Hostname: page-component-77c89778f8-9q27g Total loading time: 0 Render date: 2024-07-19T04:06:21.690Z Has data issue: false hasContentIssue false
  • English
  • Français

A multi-scale analysis of drug transport and response for a multi-phase tumour model†

Published online by Cambridge University Press:  05 October 2016

J. COLLIS ,
M. E. HUBBARD  and
R. D. O'DEA
J. COLLIS
Affiliation:
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UK email: Joe.Collis@nottingham.ac.uk, Matthew.Hubbard@nottingham.ac.uk, Reuben.O'Dea@nottingham.ac.uk
M. E. HUBBARD
Affiliation:
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UK email: Joe.Collis@nottingham.ac.uk, Matthew.Hubbard@nottingham.ac.uk, Reuben.O'Dea@nottingham.ac.uk
R. D. O'DEA
Affiliation:
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UK email: Joe.Collis@nottingham.ac.uk, Matthew.Hubbard@nottingham.ac.uk, Reuben.O'Dea@nottingham.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

In this article, we consider the spatial homogenisation of a multi-phase model for avascular tumour growth and response to chemotherapeutic treatment. The key contribution of this work is the derivation of a system of homogenised partial differential equations describing macroscopic tumour growth, coupled to transport of drug and nutrient, that explicitly incorporates details of the structure and dynamics of the tumour at the microscale. In order to derive these equations, we employ an asymptotic homogenisation of a microscopic description under the assumption of strong interphase drag, periodic microstructure, and strong separation of scales. The resulting macroscale model comprises a Darcy flow coupled to a system of reaction–advection partial differential equations. The coupled growth, response, and transport dynamics on the tissue scale are investigated via numerical experiments for simple academic test cases of microstructural information and tissue geometry, in which we observe drug- and nutrient-regulated growth and response consistent with the anticipated dynamics of the macroscale system.

Keywords

Cancer modellingmulti-scale homogenisationdrug transportporous media
Type
Papers
Information
European Journal of Applied Mathematics , Volume 28 , Issue 3 , June 2017 , pp. 499 - 534
Copyright
Copyright © Cambridge University Press 2016 

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.)

Footnotes

†

The first and second authors wish to acknowledge the support of EPSRC grant number EP/K039342/1.

References

Alarcón, T., Byrne, H. M. & Maini, P. K. (2003) A cellular automaton model for tumour growth in inhomogeneous environment. J. Theor. Biol. 225, 257–274.CrossRefGoogle ScholarPubMed
Alarcón, T., Byrne, H. M. & Maini, P. K. (2004) Towards whole-organ modelling of tumour growth. Progr. Biophys. Mol. Biol. 85, 451–472.Google Scholar
Alarcón, T., Byrne, H. M. & Maini, P. K. (2005) A multiple scale model for tumor growth. Multiscale Model. Simul. 3 (2), 440–475.CrossRefGoogle Scholar
Alarcón, T., Owen, M. R., Byrne, H. M. & Maini, P. K. (2006) Multiscale modelling of tumour growth and therapy: The influence of vessel normalisation on chemotherapy. Comput. Math. Methods Med., 7 (2–3), 85–119.Google Scholar
Ambrosi, D. & Preziosi, L. (2009) Cell adhesion mechanisms and stress relaxation in the mechanics of tumours. Biomech. Model. Mech. 8, 397–413.Google Scholar
Antonietti, P., Giani, S., Hall, E., Houston, P. & Krahl, R. (June 2015) Aptofem documentation. Available at: http://www.aptofem.com.Google Scholar
Araujo, R. P. & McElwain, D. L. S. (2004) A history of the study of solid tumour growth: The contribution of mathematical modelling. Bull. Math. Biol. 66, 1039–1091.Google Scholar
Band, L. R. & King, J. R. (2012) Multiscale modelling of auxin transport in the plant-root elongation zone. J. Math. Biol. 65 (4), 743–785.CrossRefGoogle ScholarPubMed
Breward, C. J. W., Byrne, H. M. & Lewis, C. E. (2002) The role of cell-cell interactions in a two-phase model for avascular tumour growth. J. Math. Biol. 152, 125–152.CrossRefGoogle Scholar
Breward, C. J. W., Byrne, H. M. & Lewis, C. E. (2004) A multiphase model describing vascular tumour growth. Bull. Math. Biol. 1, 1–28.Google Scholar
Brezzi, F. & Fortin, M. (1991) Mixed and Hybrid Finite Element Methods. Springer-Verlag, New York.CrossRefGoogle Scholar
Burridge, R. & Keller, J. B. (1981) Poroelasticity equations derived from microstructure. J. Acoust. Soc. Am. 70, 1140.Google Scholar
Byrne, H. M. & Owen, M. R. (2004) A new interpretation of the keller-segel model based on multiphase modelling. J. Math. Biol., 49 (6), 604–626.Google Scholar
Byrne, H. M., King, J. R., McElwain, D. L. S. & Preziosi, L. (2003) A two-phase model of solid tumour growth. Appl. Math. Lett. 20, 341–366.Google ScholarPubMed
Byrne, H. M. & Preziosi, L. (2003) Modelling solid tumour growth using the theory of mixtures. Math. Med. Biol. 16 (4), 567–573.Google Scholar
Casciari, J. J., Sotirchos, S. V. & Sutherland, R. M. (1992) Mathematical modelling of microenvironment and growth in EMT6/Ro multicellular tumour spheroids. Cell Proliferation 25, 1–22.CrossRefGoogle ScholarPubMed
Chaplain, M. A. J., McDougall, S. R. & Anderson, A. R. A. (2006) Mathematical modeling of tumour-induced angiogenesis. Annu. Rev. Biomen. Eng. 8, 233–257.Google Scholar
Collis, J., Hubbard, M. E. & O'Dea, R. D. (2016) Computational modelling of multiscale, multiphase fluid mixtures with application to tumour growth. Comput. Methods Appl. Mech. Eng. 309, 554–578.CrossRefGoogle Scholar
Davit, Y., Bell, C. G., Byrne, H. M., Chapman, L. A. C., Kimpton, L. S., Lang, G. E., Leonard, K. H. L., 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.Google Scholar
Drew, D. A. (1971) Averaged field equations for a two phase media. Stud. Appl. Math. L2, 205–231.CrossRefGoogle Scholar
Drew, D. A. (1983) Mathematical modelling of two-phase flow. Annu. Rev. Fluid Mech. 15, 261–291.CrossRefGoogle Scholar
Folkman, J. & Cotran, R. (1976) Relation of vascular proliferation to tumor growth. Int. Rev. Exp. Pathol. 16, 207–248.Google ScholarPubMed
Fozard, J. A., Byrne, H. M., Jensen, O. E. & King, J. R. (2010) Continuum approximations of individual-based models for epithelial monolayers. Math. Med. Biol. 5 (9), 39–74.Google Scholar
Franks, S. J. & King, J. R. (2003) Interactions between a uniformly proliferating tumour and its surroundings. Mathematical Medicine and Biology 20, 47–89.CrossRefGoogle ScholarPubMed
Frieboes, H. B., Lowengrub, J. S., Wise, S. M., Zheng, X., Macklin, P., Bearer, E. L. & Cristini, V. (2007) Computer simulation of glioma growth and morphology. Neuroimage 37, S58–S70.CrossRefGoogle ScholarPubMed
Gallaher, J. & Anderson, A. R. A. (2013) Evolution of intratumoral phenotypic heterogeneity: the role of trait inheritance. Interface Focus 3 (4).CrossRefGoogle ScholarPubMed
Houston, P. & Süli, E. (2001) hp-adaptive discontinuous Galerkin finite element methods for first-order hyperbolic problems. SIAM J. Sci. Comput. 23 (4), 1226–1252.Google Scholar
Hubbard, M. E. & Byrne, H. M. (2013) Multiphase modelling of vascular tumour growth in two spatial dimensions. J. Theor. Biol. 316, 70–89.Google Scholar
Jain, R. (1989) Delivery of novel therapeutic agents in tumors: physiological barriers and strategies. J. Natl. Cancer Inst. 81 (8), 570–576.CrossRefGoogle ScholarPubMed
Jain, R. (2001) Delivery of molecular and cellular medicine to solid tumors. Adv. Drug Deliv. Rev. 46, 149–168.Google Scholar
Keller, J. B. (1980) Darcy's law for flow in porous media and the two-scale method. In: Sternberg, R. L., Kalinowski, A. J., & Papadakis, J. S. (editors), Nonlinear PDE in Engineering and Applied Sciences. Marcel Dekker, New York, 429–443.Google Scholar
King, J. R. & Franks, S. J. (2007) Mathematical modelling of nutrient-limited tissue growth. In: Figueiredo, I. N., Rodrigues, J. F., & Santos, L. (editors), Free Boundary Problems. Springer, Basel, pp. 273–282.Google Scholar
Laird, A. K. (1964) Dynamics of tumour growth. British Journal of Cancer, 18 (3), 490–502.Google Scholar
Lemon, G. & King, J. R. (2007) Multiphase modelling of cell behaviour on artificial scaffolds: Effects of nutrient depletion and spatially nonuniform porosity. Mathematical Medicine and Biology, 24 (1), 57–83.Google Scholar
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. Biol. 52, 571–594.Google Scholar
Lowengrub, J. S., Frieboes, H. B., Jin, F., Chuang, Y.-L., Li, X., Macklin, P., Wise, S. M. & Cristini, V. (2010) Nonlinear modelling of cancer: Bridging the gap between cells and tumours. Nonlinearity 23 (1), R1–R91.CrossRefGoogle Scholar
Lubkin, S. R. & Jackson, T. (2002) Multiphase mechanics of capsule formation in tumors. J. Biomech. Eng. 124 (2), 237–243.Google Scholar
Macklin, P., McDougall, S., Anderson, A. R. A., Chaplain, M. A. J., Cristini, V. & Lowengrub, J. (2009) Multiscale modelling and nonlinear simulation of vascular tumour growth. J. Math. Biol., 58 (4–5), 765–798.Google Scholar
Mei, C. C. & Auriault, J.-L. (1991) The effect of weak inertia on flow through a porous medium. J. Fluid Mech. 222, 647–663.Google Scholar
Michor, F., Iwasa, Y. & Nowak, M. A. (2004) Dynamics of cancer progression. Nature Rev. Cancer, 4 (3), 197–205.Google Scholar
Morland, L. W. & Sellers, S. (2001) Multiphase mixtures and singular surfaces. Int. J. Non-Linear Mech. 36, 131–146.CrossRefGoogle Scholar
Morris, G. E., Bridge, J. C., Brace, L. A., Knox, A. J., Aylott, J. W., Brightling, C. E., Ghaemmaghami, A. M. & Rose, F. R. (2014) A novel electrospun biphasic scaffold provides optimal three-dimensional topography for in vitro co-culture of airway epithelial and fibroblast cells. Biofabrication 6 (3).Google Scholar
O'Dea, R. D. & King, J. R. (2011) Multiscale analysis of pattern formation via intercellular signalling. Math. Biosci. 231, 172–185.Google Scholar
O'Dea, R. D. & King, J. R. (2012) Continuum limits of pattern formation in hexagonal-cell monolayers. J. Math. Biol. 64, 579–610.Google Scholar
O'Dea, R. D., Nelson, M. R., El Haj, A. J., Waters, S. L. & Byrne, H. M. (2015) A multiscale analysis of nutrient transport and biological tissue growth in vitro . Math. Med. Biol. 32 (3), 345–366.CrossRefGoogle ScholarPubMed
O'Dea, R. D., Waters, S. L. & Byrne, H. M. (2008) A two-fluid model for tissue growth within a dynamic flow environment. Eur. J. Appl. Math. 19, 607–634.Google 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.Google Scholar
Owen, M. R., Alarcón, T., Maini, P. K. & Byrne, H. M. (2009) Angiogenesis and vascular remodelling in normal and cancerous tissues. J. Math. Biol. 58 (4–5), 689–721.Google Scholar
Owen, M. R., Stamper, I. J., Muthana, M., Dobson, G. W., Lewis, C. E. & Byrne, H. M. (2011) Mathematical modeling predicts synergistic antitumor effects of combining a macrophage-based, hypoxia-targeted gene therapy with chemotherapy. Cancer Res. 71 (8), 2826–2837.CrossRefGoogle ScholarPubMed
Penta, R., Ambrosi, D. & Shipley, R. J. (2014) Effective governing equations for poroelastic growing media. Q. J. Mech. Appl. Math. 67, 69–91.Google Scholar
Perfahl, H., Byrne, H. M., Chen, T., Estrella, V., Alarcón, T., Lapin, A., Gatenby, R. A., Gillies, R. J., Lloyd, M. C., Maini, P. K., Reuss, M. & Owen, M. R. (2011) Multiscale modelling of vascular tumour growth in 3d: the roles of domain size and boundary conditions. PloS one 6 (4), e14790.CrossRefGoogle ScholarPubMed
Powathil, G. G., Swat, M. & Chaplain, M. A. J. (2015) Systems oncology: Towards patient-specific treatment regimes informed by multiscale mathematical modelling. Seminars Cancer Biol. 30, 13–20.Google Scholar
Preziosi, L. & Tosin, A. (2009a) Multiphase modelling of tumour growth and extracellular matrix interaction: Mathematical tools and applications. J. Math. Biol. 58 (4–5), 625–656.Google Scholar
Preziosi, L. & Tosin, A. (2009b) Multiphase and multiscale trends in cancer modelling. Math. Model. Natural Phenom. 4 (3), 1–11.Google Scholar
Ptashnyk, M. & Chavarría-Krauser, A. (2010) Homogenization of long-range auxin transport in plant tissues. Nonlinear-Anal.: Real World Appl. 11 (6), 4524–4532.Google Scholar
Ptashnyk, M. & Roose, T. (2010) Derivation of a macroscopic model for transport of strongly sorbed solutes in the soil using homogenization theory. SIAM J. Appl. Math. 70 (7), 2097–2118.Google Scholar
Raviart, P. A. & Thomas, J. M. (1977) A mixed finite element method for second order elliptic problems. In: Galligani, I. & Magenes, E. (editors), Mathematical Aspects of the Finite Element Method, Lectures Notes in Math., Vol. 606, Springer-Verlag, New York, pp. 292–315.CrossRefGoogle Scholar
Rejniak, K. A., Estrella, V., Chen, T., Cohen, A. S., Lloyd, M. C. & Morse, D. L. (2013) The role of tumor tissue architecture in treatment penetration and efficacy: An integrative study. Frontiers Oncol. 3 (111), eCollection.Google Scholar
Roose, T., Chapman, S. J. & Maini, P. K. (2007) Mathematical models of avascular tumor growth. SIAM Rev. 49 (2), 179–208.Google Scholar
Rubinstein, J. (1987) Hydrodynamic screening in random media. In: Papanicolao, G. (editors), Hydrodynamic Behaviour and Interacting Particle Systems, IMA Volumes in Mathematics and its Application, Vol. 9, Springer, New York, pp. 137–149.Google Scholar
Rubinstein, J. & Torquato, S. (1989) Flow in random porous media: mathematical formulation, variational principles, and rigorous bounds. J. Fluid Mech. 206, 25–46.CrossRefGoogle Scholar
Shipley, R. J. & Chapman, S. J. (2010) Multiscale modelling of fluid and drug transport in vascular tumours. Bull. Math. Biol. 72 (6), 1464–1491.Google Scholar
Shipley, R. J., Chapman, S. J. & Jawad, R. (2010) Multiscale modeling of fluid transport in tumors. Bull. Math. Biol., 70 (8), 2334–2357.Google Scholar
Tartar, L. (1980) Nonhomogeneous media and vibration theory, Appendix 2. In: Lecture Notes in Physics, Vol. 127, Springer.Google Scholar
Toselli, A. (2002) hp-finite element discontinuous Galerkin approximations for the Stokes problem. M3AS 12, 1565–1616.Google Scholar
Tosin, A. & Preziosi, L. (2010) Multiphase modeling of tumor growth with matrix remodeling and fibrosis. Math. Comput. Model. 52 (7–8), 969–976.CrossRefGoogle Scholar
Tracqui, P. (2009) Biophysical models of tumour growth. Rep. Prog. Phys. 72 (5).Google Scholar
Turner, S., Sherratt, J. A., Painter, K. L. & Savill, N. J. (2004) From a discrete to a continuous model of biological cell movement. Phy. Rev. E, 69 (2), 21910/1–21910/10.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. Nature Commun. 6.Google Scholar