) 1–46; doi:10.1038/ncomms7933.Google Scholar
Wen, J. H., Choi, O., Taylor-Weiner, H., Fuhrmann, A., Karpiak, J. V., Almutairi, A. and Engler, A. J., “Haptotaxis is cell type specific and limited by substrate adhesiveness”, Cell. Mol. Bioeng. 8 (2015) 530–542; doi:10.1007/s12195-015-0398-3.Google Scholar