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A MULTIPHASE MULTISCALE MODEL FOR NUTRIENT LIMITED TISSUE GROWTH
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Mathematical biology in general
Published online by Cambridge University Press: 23 May 2018
Abstract
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We derive an effective macroscale description for the growth of tissue on a porous scaffold. A multiphase model is employed to describe the tissue dynamics; linearisation to facilitate a multiple-scale homogenisation provides an effective macroscale description, which incorporates dependence on the microscale structure and dynamics. In particular, the resulting description admits both interstitial growth and active cell motion. This model comprises Darcy flow, and differential equations for the volume fraction of cells within the scaffold and the concentration of nutrient, required for growth. These are coupled with Stokes-type cell problems on the microscale, incorporating dependence on active cell motion and pore scale structure. The cell problems provide the permeability tensors with which the macroscale flow is parameterised. A subset of solutions is illustrated by numerical simulations.
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- Research Article
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- © 2018 Australian Mathematical Society
References
Astanin, S. and Preziosi, L., “Multiphase models of tumour growth”, in: Selected topics in cancer modeling (Springer, Boston, 2008) 1–31; doi:10.1007/978-0-8176-4713-1_9.Google Scholar
Band, L. and King, J. R., “Multiscale modelling of auxin transport in the plant-root elongation zone”, J. Math. Biol.
65 (2012) 743–785; doi:10.1007/s00285-011-0472-y.Google Scholar
Barrault, M., Maday, Y., Nguyen, N. C. and Patera, A. T., “An “empirical interpolation” method: application to efficient reduced-basis discretization of partial differential equations”, C. R. Math.
339 (2004) 667–672; doi:10.1016/j.crma.2004.08.006.Google Scholar
Bazilevs, Y. and Hughes, T. J. R., “Weak imposition of Dirichlet boundary conditions in fluid mechanics”, Comput. Fluids
36 (2007) 12–26; doi:10.1016/j.compfluid.2005.07.012.Google Scholar
Bensoussan, A., Lions, J.-L. and Papanicolaou, G., Asymptotic analysis for periodic structures, Volume 5 (North Holland, Amsterdam, 1978).Google Scholar
Berman, T., Mizrahi, R. and Dosoretz, C. G., “Transparent exopolymer particles (TEP): A critical factor in aquatic biofilm initiation and fouling on filtration membranes”, Desalination
276 (2011) 184–190; doi:10.1016/j.desal.2011.03.046.Google Scholar
Bhat, Z. F. and Fayaz, H., “Prospectus of cultured meat—advancing meat alternatives”, J. Food Sci. Technol.
48 (2011) 125–140; doi:10.1007/s13197-010-0198-7.Google Scholar
Bowden, L. G., Maini, P. K., Moulton, D. E., Tang, J. B., Wang, X. T., Liu, P. Y. and Byrne, H. M., “An ordinary differential equation model for full thickness wounds and the effects of diabetes”, J. Theoret. Biol.
361 (2014) 87–100; doi:10.1016/j.jtbi.2014.07.001.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.CrossRefGoogle Scholar
Breward, C. J. W., Byrne, H. M. and Lewis, C. E., “The role of cell–cell interactions in a two-phase model for avascular tumour growth”, J. Math. Biol.
45 (2002) 125–152;doi:10.1007/s002850200149.Google Scholar
Brezzi, F. and Fortin, M., Mixed and hybrid finite element methods (Springer, New York, 1991); doi:10.1007/978-1-4612-3172-1.Google Scholar
Brown, D. L., Popov, P. and Efendiev, Y., “Effective equations for fluid-structure interaction with applications to poroelasticity”, Appl. Anal.
93 (2014) 771–790;doi:10.1080/00036811.2013.839780.CrossRefGoogle Scholar
Burridge, R. and Keller, J. B., “Poroelasticity equations derived from microstructure”, J. Acoust. Soc. Am.
70 (1981) 1140–1146; doi:10.1121/1.386945.CrossRefGoogle Scholar
Byrne, H. M., King, J. R., McElwain, D. L. S. and Preziosi, L., “A two-phase model of solid tumour growth”, Appl. Math. Lett.
16 (2003) 567–573; doi:10.1016/S0893-9659(03)00038-7.Google Scholar
Byrne, H. M. and Preziosi, L., “Modelling solid tumour growth using the theory of mixtures”, Math. Med. Biol.
20 (2003) 341–366; doi:10.1093/imammb/20.4.341.Google Scholar
Chen, Y., Zhou, S. and Li, Q., “Microstructure design of biodegradable scaffold and its effect on tissue regeneration”, Biomaterials
32 (2011) 5003–5014; doi:10.1016/j.biomaterials.2011.03.064.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. 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) 449;doi:10.1017/S0956792516000413.Google Scholar
Coombs, P., Wagner, D., Bateman, K., Harrison, H., Milodowski, A. E., Noy, D. and West, J. M., “The role of biofilms in subsurface transport processes”, Quart. J. Eng. Geol. Hydrogeol.
43 (2010) 113–139; doi:10.1144/1470-9236/08-029.CrossRefGoogle Scholar
Cristini, V., Li, X., Lowengrub, J. S. and Wise, S. M., “Nonlinear simulations of solid tumor growth using a mixture model: invasion and branching”, J. Math. Biol.
58 (2009) 723–763; doi:10.1007/s00285-008-0215-x.Google Scholar
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., Shipley, R. J., Waters, S. L., Whiteley, J. P., Wood, B. D. and Quintard, M., “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., “Averaged field equations for two-phase media”, Stud. Appl. Math.
50 (1971) 133–166; doi:10.1002/sapm1971502133.Google Scholar
Drew, D. A. and Passman, S. L., Theory of multicomponent fluids, Volume 135 (Springer Science and Business Media, 2006) 1–46; doi:10.1007/b97678.Google Scholar
El-Armouche, A., Singh, J., Naito, H., Wittkopper, K., Didie, M., Laatsch, A., Zimmermann, W. H. and Eschenhagen, T., “Adenovirus-delivered short hairpin RNA targeting
$\text{PKC}\unicode[STIX]{x1D6FC}$
improves contractile function in reconstituted heart tissue”, J. Mol. Cell. Cardiol.
43 (2007) 371–376;doi:10.1016/j.yjmcc.2007.05.021.Google Scholar
$\text{PKC}\unicode[STIX]{x1D6FC}$
improves contractile function in reconstituted heart tissue”, J. Mol. Cell. Cardiol.
43 (2007) 371–376;doi:10.1016/j.yjmcc.2007.05.021.Google ScholarFerguson, S., Bryant, J., Ganz, R. and Ito, K., “The influence of the acetabular labrum on hip joint cartilage consolidation: a poroelastic finite element model”, J. Biomech.
33 (2000) 953–960;doi:10.1016/S0021-9290(00)00042-7.Google Scholar
Franks, S. and King, J. R., “Interactions between a uniformly proliferating tumour and its surroundings: uniform material properties”, Math. Med. Biol.
20 (2003) 47–89;doi:10.1093/imammb/20.1.47.Google Scholar
Ghommem, M., Presho, M., Calo, V. M. and Efendiev, Y., “Mode decomposition methods for flows in high-contrast porous media: Global-local approach”, J. Comput. Phys.
253 (2013) 226–238; doi:10.1016/j.jcp.2013.06.033.Google Scholar
Grenier, S., Sandig, M., Holdsworth, D. W. and Mequanint, K., “Interactions of coronary artery smooth muscle cells with 3D porous polyurethane scaffolds”, J. Biomed. Mater. Res. Part A
89 (2009) 293–303; doi:10.1002/jbm.a.31972.Google Scholar
Hornung, U., Homogenization and porous media (Springer, New York, 1997);doi:10.1007/978-1-4612-1920-0.Google Scholar
Hubbard, M. and Byrne, H. M., “Multiphase modelling of vascular tumour growth in two spatial dimensions”, J. Theoret. Biol.
316 (2013) 70–89; doi:10.1016/j.jtbi.2012.09.031.Google Scholar
Hubbard, M. E. and Dodd, N., “A 2D numerical model of wave run-up and overtopping”, Coast. Engng
47 (2002) 1–26; doi:10.1016/S0378-3839(02)00094-7.Google Scholar
Ichikawa, Y., Kawamura, K., Fujii, N. and Nattavut, T., “Molecular dynamics and multiscale homogenization analysis of seepage/diffusion problem in bentonite clay”, Internat. J. Numer. Methods Engrg.
54 (2002) 1717–1749; doi:10.1002/nme.488.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
Jain, R. K., “Delivery of molecular and cellular medicine to solid tumors”, Adv. Drug Deliv. Rev.
64 (2012) 353–365; doi:10.1016/j.addr.2012.09.011.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., “Effective behavior of heterogeneous media”, in: Statistical mechanics and statistical methods in theory and application (Springer, Boston, 1977) 631–644; doi:10.1007/978-1-4613-4166-6_27.Google Scholar
Korhonen, R. K., Laasanen, M. S., Toyras, J., Lappalainen, R., Helminen, H. J. and Jurvelin, J. S., “Fibril reinforced poroelastic model predicts specifically mechanical behavior of normal, proteoglycan depleted and collagen degraded articular cartilage”, J. Biomech.
36 (2003) 1373–1379; doi:10.1016/S0021-9290(03)00069-1.Google Scholar
Larsen, E. W. and Keller, J. B., “Asymptotic solution of neutron transport problems for small mean free paths”, J. Math. Phys.
15(1) (1974) 75–81; doi:10.1063/1.1666510.Google Scholar
Lemon, G. and King, J. R., “Multiphase modelling of cell behaviour on artificial scaffolds: effects of nutrient depletion and spatially nonuniform porosity”, Math. Med. Biol.
24(1) (2007) 57–83; doi:10.1093/imammb/dql020.Google Scholar
Lemon, G. and King, J. R., “Travelling-wave behaviour in a multiphase model of a population of cells in an artificial scaffold”, J. Math. Biol.
55 (2007) 449–480; doi:10.1007/s00285-007-0091-9.CrossRefGoogle 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
Li, L., Soulhat, J., Buschmann, M. and Shirazi-Adl, A., “Nonlinear analysis of cartilage in unconfined ramp compression using a fibril reinforced poroelastic model”, Clin. Biomech.
14 (1999) 673–682; doi:10.1016/S0268-0033(99)00013-3.Google Scholar
Lin, C. Y., Kikuchi, N. and Hollister, S. J., “A novel method for biomaterial scaffold internal architecture design to match bone elastic properties with desired porosity”, J. Biomech.
37 (2004) 623–636; doi:10.1016/j.jbiomech.2003.09.029.Google Scholar
Lubkin, S. R. and Jackson, T., “Multiphase mechanics of capsule formation in tumors”, Tran. Amer. Soc. Mech. Engs J. Biomech. Eng.
124 (2002) 237–243; doi:10.1115/1.1427925.Google Scholar
Mak, A., “The apparent viscoelastic behavior of articular cartilage—the contributions from the intrinsic matrix viscoelasticity and interstitial fluid flows”, J. Biomech. Engng
108 (1986) 123–130; doi:10.1115/1.3138591.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
Matzavinos, A. and Ptashnyk, M., “Homogenization of oxygen transport in biological tissues”, Appl. Anal.
95 (2016) 1013–1049; doi:10.1080/00036811.2015.1049600.Google Scholar
Mehdi, G., Calo, V. M. and Efendiev, Y., “Mode decomposition methods for flows in high-contrast porous media. A global approach”, J. Comput. Phys.
257 (2014) 400–413;doi:10.1016/j.jcp.2013.09.031.Google Scholar
Mei, C. C. and Vernescu, B., Homogenization methods for multiscale mechanics (World Scientific, Singapore, 2010); doi:10.1142/7427.Google Scholar
Nitsche, J. A., “Uber ein Variationsprinzip zur Lösung Dirichlet-Problemen bei Verwendung von Teilräumen die keinen Randbedingungen unteworfen sind”, Abh. Math. Semin. Univ. Hambg.
36 (1971) 9–15; doi:10.1007/BF02995904.Google Scholar
O’Dea, R. D., Byrne, H. M. and Waters, S. L., “Continuum modelling of in vitro tissue engineering: a review”, in: Computational modeling in tissue engineering (Springer, Berlin, 2013) 229–266; doi:10.1007/8415_2012_140.Google Scholar
O’Dea, R. D. and King, J. R., “Multiscale analysis of pattern formation via intercellular signalling”, Math. Biosci.
231 (2011) 172–185; doi:10.1016/j.mbs.2011.03.003.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
O’Dea, R. D., Waters, S. L. and Byrne, H. M., “A two-fluid model for tissue growth within a dynamic flow environment”, European J. Appl. Math.
19 (2008) 607–634;doi:10.1017/S0956792508007687.Google Scholar
O’Dea, R. D., Waters, S. L. and Byrne, H. M., “A multiphase model for tissue construct growth in a perfusion bioreactor”, Math. Med. Biol.
27 (2010) 95–127; doi:10.1093/imammb/dqp003.Google Scholar
Osborne, J., O’Dea, R. D., Whiteley, J., Byrne, H. M. and Waters, S. L., “The influence of bioreactor geometry and the mechanical environment on engineered tissues”, J. Biomech. Engng
132 (2010) 051006; doi:10.1115/1.4001160.CrossRefGoogle ScholarPubMed
Owen, M. R., Alarcon, T., Maini, P. K. and Byrne, H. M., “Angiogenesis and vascular remodelling in normal and cancerous tissues”, J. Math. Biol.
58 (2009) 689–721;doi:10.1007/s00285-008-0213-z.CrossRefGoogle ScholarPubMed
Pavliotis, G. A. and Stuart, A., Multiscale methods: averaging and homogenization (Springer Science and Business Media, 2008); doi:10.1007/978-0-387-73829-1.Google Scholar
Pearson, N., Shipley, R. J., Waters, S. and Oliver, J., “Multiphase modelling of the influence of fluid flow and chemical concentration on tissue growth in a hollow fibre membrane bioreactor”, Math. Med. Biol.
31 (2014) 393–430; doi:10.1093/imammb/dqt015.CrossRefGoogle 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
Popov, P., Efendiev, Y. and Qin, G., “Multiscale modeling and simulations of flows in naturally fractured Karst reservoirs”, Commun. Comput. Phys.
6 (2009) 162;doi:10.4208/cicp.2009.v6.p162.Google Scholar
Ptashnyk, M. and Roose, T., “Derivation of a macroscopic model for transport of strongly sorbed solutes in the soil using homogenization theory”, SIAM J. Appl. Math.
70 (2010) 2097–2118; doi:10.1137/080729591.Google Scholar
Raviart, P. A. and Thomas, J. M., “A mixed finite element method for second order elliptic problems”, in: Mathematical aspects of the finite element method, Volume 606 of
Lecture Notes in Mathematics
(Springer, Berlin, Heidelberg, 1977) 1–46; doi:10.1007/BFb0064470.Google Scholar
Rosenberg, C., “Wound healing in the patient with diabetes mellitus”, Nurs. Clin. North Am.
25 (1990) 247–261.Google Scholar
Sanchez-Palencia, E., Non-homogeneous media and vibration theory, Volume 127 (Springer, Berlin, Heidelberg, 1980); doi:10.1007/3-540-10000-8.Google Scholar
Schaaf, S., Shibamiya, A., Mewe, M., Eder, A., Stohr, A., Hirt, M. N., Rau, T., Zimmermann, W.-H., Conradi, L., Eschenhagen, T. and Hansen, A., “Human engineered heart tissue as a versatile tool in basic research and preclinical toxicology”, PloS One
6 (2011) e26397;doi:10.1371/journal.pone.0026397.Google Scholar
Schmid, P. J., “Dynamic mode decomposition of numerical and experimental data”, J. Fluid Mech.
656 (2010) 5–28; doi:10.1017/S0022112010001217.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
Shipley, R. J., Jones, G. W., Dyson, R. J., Sengers, B. G., Bailey, C. L., Catt, C. J., Please, C. P. and Malda, J., “Design criteria for a printed tissue engineering construct: a mathematical homogenization approach”, J. Theoret. Biol.
259 (2009) 489–502; doi:10.1016/j.jtbi.2009.03.037.Google Scholar
Song, J. J., Guyette, J. P., Gilpin, S. E., Gonzalez, G., Vacanti, J. P. and Ott, H. C., “Regeneration and experimental orthotopic transplantation of a bioengineered kidney”, Nat. Med.
19 (2013) 646–651; doi:10.1038/nm.3154.Google Scholar
Taylor, C. and Hood, P., “A numerical solution of the Navier–Stokes equations using the finite element technique”, Comput. Fluids
1 (1973) 73–100; doi:10.1016/0045-7930(73)90027-3.Google Scholar
Truesdell, C., Rational thermodynamics (Springer, New York, 1984);doi:10.1007/978-1-4612-5206-1.Google Scholar
Van Leer, B., “Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov’s method”, J. Comput. Phys.
32 (1979) 101–136; doi:10.1016/0021-9991(79)90145-1.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
Ward, J. P. and King, J. R., “Mathematical modelling of avascular-tumour growth II: modelling growth”, Math. Med. Biol.
16 (1999) 171–211; doi:10.1093/imammb/16.2.171.Google Scholar
Ward, J. P. and King, J. R., “Mathematical modelling of avascular-tumour growth”, Math. Med. Biol.
14 (1997) 39–69; doi:10.1093/imammb/14.1.39.Google Scholar
Waters, S. L., Cummings, L., Shakesheff, K. and Rose, F., “Tissue growth in a rotating bioreactor. Part I: mechanical stability”, Math. Med. Biol.
23 (2006) 311–337; doi:10.1093/imammb/dql013.Google Scholar
Wei, C. and Dong, J., “Direct fabrication of high-resolution three-dimensional polymeric scaffolds using electrohydrodynamic hot jet plotting”, J. Micromech. Microengng.
23 (2013) 025017; doi:10.1088/0960-1317/23/2/025017.Google Scholar
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